Content-Type: multipart/mixed; boundary="-------------0302071939552" This is a multi-part message in MIME format. ---------------0302071939552 Content-Type: text/plain; name="03-39.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-39.keywords" Hydrodynamic limit, disordered systems, lattice gas dynamics, exclusion process. ---------------0302071939552 Content-Type: application/postscript; name="HL6.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="HL6.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: HL6.dvi %%Pages: 59 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CharterBT-Bold CharterBT-Roman CharterBT-Italic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips HL6.dvi -o HL6.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.02.07:1721 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CharterBT-Roman %!PS-AdobeFont-1.0 %%CreationDate: Fri Nov 2 19:00:13 1990 % Bitstream Type 1 Font Program % Copyright 1990 as an unpublished work by Bitstream Inc., Cambridge, MA. % All rights reserved. % Confidential and proprietary to Bitstream Inc. % U.S. GOVERNMENT RESTRICTED RIGHTS % This software typeface product is provided with RESTRICTED RIGHTS. Use, % duplication or disclosure by the Government is subject to restrictions % as set forth in the license agreement and in FAR 52.227-19 (c) (2) (May, 1987), % when applicable, or the applicable provisions of the DOD FAR supplement % 252.227-7013 subdivision (a) (15) (April, 1988) or subdivision (a) (17) % (April, 1988). Contractor/manufacturer is Bitstream Inc., % 215 First Street, Cambridge, MA 02142. % Bitstream is a registered trademark of Bitstream Inc. 11 dict begin /FontInfo 9 dict dup begin /version (2.0-1.0) readonly def /Notice (Copyright 1990 as an unpublished work by Bitstream Inc. All rights reserved. Confidential.) readonly def /FullName (Bitstream Charter) readonly def /FamilyName (Bitstream Charter) readonly def /Weight (Normal) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -109 def /UnderlineThickness 61 def end readonly def /FontName /CharterBT-Roman def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding StandardEncoding def /FontBBox {-161 -236 1193 963} readonly def /UniqueXX 15530648 def currentdict end currentfile eexec a2951840838a4133839ca9d22e2b99f2b61c767cd675080aacfcb24e19cd1336 739bb64994c56737090b4cec92c9945ff0745ef7ffc61bb0a9a3b849e7e98740 e56c0b5af787559cc6956ab31e33cf8553d55c0b0e818ef5ec6bf48162eac42e 7380ca921dae1c82b38fd6bcf2001abb5d001a56157094cfe27d8f4eac9693e8 8372d20358b47e0c3876558ebf757a1fbc5c1cddf62b3c57bf727ef1c4879422 c142a084d1c7462ac293e097fabe3a3ecfcd8271f259833bac7912707218ec9a 3063bf7385e02d8c1058ac06df00b33b8c018768b278010eb14c8bfbcb7035c1 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CharterBT-Bold %!PS-AdobeFont-1.0 %%CreationDate: Sat Nov 3 22:32:22 1990 % Bitstream Type 1 Font Program % Copyright 1990 as an unpublished work by Bitstream Inc., Cambridge, MA. % All rights reserved. % Confidential and proprietary to Bitstream Inc. % U.S. GOVERNMENT RESTRICTED RIGHTS % This software typeface product is provided with RESTRICTED RIGHTS. Use, % duplication or disclosure by the Government is subject to restrictions % as set forth in the license agreement and in FAR 52.227-19 (c) (2) (May, 1987), % when applicable, or the applicable provisions of the DOD FAR supplement % 252.227-7013 subdivision (a) (15) (April, 1988) or subdivision (a) (17) % (April, 1988). Contractor/manufacturer is Bitstream Inc., % 215 First Street, Cambridge, MA 02142. % Bitstream is a registered trademark of Bitstream Inc. 11 dict begin /FontInfo 9 dict dup begin /version (2.0-1.0) readonly def /Notice (Copyright 1990 as an unpublished work by Bitstream Inc. All rights reserved. Confidential.) readonly def /FullName (Bitstream Charter Bold) readonly def /FamilyName (Bitstream Charter) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -109 def /UnderlineThickness 90 def end readonly def /FontName /CharterBT-Bold def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding StandardEncoding def /FontBBox {-166 -236 1262 963} readonly def /UniqueXX 15530632 def currentdict end currentfile eexec a2951840838a4133839ca9d22e2b99f2b61c767cd675080aacfcb24e19cd1336 739bb64994c56737090b4cec92c9945ff0745ef7ffc61bb0a9a3b849e7e98740 e56c0b5af787559cc6956ab31e33cf8553d55c0b0e818ef5ec6bf48162eac42e 7380ca921dae1c82b38fd6bcf2001abb5d001a56157094cfe27d8f4eac9693e8 8372d20358b47e0c3876558ebf757a1fbc5c1cddf62b3c57bf727ef1c4879422 c142a084d1c7462ac293e097fabe3a3ecfcd8271f259833bac7912707218ec9a 3063bf7385e02d8c1058ac06df00b33b8c018768b278010eb14bd7a94f4c669c 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(jumps)g(between)g(nearest)f(neighbors)h(sites)i(are)e(al-)118 3673 y(lowed.)31 b(In)22 b(the)i(conduction)e(models)g(the)h(hard)h (core)d(exclusion)h(condition)h(re\003ects)f(the)h(underlying)118 3781 y(P)-5 b(auli)31 b(exclusion)g(principle)g(for)g(electrons.)49 b(The)32 b(main)g(focus)f(of)h(the)g(mathematical)g(and)f(physics)118 3889 y(literature)d(on)h(hopping)g(motion)g(models)f(has)i(been)e(the)h (understanding)f(of)h(transport)g(properties)118 3997 y(and)c(particularly)g(of)g(the)h(collective)d(diffusive)k(behavior)e (\(see)g(for)g(instance)g([34]\).)118 4105 y(In)f(this)i(paper)e(we)g (consider)f(an)h(interacting)g(particle)f(system)i(related)e(to)i (conduction)e(of)i(free)e(elec-)118 4213 y(trons)g(in)g(doped)g (crystals)g(that)h(can)e(be)h(described)g(as)g(follows.)31 b(A)23 b(particle)f(sitting)i(on)f(a)g(site)g Fq(x)g Fu(of)g(the)118 4321 y(cubic)h(lattice)i Fs(Z)684 4288 y Fr(d)745 4321 y Fu(waits)g(an)f(exponential)f(time)h(and)g(then)g (attempts)h(to)g(jump)e(to)i(a)f(neighbor)f(site)i Fq(y)s Fu(.)118 4429 y(If)k(the)f(site)i Fq(y)h Fu(is)e(occupied)e(then)h(the) h(jump)f(is)h(canceled)d(otherwise)j(it)g(is)g(realized)e(with)j(a)e (rate)g Fq(c)3580 4396 y Fr(\013)3580 4451 y(xy)118 4541 y Fu(depending)24 b(only)f(on)h(the)h(values)e Fp(\()p Fq(\013)1398 4555 y Fr(x)1443 4541 y Fq(;)30 b(\013)1556 4555 y Fr(y)1598 4541 y Fp(\))25 b Fu(of)f(some)g(external)f(quenched)g (disorder)i(\002eld)f Fo(f)p Fq(\013)3400 4555 y Fr(x)3445 4541 y Fo(g)3490 4563 y Fr(x)p Fn(2)p Fm(Z)3627 4544 y FA(d)118 4649 y Fu(that,)29 b(for)e(simplicity)-12 b(,)29 b(is)f(assumed)f(to)h(be)f(a)g(collection)e(of)j(i.i.d.)39 b(bounded)27 b(random)f(variables.)38 b(Our)118 4757 y(assumptions)31 b(on)e(the)h(transition)g(rates)g(are)f(quite)h (general.)42 b(W)-8 b(e)30 b(require)f(them)g(to)h(be)f(translation)118 4865 y(covariant,)39 b(strictly)d(bounded)g(and)g(positive)h(\(to)f (avoid)h(trapping)f(phenomena\),)h(and)f(to)h(satisfy)118 4976 y(the)f(detailed)f(balance)f(condition)i(w)-11 b(.r)h(.t.)62 b(to)35 b(the)h(\(product\))f(Gibbs)h(measure)e Fq(\026)3012 4943 y Fr(\013)3061 4976 y Fp(\()p Fq(\021)s Fp(\))43 b Fo(/)f Fq(e)3377 4943 y Fn(\000)p Fr(H)3495 4919 y FA(\013)3539 4943 y Fl(\()p Fr(\021)r Fl(\))3636 4976 y Fu(,)118 5084 y Fq(H)194 5098 y Fr(\013)243 5084 y Fp(\()p Fq(\021)s Fp(\))28 b(=)e Fo(\000)572 5016 y Fk(P)668 5111 y Fr(x)727 5084 y Fq(\013)785 5098 y Fr(x)829 5084 y Fq(\021)874 5098 y Fr(x)918 5084 y Fu(,)h(where)e Fq(\021)1286 5098 y Fr(x)1356 5084 y Fu(is)i(the)f(particle)g(occupation)f(number)g (at)h(site)h Fq(x)p Fu(.)34 b(These)26 b(require-)118 5192 y(ments)g(are)f(general)f(enough)h(to)h(include)f(some)h(popular)f (models)g(like)h(the)g(Random)g(T)-7 b(rap)26 b(and)g(the)118 5300 y(Miller\226Abrahams)j(models,)i(but)f(not)f(other)h(models)f (like)h(the)g(Random)f(Barrier)g(Model)g(in)h(which)p 118 5389 499 4 v 218 5480 a Fw(Date)p FD(:)20 b(F)-5 b(ebruary)20 b(7,)g(2003.)1871 5580 y Fj(1)p eop %%Page: 2 2 2 1 bop 118 45 a Fj(2)1188 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(the)37 b(jumps)f(rates)g(between)g Fq(x;)15 b(y)39 b Fu(is)e(assumed)g(to)f (depend)g(only)g(on)g(the)g(unoriented)f(bond)i Fp([)p Fq(x;)15 b(y)s Fp(])118 474 y Fu([20)o(].)33 b(F)-6 b(or)25 b(a)h(detailed)g(derivation)g(of)g(the)g(Hamiltonian)f Fq(H)2210 441 y Fr(\013)2285 474 y Fu(in)h(the)f(tight-binding)i (approximation)118 582 y(and)e(a)h(discussion)g(of)g(the)f(regime)f(of) i(its)g(validity)h(we)e(refer)f(to)i([5)o(].)218 798 y(Since)d(in)i(the)g(linear\226response)f(regime)f(the)i(conductivity)h (in)e(a)h(solid)h(is)f(linked)g(to)g(the)g(diffusion)118 906 y(matrix)g(via)h(the)f(Einstein)h(relation)e(\(see)h([34)o(]\),)h (our)e(main)h(target)g(has)h(been)d(the)j(study)g(of)f(the)g(bulk)118 1014 y(diffusion)35 b(of)f(the)g(disordered)g(lattice)g(gas)g (discussed)h(above.)56 b(Our)34 b(main)f(result)g(states)i(that,)j(for) 118 1122 y Fq(d)26 b Fo(\025)f Fp(3)p Fu(,)f(for)g(almost)f(any)g (realization)g(of)h(the)g(random)f(\002eld)g Fq(\013)p Fu(,)h(the)g(diffusively)g(rescaled)f(system)h(has)118 1230 y(hydrodynamical)h(limit)h(given)e(by)i(a)f(non)g(linear)f (differential)i(equation)1452 1407 y Fq(@)1500 1421 y Fr(t)1530 1407 y Fq(m)f Fp(=)g Fo(r)20 b(\001)h Fp(\()p Fq(D)s Fp(\()p Fq(m)p Fp(\))p Fo(r)p Fq(m)p Fp(\))118 1584 y Fu(where)28 b Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))28 b Fu(denotes)g(the)h(macroscopic)e(density)i(function)f(at)h (time)g Fq(t)f Fu(at)h(the)f(point)h Fq(\022)i Fu(of)d(the)h Fq(d)p Fu(\226)118 1695 y(dimensional)23 b(torus)f(in)h Fs(R)1009 1662 y Fr(d)1077 1695 y Fu(with)h(unit)f(volume)e(and)i(the)f (non)g(random)g(matrix)h Fq(D)s Fp(\()p Fo(\001)p Fp(\))g Fu(is)g(the)g(diffusion)118 1802 y(matrix.)46 b(Moreover)-10 b(,)29 b(we)h(give)f(a)h(variational)h(characterization)e(of)h(the)g (matrix)g Fq(D)s Fp(\()p Fq(m)p Fp(\))h Fu(in)f(terms)f(of)118 1910 y(the)k(distribution)i(of)e(the)g(random)f(\002eld)g Fq(\013)h Fu(similar)g(to)h(the)f(usual)f(Green\226K)n(ubo)f(formula)h (and)h(we)118 2018 y(prove)25 b(that)h Fp(inf)661 2032 y Fr(m)742 2018 y Fq(D)s Fp(\()p Fq(m)p Fp(\))g Fq(>)f Fp(0)h Fu(and)f(that)h Fq(D)s Fp(\()p Fo(\001)p Fp(\))g Fu(is)h(continuous)d(in)i(the)f(open)g(interval)f Fp(\(0)p Fq(;)15 b Fp(1\))p Fu(.)118 2126 y(W)-8 b(e)29 b(remark)f(that)h(the)g (above)f(result)g(without)i(the)f(restriction)g(on)f(the)h(dimension)g Fq(d)p Fu(,)h(was)f(already)118 2234 y(announced)23 b(in)i([29])g (several)f(years)g(ago)h(together)f(with)i(some)f(sketchy)g(ideas)g (for)g(its)h(proof)m(.)31 b(How-)118 2342 y(ever)g(the)g(details)i(of)f (the)g(proof)f(have)h(never)e(since)h(been)g(published)h(and)f(some)h (of)f(the)h(technical)118 2450 y(estimates)27 b(suggested)e(in)h([29)o (])g(turned)f(out)h(to)g(be)f(troublesome)f(even)h(in)h(the)f(absence)g (of)h(disorder)118 2558 y(\(symmetric)33 b(simple)h(exclusion)f (model\))g(as)h(explained)f(in)h([15],)i(chapter)e(6.)57 b(Therefore)33 b(we)g(de-)118 2666 y(cided)c(to)f(tackle)f(again)h(the) h(problem)d(but)j(we)e(were)h(forced)f(to)i(take)f(a)g(different)g (route)g(w)-11 b(.r)h(.t.)40 b(that)118 2774 y(indicated)26 b(in)g([29)o(].)118 2882 y(W)-8 b(e)26 b(also)f(observe)g(that)i(the)e (problem)f(of)i(collective)e(behavior)h(in)h(disordered)g(lattice)f (gas)g(has)i(been)118 2990 y(discussed)40 b(mathematically)d(in)i (other)f(papers,)k(but,)g(to)c(the)h(best)f(of)h(our)f(knowledge,)i (only)e(for)118 3098 y(models)33 b(with)h(either)f(homogeneous)f (equilibrium)h(measures)f(\(see)h(for)g(example)f([28)o(],)k([16)o(])e (for)118 3206 y(the)23 b(one\226dimensional)g(Random)g(Barrier)f(model) g(and)i(its)g(Brownian)f(version\))f(or)h(with)h(periodicity)118 3313 y(in)30 b(the)g(random)f(\002eld)h Fq(\013)g Fu(allowing)f(to)h (solve)f(directly)g(the)h(generalized)e(Fick's)i(law)g(\(see)f([32)o(]) h(and)118 3421 y([38)o(])e(for)f(the)g(one\226dimensional)g(Random)f(T) -7 b(rap)28 b(model)e(having)h(random)f(\002eld)h Fq(\013)h Fu(of)f(period)g(2\))g(or)118 3529 y(\002nally)35 b(for)g(models)g (satisfying)i(the)e(so)h(called)e(\223gradient)h(condition\224)g(\(see) g(below\))g([25)o(].)62 b(From)118 3637 y(the)26 b(physical)g(point)g (of)g(view)-11 b(,)26 b(diffusion)g(of)g(lattice)g(gases)f(in)h (systems)g(with)h(site)f(disorder)g(has)h(been)118 3745 y(studied)e(mainly)e(by)g(means)g(of)h(simulations)g(and)g(more)e(or)i (less)f(rough)g(approximations)h(like)f(mean)118 3853 y(\002eld)32 b(.)52 b(W)-8 b(e)33 b(refer)e(the)h(interested)h(reader)e (to)h([19],)i([20)o(],)h([21)o(],)g([22)o(],)f([24])e(and)g(to)h([17)o (])g(for)f(an)118 3961 y(iterative)26 b(procedure)e(to)h(compute)g (corrections)f(to)i(the)f(mean\226\002eld)g(approximation.)118 4177 y(The)30 b(main)g(technical)g(features)g(of)g(the)h(model)e (considered)h(here)f(are)h(the)h(absence)e(of)h(translation)118 4285 y(invariance)24 b(\(for)h(a)f(given)g(disorder)i (con\002guration\))d(and)i(the)g(non)f(validity)i(of)f(the)g(so)g (called)e(gradi-)118 4393 y(ent)30 b(condition.)45 b(This)32 b(condition)e(corresponds)f(to)h(the)g(Fick's)h(law)e(of)h(\003uid)g (mechanics)g(according)118 4501 y(to)25 b(which)g(the)f(current)f(can)g (be)h(written)h(as)g(the)f(gradient)g(of)h(some)f(function.)31 b(Since)23 b(the)h(continuity)118 4609 y(equation)g(states)i(that)f Fq(@)968 4623 y Fr(t)998 4609 y Fq(m)g Fp(=)g Fo(r)16 b(\001)g Fq(J)9 b Fu(,)25 b Fq(J)33 b Fu(being)24 b(the)g(macroscopic)f (current,)g(the)h(main)g(problem)f(is)i(to)118 4717 y(derive)e Fq(J)32 b Fu(from)23 b(the)g(family)g(of)h(microscopic)e(instantaneous) i(currents)e Fq(j)2660 4684 y Fr(\013)2655 4739 y(x;y)2756 4717 y Fp(\()p Fq(\021)s Fp(\))k(:=)f Fq(c)3060 4684 y Fr(\013)3060 4739 y(x;y)3161 4717 y Fp(\()p Fq(\021)s Fp(\))3279 4643 y Fk(\000)3322 4717 y Fq(\021)3367 4731 y Fr(x)3424 4717 y Fo(\000)13 b Fq(\021)3553 4731 y Fr(y)3594 4643 y Fk(\001)3636 4717 y Fu(,)118 4827 y(de\002ned)28 b(as)h(the)g(difference)f(between)f(the)i(rate)f(at)h(which)g(a)g (particle)f(jumps)g(from)g Fq(x)g Fu(to)h Fq(y)j Fu(and)c(the)118 4935 y(rate)37 b(at)g(which)g(a)g(particle)f(jumps)h(from)g Fq(y)i Fu(to)e Fq(x)p Fu(.)66 b(The)37 b(gradient)g(condition)g(\(the)g (Fick's)g(law\))g(is)118 5043 y(satis\002ed)28 b(if)m(,)e(for)g(each)f (disorder)i(con\002guration)d Fq(\013)p Fu(,)j(there)e(exists)i(a)f (local)f(function)g Fq(h)3103 5010 y Fr(\013)3153 5043 y Fp(\()p Fq(\021)s Fp(\))i Fu(such)e(that)118 5153 y Fq(j)160 5121 y Fr(\013)155 5176 y(x;x)p Fl(+)p Fr(e)346 5153 y Fp(\()p Fq(\021)s Fp(\))i(=)e Fq(\034)627 5167 y Fr(x)p Fl(+)p Fr(e)758 5153 y Fq(h)810 5121 y Fr(\013)860 5153 y Fp(\()p Fq(\021)s Fp(\))20 b Fo(\000)e Fq(\034)1127 5167 y Fr(x)1171 5153 y Fq(h)1223 5121 y Fr(\013)1273 5153 y Fp(\()p Fq(\021)s Fp(\))26 b Fu(for)f(any)f Fq(x)i Fo(2)e Fs(Z)1950 5121 y Fr(d)1987 5153 y Fu(,)h(where)f Fq(\034)2347 5167 y Fr(x)2391 5153 y Fq(h)2443 5121 y Fr(\013)2492 5153 y Fp(\()p Fq(\021)s Fp(\))j(:=)e Fq(h)2810 5121 y Fr(\034)2841 5129 y FA(x)2881 5121 y Fr(\013)2930 5153 y Fp(\()p Fq(\034)3005 5167 y Fr(x)3049 5153 y Fq(\021)s Fp(\))h Fu(and)f Fq(\034)3372 5167 y Fr(x)3415 5153 y Fq(\021)s(;)31 b(\034)3559 5167 y Fr(x)3603 5153 y Fq(\013)118 5264 y Fu(denote)25 b(the)h(particle)e(and)i(disorder)g (con\002gurations)e Fq(\021)s(;)31 b(\013)26 b Fu(translated)f(by)h (the)f(vector)g Fq(x)p Fu(.)118 5372 y(If)32 b(the)f(system)g (satis\002es)i(the)e(gradient)g(condition,)i(the)e(derivation)g(of)h Fq(J)40 b Fu(is)31 b(not)g(too)h(dif\002cult)f(\(see)118 5480 y([23)o(])g(and)g(reference)c(therein\).)47 b(It)31 b(is)g(however)f(simple)g(to)h(check)e(\(as)i(in)f([34],)i(p.)47 b(182\))29 b(that)i(our)p eop %%Page: 3 3 3 2 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g(DISORDERED)f (LA)-7 b(T)q(TICE)23 b(GAS)860 b(3)118 366 y Fu(system)26 b(never)d(satis\002es)k(the)e(gradient)g(condition)h(except)e(for)h (constant)g(disorder)g(\002eld)g Fq(\013)p Fu(.)32 b(W)-8 b(e)26 b(thus)118 474 y(have)i(to)g(appeal)f(to)i(the)f(methods)g (developed)f(by)h(V)-10 b(aradhan)28 b([36],)h(Quastel)f([30)o(])g(and) g(V)-10 b(aradhan-)111 582 y(Y)h(au)35 b([37)o(])g(\(see)f(also)h([23)o (])g(and)f(references)f(therein\))h(for)g(studying)h(the)g (hydrodynamic)f(limit)h(of)118 690 y(non)23 b(disordered)i(non)e (gradient)h(systems.)32 b(There)24 b(the)g(main)g(idea)g(is)h(to)g (prove)e(a)h(generalized)e(Fick's)118 798 y(law)j(of)h(the)f(form)1179 942 y Fq(j)1221 904 y Fr(\013)1216 964 y Fl(0)p Fr(;e)1333 942 y Fo(\031)1436 855 y Fk(X)1429 1053 y Fr(e)1462 1034 y Fi(0)1484 1053 y Fn(2E)1590 942 y Fq(D)1665 957 y Fr(e;e)1751 938 y Fi(0)1776 942 y Fp(\()p Fq(m)1891 957 y Fr(`)1925 942 y Fp(\)\()p Fq(\021)2040 957 y Fr(e)2073 938 y Fi(0)2120 942 y Fo(\000)20 b Fq(\021)2256 956 y Fl(0)2295 942 y Fp(\))h(+)f Fo(L)2505 904 y Fr(\013)2554 942 y Fq(g)862 b Fu(\(1.1\))118 1197 y(for)31 b(a)g(suitable)g(non)g(random)f(matrix)h Fq(D)s Fp(\()p Fq(m)p Fp(\))p Fu(,)i(where)d Fq(m)2135 1212 y Fr(`)2199 1197 y Fu(is)i(the)f(particle)g(density)h(in)f(a)g (cube)f(cen-)118 1305 y(tered)e(in)g(the)h(origin)f(of)g(mesoscopic)g (side)h Fq(`)p Fu(,)g Fq(g)s Fp(\()p Fq(\013;)15 b(\021)s Fp(\))31 b Fu(is)e(a)f(local)g(function)f(and)i Fo(L)f Fu(is)h(the)f(generator)118 1413 y(of)e(the)f(dynamics.)118 1521 y(One)30 b(\(among)f(many)h(others\))g(main)g(dif\002culty)h(in)g (proving)e(such)h(an)g(approximation)h(for)f(a)h(disor-)118 1629 y(dered)g(system)g(is)g(due)g(to)g(the)g(fact)g(that)h(the)f (disorder)g(itself)h(induces)e(strong)h(\003uctuations)f(in)h(the)118 1737 y(gradient)26 b(density)h(\002eld)f(as)h(it)g(is)g(easily)g(seen)e (by)h(taking,)h(for)f(any)g(\002xed)h(disorder)f(con\002guration)g Fq(\013)p Fu(,)118 1845 y(the)j(average)f(w)-11 b(.r)h(.t.)43 b(to)30 b(the)f(Gibbs)h(measure)d Fq(\026)1777 1812 y Fr(\013)1856 1845 y Fu(of)i(\(1.1\).)41 b(By)29 b(construction)g(the)g (current)e Fq(j)3396 1812 y Fr(\013)3391 1869 y Fl(0)p Fr(;e)3512 1845 y Fu(and)118 1958 y(the)d(\003uctuation)e(term)h Fo(L)p Fq(g)j Fu(have)e(in)f(fact)g(zero)f(average)g(while)h(the)h (average)e(of)h Fq(\021)2936 1973 y Fr(e)2969 1954 y Fi(0)3009 1958 y Fo(\000)13 b Fq(\021)3138 1972 y Fl(0)3201 1958 y Fu(\(we)22 b(neglect)118 2066 y(the)28 b(factor)f Fq(D)s Fp(\()p Fq(m)722 2081 y Fr(`)755 2066 y Fp(\))g Fu(for)h(simplicity\))g(is)g(in)f(general)f Fq(O)s Fp(\(1\))i Fu(because)e(of)i(the)f(disorder)-10 b(.)38 b(However)-10 b(,)27 b(and)118 2174 y(this)38 b(is)e(a)g(key)g(input,)j(the)e (average)d(over)h(the)i(disorder)f(of)g(the)h(Gibbs)g(average)d(of)i Fq(\026)3214 2141 y Fr(\013)3264 2100 y Fk(\000)3305 2174 y Fq(\021)3350 2189 y Fr(e)3383 2170 y Fi(0)3437 2174 y Fo(\000)27 b Fq(\021)3580 2188 y Fl(0)3620 2100 y Fk(\001)118 2282 y Fu(vanishes)e(and)g(therefore)e(one)h(can)g(hope)g (to)h(tame)f(the)g(disorder)i(induced)e(\003uctuations)g(in)g(the)h (gra-)118 2390 y(dient)33 b(of)g(the)g(density)g(\002eld)f(by)h (\002rst)g(smearing)f(them)g(out)h(using)f(suitable)h(spatial)h (averages)d(and)118 2498 y(then)24 b(by)g(appealing)g(to)g(the)h (ergodic)e(properties)i(of)f(the)h(disorder)f(\002eld)g Fq(\013)p Fu(,)i(at)e(least)h(in)f(high)h(enough)118 2606 y(dimension.)32 b(It)26 b(turns)f(out)g(that)i(the)e(above)g (sketchy)h(plan)e(works)i(as)g(soon)f(as)h Fq(d)f Fo(\025)g Fp(3)p Fu(.)118 2858 y(W)-8 b(e)26 b(conclude)d(this)k(short)e (introduction)h(with)g(a)f(plan)g(of)g(the)h(paper)-10 b(.)31 b(In)25 b(section)g(2)g(we)g(\002x)g(the)g(nota-)118 2966 y(tion,)j(describe)e(the)h(model)e(and)i(state)g(the)g(main)g (results.)35 b(In)26 b(section)h(3)f(and)g(section)h(4)f(we)g(discuss) 118 3074 y(most)32 b(of)g(the)g(\223high)g(level\224)e(technical)h (tools)h(\(entropy)f(estimates,)j(perturbation)e(theory)-12 b(,)33 b(spectral)118 3182 y(gap)d(bounds\))g(and)h(complete)e(the)h (proof)h(of)f(the)h(main)f(theorems)g(following)f(the)i(standard)g (route)118 3290 y(of)i(non)f(gradient)h(systems,)j(modulo)c(some)g(key) h(technical)f(results.)54 b(In)33 b(section)f(5)h(we)f(discuss)i(in)118 3398 y(detail)23 b(the)g(problem)f(of)h(the)g(\003uctuations)g(of)g (the)g(gradient)f(density)i(\002eld)e(induced)h(by)g(the)g(disorder)-10 b(.)118 3506 y(Section)24 b(6)f(is)j(devoted)e(to)h(the)f(proof)h(of)f (several)f(technical)h(bounds)g(while)h(in)f(section)g(7)g(we)g (discuss)118 3614 y(at)30 b(length)f(central)g(limit)h(variance,)h (closed)e(and)h(exact)f(forms)h(in)g(our)f(context)h(together)f(with)i (our)118 3722 y(own)21 b(interpretation)g(of)g(the)g(long)f(jump)g (method)i(described)f(in)g([29)o(].)30 b(Finally)21 b(some)f(very)g (technical)118 3830 y(estimates)26 b(are)f(collected)f(in)h(an)g (appendix)g(at)h(the)g(end.)118 3938 y(W)-8 b(e)28 b(\002nish)g(by)f (saying)g(that)h(most)f(of)g(the)h(material)e(presented)h(here)f(is)i (based)g(on)e(the)i(unpublished)118 4045 y(thesis)i([15)o(])f(written)g (by)g(one)e(of)i(us)g(\(A.F\))f(where)g(an)g(expanded)g(version)h(of)f (several)g(of)h(the)f(argu-)118 4153 y(ments)d(used)g(in)h(this)g (paper)f(can)g(be)g(found.)118 4386 y FE(Acknowledgments.)45 b Fu(P)-5 b(art)30 b(of)g(this)i(work)d(was)i(done)f(while)g(both)g (authors)h(were)e(visiting)i(the)f(In-)118 4494 y(stitute)f(H.)f(P)-5 b(oincar\264)-45 b(e)27 b(during)g(the)h(special)f(semester)g(on)h (\223Hydrodynamic)f(limits\224.)39 b(W)-8 b(e)28 b(would)g(like)118 4602 y(to)j(thank)g(the)f(organizers)g(F)-17 b(.)30 b(Golse)g(and)h(S.) f(Olla)g(for)g(their)h(kind)g(invitation)h(and)e(the)h(stimulating)118 4710 y(scienti\002c)e(atmosphere)g(there.)42 b(W)-8 b(e)29 b(are)f(also)h(grateful)f(to)h(J.)h(Quastel)f(for)f(providing)i (unpublished)118 4818 y(notes)d(on)f(the)h(problem)f(and)g(for)h (sharing)g(his)h(insight)f(of)g(the)g(subject.)36 b(W)-8 b(e)27 b(are)f(also)h(grateful)f(to)h(S.)118 4926 y(Olla,)e(C.)g (Landim,)g(G.B.)f(Giacomin)h(for)f(many)f(enlightening)g(discussions)j (and)e(to)h(P)-24 b(.)24 b(Caputo)h(for)f(his)118 5034 y(proof)h(of)h(the)f(spectral)g(gap)g(bound.)995 5318 y(2.)47 b(N)t Ft(O)t(T)l(A)l(T)t(I)t(O)t(N)t Fu(,)35 b Ft(T)t(H)t(E)30 b(M)t(O)t(D)t(E)t(L)i(A)t(N)t(D)g(M)t(A)t(I)t(N)f(R)t (E)t(S)t(U)t(L)n(T)t(S)218 5480 y Fu(In)25 b(this)i(section)e(we)g (\002x)g(the)h(notation,)g(we)f(de\002ne)g(the)g(model)g(and)g(state)h (our)f(main)g(result.)p eop %%Page: 4 4 4 3 bop 118 45 a Fj(4)1188 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(2.1.)45 b FE(Notation.)118 474 y Fh(Geometric)32 b(setting.)51 b Fu(W)-8 b(e)33 b(consider)f(the)g Fq(d)h Fu(dimensional)f(lattice)g Fs(Z)2480 441 y Fr(d)2548 474 y Fu(with)i(sites)f Fq(x)k Fp(=)f Fo(f)p Fq(x)3256 488 y Fl(1)3296 474 y Fq(;)15 b(:)g(:)g(:)i(;)e(x)3550 489 y Fr(d)3591 474 y Fo(g)p Fu(,)118 585 y(canonical)28 b(basis)j Fo(E)38 b Fu(and)30 b(norm)e Fo(j)p Fq(x)p Fo(j)33 b Fp(=)f(max)p Fo(fj)p Fq(x)1782 599 y Fl(1)1822 585 y Fo(j)p Fq(;)15 b(:)g(:)g(:)i(;)e Fo(j)p Fq(x)2126 600 y Fr(d)2167 585 y Fo(jg)p Fu(.)46 b(The)29 b(bonds)h(of)g Fs(Z)2927 552 y Fr(d)2993 585 y Fu(are)f(non)g(oriented)118 692 y(couple)24 b(of)h(adjacent)h(sites)g (and)f(a)h(generic)d(bond)j(will)f(be)g(denoted)g(by)g Fq(b)p Fu(.)118 800 y(The)33 b(cardinality)h(of)f(a)h(\002nite)f (subset)h Fp(\003)k Fo(\032)g Fs(Z)1742 767 y Fr(d)1812 800 y Fu(is)c(denoted)f(by)g Fo(j)p Fp(\003)p Fo(j)h Fu(and)f Fs(F)44 b Fu(denotes)33 b(the)g(set)h(of)f(all)118 911 y(nonempty)24 b(\002nite)i(subsets)g(of)f Fs(Z)1250 878 y Fr(d)1287 911 y Fu(.)118 1018 y(Given)i Fq(`)i Fo(2)f Fs(N)40 b Fu(we)27 b(denote)g(by)h Fp(\003)1250 1033 y Fr(`)1310 1018 y Fu(the)g(cube)e(centered)g(at)i(the)f(origin)h (of)f(side)h Fp(2)p Fq(`)22 b Fp(+)g(1)p Fu(.)38 b(If)28 b Fq(`)g Fp(=)h(2)p Fq(j)f Fp(+)21 b(1)118 1126 y Fu(we)36 b(also)g(set)g Fq(Q)681 1141 y Fr(`)757 1126 y Fp(=)42 b(\003)933 1140 y Fr(j)970 1126 y Fu(.)63 b(The)36 b(same)g(cubes)f (centered)f(at)i Fq(x)g Fu(will)g(be)f(denoted)h(by)g Fp(\003)3184 1141 y Fr(x;`)3312 1126 y Fu(and)g Fq(Q)3569 1141 y Fr(x;`)118 1242 y Fu(respectively)-12 b(.)30 b(More)25 b(generally)-12 b(,)23 b(for)i(any)g Fq(V)45 b Fo(\032)25 b Fs(Z)1831 1209 y Fr(d)1893 1242 y Fu(and)g Fq(x)g Fo(2)g Fs(Z)2295 1209 y Fr(d)2331 1242 y Fu(,)h(we)f(will)g(set)g Fq(V)2883 1256 y Fr(x)2953 1242 y Fp(:=)g Fq(V)40 b Fp(+)20 b Fq(x)p Fu(.)118 1349 y(Next,)25 b(given)g Fq(e)g Fo(2)g(E)33 b Fu(and)26 b Fq(`)f Fp(=)g(2)p Fq(`)1245 1316 y Fn(0)1289 1349 y Fp(+)20 b(1)25 b Fu(with)i Fq(`)1691 1316 y Fn(0)1739 1349 y Fo(2)e Fs(N)7 b Fu(,)31 b(we)25 b(let)779 1535 y Fp(\003)842 1490 y Fl(1)p Fr(;e)842 1564 y(`)959 1535 y Fp(:=)g(\003)1143 1553 y Fn(\000)p Fl(\()p Fr(`)1254 1534 y Fi(0)1277 1553 y Fl(+1\))p Fr(e;`)1476 1534 y Fi(0)1517 1535 y Fq(;)106 b Fp(\003)1711 1490 y Fl(2)p Fr(;e)1711 1564 y(`)1829 1535 y Fp(:=)25 b(\003)2013 1550 y Fr(`)2042 1531 y Fi(0)2064 1550 y Fr(e;`)2146 1531 y Fi(0)2187 1535 y Fq(;)106 b Fp(\003)2381 1497 y Fr(e)2381 1558 y(`)2444 1535 y Fp(:=)25 b(\003)2628 1490 y Fl(1)p Fr(;e)2628 1564 y(`)2740 1535 y Fo([)20 b Fp(\003)2884 1490 y Fl(2)p Fr(;e)2884 1564 y(`)2976 1535 y Fq(:)458 b Fu(\(2.1\))118 1723 y(Finally)-12 b(,)35 b(given)d Fq(\017)39 b Fo(2)f Fp(\(0)p Fq(;)15 b Fp(1\))35 b Fu(such)f(that)g Fq(\017)1534 1690 y Fn(\000)p Fl(1)1667 1723 y Fo(2)k Fs(N)7 b Fu(,)41 b(we)34 b(de\002ne)e(the)i(discrete)f (torus)h(of)f(spacing)g Fq(\017)g Fu(by)118 1834 y Fs(T)177 1801 y Fr(d)177 1856 y(\017)251 1834 y Fp(:=)e Fs(Z)443 1801 y Fr(d)479 1834 y Fq(=\017)561 1801 y Fn(\000)p Fl(1)656 1834 y Fs(Z)721 1801 y Fr(d)757 1834 y Fu(.)43 b(The)29 b(usual)g Fq(d)p Fu(\226dimensional)g(torus)h Fs(R)2141 1801 y Fr(d)2188 1834 y Fq(=)p Fs(Z)2298 1801 y Fr(d)2363 1834 y Fu(\(with)g(unite)f(volume\))f(will)h(instead)118 1944 y(be)c(denoted)g(by)g Fs(T)763 1911 y Fr(d)805 1944 y Fu(.)31 b Fo(M)970 1958 y Fl(1)1010 1944 y Fp(\()p Fs(T)1104 1911 y Fr(d)1146 1944 y Fp(\))25 b Fu(will)g(denote)g(the)g (set)g(of)g(positive)h(Borel)e(measures)g(on)h Fs(T)3211 1911 y Fr(d)3278 1944 y Fu(with)h(total)118 2052 y(mass)d(bounded)e(by) h Fp(1)p Fu(,)i(endowed)d(of)h(the)h(weak)e(topology)-12 b(,)23 b(while)f Fo(M)2508 2066 y Fl(2)2573 2052 y Fo(\032)j(M)2778 2066 y Fl(1)2839 2052 y Fu(will)d(denote)g(the)g(set)g(of)118 2160 y(measures)e(in)i Fo(M)718 2174 y Fl(1)778 2160 y Fu(which)g(are)f(absolutely)f(continuous)h(w)-11 b(.r)h(.t.)31 b(the)21 b(Lebesgue)f(measure)g(with)j(density)118 2267 y Fq(\032)i Fu(satisfying)i Fo(k)p Fq(\032)p Fo(k)726 2281 y Fn(1)827 2267 y Fo(\024)e Fp(1)p Fu(.)118 2483 y Fh(Spatial)d(averages.)29 b Fu(W)-8 b(e)24 b(will)e(make)g(heavy)h (use)f(of)h(spatial)h(averages)e(and)h(it)g(is)h(better)e(to)h(\002x)g (from)f(the)118 2591 y(beginning)27 b(some)g(handy)i(notation.)40 b(Given)27 b Fp(\003)j Fo(2)g Fs(F)38 b Fu(and)28 b Fq(`)i Fo(2)f Fs(N)7 b Fu(,)35 b(the)28 b(spatial)h(average)e(of)h Fo(f)p Fq(f)3401 2605 y Fr(x)3445 2591 y Fo(g)3490 2613 y Fr(x)p Fn(2)p Fm(Z)3627 2595 y FA(d)118 2724 y Fu(in)d Fp(\003)c Fo(\\)f Fq(`)15 b Fs(Z)21 b Fu(will)k(be)g(denoted)g(by)g Fp(Av)1399 2676 y Fl(\()p Fr(`)p Fl(\))1399 2752 y Fr(x)p Fn(2)p Fl(\003)1539 2724 y Fq(f)1584 2738 y Fr(x)1628 2724 y Fu(.)31 b(When)25 b Fq(`)h Fp(=)f(1)g Fu(we)g(will)g(simply)h (write)f Fp(Av)3116 2738 y Fr(x)p Fn(2)p Fl(\003)3271 2724 y Fq(f)3316 2738 y Fr(x)3359 2724 y Fu(.)118 2869 y(Next,)31 b(given)d Fq(e)33 b Fo(2)f(E)38 b Fu(and)29 b(two)h(odd)h(integers)d Fq(`)33 b Fp(=)f(2)p Fq(`)1997 2836 y Fn(0)2044 2869 y Fp(+)22 b(1)p Fu(,)32 b Fq(s)g Fp(=)g(2)p Fq(s)2505 2836 y Fn(0)2551 2869 y Fp(+)23 b(1)30 b Fu(such)f(that)3127 2834 y Fr(s)p 3127 2849 33 4 v 3129 2901 a(`)3202 2869 y Fo(2)j Fs(N)7 b Fu(,)37 b(we)29 b(let)118 3007 y Fq(Q)190 2959 y Fl(\()p Fr(`)p Fl(\))190 3019 y Fr(s)306 3007 y Fp(:=)e Fq(`)15 b Fs(Z)547 2974 y Fr(d)605 3007 y Fo(\\)20 b Fq(Q)758 3021 y Fr(s)795 3007 y Fu(.)36 b(Notice)26 b(that,)j(if)e(we)g(divide)h(the)f(cube)e Fp(\003)2269 2963 y Fl(1)p Fr(;e)2269 3019 y(s)2388 3007 y Fu(in)i(cubes)f(of)h(side)g Fq(`)p Fu(,)h(the)f(centers)e(of)118 3131 y(these)h(cubes)e(form)h(the)h(set)f Fq(Q)1173 3083 y Fl(\()p Fr(`)p Fl(\))1173 3142 y Fr(x;s)1294 3131 y Fu(with)i Fq(x)e Fp(=)g Fo(\000)p Fp(\()p Fq(s)1819 3098 y Fn(0)1862 3131 y Fp(+)20 b(1\))p Fq(e)p Fu(.)118 3243 y(W)n(ith)27 b(these)f(notation)f(we)g(de\002ne)g(the)h Fp(\()p Fq(`;)15 b(s;)g(e)p Fp(\))26 b Fu(spatial)h(average)d(around)g Fq(y)k Fo(2)d Fs(Z)2938 3210 y Fr(d)3000 3243 y Fu(by)928 3524 y Fp(Av)1046 3487 y Fr(`;s)1046 3547 y(z)s(;y)1158 3524 y Fq(f)1203 3538 y Fr(z)1267 3524 y Fp(:=)1474 3463 y(1)p 1398 3504 197 4 v 1398 3587 a(\()p Fq(s=`)p Fp(\))1620 3402 y Fl(\()p Fr(s=`)p Fl(\))p Fn(\000)p Fl(1)1675 3438 y Fk(X)1684 3634 y Fr(i)p Fl(=0)1877 3524 y Fp(Av)1994 3566 y Fr(x)p Fn(2)p Fr(Q)2137 3530 y Fg(\()p FA(`)p Fg(\))2137 3576 y FA(s)2234 3524 y Fq(f)2279 3543 y Fr(y)r Fl(+)p Fr(x)p Fl(+\()p Fr(`)2522 3524 y Fi(0)2544 3543 y Fl(+)p Fr(i`)p Fn(\000)p Fr(s)2740 3524 y Fi(0)2762 3543 y Fl(\))p Fr(e)2826 3524 y Fq(:)608 b Fu(\(2.2\))118 3788 y(The)26 b(motivation)g(of)f(introducing)g(such)g(a)g(spatial)i (average)d(will)h(be)g(discussed)h(in)g(subsection)f(4.2.)118 4004 y Fh(The)37 b(disorder)e(\002eld.)65 b Fu(W)-8 b(e)38 b(assume)e(the)h(disorder)h(to)f(be)f(described)h(by)g(a)g(collection)e (of)i(real)f(i.i.d)118 4112 y(random)27 b(variables)g Fq(\013)i Fp(:=)f Fo(f)p Fq(\013)1149 4126 y Fr(x)1194 4112 y Fo(g)1239 4134 y Fr(x)p Fn(2)p Fm(Z)1376 4115 y FA(d)1438 4112 y Fu(such)f(that)h Fp(sup)1967 4133 y Fr(x)2026 4112 y Fo(j)p Fq(\013)2109 4126 y Fr(x)2153 4112 y Fo(j)h(\024)f Fq(B)k Fu(for)27 b(some)g(\002nite)g(constant)g Fq(B)5 b Fu(.)37 b(The)118 4237 y(corresponding)31 b(product)h(measure) f(on)h Fp(\012)1623 4251 y Fl(D)1718 4237 y Fp(:=)k([)p Fo(\000)p Fq(B)5 b(;)15 b(B)5 b Fp(])2159 4204 y Fm(Z)2209 4181 y FA(d)2276 4237 y Fu(will)32 b(be)g(denoted)g(by)g Fs(P)p Fu(.)53 b(Expectation)118 4345 y(w)-11 b(.r)h(.t.)32 b Fs(P)26 b Fu(will)f(be)g(denoted)g(by)h Fs(E)10 b Fu(.)118 4488 y(Notice)32 b(that,)k(for)c(any)g(given)f Fq(\017)37 b Fo(2)f Fp(\(0)p Fq(;)15 b Fp(1\))35 b Fu(such)d(that)h Fq(\017)2034 4455 y Fn(\000)p Fl(1)2161 4488 y Fu(is)g(an)f(odd)h (integer)-10 b(,)33 b(the)f(random)g(\002eld)g Fq(\013)118 4598 y Fu(induces)e(in)f(a)h(natural)f(way)h(a)g(random)f(\002eld)g(on) g Fs(T)1946 4565 y Fr(d)1946 4621 y(\017)2018 4598 y Fu(via)h(the)g(identi\002cation)h(of)f Fs(T)3041 4565 y Fr(d)3041 4621 y(\017)3113 4598 y Fu(with)h(the)f(cube)118 4706 y Fq(Q)190 4725 y Fl(1)p Fr(=\017)293 4706 y Fu(.)i(F)-6 b(or)25 b(notation)g(convenience)e(the)j(induced)f(random)f(\002eld)h (will)g(always)h(be)f(denoted)g(by)g Fq(\013)p Fu(.)118 4829 y(Finally)-12 b(,)25 b(given)f Fq(\013)i Fo(2)f Fp(\012)900 4843 y Fl(D)983 4829 y Fu(and)g Fp(\003)g Fo(\032)g Fs(Z)1406 4796 y Fr(d)1443 4829 y Fu(,)g(we)g(de\002ne)g Fq(\013)1961 4843 y Fl(\003)2039 4829 y Fp(:=)g Fo(f)p Fq(\013)2263 4843 y Fr(x)2308 4829 y Fo(g)2353 4843 y Fr(x)p Fn(2)p Fl(\003)2493 4829 y Fu(.)118 5044 y Fh(The)30 b(particle)g(con\002guration)f(space.)45 b Fu(Our)30 b(particle)g(con\002guration)f(space)h(is)h Fp(\012)i(=)g Fq(S)3099 5011 y Fm(Z)3149 4988 y FA(d)3184 5044 y Fu(,)e Fq(S)39 b Fp(=)33 b Fo(f)p Fp(0)p Fq(;)15 b Fp(1)p Fo(g)118 5155 y Fu(endowed)28 b(with)i(the)f(discrete)g(topology)-12 b(,)29 b(or)f Fp(\012)1773 5169 y Fl(\003)1857 5155 y Fp(=)i Fq(S)2019 5122 y Fl(\003)2101 5155 y Fu(for)f(some)f Fp(\003)j Fo(2)f Fs(F)5 b Fu(.)48 b(When)28 b Fp(\003)j(=)g Fs(T)3301 5122 y Fr(d)3301 5177 y(\017)3371 5155 y Fu(we)e(will)118 5264 y(simply)j(write)g Fp(\012)712 5278 y Fr(\017)744 5264 y Fu(.)50 b(Given)32 b Fq(\021)39 b Fo(2)c Fp(\012)c Fu(and)h Fp(\003)k Fo(\032)f Fs(Z)1809 5231 y Fr(d)1877 5264 y Fu(we)c(denote)g(by)h Fq(\021)2497 5278 y Fl(\003)2581 5264 y Fu(the)g(natural)f(projection)g(over)118 5372 y Fp(\012)184 5386 y Fl(\003)237 5372 y Fu(.)j(Given)25 b(two)i(sites)g Fq(x;)15 b(y)30 b Fo(2)c Fs(Z)1245 5339 y Fr(d)1307 5372 y Fu(and)g(a)g(particle)g(con\002guration)f Fq(\021)k Fu(we)d(denote)g(by)g Fq(\021)3115 5339 y Fr(x;y)3242 5372 y Fu(and)g Fq(\021)3465 5339 y Fr(x)3535 5372 y Fu(the)118 5480 y(con\002gurations)f(obtained)h(from)f Fq(\021)j Fu(by)e(exchanging)d(the)j(values)f(of)g Fq(\021)k Fu(at)d Fq(x;)15 b(y)29 b Fu(and)c(by)h(\223\003ipping\224)e(the)p eop %%Page: 5 5 5 4 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g(DISORDERED)f (LA)-7 b(T)q(TICE)23 b(GAS)860 b(5)118 366 y Fu(value)24 b(of)i Fq(\021)j Fu(at)c Fq(x)g Fu(respectively)-12 b(.)31 b(More)24 b(precisely)-12 b(,)643 605 y Fk(\000)684 679 y Fq(\021)732 642 y Fr(x;y)834 605 y Fk(\001)875 715 y Fr(z)940 679 y Fp(:=)1061 465 y Fk(8)1061 547 y(>)1061 575 y(<)1061 738 y(>)1061 765 y(:)1142 553 y Fq(\021)1187 567 y Fr(y)1322 553 y Fu(if)26 b Fq(z)k Fp(=)25 b Fq(x)1142 682 y(\021)1187 696 y Fr(x)1322 682 y Fu(if)h Fq(z)k Fp(=)25 b Fq(y)1142 812 y(\021)1187 826 y Fr(z)1322 812 y Fu(otherwise)1734 679 y Fq(;)197 b Fp(\()p Fq(\021)2039 642 y Fr(x)2084 679 y Fp(\))2119 693 y Fr(z)2184 679 y Fp(:=)2305 524 y Fk(\()2379 618 y Fp(1)20 b Fo(\000)g Fq(\021)2580 632 y Fr(x)2715 618 y Fu(if)26 b Fq(z)j Fp(=)c Fq(x)2379 747 y(\021)2424 761 y Fr(z)2715 747 y Fu(otherwise)q Fq(:)118 993 y Fu(Sometimes)i(we)g(will)g(write)g Fq(\021)1176 961 y Fr(x;y)1306 993 y Fp(:=)h Fq(S)1486 1007 y Fr(x;y)1587 993 y Fq(\021)i Fu(and)e(call)e Fq(S)2060 1007 y Fr(x;y)2188 993 y Fu(the)h Fh(exchange)f(operator)g Fu(between)g Fq(x)h Fu(and)118 1101 y Fq(y)s Fu(.)k(Finally)-12 b(,)24 b(given)f(a)i(probability)g(measure)e Fq(\026)h Fu(on)g Fp(\012)1920 1115 y Fl(\003)1973 1101 y Fu(,)h(we)f(will)g (denote)g(by)g Fp(V)-8 b(ar)2889 1115 y Fr(\026)2935 1101 y Fp(\()p Fq(\030)t Fp(\))25 b Fu(the)g(variance)e(of)118 1209 y(the)29 b(random)f(variable)g Fq(\030)33 b Fu(w)-11 b(.r)h(.t.)43 b Fq(\026)p Fu(,)29 b(by)g Fq(\026)p Fp(\()p Fq(\030)t Fp(;)15 b Fq(\030)1711 1176 y Fn(0)1735 1209 y Fp(\))29 b Fu(its)h(covariance)d(with)j(the)f(random)f(variable)g Fq(\030)3460 1176 y Fn(0)3512 1209 y Fu(and)118 1317 y(by)d Fq(\026)p Fp(\()p Fq(\030)t(;)15 b(\030)455 1284 y Fn(0)479 1317 y Fp(\))26 b Fu(the)f(scalar)g(product)g(between)f Fq(\030)30 b Fu(and)25 b Fq(\030)1927 1284 y Fn(0)1975 1317 y Fu(in)h(the)f(Hilbert)h(space)f Fq(L)2837 1284 y Fl(2)2876 1317 y Fp(\(\012)2977 1331 y Fl(\003)3030 1317 y Fq(;)15 b(d\026)p Fp(\))p Fu(.)118 1533 y Fh(Local)27 b(functions.)38 b Fu(If)29 b Fq(f)37 b Fu(is)29 b(a)f(measurable)e (function)i(on)2071 1510 y Fp(~)2061 1533 y(\012)h(:=)h(\012)2348 1547 y Fl(D)2428 1533 y Fo(\002)21 b Fp(\012)p Fu(,)29 b(the)f(support)g(of)g Fq(f)10 b Fu(,)28 b(denoted)118 1641 y(by)j Fp(\001)319 1656 y Fr(f)364 1641 y Fu(,)i(is)f(the)g (smallest)f(subset)g(of)h Fs(Z)1480 1608 y Fr(d)1547 1641 y Fu(such)f(that)i Fq(f)10 b Fp(\()p Fq(\013;)15 b(\021)s Fp(\))32 b Fu(depends)f(only)g(on)g Fq(\013)3005 1655 y Fl(\001)3064 1667 y FA(f)3108 1641 y Fq(;)15 b(\033)3200 1655 y Fl(\001)3259 1667 y FA(f)3334 1641 y Fu(and)31 b Fq(f)41 b Fu(is)118 1753 y(called)29 b Fh(local)g Fu(if)i Fp(\001)752 1768 y Fr(f)827 1753 y Fu(is)f(\002nite.)46 b(By)30 b Fo(k)p Fq(f)10 b Fo(k)1468 1767 y Fn(1)1573 1753 y Fu(we)29 b(mean)g(the)h(supremum)f(norm)g(of)h Fq(f)10 b Fu(.)44 b(Given)30 b(two)g(sites)118 1868 y Fq(x;)15 b(y)29 b Fo(2)24 b Fs(Z)434 1835 y Fr(d)496 1868 y Fu(we)h(de\002ne)1201 2043 y Fo(r)1277 2057 y Fr(x;y)1378 2043 y Fq(f)10 b Fp(\()p Fq(\013;)15 b(\021)s Fp(\))26 b(:=)g Fq(f)10 b Fp(\()p Fq(\013;)15 b(\021)2033 2006 y Fr(x;y)2134 2043 y Fp(\))21 b Fo(\000)f Fq(f)10 b Fp(\()p Fq(\013;)15 b(\021)s Fp(\))p Fq(;)1258 2183 y Fo(r)1334 2197 y Fr(x)1378 2183 y Fq(f)10 b Fp(\()p Fq(\013;)15 b(\021)s Fp(\))26 b(:=)g Fq(f)10 b Fp(\()p Fq(\013;)15 b(\021)2033 2146 y Fr(x)2077 2183 y Fp(\))21 b Fo(\000)f Fq(f)10 b Fp(\()p Fq(\013;)15 b(\021)s Fp(\))p Fq(:)118 2379 y Fu(W)-8 b(e)34 b(write)f Fs(G)58 b Fu(for)34 b(the)f(set)h(of)f(measurable,)h(local)f(and)g(bounded)g(functions)g Fq(g)k Fu(on)3101 2356 y Fp(~)3091 2379 y(\012)c Fu(and)g(for)h(any)118 2487 y Fq(g)29 b Fo(2)c Fs(G)50 b Fu(we)25 b(introduce)g(the)g(formal)g (series)g Fq(g)p 1602 2519 47 4 v 1637 2694 a(g)p 1637 2726 V 29 w Fp(:=)1848 2608 y Fk(X)1830 2813 y Fr(x)p Fn(2)p Fm(Z)1967 2794 y FA(d)2012 2694 y Fq(\034)2052 2708 y Fr(x)2096 2694 y Fq(g)118 2970 y Fu(where)30 b Fq(\034)434 2984 y Fr(x)478 2970 y Fq(f)10 b Fp(\()p Fq(\013;)15 b(\021)s Fp(\))35 b(:=)g Fq(f)10 b Fp(\()p Fq(\034)1045 2984 y Fr(x)1088 2970 y Fq(\013;)15 b(\034)1226 2984 y Fr(x)1270 2970 y Fq(\021)s Fp(\))32 b Fu(and)e Fq(\034)1604 2984 y Fr(x)1648 2970 y Fq(\013)h Fu(and)g Fq(\034)1957 2984 y Fr(x)2001 2970 y Fq(\021)j Fu(are)c(the)h(disorder) g(and)g(particle)f(con\002gura-)118 3080 y(tions)c(translated)g(by)f Fq(x)g Fo(2)g Fs(Z)1106 3047 y Fr(d)1167 3080 y Fu(respectively:)1231 3255 y Fp(\()p Fq(\034)1306 3269 y Fr(x)1350 3255 y Fq(\013)p Fp(\))1443 3269 y Fr(z)1509 3255 y Fp(:=)g Fq(\013)1688 3269 y Fr(x)p Fl(+)p Fr(z)1823 3255 y Fq(;)106 b Fp(\()p Fq(\034)2029 3269 y Fr(x)2073 3255 y Fq(\021)s Fp(\))2156 3269 y Fr(z)2222 3255 y Fp(:=)25 b Fq(\021)2388 3269 y Fr(x)p Fl(+)p Fr(z)2523 3255 y Fq(:)118 3430 y Fu(Although)h(the)g (above)f(series)h(is)h(only)e(formal,)g(by)h(the)g(locality)f(of)h Fq(g)s Fu(,)h(the)f(gradient)g Fo(r)3138 3444 y Fr(x;y)3253 3430 y Fq(g)p 3253 3462 V 30 w Fu(is)g(mean-)118 3553 y(ingful)f(for)g(any)g Fq(x;)15 b(y)28 b Fo(2)d Fs(Z)997 3520 y Fr(d)1033 3553 y Fu(.)118 3769 y Fh(Limits.)32 b Fu(Given)24 b Fq(n)g Fu(parameters)g Fq(`)1250 3783 y Fl(1)1289 3769 y Fq(;)15 b(:)g(:)g(:)i(`)1489 3783 y Fr(n)1560 3769 y Fu(we)24 b(use)g(the)h(compact)f(notation)h Fp(lim)2849 3788 y Fr(`)2878 3796 y FA(n)2920 3788 y Fn(!)p Fr(`)3020 3770 y Fi(0)3020 3805 y FA(n)3062 3788 y Fr(;:::)n(;)12 b(`)3201 3797 y Fg(1)3235 3788 y Fn(!)p Fr(`)3335 3766 y Fi(0)3335 3810 y Fg(1)3398 3769 y Fu(for)24 b(the)118 3885 y(ordered)j(limits)i Fp(lim)830 3900 y Fr(`)859 3908 y FA(n)901 3900 y Fn(!)p Fr(`)1001 3882 y Fi(0)1001 3917 y FA(n)1063 3885 y Fq(:)15 b(:)g(:)h Fp(lim)1310 3904 y Fr(`)1339 3913 y Fg(1)1373 3904 y Fn(!)p Fr(`)1473 3882 y Fi(0)1473 3926 y Fg(1)1512 3885 y Fu(.)38 b(The)28 b(same)g(convention)e(is)j(valid)f(when)f(\223)p Fp(lim)n Fu(\224)g(is)i(replaced)118 4001 y(by)c(\223)p Fp(lim)15 b(sup)m Fu(\224)25 b(or)g(\223)p Fp(lim)15 b(inf)t Fu(\224.)118 4222 y(2.2.)45 b FE(The)31 b(model.)46 b Fu(In)29 b(this)i(subsection)f(we)f(describe)g(the)h(lattice)g(gas)f (model)g(at)h(the)g(microscopic)118 4330 y(scale)25 b Fq(\017)g Fu(for)g(a)g(given)g(disorder)h(con\002guration)e Fq(\013)p Fu(.)118 4546 y Fh(Gibbs)c(measures.)28 b Fu(Given)20 b(an)f(external)g(chemical)f(potential)i Fq(\025)26 b Fo(2)e Fs(R)s Fu(,)j(the)20 b(Hamiltonian)g(of)g(the)g(system)118 4654 y(in)25 b(the)h(set)g Fp(\003)f Fo(\032)g Fs(Z)758 4621 y Fr(d)819 4654 y Fu(is)h(de\002ned)g(as)1380 4829 y Fq(H)1463 4791 y Fr(\013)1456 4851 y Fl(\003)1512 4829 y Fp(\()p Fq(\021)s Fp(\))g(=)f Fo(\000)1840 4743 y Fk(X)1838 4939 y Fr(x)p Fn(2)p Fl(\003)1974 4829 y Fp(\()p Fq(\013)2067 4843 y Fr(x)2132 4829 y Fp(+)19 b Fq(\025)p Fp(\))p Fq(\021)2355 4843 y Fr(x)118 5119 y Fu(and)28 b(the)g(corresponding)f(grand)g (canonical)f(Gibbs)j(measure)d(on)i Fp(\012)2515 5133 y Fl(\003)2568 5119 y Fu(,)g(denoted)g(by)g Fq(\026)3149 5074 y Fr(\013;\025)3149 5147 y Fl(\003)3259 5119 y Fu(,)h(is)f(simply) 118 5227 y(the)e(product)f(measure)1302 5405 y Fq(\026)1357 5360 y Fr(\013;\025)1357 5433 y Fl(\003)1467 5405 y Fp(\()p Fq(\021)s Fp(\))h(:=)1809 5343 y(1)p 1742 5384 179 4 v 1742 5485 a Fq(Z)1811 5441 y Fr(\013;\025)1804 5514 y Fl(\003)1946 5405 y Fp(exp\()p Fo(\000)p Fq(H)2274 5367 y Fr(\013)2267 5427 y Fl(\003)2323 5405 y Fp(\()p Fq(\021)s Fp(\)\))983 b Fu(\(2.3\))p eop %%Page: 6 6 6 5 bop 118 47 a Fj(6)1188 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 374 y Fu(where)25 b Fq(Z)458 330 y Fr(\013;\025)451 402 y Fl(\003)593 374 y Fu(is)h(such)f(that)h Fq(\026)1123 330 y Fr(\013;\025)1123 402 y Fl(\003)1233 374 y Fp(\(\012)1334 388 y Fl(\003)1388 374 y Fp(\))f(=)g(1)p Fu(.)118 482 y(F)-6 b(or)43 b(our)f(purposes)h (it)h(is)f(important)g(to)h(introduce)e(also)h(the)g(canonical)f (measures)g Fq(\027)3301 449 y Fr(\013)3295 509 y Fl(\003)p Fr(;m)3430 482 y Fu(.)84 b(Let)118 608 y Fq(N)191 622 y Fl(\003)244 608 y Fp(\()p Fq(\021)s Fp(\))27 b(=)485 540 y Fk(P)581 635 y Fr(x)p Fn(2)p Fl(\003)736 608 y Fq(\021)781 622 y Fr(x)850 608 y Fu(and)e(let)g Fq(m)g Fo(2)g Fp([0)p Fq(;)1489 573 y Fl(1)p 1462 588 89 4 v 1462 641 a Fn(j)p Fl(\003)p Fn(j)1560 608 y Fq(;)15 b(:)g(:)g(:)i(;)e Fp(1])p Fu(.)32 b(Then)1313 809 y Fq(\027)1364 772 y Fr(\013)1358 832 y Fl(\003)p Fr(;m)1493 809 y Fp(\()p Fo(\001)p Fp(\))26 b(=)f Fq(\026)1765 765 y Fr(\013;\025)1765 838 y Fl(\003)1875 809 y Fp(\()p Fo(\001)15 b(j)g Fq(N)2063 823 y Fl(\003)2142 809 y Fp(=)25 b Fq(m)p Fo(j)p Fp(\003)p Fo(j)p Fp(\))993 b Fu(\(2.4\))118 973 y(The)46 b(random)f(variable)h Fq(N)1105 987 y Fl(\003)1204 973 y Fu(will)g(usually)f(be)g(referred)g (to)h(as)g(the)g(number)f(of)h(particles)f(and)118 1081 y Fq(m)198 1095 y Fl(\003)276 1081 y Fp(:=)25 b Fq(N)470 1095 y Fl(\003)524 1081 y Fq(=)p Fo(j)p Fp(\003)p Fo(j)k Fu(as)g(the)g(particle)f(density)h(or)f(simply)h(the)f(density)-12 b(.)42 b(The)29 b(set)g(of)f(all)g(canonical)f(mea-)118 1191 y(sure)k Fq(\027)368 1158 y Fr(\013)362 1218 y Fl(\003)p Fr(;m)528 1191 y Fu(as)g Fq(m)g Fu(varies)g(in)g Fp([0)p Fq(;)1271 1155 y Fl(1)p 1244 1170 V 1244 1223 a Fn(j)p Fl(\003)p Fn(j)1342 1191 y Fq(;)15 b(:)g(:)g(:)i(;)e Fp(1])32 b Fu(will)f(be)g(denoted)g(by)g Fo(M)2537 1158 y Fr(\013)2586 1191 y Fp(\(\003\))p Fu(.)50 b(Notice)31 b(that)h Fq(\027)3320 1158 y Fr(\013)3314 1218 y Fl(\003)p Fr(;m)3480 1191 y Fu(does)118 1316 y(not)i(depend)f(on)g(the)h (chemical)f(potential)h Fq(\025)p Fu(.)56 b(However)-10 b(,)35 b(as)f(it)h(is)f(well)f(known)g([6],)j(the)e(canoni-)118 1423 y(cal)29 b(and)g(grand)g(canonical)f(Gibbs)i(measures)e(are)h (closely)f(related)h(if)h(the)g(chemical)e(potential)h Fq(\025)h Fu(is)118 1542 y(canonically)e(conjugate)g(to)i(the)g (density)g Fq(m)g Fu(in)g(the)f(sense)h(that)g(the)g Fh(average)e(density)h Fu(w)-11 b(.r)h(.t.)45 b Fq(\026)3460 1498 y Fr(\013;\025)3460 1571 y Fl(\003)3600 1542 y Fu(is)118 1650 y(equal)27 b(to)h Fq(m)p Fu(.)38 b(W)n(ith)29 b(this)g(in)f(mind,) g(for)g(any)f Fq(m)i Fo(2)g Fp([0)p Fq(;)15 b Fp(1])p Fu(,)30 b(we)d(de\002ne)g(the)h Fh(empirical)f(chemical)g(poten-)118 1771 y(tial)i Fq(\025)329 1785 y Fl(\003)382 1771 y Fp(\()p Fq(\013;)15 b(m)p Fp(\))30 b Fu(as)g(the)f(unique)f(value)g(of)h Fq(\025)f Fu(such)h(that)h Fq(\026)2110 1727 y Fr(\013;\025)2110 1800 y Fl(\003)2220 1771 y Fp(\()p Fq(N)2328 1785 y Fl(\003)2381 1771 y Fp(\))i(=)f Fq(m)p Fo(j)p Fp(\003)p Fo(j)p Fu(,)f(the)f Fh(annealed)d(chemical)118 1888 y(potential)j Fq(\025)542 1902 y Fl(0)582 1888 y Fp(\()p Fq(m)p Fp(\))h Fu(as)h(the)f(unique)g Fq(\025)g Fu(such)g(that)h Fs(E)1871 1814 y Fk(\002)1914 1888 y Fq(\026)1969 1855 y Fr(\013;\025)2080 1888 y Fp(\()p Fq(\021)2160 1902 y Fl(0)2199 1888 y Fp(\))2234 1814 y Fk(\003)2306 1888 y Fp(=)i Fq(m)d Fu(and)g(the)g(corresponding)f (static)118 2018 y(compressibility)34 b Fq(\037)p Fp(\()p Fq(m)p Fp(\))g Fu(as)g Fq(\037)p Fp(\()p Fq(m)p Fp(\))39 b(=)f Fs(E)1513 1944 y Fk(\002)1557 2018 y Fq(\026)1612 1985 y Fr(\013;\025)1718 1994 y Fg(0)1752 1985 y Fl(\()p Fr(m)p Fl(\))1873 2018 y Fp(\()p Fq(\021)1953 2032 y Fl(0)1993 2018 y Fp(;)15 b Fq(\021)2078 2032 y Fl(0)2118 2018 y Fp(\))2153 1944 y Fk(\003)2192 2018 y Fu(.)55 b(Since)2549 1982 y Fr(@)p 2528 1997 83 4 v 2528 2049 a(@)t(\025)2621 2018 y Fq(\026)2676 1974 y Fr(\013;\025)2676 2047 y Fl(\003)2786 2018 y Fp(\()p Fq(f)10 b Fp(\))38 b(=)h Fq(\026)3114 1974 y Fr(\013;\025)3114 2047 y Fl(\003)3224 2018 y Fp(\()p Fq(f)10 b Fp(;)15 b Fq(N)3427 2032 y Fl(\003)3480 2018 y Fp(\))34 b Fu(for)118 2126 y(any)25 b(local)f(function)h Fq(f)10 b Fu(,)25 b(we)g(get)g(the)g(following)g(thermodynamic)f (relations:)538 2270 y Fq(@)p 498 2311 134 4 v 498 2394 a(@)5 b(m)641 2332 y(\025)694 2346 y Fl(\003)747 2332 y Fp(\()p Fq(\013;)15 b(m)p Fp(\))27 b(=)1118 2258 y Fk(\002)1171 2332 y Fq(\026)1226 2284 y Fr(\013;\025)1332 2295 y Fg(\003)1377 2284 y Fl(\()p Fr(\013;m)p Fl(\))1226 2361 y(\003)1564 2332 y Fp(\()p Fq(m)1679 2346 y Fl(\003)1732 2332 y Fp(;)15 b Fq(N)1845 2346 y Fl(\003)1898 2332 y Fp(\))1933 2258 y Fk(\003)1972 2281 y Fn(\000)p Fl(1)2182 2332 y Fu(and)2497 2270 y Fq(@)p 2457 2311 V 2457 2394 a(@)5 b(m)2600 2332 y(\025)2653 2346 y Fl(0)2693 2332 y Fp(\()p Fq(m)p Fp(\))26 b(=)f Fq(\037)p Fp(\()p Fq(m)p Fp(\))3172 2294 y Fn(\000)p Fl(1)3266 2332 y Fq(:)118 2575 y FE(Notation)30 b(warning.)81 b Fh(From)28 b(now)g(on,)g(in)g (order)g(to)g(keep)f(the)h(notation)f(to)h(an)g(acceptable)e(level,)j (we)118 2683 y(need)24 b(to)h(adopt)f(the)h(following)f(shortcuts)h (whenever)e(no)i(confusion)e(arises.)185 2818 y(i\))42 b(Most)26 b(of)e(the)i(times)f(the)g(label)g Fq(\013)h Fh(will)g(be)f(omitted.)32 b(That)26 b(means)e(that)i(quantities)e (like)h Fq(\026)3317 2785 y Fr(\025)3317 2845 y Fl(\003)3370 2818 y Fp(\()p Fq(f)10 b Fp(\))26 b Fh(will)291 2926 y(actually)i(be)g(random)f(variables)h(w)-5 b(.r)-12 b(.t)29 b(the)f(disorder)e Fq(\013)p Fh(.)41 b(Moreover)-12 b(,)29 b(the)f(label)g Fq(\025)g Fh(of)f(the)h(chemical)291 3034 y(potential)c(will)i(be)f(omitted)f(when)h Fq(\025)g Fp(=)g(0)p Fh(.)159 3142 y(ii\))42 b(If)28 b(the)f(region)f Fp(\003)i Fh(on)e(which)i(the)f(Gibbs)h(measures)e(or)-12 b(,)28 b(later)-12 b(,)29 b(the)e(generator)f(of)g(the)i(dynamics)e (are)291 3250 y(de\002ned)f(coincides)g(with)i Fs(T)1217 3217 y Fr(d)1217 3272 y(\017)1260 3250 y Fh(,)g(then)g(the)g(suf\002x)g Fp(\003)h Fh(will)f(be)f(simply)h(replaced)f(by)h Fq(\017)g Fh(while)g(if)g Fp(\003)h(=)g Fs(Z)3625 3217 y Fr(d)291 3357 y Fh(it)e(will)f(simply)g(be)g(dropped)f(\(i.e.)31 b Fq(\026)1497 3371 y Fr(\017)1555 3357 y Fp(:=)25 b Fq(\026)1731 3325 y Fr(\013)1731 3393 y Fm(T)1781 3374 y FA(d)1781 3409 y(\017)1815 3357 y Fh(\).)133 3515 y(iii\))42 b(The)36 b(symbol)f Fq(\026)825 3467 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))825 3544 y(\003)1023 3515 y Fh(will)h(always)f(denote)e (the)j(grand)f(canonical)e(Gibbs)j(measure)f(on)g Fp(\012)3402 3529 y Fl(\003)3490 3515 y Fh(with)291 3623 y(empirical)25 b(chemical)f(potential)h Fq(\025)1451 3637 y Fl(\003)1504 3623 y Fp(\()p Fq(\013;)15 b(m)p Fp(\))p Fh(.)144 3731 y(iv\))42 b(The)27 b(letter)g Fq(c)g Fh(will)g(denote)e(a)h(generic)f (positive)i(constant)e(depending)f(only)i(on)g Fq(d)h Fh(and)f Fq(B)31 b Fh(that)c(may)291 3839 y(vary)f(from)f(estimate)f (to)h(estimate.)118 4081 y(The)d(dynamics.)28 b Fu(The)22 b(lattice)g(gas)f(dynamics)g(we)h(are)f(interested)g(in)h(is)g(the)g (continuous)e(time)i(Markov)118 4189 y(chain)f(on)g Fp(\012)542 4203 y Fr(\017)595 4189 y Fu(described)h(by)f(the)g(Markov)g(generator) e Fq(\017)2020 4156 y Fn(\000)p Fl(2)2115 4189 y Fo(L)2178 4203 y Fr(\017)2231 4189 y Fu(where)i Fo(L)2561 4203 y Fr(\017)2618 4189 y Fp(:=)k Fo(L)2802 4211 y Fm(T)2852 4193 y FA(d)2852 4228 y(\017)2908 4189 y Fu(and)c(for)g(any)g Fp(\003)k Fo(\032)g Fs(Z)3625 4156 y Fr(d)1475 4385 y Fo(L)1538 4399 y Fl(\003)1591 4385 y Fq(f)10 b Fp(\()p Fq(\021)s Fp(\))25 b(=)1887 4298 y Fk(X)1885 4496 y Fr(b)p Fn(\032)p Fl(\003)2034 4385 y Fo(L)2097 4400 y Fr(b)2131 4385 y Fq(f)10 b Fp(\()p Fq(\021)s Fp(\))118 4639 y Fu(where,)25 b(for)g(any)g(bond)g Fq(b)h Fp(=)f Fo(f)p Fq(x;)15 b(y)s Fo(g)p Fu(,)1344 4800 y Fo(L)1407 4814 y Fr(x;y)1507 4800 y Fq(f)10 b Fp(\()p Fq(\021)s Fp(\))26 b(:=)f Fq(c)1866 4762 y Fr(\013)1866 4822 y(x;y)1967 4800 y Fp(\()p Fq(\021)s Fp(\))p Fo(r)2161 4814 y Fr(x;y)2263 4800 y Fq(f)10 b Fp(\()p Fq(\021)s Fp(\))118 4965 y Fu(The)29 b(non-negative)e(real)h (quantities)i Fq(c)1480 4932 y Fr(\013)1480 4987 y(x;y)1581 4965 y Fp(\()p Fq(\021)s Fp(\))f Fu(are)f(the)h Fh(transition)f(rates)g Fu(for)g(the)h(process.)41 b(They)29 b(are)118 5076 y(de\002ned)c(as) 899 5199 y Fq(c)938 5161 y Fr(\013)938 5221 y(x;x)p Fl(+)p Fr(e)1129 5199 y Fp(\()p Fq(\021)s Fp(\))h(=)f Fq(f)1414 5213 y Fr(e)1451 5199 y Fp(\()p Fq(\013)1544 5213 y Fr(x)1588 5199 y Fq(;)15 b(\021)1673 5213 y Fr(x)1718 5199 y Fq(;)g(\013)1816 5213 y Fr(x)p Fl(+)p Fr(e)1948 5199 y Fq(;)g(\021)2033 5213 y Fr(x)p Fl(+)p Fr(e)2165 5199 y Fp(\))91 b Fo(8)p Fq(x)25 b Fo(2)f Fs(Z)2569 5161 y Fr(d)2606 5199 y Fq(;)40 b(e)26 b Fo(2)f(E)118 5370 y Fu(where)41 b Fq(f)450 5384 y Fr(e)527 5370 y Fu(is)h(a)g(generic)d(bounded)i(function)g(on)1951 5297 y Fk(\000)1993 5370 y Fp([)p Fo(\000)p Fq(B)5 b(;)15 b(B)5 b Fp(])30 b Fo(\002)h Fq(S)2495 5297 y Fk(\001)2536 5319 y Fl(2)2617 5370 y Fu(such)41 b(that)h Fq(f)3082 5384 y Fr(e)3119 5370 y Fp(\()p Fq(a;)15 b(s;)g(a)3373 5337 y Fn(0)3397 5370 y Fq(;)g(s)3480 5337 y Fn(0)3503 5370 y Fp(\))53 b(=)118 5480 y Fq(f)163 5494 y Fr(e)200 5480 y Fp(\()p Fq(a)283 5447 y Fn(0)306 5480 y Fq(;)15 b(s)389 5447 y Fn(0)413 5480 y Fq(;)g(a;)g(s)p Fp(\))40 b Fu(and)f Fq(f)892 5494 y Fr(e)977 5480 y Fo(\025)48 b Fq(c)h(>)f Fp(0)40 b Fu(for)f(a)h(suitable)f(constant)g Fq(c)p Fu(.)74 b(Thanks)40 b(to)g(this)g(de\002nition)g(the)p eop 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Fp(\))26 b(=)f Fq(f)1011 966 y Fr(e)1047 952 y Fp(\()p Fq(a;)15 b(s)1213 915 y Fn(0)1237 952 y Fq(;)g(a)1325 915 y Fn(0)1348 952 y Fq(;)g(s)p Fp(\))p Fq(e)1508 915 y Fn(\000)p Fl(\()p Fr(s)1623 891 y Fi(0)1646 915 y Fn(\000)p Fr(s)p Fl(\)\()p Fr(a)1825 891 y Fi(0)1849 915 y Fn(\000)p Fr(a)p Fl(\))2155 952 y Fo(8)p Fq(e)25 b Fo(2)f(E)8 b Fq(;)41 b(a;)15 b(a)2616 915 y Fn(0)2665 952 y Fo(2)25 b Fp([)p Fo(\000)p Fq(B)5 b(;)15 b(B)5 b Fp(])p Fq(;)30 b(s;)15 b(s)3241 915 y Fn(0)3290 952 y Fo(2)24 b Fq(S)118 1107 y Fu(which)k(implies)g(that)g (the)g(generator)e Fo(L)1509 1121 y Fl(\003)1589 1107 y Fu(becomes)g(a)h(selfadjoint)i(operator)e(on)g Fq(L)3028 1074 y Fl(2)3067 1107 y Fp(\()p Fq(\026)3157 1074 y Fr(\025)3157 1134 y Fl(\003)3210 1107 y Fp(\))h Fu(for)f(any)g Fq(\025)p Fu(.)118 1215 y(Actually)-12 b(,)23 b(since)h(the)g(moves)f(of)h(the)g (Markov)g(chain)g(generated)e(by)i Fo(L)2542 1229 y Fl(\003)2618 1215 y Fu(do)h(not)e(change)g(the)h(number)118 1323 y(of)j(particles,)g (for)g(any)g(canonical)e(Gibbs)j(measure)e Fq(\027)33 b Fo(2)27 b(M)p Fp(\(\003\))h Fu(the)f(operator)g Fo(L)2952 1337 y Fl(\003)3031 1323 y Fu(is)h(selfadjoint)g(on)118 1431 y Fq(L)180 1398 y Fl(2)219 1431 y Fp(\()p Fq(\027)6 b Fp(\))26 b Fu(with)h(a)e(positive)h(spectral)f(gap)974 1637 y Fp(gap\()p Fo(L)1213 1651 y Fl(\003)1267 1637 y Fq(;)15 b(\027)6 b Fp(\))25 b(:=)h(inf)1650 1536 y Fk(n)1736 1575 y Fq(\027)6 b Fp(\()p Fq(f)f(;)15 b Fo(\000L)2046 1589 y Fl(\003)2098 1575 y Fq(f)10 b Fp(\))p 1736 1616 452 4 v 1807 1699 a(V)-8 b(ar)1949 1713 y Fr(\027)1992 1699 y Fp(\()p Fq(f)10 b Fp(\))2198 1637 y(;)30 b(V)-8 b(ar)2395 1651 y Fr(\027)2438 1637 y Fp(\()p Fq(f)10 b Fp(\))25 b Fo(6)p Fp(=)g(0)2744 1536 y Fk(o)3459 1637 y Fu(\(2.5\))118 1843 y(and)34 b(the)h(corresponding)e(Markov)h(chain)g (is)h(irreducible)e(on)h Fo(f)p Fq(\021)45 b Fo(2)39 b Fp(\012)2641 1857 y Fl(\003)2735 1843 y Fp(:)81 b Fq(N)2914 1857 y Fl(\003)2967 1843 y Fp(\()p Fq(\021)s Fp(\))41 b(=)f Fq(n)p Fo(g)34 b Fu(for)h(any)118 1951 y Fq(n)25 b Fo(2)g Fp([0)p Fq(;)15 b Fp(1)p Fq(;)g(:)g(:)g(:)j(;)d Fo(j)p Fp(\003)p Fo(j)p Fp(])p Fu(.)118 2082 y(Given)31 b Fq(g)38 b Fo(2)d Fs(G)56 b Fu(we)31 b(denote)f(by)h Fo(L)p Fq(g)k Fu(the)c(function)1890 2013 y Fk(P)1986 2108 y Fr(b)p Fn(\032)p Fm(Z)2121 2089 y FA(d)2170 2082 y Fo(L)2233 2097 y Fr(b)2283 2082 y Fq(g)s Fu(.)49 b(Given)31 b Fp(\001)j Fo(\032)h Fp(\003)c Fu(and)g(a)h(probability)118 2189 y(measure)24 b Fq(\026)h Fu(on)g Fp(\012)750 2203 y Fl(\003)803 2189 y Fu(,)g(for)h(any)f Fq(f)34 b Fu(with)27 b(support)e(inside)h Fp(\003)g Fu(we)f(will)g(set)1264 2383 y Fo(D)1334 2397 y Fl(\001)1397 2383 y Fp(\()p Fq(f)10 b Fp(;)15 b Fq(\026)p Fp(\))26 b(:=)1774 2321 y(1)p 1774 2362 46 4 v 1774 2445 a(2)1851 2297 y Fk(X)1845 2494 y Fr(b)p Fn(\032)p Fl(\001)2004 2383 y Fq(\026)2059 2309 y Fk(\000)2100 2383 y Fq(c)2139 2398 y Fr(b)2174 2383 y Fp(\()p Fo(r)2285 2398 y Fr(b)2319 2383 y Fq(f)10 b Fp(\))2409 2345 y Fl(2)2448 2309 y Fk(\001)2490 2383 y Fq(:)118 2628 y Fu(Notice)25 b(that,)i(if)f Fp(\001)f(=)g(\003)h Fu(and)f Fq(\026)g Fu(is)h(either)g(a)f(grand)g(canonical)f(or)h(a)h (canonical)e(measure)g(on)h Fp(\003)p Fu(,)h(then)118 2736 y(the)j(above)f(expression)g(is)h(nothing)g(but)f(the)h(Dirichlet) g(form)f(of)g(the)h(Markov)f(chain)g(generated)f(by)118 2844 y Fo(L)181 2858 y Fl(\003)259 2844 y Fu(w)-11 b(.r)h(.t.)32 b Fq(\026)p Fu(.)118 2974 y(Finally)-12 b(,)22 b(given)f(a)h (probability)h(measure)e Fq(\026)g Fu(on)h Fp(\012)1800 2988 y Fr(\017)1854 2974 y Fu(and)g Fq(T)38 b(>)25 b Fp(0)p Fu(,)e(we)f(denote)g(by)g Fs(P)2919 2930 y Fr(\013;\026)2907 3003 y(T)3052 2974 y Fu(the)h(distribution)118 3090 y(at)k(time)g Fq(T)40 b Fu(of)27 b(the)g(Markov)f(chain)h(on)f Fs(T)1531 3057 y Fr(d)1531 3113 y(\017)1600 3090 y Fu(with)i(generator)d Fq(\017)2251 3057 y Fn(\000)p Fl(2)2346 3090 y Fo(L)2409 3057 y Fr(\013)2409 3113 y(\017)2485 3090 y Fu(and)h(initial)i (distribution)h Fq(\026)p Fu(,)e(and)118 3198 y(by)33 b Fs(P)312 3165 y Fr(\013;\026)456 3198 y Fu(the)h(induced)e (probability)i(measure)d(on)i(the)g(Skorohod)f(space)h Fq(D)s Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fp(\012)3173 3212 y Fr(\017)3207 3198 y Fp(\))33 b Fu(\(see)f([4]\).)118 3306 y(The)e(expectation)f(w)-11 b(.r)h(.t.)45 b Fs(P)1087 3273 y Fr(\013;\026)1229 3306 y Fu(will)29 b(be)g(denoted)h(by)f Fs(E)2059 3273 y Fr(\013;\026)2177 3306 y Fu(.)44 b(Notice)29 b(that,)k(in)c(turn,)i Fs(P)3143 3273 y Fr(\013;\026)3284 3306 y Fu(induces)e(a)118 3414 y(probability)d(measure)e Fq(Q)1017 3381 y Fr(\013;\026)1154 3414 y Fu(on)h Fq(D)s Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)1743 3428 y Fl(1)1784 3414 y Fp(\))26 b Fu(by)f(the)g(formula)g Fs(P)2521 3381 y Fr(\013;\026)2653 3414 y Fo(\016)20 b Fq(\031)2773 3381 y Fn(\000)p Fl(1)2770 3437 y Fr(\017)2868 3414 y Fu(,)25 b(where)1239 3574 y Fq(\031)1291 3588 y Fr(\017)1323 3574 y Fp(\()p Fq(\021)s Fp(\))h(:=)g(Av)1706 3596 y Fr(x)p Fn(2)p Fm(T)1843 3577 y FA(d)1843 3612 y(\017)1893 3574 y Fq(\021)1938 3588 y Fr(x)1997 3574 y Fq(\016)2037 3588 y Fr(\017x)2135 3574 y Fo(2)f(M)2330 3588 y Fl(1)2369 3574 y Fp(\()p Fs(T)2463 3536 y Fr(d)2506 3574 y Fp(\))118 3733 y Fu(denotes)g(the)h(empirical)e(measure.)118 3971 y FE(W)-8 b(arning)28 b Fu(In)g(all)f(the)h(above)g(measures,)f (the)h(crucial)f(dependence)f(on)h(the)h(parameter)f Fq(\017)i(>)g Fp(0)f Fu(does)118 4079 y(not)d(appear)g(in)h(the)f (various)h(symbols)f(in)g(order)g(to)h(keep)e(the)i(notation)g(to)f(an) g(acceptable)f(level.)118 4256 y(2.3.)45 b FE(Main)31 b(results.)44 b Fu(Our)29 b(\002rst)h(result)e(concerns)f(the)j (existence)e(and)h(regularity)f(of)i(the)f(diffusion)118 4363 y(matrix)37 b Fq(D)s Fp(\()p Fq(m)p Fp(\))g Fu(corresponding)f(to) h(the)g(usual)f(Green-K)n(ubo)f(matrix)i(\(see)g([34)o(],)j (proposition)e Fp(2)p Fq(:)p Fp(2)118 4471 y Fu(page)25 b Fp(180)p Fu(\).)118 4602 y FE(Theorem)32 b(2.1.)44 b Fh(Let)30 b Fq(d)j Fo(\025)g Fp(3)p Fh(.)46 b(Then)30 b(for)f(any)g(density)g Fq(m)k Fo(2)g Fp(\(0)p Fq(;)15 b Fp(1\))32 b Fh(there)d(exists)h(a)f(unique)h(symmetric)118 4710 y Fq(d)21 b Fo(\002)f Fq(d)25 b Fh(matrix)h Fq(D)s Fp(\()p Fq(m)p Fp(\))p Fh(,)g(such)f(that)173 4832 y Fk(\000)215 4906 y Fq(a;)15 b(D)s Fp(\()p Fq(m)p Fp(\))p Fq(a)579 4832 y Fk(\001)647 4906 y Fp(=)856 4844 y(1)p 753 4885 253 4 v 753 4968 a(2)p Fq(\037)p Fp(\()p Fq(m)p Fp(\))1043 4906 y(inf)1031 4967 y Fr(g)r Fn(2)p Fm(G)1196 4819 y Fk(X)1200 5016 y Fr(e)p Fn(2E)1357 4906 y Fs(E)1412 4805 y Fk(h)1476 4906 y Fq(\026)1531 4868 y Fr(\013;\025)1637 4877 y Fg(0)1671 4868 y Fl(\()p Fr(m)p Fl(\))1793 4805 y Fk(\020)1847 4906 y Fq(c)1886 4868 y Fr(\013)1886 4928 y Fl(0)p Fr(;e)1978 4832 y Fk(\000)2020 4906 y Fq(a)2068 4920 y Fr(e)2105 4906 y Fp(\()p Fq(\021)2185 4920 y Fr(e)2242 4906 y Fo(\000)20 b Fq(\021)2378 4920 y Fl(0)2418 4906 y Fp(\))h(+)e Fo(r)2640 4920 y Fl(0)p Fr(;e)2732 4906 y Fq(g)p 2732 4938 47 4 v 2779 4832 a Fk(\001)2820 4855 y Fl(2)2860 4805 y Fk(\021)c(i)3063 4906 y Fo(8)p Fq(a)25 b Fo(2)g Fs(R)3332 4868 y Fr(d)3379 4906 y Fq(:)55 b Fu(\(2.6\))118 5151 y Fh(Moreover)25 b Fq(D)s Fp(\()p Fq(m)p Fp(\))g Fh(is)g(continuous)f(in)h(the)g(open)f(interval)h Fp(\(0)p Fq(;)15 b Fp(1\))27 b Fh(and)1164 5307 y Fp(0)f Fq(<)f(c)1370 5270 y Fn(\000)p Fl(1)1465 5307 y Fp(1)-25 b(I)25 b Fo(\024)g Fq(D)s Fp(\()p Fq(m)p Fp(\))g Fo(\024)g Fq(c)p Fp(1)-25 b(I)92 b Fo(8)p Fq(m)24 b Fo(2)h Fp(\(0)p Fq(;)15 b Fp(1\))118 5460 y Fh(for)25 b(some)f(positive)h(constant)e Fq(c)p Fh(.)p eop %%Page: 8 8 8 7 bop 118 45 a Fj(8)1188 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y FE(R)m(emark)35 b(2.2.)46 b Fh(W)-7 b(e)33 b(actually)g(expect)g(the)g(matrix)h Fq(D)i Fh(to)d(be)g(continuously)f(extendable)g(to)h(the)g(closed)118 474 y(interval)i Fp([0)p Fq(;)15 b Fp(1])p Fh(.)62 b(In)35 b(particular)g(we)g(expect)f(that)i Fq(D)s Fp(\()p Fq(m)p Fp(\))f Fh(converges)e(to)i(the)g(diffusion)e(matrix)i(of)g(the)118 582 y(random)19 b(walk)h(of)g(a)g(single)f(particle)h(in)g(the)g (random)g(environment)e Fq(\013)j Fh(as)f Fq(m)h Fh(goes)e(to)h(zero,)h (as)f(con\002rmed)118 690 y(by)26 b(simulations)e(\(see)g Fu([22)o(])p Fh(\).)218 822 y Fu(In)h(order)g(to)g(state)i(the)e(next)g (main)g(result)g(we)g(need)f(the)i(following)e(de\002nition.)118 953 y FE(De\002nition)32 b(2.3.)45 b Fh(Given)31 b(a)g(Lebesgue)f (absolutely)g(continuous)g(measure)g Fq(m)p Fp(\()p Fq(\022)s Fp(\))p Fq(d\022)38 b Fo(2)d(M)3212 967 y Fl(2)3251 953 y Fp(\()p Fs(T)3345 920 y Fr(d)3388 953 y Fp(\))p Fh(,)e(a)e(se-)118 1061 y(quence)23 b(of)h(probability)h(measures)e Fq(\026)1376 1028 y Fr(\017)1434 1061 y Fh(on)h Fp(\012)1621 1075 y Fr(\017)1678 1061 y Fh(is)g(said)g(to)h(correspond)d(to)j(the)f (macroscopic)f(pro\002le)i Fq(m)p Fp(\()p Fo(\001)p Fp(\))118 1171 y Fh(if)-8 b(,)28 b(under)f Fq(\026)522 1138 y Fr(\017)554 1171 y Fh(,)h(the)f(random)f(variable)h Fq(\031)1470 1185 y Fr(\017)1530 1171 y Fh(in)g Fo(M)1743 1185 y Fl(1)1783 1171 y Fp(\()p Fs(T)1877 1138 y Fr(d)1919 1171 y Fp(\))h Fh(converges)d(in)i(probability)h(to)f Fq(m)p Fp(\()p Fq(\022)s Fp(\))p Fq(d\022)i Fh(as)e Fq(\017)i Fo(#)g Fp(0)p Fh(,)118 1281 y(i.e.)j(for)24 b(any)h(smooth)g(function)f Fq(H)32 b Fh(on)24 b Fs(T)1499 1248 y Fr(d)1567 1281 y Fh(and)g(any)g Fq(\016)29 b(>)c Fp(0)818 1480 y(lim)831 1542 y Fr(\017)p Fn(#)p Fl(0)959 1480 y Fq(\026)1014 1443 y Fr(\017)1047 1407 y Fk(\000)1088 1403 y(\014)1088 1458 y(\014)1134 1480 y Fp(Av)1251 1502 y Fr(x)p Fn(2)p Fm(T)1388 1484 y FA(d)1388 1519 y(\017)1423 1480 y Fq(H)7 b Fp(\()p Fq(\017x)p Fp(\))p Fq(\021)1710 1494 y Fr(x)1774 1480 y Fo(\000)1865 1356 y Fk(Z)1916 1563 y Fm(T)1966 1544 y FA(d)2015 1480 y Fq(H)g Fp(\()p Fq(\022)s Fp(\))p Fq(m)p Fp(\()p Fq(\022)s Fp(\))p Fq(d\022)2518 1403 y Fk(\014)2518 1458 y(\014)2573 1480 y Fq(>)25 b(\016)2712 1407 y Fk(\001)2780 1480 y Fp(=)g(0)15 b Fq(:)118 1709 y Fu(W)n(ith)24 b(the)f(above)f(de\002nition)h(the)g(existence)e(of)i (the)g(hydrodynamical)f(limit)h(for)f(almost)h(all)f(disorder)118 1817 y(con\002gurations)j(reads)g(as)h(follows.)118 1949 y FE(Theorem)34 b(2.4.)45 b Fh(Let)32 b Fq(d)37 b Fo(\025)g Fp(3)p Fh(,)d(let)e Fq(T)49 b(>)37 b Fp(0)32 b Fh(and)f(assume)g(that)h Fq(D)s Fp(\()p Fq(m)p Fp(\))h Fh(can)e(be)h(continuously)f(extended)118 2057 y(to)i(the)f(closed)f(interval)h Fp([0)p Fq(;)15 b Fp(1])p Fh(.)55 b(Then)32 b(almost)g(all)h(disorder)d (con\002gurations)h Fq(\013)i Fh(satisfy)f(the)g(following)118 2165 y(property)-5 b(.)31 b(Let)25 b Fq(m)712 2179 y Fl(0)752 2165 y Fp(\()p Fq(\022)s Fp(\))g Fo(2)g(M)1088 2179 y Fl(2)1153 2165 y Fh(and)f(suppose)f(that)j(the)f(Cauchy)h (problem)1042 2256 y Fk(\()1116 2347 y Fq(@)1164 2361 y Fr(t)1194 2347 y Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))25 b(=)g Fo(r)1660 2362 y Fr(\022)1699 2246 y Fk(\020)1753 2347 y Fq(D)1831 2274 y Fk(\000)1872 2347 y Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))2141 2274 y Fk(\001)2183 2347 y Fo(r)2259 2362 y Fr(\022)2313 2347 y Fq(m)p Fp(\()p Fq(t;)g(\022)s Fp(\))2582 2246 y Fk(\021)1116 2497 y Fq(m)p Fp(\(0)p Fq(;)g(\022)s Fp(\))26 b(=)f Fq(m)1599 2511 y Fl(0)1638 2497 y Fp(\()p Fq(\022)s Fp(\))3459 2412 y Fu(\(2.7\))118 2659 y Fh(has)g(a)g(unique)f(weak)g(solution)h Fq(m)g Fo(2)g Fq(C)7 b Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)1839 2673 y Fl(2)1880 2659 y Fp(\))25 b Fh(satisfying)f (the)h(energy)f(estimate)1272 2756 y Fk(Z)1363 2782 y Fr(T)1323 2962 y Fl(0)1433 2880 y Fq(ds)1538 2756 y Fk(Z)1589 2962 y Fm(T)1639 2943 y FA(d)1689 2880 y Fq(d\022)17 b Fo(jr)1897 2895 y Fr(\022)1936 2880 y Fq(m)p Fp(\()p Fq(t;)e(\022)s Fp(\))p Fo(j)2230 2842 y Fl(2)2295 2880 y Fq(<)25 b Fo(1)p Fq(:)952 b Fu(\(2.8\))118 3086 y Fh(Let)20 b(also)f Fo(f)p Fq(\026)524 3053 y Fr(\017)557 3086 y Fo(g)602 3100 y Fr(\017>)p Fl(0)745 3086 y Fh(be)g(a)h(sequence)d(of)i (probability)h(measures)f(on)g Fp(\012)2355 3100 y Fr(\017)2407 3086 y Fh(corresponding)e(to)i(the)h(macroscopic)118 3195 y(density)27 b(pro\002le)h Fq(m)769 3209 y Fl(0)808 3195 y Fp(\()p Fq(\022)s Fp(\))p Fh(.)41 b(Then)28 b(the)g(measure)f Fq(Q)1775 3162 y Fr(\013;)12 b(\026)1894 3138 y FA(\017)1957 3195 y Fh(converges)26 b(weakly)h(to)i(the)f(probability)g(measure)118 3303 y(on)37 b Fq(D)s Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)715 3317 y Fl(1)756 3303 y Fp(\))38 b Fh(concentrated)d(on) h(the)h(path)h Fo(f)p Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))p Fq(d\022)s Fo(g)2308 3321 y Fr(t)p Fn(2)p Fl([0)p Fr(;T)10 b Fl(])2530 3303 y Fh(.)68 b(In)37 b(particular)-12 b(,)41 b(for)c(any)g Fp(0)46 b Fo(\024)118 3435 y Fq(t)38 b Fo(\024)f Fq(T)13 b Fh(,)34 b(the)f(sequence)d(of)i(time)g(dependent) e(probability)j(measures)e Fo(f)p Fs(P)2604 3391 y Fr(\013;\026)2711 3367 y FA(")2604 3459 y Fr(t)2750 3435 y Fo(g)2795 3449 y Fr(\017>)p Fl(0)2951 3435 y Fh(corresponds)f(to)j(the)118 3547 y(macroscopic)24 b(density)g(pro\002le)h Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))p Fh(,)25 b(i.e.)32 b(for)24 b(any)h(smooth)f(function)g Fq(H)33 b Fh(on)24 b Fs(T)2872 3514 y Fr(d)2940 3547 y Fh(and)g(any)g Fq(\016)29 b(>)c Fp(0)734 3748 y(lim)747 3810 y Fr(\017)p Fn(#)p Fl(0)875 3748 y Fs(P)930 3704 y Fr(\013;\026)1037 3712 y FA(\017)930 3772 y Fr(t)1073 3675 y Fk(\000)1114 3671 y(\014)1114 3726 y(\014)1160 3748 y Fp(Av)1277 3771 y Fr(x)p Fn(2)p Fm(T)1414 3752 y FA(d)1414 3787 y(\017)1448 3748 y Fq(H)7 b Fp(\()p Fq(\017x)p Fp(\))p Fq(\021)1735 3762 y Fr(x)1800 3748 y Fo(\000)1891 3625 y Fk(Z)1942 3831 y Fm(T)1992 3812 y FA(d)2041 3748 y Fq(H)g Fp(\()p Fq(\022)s Fp(\))p Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))p Fq(d\022)2617 3671 y Fk(\014)2617 3726 y(\014)2672 3748 y Fq(>)25 b(\016)2811 3675 y Fk(\001)2879 3748 y Fp(=)g(0)p Fq(:)414 b Fu(\(2.9\))118 3955 y Fh(The)23 b(thesis)g(remains)f(valid)h(also)f(if)h Fq(D)s Fp(\()p Fq(m)p Fp(\))h Fh(has)e(no)h(continuous)f(extension)f (provided)h(that)i(one)e(assumes)118 4063 y(instead)27 b(that,)i(for)e(some)g(\002xed)h Fq(\032)i Fo(2)g Fp(\(0)p Fq(;)15 b Fp(1\))p Fh(,)31 b(there)d(exists)f(a)h(sequence)e(of)h (probability)h(measures)f Fq(\026)3498 4030 y Fr(\017)3498 4085 y Fn(\003)3565 4063 y Fh(on)118 4171 y Fp(\012)184 4185 y Fr(\017)242 4171 y Fh(such)e(that)426 4326 y Fq(H)7 b Fp([)p Fq(\026)589 4288 y Fr(\017)622 4326 y Fo(j)p Fq(\026)702 4288 y Fr(\017)702 4348 y Fn(\003)741 4326 y Fp(])26 b(=)f Fq(o)p Fp(\()p Fq(\017)1004 4288 y Fn(\000)p Fr(d)1099 4326 y Fp(\))183 b Fh(and)195 b Fp(inf)1700 4380 y Fr(\017)1813 4326 y Fp(inf)1785 4394 y Fr(x)p Fn(2)p Fm(T)1922 4375 y FA(d)1922 4411 y(\017)1967 4326 y Fp(min)2119 4252 y Fk(\000)2161 4326 y Fq(\026)2216 4288 y Fr(\017)2216 4348 y Fn(\003)2255 4326 y Fp(\()p Fq(\021)2335 4340 y Fr(x)2379 4326 y Fp(\))p Fq(;)15 b Fp(1)22 b Fo(\000)e Fq(\026)2667 4288 y Fr(\017)2667 4348 y Fn(\003)2706 4326 y Fp(\()p Fq(\021)2786 4340 y Fr(x)2830 4326 y Fp(\))2865 4252 y Fk(\001)2932 4326 y Fo(\025)25 b Fq(\032;)309 b Fu(\(2.10\))118 4543 y Fh(where)25 b Fq(H)7 b Fp([)p Fo(\001j\001)p Fp(])26 b Fh(denotes)d(the)i(relative)g(entropy)-5 b(.)118 4675 y FE(R)m(emark)27 b(2.5.)42 b Fh(Notice)25 b(that)h(condition)e (\(2.10\))h(becomes)f(rather)h(natural)h(if)f(the)h(initial)f (pro\002le)g Fq(m)3526 4689 y Fl(0)3565 4675 y Fp(\()p Fo(\001)p Fp(\))118 4785 y Fh(satis\002es)f Fq(\032)h Fo(\024)g Fq(m)687 4799 y Fl(0)727 4785 y Fp(\()p Fq(\022)s Fp(\))g Fo(\024)g Fp(1)20 b Fo(\000)g Fq(\032)26 b Fh(for)e(any)h Fq(\022)i Fo(2)e Fs(T)1708 4752 y Fr(d)1750 4785 y Fh(.)844 4995 y Fu(3.)47 b(P)t Ft(L)t(A)t(N)31 b(O)t(F)g(T)t(H)t(E)g(P)t(R)t(O)t (O)t(F)g(O)t(F)g(T)t(H)t(E)f(T)t(W)t(O)i(M)t(A)t(I)t(N)f(T)t(H)t(E)t(O) t(R)t(E)t(M)t(S)218 5157 y Fu(The)h(proof)h(of)f(theorem)g(2.1)g(will)g (be)g(given)g(in)g(section)g(7.4)g(and)h(it)g(is)g(based)g(on)f(more)f (or)h(less)118 5264 y(standard)24 b(techniques.)31 b(The)23 b(proof)h(of)f(theorem)f(2.4)h(is)h(more)e(involved)h(and)g(it)h(can)e (be)h(divided)i(into)118 5372 y(several)j(steps)h(that)g(we)f (illustrate)h(in)f(what)i(follows.)40 b(In)29 b(order)f(to)h(work)f(in) g(the)h(simplest)g(possible)118 5480 y(setting,)24 b(in)e(the)h(sequel) f(we)g(assume)h(that)g(the)g(diffusion)h(matrix)e Fq(D)k Fu(can)21 b(be)h(continuously)g(extended)p eop %%Page: 9 9 9 8 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g(DISORDERED)f (LA)-7 b(T)q(TICE)23 b(GAS)860 b(9)118 366 y Fu(to)31 b(the)f(closed)g(interval)f Fp([0)p Fq(;)15 b Fp(1])p Fu(.)48 b(Only)30 b(at)g(the)h(end)e(\(see)h(subsection)g(4.8\))g(we)f (will)h(explain)g(how)g(to)118 474 y(treat)c(the)f(other)g(case.)218 582 y(Let)37 b(us)g(begin)f(with)j(some)d(remarks)h(on)f(the)i(weak)e (interpretation)i(of)f(\(2.7\))f(and)h(\(2.8\).)66 b(Let)118 690 y Fq(A)p Fp(\()p Fq(m)p Fp(\))p Fu(,)26 b Fq(m)f Fo(2)g Fp([0)p Fq(;)15 b Fp(1])p Fu(,)28 b(be)d(a)g Fq(d)20 b Fo(\002)g Fq(d)26 b Fu(matrix)f(such)g(that)i Fq(A)1973 657 y Fn(0)1996 690 y Fp(\()p Fq(m)p Fp(\))f(=)f Fq(D)s Fp(\()p Fq(m)p Fp(\))h Fu(so)f(that)742 788 y Fk(\000)784 862 y Fq(D)s Fp(\()p Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\)\))p Fo(r)1277 877 y Fr(\022)1316 862 y Fq(m)p Fp(\()p Fq(t;)g(\022)s Fp(\))1585 788 y Fk(\001)1627 898 y Fr(e)1689 862 y Fp(=)1792 776 y Fk(X)1785 974 y Fr(e)1818 955 y Fi(0)1840 974 y Fn(2E)1945 862 y Fq(@)1993 877 y Fr(\022)2026 896 y FA(e)2055 882 y Fi(0)2086 862 y Fq(A)2154 877 y Fr(e;e)2240 858 y Fi(0)2266 862 y Fp(\()p Fq(m)p Fp(\()p Fq(t;)g(\022)s Fp(\)\))g Fq(;)107 b Fo(8)p Fq(e)25 b Fo(2)g(E)8 b Fq(:)118 1109 y Fu(It)29 b(is)f(simple)g(to)g(prove)f (\(see)g(appendix)h(of)g([15]\))g(that)h(given)d Fq(m)k Fo(2)f Fq(D)s Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)2964 1123 y Fl(2)3005 1109 y Fp(\))28 b Fu(there)g(is)g(a)g(mea-)118 1217 y(surable)d(function)h Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))26 b Fu(univocally)f(de\002ned)g(up)h(to)h(sets)f(of)h(zero)e (Lebesgue)g(measure)f(such)i(that)118 1324 y Fq(m)198 1338 y Fr(t)261 1324 y Fp(=)34 b Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))p Fq(d\022)33 b Fu(for)d(any)g Fq(t)k Fo(2)f Fp([0)p Fq(;)15 b(T)e Fp(])32 b Fu(\(see)d(appendix)i(of)f ([15]\).)47 b(In)30 b(what)h(follows,)h(we)e(will)h(often)118 1432 y(identify)26 b Fq(m)g Fu(with)g(the)g(funtion)f Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))p Fu(.)118 1540 y(A)25 b(path)i Fq(m)e Fo(2)f Fq(D)s Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)1058 1554 y Fl(2)1100 1540 y Fp(\))25 b Fu(is)h(called)f(a)g(weak)g(solution)h(of)f(\(2.7\))f(if)i Fq(m)p Fp(\(0)p Fq(;)15 b Fo(\001)p Fp(\))27 b(=)e Fq(m)3029 1554 y Fl(0)3068 1540 y Fp(\()p Fo(\001)p Fp(\))i Fu(and)1144 1700 y Fp(\010\()p Fq(m;)15 b(H)7 b Fp(\))25 b(=)g(0)92 b Fo(8)p Fq(H)31 b Fo(2)25 b Fq(C)2057 1663 y Fl(1)p Fr(;)p Fl(2)2151 1700 y Fp(\([0)p Fq(;)15 b(T)e Fp(])22 b Fo(\002)e Fs(T)2559 1663 y Fr(d)2601 1700 y Fp(\))118 1853 y Fu(where)118 2050 y Fp(\010\()p Fq(m;)15 b(H)7 b Fp(\))26 b(:)f(=)629 1927 y Fk(Z)680 2133 y Fm(T)730 2114 y FA(d)780 2050 y Fq(m)p Fp(\()p Fq(T)8 b(;)15 b(\022)s Fp(\))p Fq(H)7 b Fp(\()p Fq(T)h(;)15 b(\022)s Fp(\))g Fq(d\022)22 b Fo(\000)1595 1927 y Fk(Z)1646 2133 y Fm(T)1696 2114 y FA(d)1745 2050 y Fq(m)p Fp(\(0)p Fq(;)15 b(\022)s Fp(\))p Fq(H)7 b Fp(\(0)p Fq(;)15 b(\022)s Fp(\))g Fq(d\022)25 b Fo(\000)2531 1927 y Fk(Z)2621 1953 y Fr(T)2581 2133 y Fl(0)2692 1927 y Fk(Z)2742 2133 y Fm(T)2792 2114 y FA(d)2842 2050 y Fq(m)p Fp(\()p Fq(s;)15 b(\022)s Fp(\))p Fq(@)3169 2064 y Fr(s)3206 2050 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\022)s Fp(\))g Fq(d\022)i(ds)528 2311 y Fo(\000)619 2224 y Fk(X)631 2423 y Fr(e;e)717 2404 y Fi(0)766 2187 y Fk(Z)857 2213 y Fr(T)816 2393 y Fl(0)927 2187 y Fk(Z)977 2393 y Fm(T)1027 2374 y FA(d)1077 2311 y Fq(A)1145 2326 y Fr(e;e)1231 2307 y Fi(0)1257 2237 y Fk(\000)1299 2311 y Fq(m)p Fp(\()p Fq(s;)e(\022)s Fp(\))1578 2237 y Fk(\001)1634 2311 y Fq(@)1687 2273 y Fl(2)1682 2334 y Fr(\022)1715 2342 y FA(e)1749 2334 y Fr(;\022)1802 2353 y FA(e)1831 2339 y Fi(0)1861 2311 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\022)s Fp(\))g Fq(d\022)j(ds:)3459 2518 y Fu(\(3.1\))118 2676 y(Moreover)-10 b(,)24 b Fq(m)h Fo(2)g Fq(D)s Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)1193 2690 y Fl(2)1234 2676 y Fp(\))26 b Fu(satis\002es)h(the)e(energy)f(estimate)i(\(2.8\))e (if)836 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3369 y Fu(and)1996 3295 y Fk(\002)2044 3333 y Fr(b)p 2044 3348 31 4 v 2045 3400 a(\017)2084 3295 y Fk(\003)2122 3369 y Fu(,)h(where)f Fp([)p Fo(\001)p Fp(])i Fu(denotes)e(the)h(integer)f(part.)31 b(F)-6 b(or)118 3491 y(simplicity)24 b(of)g(notation)g(these)f(new)g(scales)g(will)g (be)f(denoted)i(only)e(by)2608 3455 y Fr(a)p 2608 3470 38 4 v 2612 3522 a(\017)2679 3491 y Fu(and)2861 3455 y Fr(b)p 2861 3470 31 4 v 2862 3522 a(\017)2902 3491 y Fu(.)30 b(Moreover)-10 b(,)23 b(we)g(will)118 3598 y(introduce)29 b(the)g(scale)f Fq(n)h Fu(where)f Fq(n)h Fu(is)h(a)f(positive)h(odd)g(integer)-10 b(.)42 b(The)29 b(property)g(of)g Fq(n)g Fu(to)g(be)g(odd)h(will)118 3706 y(be)25 b(always)h(understood.)118 3883 y(3.1.)45 b FE(T)n(ightness.)g Fu(The)36 b(\002rst)h(step)g(toward)f(the)h(proof) f(of)h(theorem)e(2.4)h(is)h(to)g(show)g(that,)j(for)c(all)118 3991 y(disorder)26 b(con\002gurations)f Fq(\013)p Fu(,)h(if)g Fo(f)p Fq(\026)1352 3958 y Fr(\017)1385 3991 y Fo(g)1430 4005 y Fr(\017>)p Fl(0)1579 3991 y Fu(is)h(a)e(sequence)f(of)i (probability)g(measures)f(on)g Fp(\012)3274 4005 y Fr(\017)3332 3991 y Fu(then)g(the)118 4100 y(sequence)18 b(of)i(measures)f(on)g Fq(D)s Fp(\([0)p Fq(;)c(T)e Fp(])p Fq(;)i Fo(M)1572 4114 y Fl(1)1613 4100 y Fp(\))p Fu(,)21 b Fo(f)p Fq(Q)1811 4067 y Fr(\013;)12 b(\026)1930 4043 y FA(\017)1965 4100 y Fo(g)2010 4114 y Fr(\017>)p Fl(0)2134 4100 y Fu(,)21 b(is)f(relatively)f(compact.)29 b(F)-6 b(or)19 b(this)i(purpose)118 4208 y(it)36 b(is)g(enough)d(to)i(use)g(the)g(Garsia-R)m(odemich-R)m (umsey)f(inequality)h(as)g(done)g(in)g([23)o(],)j(chapter)d Fp(7)p Fu(,)118 4316 y(section)25 b Fp(6)p Fu(.)118 4492 y(3.2.)45 b FE(R)m(egularity)32 b(properties)g(of)h(the)g(limit)f (points.)44 b Fu(In)32 b(the)g(second)f(step)i(one)d(proves)i(that,)j (for)118 4600 y(almost)21 b(all)g Fq(\013)p Fu(,)h(given)f(a)g (sequence)e Fo(f)p Fq(\026)1413 4567 y Fr(\017)1446 4600 y Fo(g)1491 4614 y Fr(\017>)p Fl(0)1635 4600 y Fu(of)j(probability)g (measures)e(on)g Fp(\012)2765 4614 y Fr(\017)2798 4600 y Fu(,)i(any)f(limit)g(point)h Fq(Q)3513 4567 y Fr(\013)3583 4600 y Fu(of)118 4709 y(the)30 b(sequence)e Fo(f)p Fq(Q)783 4676 y Fr(\013;)12 b(\026)902 4653 y FA(\017)937 4709 y Fo(g)982 4723 y Fr(\017>)p Fl(0)1135 4709 y Fu(is)31 b(concentrated)d(on)h(paths)i(enjoying)e(a)h(certain)f(regularity)g (property)-12 b(.)118 4818 y(F)-6 b(or)26 b(this)h(purpose)e(we)h (\002rst)g(observe)f(that,)i(for)f(any)f Fq(\013)p Fu(,)i Fq(Q)2129 4785 y Fr(\013)2204 4818 y Fu(must)f(satisfy)i Fq(Q)2772 4785 y Fr(\013)2821 4744 y Fk(\000)2878 4818 y Fq(C)7 b Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)3335 4832 y Fl(2)3376 4818 y Fp(\))3426 4744 y Fk(\001)3494 4818 y Fp(=)26 b(1)p Fu(,)118 4937 y(since)f(for)g(any)g Fq(\021)k Fo(2)c Fp(\012)874 4951 y Fr(\017)906 4937 y Fu(,)h Fq(H)32 b Fo(2)25 b Fq(C)7 b Fp(\()p Fs(T)1317 4904 y Fr(d)1359 4937 y Fp(\))25 b Fu(and)h Fq(b)f Fo(\032)g Fs(T)1813 4904 y Fr(d)1813 4959 y(\017)437 5019 y Fk(\014)437 5074 y(\014)483 5097 y Fq(\031)535 5111 y Fr(\017)567 5097 y Fp(\()p Fq(\021)s Fp(\)[)p Fq(H)7 b Fp(])833 5019 y Fk(\014)833 5074 y(\014)890 5097 y Fo(\024)25 b Fp(Av)1104 5119 y Fr(x)p Fn(2)p Fm(T)1241 5100 y FA(d)1241 5135 y(\017)1275 5097 y Fo(j)p Fq(H)7 b Fp(\()p Fq(\017x)p Fp(\))p Fo(j)92 b Fu(and)1899 5019 y Fk(\014)1899 5074 y(\014)1944 5097 y Fq(\031)1996 5111 y Fr(\017)2029 5097 y Fp(\()p Fq(\021)2112 5059 y Fr(b)2147 5097 y Fp(\)[)p Fq(H)7 b Fp(])21 b Fo(\000)f Fq(\031)2479 5111 y Fr(\017)2511 5097 y Fp(\()p Fq(\021)s Fp(\)[)p Fq(H)7 b Fp(])2777 5019 y Fk(\014)2777 5074 y(\014)2835 5097 y Fo(\024)25 b Fp(2)15 b Fo(k)p Fq(H)7 b Fo(k)3164 5111 y Fn(1)3240 5097 y Fq(\017)3277 5059 y Fr(d)3317 5097 y Fq(:)118 5257 y Fu(Moreover)-10 b(,)29 b(if)i(the)e(sequence)f(of)i Fo(f)p Fq(\026)1384 5224 y Fr(\017)1417 5257 y Fo(g)1462 5271 y Fr(\017>)p Fl(0)1615 5257 y Fu(corresponds)f(to)h(the)g (macroscopic)f(pro\002le)f Fq(m)3272 5271 y Fl(0)3312 5257 y Fp(\()p Fq(\022)s Fp(\))p Fu(,)j(then)118 5365 y(necessarily)837 5480 y Fq(Q)909 5443 y Fr(\013)959 5407 y Fk(\000)1015 5480 y Fq(m)25 b Fo(2)g Fq(C)7 b Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)1663 5494 y Fl(2)1704 5480 y Fp(\))41 b(:)51 b Fq(m)p Fp(\(0)p Fq(;)15 b(\022)s Fp(\))26 b(=)f Fq(m)2339 5494 y Fl(0)2378 5480 y Fp(\()p Fq(\022)s Fp(\))2509 5407 y Fk(\001)2576 5480 y Fp(=)g(1)91 b Fo(8)p Fq(\013:)517 b Fu(\(3.3\))p eop %%Page: 10 10 10 9 bop 118 45 a Fj(10)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(The)26 b(key)e(result)h(here,)g(whose)h(proof)f(will)g(be)g(given)f(later)h (in)g(section)h(4.7,)f(is)h(the)f(following.)118 474 y(Given)k(a)f(path)i Fq(\021)s Fp(\()p Fo(\001)p Fp(\))i Fo(2)f Fq(D)s Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fp(\012)1345 488 y Fr(\017)1379 474 y Fp(\))p Fu(,)30 b Fq(x)h Fo(2)g Fs(T)1703 441 y Fr(d)1703 497 y(\017)1774 474 y Fu(and)d Fq(`)j Fo(2)g Fs(N)7 b Fu(,)35 b(let)29 b Fq(m)2442 489 y Fr(x;`)2534 474 y Fp(\()p Fq(t)p Fp(\))g Fu(be)f(the)h(particle)g (density)g(of)118 582 y Fq(\021)s Fp(\()p Fq(t)p Fp(\))d Fu(in)g(the)f(cube)f Fq(Q)832 597 y Fr(x;`)925 582 y Fu(.)31 b(Then)25 b(we)g(have)118 712 y FE(L)r(emma)39 b(3.1.)49 b(\(Energy)39 b(estimate\).)65 b Fh(Let)37 b Fq(d)46 b Fo(\025)f Fp(3)p Fh(,)c(let)c Fq(T)58 b(>)45 b Fp(0)37 b Fh(and)f(assume)g(that)i Fq(D)s Fp(\()p Fq(m)p Fp(\))f Fh(can)g(be)118 820 y(continuously)29 b(extended)g(to)h(the)g (closed)f(interval)h Fp([0)p Fq(;)15 b Fp(1])p Fh(.)48 b(Then)30 b(almost)g(any)f(disorder)g(con\002gurations)118 928 y Fq(\013)g Fh(have)f(the)g(following)e(property)-5 b(.)40 b(F)l(or)29 b(any)e(sequence)f Fo(f)p Fq(\026)2106 895 y Fr(\017)2139 928 y Fo(g)2184 942 y Fr(\017>)p Fl(0)2336 928 y Fh(of)h(probability)i(measures)d(on)i Fp(\012)3455 942 y Fr(\017)3516 928 y Fh(and)118 1036 y(any)d Fq(e)g Fo(2)g(E)559 1267 y Fp(sup)567 1346 y Fr(b>)p Fl(0)711 1267 y Fp(lim)15 b(sup)737 1346 y Fr(a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)1005 1267 y Fs(E)1059 1229 y Fr(\013)q(;\026) 1167 1206 y FA(\017)1207 1166 y Fk(\020)1261 1267 y Fp(Av)1379 1289 y Fr(x)p Fn(2)p Fm(T)1516 1270 y FA(d)1516 1306 y(\017)1565 1143 y Fk(Z)1656 1169 y Fr(T)1616 1349 y Fl(0)1711 1166 y Fk(h)1779 1175 y Fq(m)1859 1205 y Fr(x)p Fl(+)1964 1178 y FA(b)p 1964 1190 28 3 v 1965 1231 a(\017)2001 1205 y Fr(e;)2064 1178 y FA(a)p 2063 1190 34 3 v 2067 1231 a(\017)2111 1175 y Fp(\()p Fq(s)p Fp(\))20 b Fo(\000)g Fq(m)2415 1192 y Fr(x;)2485 1165 y FA(a)p 2484 1177 V 2488 1218 a(\017)2532 1175 y Fp(\()p Fq(s)p Fp(\))p 1779 1246 867 4 v 2193 1329 a Fq(b)2671 1166 y Fk(i)2713 1189 y Fl(2)2768 1267 y Fq(ds)2858 1166 y Fk(\021)2938 1267 y Fq(<)25 b Fp(+)p Fo(1)p Fq(:)238 b Fu(\(3.4\))118 1497 y Fh(Moreover)25 b(any)f(limit)i(point)f Fq(Q)1172 1464 y Fr(\013)1247 1497 y Fh(of)f(the)h(sequence)e Fo(f)p Fq(Q)1963 1464 y Fr(\013;)12 b(\026)2082 1441 y FA(\017)2117 1497 y Fo(g)2162 1511 y Fr(\017>)p Fl(0)2310 1497 y Fh(satis\002es)877 1676 y Fq(Q)949 1638 y Fr(\013)998 1575 y Fk(n)1074 1676 y Fq(m)25 b Fo(2)g Fq(C)7 b Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)1722 1690 y Fl(2)1763 1676 y Fp(\))41 b(:)25 b(l.h.s.)40 b(of)31 b(\(3.2)g(\))26 b Fq(<)f Fo(1)2650 1575 y Fk(o)2736 1676 y Fp(=)g(1)p Fq(:)557 b Fu(\(3.5\))118 1875 y(3.3.)45 b FE(Microscopic)27 b(identi\002cation)d(of)i(the)f (hydrodynamic)i(equation.)46 b Fu(In)24 b(the)h(third)h(step)f(of)f (the)118 1983 y(proof)k(one)f(identi\002es)i(at)f(the)g(microscopic)f (level)g(the)h(hydrodynamic)f(equation.)40 b(It)28 b(is)h(convenient) 118 2091 y(to)c(introduce)f(some)h(more)e(notation.)32 b(Given)25 b Fq(e;)15 b(e)1856 2058 y Fn(0)1905 2091 y Fo(2)25 b(E)8 b Fu(,)25 b(two)h(positive)g(numbers)d Fq(a;)15 b(b)25 b Fu(and)g(a)g(smooth)118 2199 y(function)g Fq(H)32 b Fu(on)25 b Fp([0)p Fq(;)15 b(T)e Fp(])21 b Fo(\002)f Fs(T)1079 2166 y Fr(d)1121 2199 y Fu(,)26 b(we)f(set)266 2393 y Fp(\026)243 2416 y Fq(H)319 2431 y Fr(b;a;\017)483 2416 y Fp(:=)g(Av)722 2438 y Fr(x)p Fn(2)p Fm(T)859 2419 y FA(d)859 2454 y(\017)893 2315 y Fk(h)936 2416 y Fq(H)7 b Fp(\()p Fq(T)h(;)15 b(\017x)p Fp(\))p Fq(\021)1324 2430 y Fr(x)1369 2416 y Fp(\()p Fq(T)e Fp(\))20 b Fo(\000)g Fq(H)7 b Fp(\(0)p Fq(;)15 b(\017x)p Fp(\))p Fq(\021)1988 2430 y Fr(x)2034 2416 y Fp(\(0\))21 b Fo(\000)2261 2292 y Fk(Z)2352 2318 y Fr(T)2311 2498 y Fl(0)2422 2416 y Fq(ds)15 b(\021)2572 2430 y Fr(x)2616 2416 y Fp(\()p Fq(s)p Fp(\))p Fq(@)2777 2430 y Fr(s)2814 2416 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))3139 2315 y Fk(i)478 2828 y Fp(+)602 2742 y Fk(X)569 2940 y Fr(e;e)655 2921 y Fi(0)676 2940 y Fn(2E)781 2705 y Fk(Z)872 2731 y Fr(T)832 2911 y Fl(0)943 2828 y Fq(ds)g Fp(Av)1165 2850 y Fr(x)p Fn(2)p Fm(T)1302 2832 y FA(d)1302 2867 y(\017)1336 2828 y Fo(r)1412 2791 y Fr(\017)1412 2851 y(e)1449 2828 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))p Fq(D)1849 2844 y Fr(e;e)1935 2825 y Fi(0)1961 2755 y Fk(\000)2003 2828 y Fq(m)2083 2846 y Fr(x;)2153 2818 y FA(a)p 2152 2830 34 3 v 2156 2872 a(\017)2200 2828 y Fp(\()p Fq(s)p Fp(\))2313 2755 y Fk(\001)2355 2727 y(h)2423 2736 y Fq(m)2503 2767 y Fr(x)p Fl(+)2608 2740 y FA(b)p 2607 2752 28 3 v 2608 2793 a(\017)2645 2767 y Fr(e)2678 2748 y Fi(0)2700 2767 y Fr(;)2730 2740 y FA(a)p 2729 2752 34 3 v 2733 2793 a(\017)2777 2736 y Fp(\()p Fq(s)p Fp(\))21 b Fo(\000)f Fq(m)3082 2767 y Fr(x)p Fn(\000)3187 2740 y FA(b)p 3186 2752 28 3 v 3187 2793 a(\017)3223 2767 y Fr(e)3256 2748 y Fi(0)3278 2767 y Fr(;)3308 2740 y FA(a)p 3308 2752 34 3 v 3312 2793 a(\017)3356 2736 y Fp(\()p Fq(s)p Fp(\))p 2423 2807 1046 4 v 2904 2891 a(2)p Fq(b)3494 2727 y Fk(i)3459 3035 y Fu(\(3.6\))118 3202 y(where)25 b Fo(r)465 3169 y Fr(\017)465 3224 y(e)501 3202 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))26 b(:=)983 3166 y Fl(1)p 983 3181 36 4 v 986 3233 a Fr(\017)1029 3128 y Fk(\002)1066 3202 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)21 b Fp(+)f Fq(\017e)p Fp(\))h Fo(\000)e Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))2018 3128 y Fk(\003)2057 3202 y Fu(.)118 3331 y(The)26 b(following)e(theorem,)h(whose)g(proof)h(will)f(be)g(discussed)h(in)f (a)h(little)f(while,)h(corresponds)f(to)g(the)118 3439 y(microscopic)g(identi\002cation)h(of)g(the)f(hydrodynamical)g (equation.)118 3568 y FE(Theorem)34 b(3.2.)45 b Fh(Let)32 b Fq(d)37 b Fo(\025)g Fp(3)p Fh(,)d(let)e Fq(T)49 b(>)37 b Fp(0)32 b Fh(and)f(assume)g(that)h Fq(D)s Fp(\()p Fq(m)p Fp(\))h Fh(can)e(be)h(continuously)f(extended)118 3676 y(to)37 b(the)g(closed)d(interval)j Fp([0)p Fq(;)15 b Fp(1])p Fh(.)68 b(Then)36 b(almost)g(all)h(disorder)e(con\002gurations) g Fq(\013)i Fh(have)g(the)f(following)118 3784 y(property)-5 b(.)73 b(F)l(or)40 b(any)f(sequence)d Fo(f)p Fq(\026)1351 3751 y Fr(\017)1384 3784 y Fo(g)1429 3798 y Fr(\017>)p Fl(0)1592 3784 y Fh(of)j(probability)g(measures)f(on)h Fp(\012)2756 3798 y Fr(\017)2788 3784 y Fh(,)44 b(any)38 b Fq(\016)53 b(>)c Fp(0)40 b Fh(and)e(any)118 3894 y Fq(H)32 b Fo(2)25 b Fq(C)384 3861 y Fl(1)p Fr(;)p Fl(2)478 3894 y Fp(\([0)p Fq(;)15 b(T)e Fp(])22 b Fo(\002)e Fs(T)886 3861 y Fr(d)928 3894 y Fp(\))1275 4013 y(lim)15 b(sup)1229 4092 y Fr(b)p Fn(#)p Fl(0)p Fr(;)d(a)p Fn(#)p Fl(0)p Fr(;)g(\017)p Fn(#)p Fl(0)1614 4013 y Fs(P)1669 3976 y Fr(\013;\026)1776 3952 y FA(")1815 3939 y Fk(\000)1872 4013 y Fo(j)1921 3990 y Fp(\026)1897 4013 y Fq(H)1973 4028 y Fr(b;a;\017)2112 4013 y Fo(j)25 b Fq(>)g(\016)2317 3939 y Fk(\001)2384 4013 y Fp(=)g(0)p Fq(:)909 b Fu(\(3.7\))118 4214 y(The)24 b(proof)f(of)h(theorem)e(2.4,)i(given)e(Lemma)h(3.1)g (and)g(theorem)g(3.2,)g(now)g(follows)h(by)f(more)f(or)h(less)118 4322 y(standard)j(arguments)e(and)i(it)g(can)e(be)h(found)g(in)h (section)f(1.5)g(of)g([15].)694 4520 y(4.)48 b(P)t Ft(R)t(O)t(O)t(F)32 b(O)t(F)e(T)t(H)t(E)t(O)t(R)t(E)t(M)j Fu(3)t(.)t(2)d Ft(M)t(O)t(D)t(U)t(L)t(O)j(S)t(O)t(M)t(E)f(T)t(E)t(C)t(H)t(N)t(I)t(C)s (A)t(L)h(S)t(T)t(E)t(P)t(S)218 4682 y Fu(In)c(this)i(section)f(we)f (prove)f(theorem)h(3.2)g(modulo)g(certain)g(technical)f(results)h(that) i(will)e(be)g(dis-)118 4790 y(cussed)c(in)g(the)h(remaining)e (sections.)32 b(F)-6 b(ollowing)24 b([37)o(])h(the)h(\002rst)f(main)g (step)h(is)g(to)f(reduce)f(the)h(proof)118 4898 y(of)e(the)g(theorem)e (to)i(the)g(eigenvalue)e(estimates)i(of)g(certain)f(symmetric)g (operators,)h(via)g(the)g(entropy)118 5006 y(inequality)34 b(and)g(the)g(F)-6 b(eynman\226Kac)31 b(formula.)56 b(T)-9 b(o)34 b(this)h(aim)f(we)f(de\002ne)h Fq(j)2814 5020 y Fr(x;x)p Fl(+)p Fr(e)3038 5006 y Fu(as)h(the)e(instanta-)118 5114 y(neous)22 b(current)g(through)h(the)g(oriented)f(bond)h Fo(f)p Fq(x;)15 b(x)d Fp(+)g Fq(e)p Fo(g)p Fu(,)26 b(i.e.)31 b(as)23 b(the)h(difference)d(between)h(the)i(rate)118 5221 y(at)g(which)g(a)f(particle)g(jumps)h(from)f Fq(x)g Fu(to)h Fq(x)14 b Fp(+)g Fq(e)23 b Fu(and)g(the)h(rate)f(at)h(which)g (a)f(particle)g(jumps)h(from)f Fq(x)14 b Fp(+)g Fq(e)118 5329 y Fu(to)26 b Fq(x)p Fu(.)31 b(It)26 b(is)g(simple)g(to)f(check)f (that)1257 5480 y Fq(j)1294 5494 y Fr(x;x)p Fl(+)p Fr(e)1485 5480 y Fp(\()p Fq(\021)s Fp(\))j(=)e Fq(c)1765 5494 y Fr(x;x)p Fl(+)p Fr(e)1956 5480 y Fp(\()p Fq(\021)s Fp(\)\()p Fq(\021)2154 5494 y Fr(x)2220 5480 y Fo(\000)19 b Fq(\021)2355 5494 y Fr(x)p Fl(+)p Fr(e)2487 5480 y Fp(\))p eop %%Page: 11 11 11 10 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(11)118 366 y Fu(and)1185 497 y Fo(L)1248 511 y Fr(\017)1280 497 y Fq(\021)1325 511 y Fr(x)1395 497 y Fp(=)1491 410 y Fk(X)1495 607 y Fr(e)p Fn(2E)1622 497 y Fp(\()p Fo(\000)p Fq(j)1765 511 y Fr(x;x)p Fl(+)p Fr(e)1956 497 y Fp(\()p Fq(\021)s Fp(\))22 b(+)e Fq(j)2224 511 y Fr(x)p Fn(\000)p Fr(e;x)2415 497 y Fp(\()p Fq(\021)s Fp(\)\))p Fq(:)118 738 y Fu(In)30 b(particular)g(\(see)f(lemma)g Fp(5)p Fq(:)p Fp(1)p Fu(,)j(appendix)e Fp(1)g Fu(in)g([23],)h(or)f([15]\),)h (for)f(any)g(smooth)g Fq(H)7 b Fp(\()p Fq(t;)15 b(x)p Fp(\))p Fu(,)32 b(inte-)118 846 y(gration)25 b(by)g(parts)h(and)g (stochastic)g(calculus)e(show)i(that)769 1012 y Fp(Av)887 1034 y Fr(x)p Fn(2)p Fm(T)1024 1015 y FA(d)1024 1051 y(\017)1058 939 y Fk(\002)1096 1012 y Fq(H)7 b Fp(\()p Fq(T)h(;)15 b(\017x)p Fp(\))p Fq(\021)1484 1026 y Fr(x)1529 1012 y Fp(\()p Fq(T)e Fp(\))20 b Fo(\000)g Fq(H)7 b Fp(\(0)p Fq(;)15 b(\017x)p Fp(\))p Fq(\021)2148 1026 y Fr(x)2194 1012 y Fp(\(0\))2309 939 y Fk(\003)2373 1012 y Fp(=)219 1229 y(Av)337 1251 y Fr(x)p Fn(2)p Fm(T)474 1232 y FA(d)474 1268 y(\017)523 1105 y Fk(Z)614 1132 y Fr(T)574 1311 y Fl(0)684 1229 y Fq(@)732 1243 y Fr(s)769 1229 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))p Fq(\021)1139 1243 y Fr(x)1184 1229 y Fp(\()p Fq(s)p Fp(\))p Fq(ds)20 b Fp(+)g Fq(\017)1535 1192 y Fn(\000)p Fl(1)1645 1143 y Fk(X)1649 1340 y Fr(e)p Fn(2E)1791 1229 y Fp(Av)1908 1251 y Fr(x)p Fn(2)p Fm(T)2045 1232 y FA(d)2045 1268 y(\017)2095 1105 y Fk(Z)2186 1132 y Fr(T)2145 1311 y Fl(0)2256 1229 y Fo(r)2332 1192 y Fr(\017)2332 1252 y(e)2369 1229 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))p Fq(j)2731 1243 y Fr(x;x)p Fl(+)p Fr(e)2923 1229 y Fq(ds)20 b Fp(+)g Fq(M)10 b Fp(\()p Fq(T)j Fp(\))3459 1167 y Fu(\(4.1\))118 1499 y(where)25 b Fq(M)10 b Fp(\()p Fo(\001)p Fp(\))26 b Fu(is)g(a)g(martingale)e(w)-11 b(.r)h(.t)26 b Fs(P)1466 1466 y Fr(\026)1508 1443 y FA(\017)1568 1499 y Fu(satisfying)1039 1674 y Fs(P)1094 1637 y Fr(\026)1136 1613 y FA(\017)1171 1600 y Fk(\002)1224 1674 y Fo(j)p Fq(M)10 b Fp(\()p Fq(T)j Fp(\))p Fo(j)27 b Fq(>)e(\016)1690 1600 y Fk(\003)1753 1674 y Fo(\024)g Fq(c)p Fp(\()p Fq(H)7 b Fp(\))15 b Fq(\016)2099 1637 y Fn(\000)p Fl(2)2195 1674 y Fq(\017)2232 1637 y Fr(d)2454 1674 y Fo(8)p Fq(\016)29 b(>)c Fp(0)p Fq(:)719 b Fu(\(4.2\))118 1842 y(In)29 b(order)g(to)g(bene\002t)g(of)g(the)g (ergodicity)g(of)h(the)f(system,)i(it)f(is)f(convenient)f(to)h(replace) f(the)h(current)118 1950 y Fq(j)155 1964 y Fr(x;x)p Fl(+)p Fr(e)382 1950 y Fu(in)36 b(\(4.1\))f(by)g(its)i(local)d(average)h (around)g Fq(x)p Fu(.)62 b(T)-9 b(o)37 b(this)g(aim)e(let)h(us)f (introduce)g(a)h(new)f(scale)118 2058 y(parameter)26 b Fq(`)p Fu(,)g(that)i(will)e(be)g(sent)h(to)g Fo(1)f Fu(after)g(the)h(limit)g Fq(\017)g Fo(#)h Fp(0)p Fu(.)35 b(Then,)27 b(because)e(of)i(the)g(smoothness)118 2166 y(of)j(the)g(function)f Fq(H)7 b Fu(,)32 b(for)d(any)h Fq(`)i Fo(\035)h Fp(1)d Fu(one)f(can)g(safely)h(replace)e(in)i(the)g(r) -10 b(.h.s.)46 b(of)30 b(\(4.1\))f(the)h(current)118 2279 y Fq(j)155 2293 y Fr(x;x)p Fl(+)p Fr(e)372 2279 y Fu(by)25 b(a)g(local)f(average)g Fp(Av)1226 2298 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)1539 2307 y Fg(1)1576 2279 y Fq(j)1613 2293 y Fr(y)r(;y)r Fl(+)p Fr(e)1800 2279 y Fu(,)h Fq(`)1888 2293 y Fl(1)1953 2279 y Fp(:=)g Fq(`)20 b Fo(\000)2223 2202 y(p)p 2299 2202 38 4 v 77 x Fq(`)o Fu(,)26 b(in)f(the)h(sense)f(that,)h(for)g(any) f Fq(\016)k(>)c Fp(0)254 2526 y(lim)268 2587 y Fr(\017)p Fn(#)p Fl(0)396 2526 y Fs(P)451 2488 y Fr(\026)493 2465 y FA(\017)528 2425 y Fk(h)586 2448 y(\014)586 2503 y(\014)617 2526 y Fq(\017)654 2488 y Fn(\000)p Fl(1)748 2526 y Fp(Av)865 2548 y Fr(x)p Fn(2)p Fm(T)1002 2529 y FA(d)1002 2564 y(\017)1052 2402 y Fk(Z)1143 2428 y Fr(T)1102 2608 y Fl(0)1213 2526 y Fo(r)1289 2488 y Fr(\017)1289 2548 y(e)1325 2526 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))1650 2452 y Fk(\002)1689 2526 y Fq(j)1726 2540 y Fr(x;x)p Fl(+)p Fr(e)1937 2526 y Fo(\000)20 b Fp(Av)2146 2544 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)2459 2553 y Fg(1)2496 2526 y Fq(j)2533 2540 y Fr(y)r(;y)r Fl(+)p Fr(e)2719 2452 y Fk(\003)2772 2526 y Fq(ds)2877 2448 y Fk(\014)2877 2503 y(\014)2933 2526 y Fq(>)25 b(\016)3088 2425 y Fk(i)3156 2526 y Fp(=)g(0)p Fq(:)137 b Fu(\(4.3\))118 2745 y(The)23 b(key)f(observation)h(in)g(the)f(theory)h(of)g (non-gradient)e(systems)j(is)f(that,)i(thanks)e(again)f(to)h(stochas-) 118 2852 y(tic)j(calculus,)340 3079 y Fp(lim)354 3141 y Fr(\017)p Fn(#)p Fl(0)482 3079 y Fs(P)537 3042 y Fr(\026)579 3019 y FA(\017)614 3006 y Fk(\002)667 3002 y(\014)667 3057 y(\014)697 3079 y Fq(\017)734 3042 y Fn(\000)p Fl(1)828 3079 y Fp(Av)946 3102 y Fr(x)p Fn(2)p Fm(T)1083 3083 y FA(d)1083 3118 y(\017)1132 2956 y Fk(Z)1223 2982 y Fr(T)1183 3162 y Fl(0)1293 3079 y Fo(r)1369 3042 y Fr(\017)1369 3102 y(e)1406 3079 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))p Fq(\034)1771 3093 y Fr(x)1815 3079 y Fo(L)p Fq(g)k(ds)2045 3002 y Fk(\014)2045 3057 y(\014)2100 3079 y Fq(>)25 b(\016)2255 3006 y Fk(\003)2318 3079 y Fp(=)g(0)183 b Fo(8)p Fq(\016)28 b(>)d Fp(0)p Fq(;)31 b Fo(8)p Fq(g)e Fo(2)c Fs(G)246 b Fu(\(4.4\))118 3299 y(and)25 b(similarly)h(for)f Fp(Av)914 3317 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)1227 3326 y Fg(1)1265 3299 y Fq(\034)1305 3313 y Fr(y)1346 3299 y Fo(L)p Fq(g)k Fu(in)c(place)f(of)i Fq(\034)1959 3313 y Fr(x)2002 3299 y Fo(L)p Fq(g)s Fu(.)118 3439 y(In)h(conclusion,) f(thanks)h(to)h(\(4.1\),)e(\(4.2\),)h(\(4.3\))f(and)h(\(4.4\),)f(in)h (order)f(to)h(prove)f(\(3.7\))g(it)i(is)g(enough)118 3547 y(to)e(show)g(that)g(for)f(almost)h(all)e(disorder)i (con\002guration)e Fq(\013)i Fu(and)f(for)h(any)f Fq(e)g Fo(2)g(E)268 3779 y Fp(inf)256 3839 y Fr(g)r Fn(2)p Fm(G)535 3779 y Fp(lim)15 b(sup)406 3858 y Fr(b)p Fn(#)p Fl(0)p Fr(;)d(a)p Fn(#)p Fl(0)p Fr(;)h(`)p Fn("1)p Fr(;)e(\017)p Fn(#)p Fl(0)958 3779 y Fs(E)1013 3741 y Fr(\026)1055 3718 y FA(\017)1095 3678 y Fk(\020)1165 3701 y(\014)1165 3756 y(\014)1195 3655 y(Z)1286 3681 y Fr(T)1246 3861 y Fl(0)1356 3779 y Fq(\017)1393 3741 y Fn(\000)p Fl(1)1488 3779 y Fp(Av)1605 3801 y Fr(x)p Fn(2)p Fm(T)1742 3782 y FA(d)1742 3817 y(\017)1776 3779 y Fo(r)1852 3741 y Fr(\017)1852 3801 y(e)1889 3779 y Fq(H)c Fp(\()p Fq(s;)15 b(\017x)p Fp(\))2214 3678 y Fk(h)2257 3779 y Fp(Av)2375 3797 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)2688 3806 y Fg(1)2725 3779 y Fp(\()p Fq(j)2797 3793 y Fr(y)r(;y)r Fl(+)p Fr(e)3004 3779 y Fp(+)20 b Fq(\034)3135 3793 y Fr(y)3176 3779 y Fo(L)p Fq(g)s Fp(\))1116 4060 y(+)1214 3973 y Fk(X)1207 4172 y Fr(e)1240 4153 y Fi(0)1262 4172 y Fn(2E)1367 4060 y Fq(D)1442 4075 y Fr(e;e)1528 4056 y Fi(0)1554 4060 y Fp(\()p Fq(m)1669 4077 y Fr(x;)1739 4050 y FA(a)p 1738 4062 34 3 v 1742 4103 a(\017)1786 4060 y Fp(\))1821 3986 y Fk(\002)1869 3968 y Fq(m)1949 3998 y Fr(x)p Fl(+)2054 3971 y FA(b)p 2054 3983 28 3 v 2055 4024 a(\017)2091 3998 y Fr(e)2124 3979 y Fi(0)2146 3998 y Fr(;)2176 3971 y FA(a)p 2175 3983 34 3 v 2179 4024 a(\017)2243 3968 y Fo(\000)g Fq(m)2414 3998 y Fr(x)p Fn(\000)2519 3971 y FA(b)p 2519 3983 28 3 v 2520 4024 a(\017)2556 3998 y Fr(e)2589 3979 y Fi(0)2611 3998 y Fr(;)2641 3971 y FA(a)p 2641 3983 34 3 v 2645 4024 a(\017)p 1869 4039 820 4 v 2195 4122 a Fp(2)p Fq(b=\017)2698 3986 y Fk(\003)2736 3959 y(i)2794 4060 y Fq(ds)2899 3983 y Fk(\014)2899 4037 y(\014)2930 3959 y(\021)3009 4060 y Fp(=)25 b(0)p Fq(:)3459 3933 y Fu(\(4.5\))118 4321 y(W)-8 b(e)24 b(next)g(reduce)e(\(4.5\))h(to)i(certain)e(equilibrium)h (eigenvalue)e(estimates)i(by)g(means)g(of)g(the)g(entropy)118 4428 y(inequality)30 b(and)f(the)h(F)-6 b(eynman-Kac)27 b(formula)i(\(see)g(proposition)h(A.8\).)44 b(Let)30 b(us)f(recall)f(the)i(former:)118 4536 y(given)23 b(two)h(probability)g (measures)f Fq(\031)s(;)15 b(\031)1527 4503 y Fn(0)1575 4536 y Fu(on)23 b(the)h(same)f(probability)i(space,)f(for)f(any)g Fq(\014)31 b(>)25 b Fp(0)f Fu(and)g(any)118 4644 y(bounded)h(and)g (measurable)f(function)h Fq(f)10 b Fu(,)1164 4814 y Fq(\031)s Fp(\()p Fq(f)g Fp(\))26 b Fo(\024)f Fq(\014)1522 4777 y Fn(\000)p Fl(1)1616 4741 y Fk(\010)1669 4814 y Fq(H)7 b Fp(\()p Fq(\031)19 b Fo(j)c Fq(\031)1953 4777 y Fn(0)1977 4814 y Fp(\))20 b(+)g(ln)2199 4741 y Fk(\000)2241 4814 y Fq(\031)2296 4777 y Fn(0)2319 4814 y Fp(\()p Fq(e)2396 4777 y Fr(\014)s(f)2485 4814 y Fp(\))2520 4741 y Fk(\001\011)3459 4814 y Fu(\(4.6\))118 4984 y(where)k Fq(H)7 b Fp(\()p Fq(\031)19 b Fo(j)c Fq(\031)672 4951 y Fn(0)696 4984 y Fp(\))25 b Fu(denotes)g(the)g(entropy)f(of)h Fq(\031)j Fu(w)-11 b(.r)h(.t.)33 b Fq(\031)2030 4951 y Fn(0)2053 4984 y Fu(.)e(It)26 b(is)g(simple)e(to)i(verify)f(that,)h(for)f(any)f (initial)118 5092 y(distribution)34 b Fq(\026)e Fu(on)g Fp(\012)897 5106 y Fr(\017)929 5092 y Fu(,)i(the)f(relative)f(entropy)f (between)g(the)i(path)g(measure)e Fs(P)2971 5059 y Fr(\026)3050 5092 y Fu(starting)i(from)f Fq(\026)118 5200 y Fu(and)26 b(the)h(equilibrium)e(path)i(measure)e Fs(P)1550 5167 y Fr(\026)1592 5175 y FA(\017)1653 5200 y Fu(starting)i(from)f(the)g (grand)f(canonical)g(measure)g Fq(\026)3426 5214 y Fr(\017)3484 5200 y Fu(with)118 5308 y(zero)f(chemical)g(potential,)i(satis\002es) 1495 5480 y Fq(H)1578 5407 y Fk(\000)1620 5480 y Fs(P)1675 5443 y Fr(\026)1737 5480 y Fo(j)15 b Fs(P)1832 5443 y Fr(\026)1874 5451 y FA(\017)1910 5407 y Fk(\001)1977 5480 y Fo(\024)25 b Fq(c)15 b(\017)2164 5443 y Fn(\000)p Fr(d)2259 5480 y Fq(:)p eop %%Page: 12 12 12 11 bop 118 45 a Fj(12)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(Therefore,)25 b(for)g(any)g Fq(\015)30 b(>)25 b Fp(0)h Fu(and)f(any)g(function)g Fq(h)h Fu(on)f Fp([0)p Fq(;)15 b(T)e Fp(])21 b Fo(\002)f Fp(\012)2386 380 y Fr(\017)395 599 y Fs(E)449 561 y Fr(\026)502 525 y Fk(\000)559 522 y(\014)559 576 y(\014)589 475 y(Z)680 502 y Fr(T)639 681 y Fl(0)750 599 y Fq(h)p Fp(\()p Fq(s;)15 b(\021)s Fp(\()p Fq(s)p Fp(\)\))p Fq(ds)1206 522 y Fk(\014)1206 576 y(\014)1253 525 y(\001)1320 599 y Fo(\024)1433 538 y Fq(c)p 1426 578 53 4 v 1426 661 a(\015)1508 599 y Fp(+)1609 538 y Fq(\017)1646 505 y Fr(d)p 1609 578 78 4 v 1622 661 a Fq(\015)1712 599 y Fp(ln)f Fs(E)1857 561 y Fr(\026)1900 569 y FA(\017)1940 498 y Fk(\020)1994 599 y Fp(exp)2133 525 y Fk(\010)2186 599 y Fq(\015)5 b(\017)2275 561 y Fn(\000)p Fr(d)2371 522 y Fk(\014)2371 576 y(\014)2401 475 y(Z)2492 502 y Fr(T)2451 681 y Fl(0)2562 599 y Fq(h)p Fp(\()p Fq(s;)15 b(\021)s Fp(\()p Fq(s)p Fp(\)\))p Fq(ds)3018 522 y Fk(\014)3018 576 y(\014)3050 525 y(\011)3103 498 y(\021)3157 599 y Fq(:)277 b Fu(\(4.7\))118 816 y(The)26 b(F)-6 b(eynman\226Kac)23 b(formula)h(\(see)h(proposition)i(A.8\))e (now)g(shows)h(that,)1018 984 y Fq(\017)1055 951 y Fr(d)p 1018 1025 V 1031 1108 a Fq(\015)1121 1046 y Fp(ln)14 b Fs(E)1266 1008 y Fr(\026)1309 1016 y FA(\017)1349 945 y Fk(\020)1403 1046 y Fp(exp)1542 972 y Fk(\010)1595 1046 y Fq(\015)5 b(\017)1684 1008 y Fn(\000)p Fr(d)1780 972 y Fk(\000)1821 1046 y Fo(\006)1907 922 y Fk(Z)1998 948 y Fr(T)1958 1128 y Fl(0)2068 1046 y Fq(h)p Fp(\()p Fq(s;)15 b(\021)s Fp(\()p Fq(s)p Fp(\)\))p Fq(ds)2524 972 y Fk(\001)r(\011)2621 945 y(\021)2700 1046 y Fo(\024)999 1182 y Fk(Z)1090 1209 y Fr(T)1049 1389 y Fl(0)1160 1306 y Fp(sup)f Fq(spec)1482 1326 y Fr(L)1530 1307 y Fg(2)1565 1326 y Fl(\()p Fr(\026)1634 1334 y FA(\017)1665 1326 y Fl(\))1696 1233 y Fk(\010)1749 1306 y Fo(\006)p Fq(h)p Fp(\()p Fq(s;)h Fo(\001)p Fp(\))22 b(+)e Fq(\015)2215 1269 y Fn(\000)p Fl(1)2309 1306 y Fq(\017)2346 1269 y Fr(d)p Fn(\000)p Fl(2)2477 1306 y Fo(L)2540 1320 y Fr(\017)2572 1233 y Fk(\011)2640 1306 y Fq(ds)25 b(:)679 b Fu(\(4.8\))118 1524 y(W)-8 b(e)26 b(now)f(apply)g(the)h(above)f(reasoning)f(to)i(the)g (function)f Fq(h)p Fp(\()p Fq(s;)15 b(\021)s Fp(\))27 b(=)50 b Fu(integrand)25 b(of)g(\(4.5\))o(.)32 b(Since)25 b(for)118 1632 y(any)k Fq(\017)i(>)g Fp(0)e(sup)671 1653 y Fr(s)p Fn(2)p Fl([0)p Fr(;T)10 b Fl(])915 1632 y Fp(sup)1052 1654 y Fr(x)p Fn(2)p Fm(T)1189 1635 y FA(d)1238 1632 y Fo(jr)1339 1599 y Fr(\017)1339 1654 y(e)1376 1632 y Fq(H)d Fp(\()p Fq(s;)15 b(\017x)p Fp(\))p Fo(j)32 b(\024)f Fq(c)p Fp(\()p Fq(H)7 b Fp(\))p Fu(,)31 b(after)e(a)g(suitable)g (reparametrization)f(of)h Fq(\015)5 b Fu(,)118 1746 y(in)25 b(order)g(to)h(prove)e(\(4.5\))h(it)h(is)g(enough)e(to)i(prove)e(the)i (following)e(key)h(eigenvalue)f(estimate.)118 1883 y FE(Proposition)32 b(4.1.)44 b Fh(Let)30 b Fq(d)k Fo(\025)f Fp(3)p Fh(.)46 b(Then,)31 b(almost)e(all)h(disorder)e(con\002gurations) g Fq(\013)j Fh(have)f(the)f(following)118 1991 y(property)-5 b(.)31 b(F)l(or)25 b(all)g Fq(\015)31 b(>)25 b Fp(0)674 2161 y(inf)662 2222 y Fr(g)r Fn(2)p Fm(G)941 2161 y Fp(lim)15 b(sup)812 2240 y Fr(b)p Fn(#)p Fl(0)p Fr(;)d(a)p Fn(#)p Fl(0)p Fr(;)g(`)p Fn("1)p Fr(;)f(\017)p Fn(#)p Fl(0)1379 2161 y Fp(sup)1425 2240 y Fr(J)1546 2161 y Fp(sup)j Fq(spec)1868 2181 y Fr(L)1916 2162 y Fg(2)1950 2181 y Fl(\()p Fr(\026)2019 2189 y FA(\017)2050 2181 y Fl(\))2082 2088 y Fk(\010)2135 2161 y Fq(\017)2172 2124 y Fn(\000)p Fl(1)2288 2139 y Fp(\026)2266 2161 y Fq(J)2325 2117 y Fr(g)2316 2191 y(b;a;`;\017)2525 2161 y Fp(+)20 b Fq(\015)5 b(\017)2705 2124 y Fr(d)p Fn(\000)p Fl(2)2835 2161 y Fo(L)2898 2088 y Fk(\011)2976 2161 y Fo(\024)25 b Fp(0)342 b Fu(\(4.9\))118 2396 y Fh(where)912 2546 y Fp(\026)890 2569 y Fq(J)949 2524 y Fr(g)940 2598 y(b;a;`;\017)1154 2569 y Fp(:=)25 b(Av)1393 2591 y Fr(x)p Fn(2)p Fm(T)1530 2572 y FA(d)1530 2607 y(\017)1564 2569 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))1782 2468 y Fk(h)1826 2569 y Fp(Av)1943 2587 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)2256 2596 y Fg(1)2294 2569 y Fp(\()p Fq(j)2366 2583 y Fr(y)r(;y)r Fl(+)p Fr(e)2572 2569 y Fp(+)20 b Fq(\034)2703 2583 y Fr(y)2744 2569 y Fo(L)p Fq(g)s Fp(\))911 2808 y(+)1008 2721 y Fk(X)1001 2920 y Fr(e)1034 2901 y Fi(0)1057 2920 y Fn(2E)1162 2808 y Fq(D)1237 2823 y Fr(e;e)1323 2804 y Fi(0)1349 2808 y Fp(\()p Fq(m)1464 2825 y Fr(x;)1534 2798 y FA(a)p 1533 2810 34 3 v 1537 2851 a(\017)1581 2808 y Fp(\))1616 2734 y Fk(\002)1664 2716 y Fq(m)1744 2746 y Fr(x)p Fl(+)1849 2719 y FA(b)p 1848 2731 28 3 v 1849 2772 a(\017)1886 2746 y Fr(e)1919 2727 y Fi(0)1941 2746 y Fr(;)1971 2719 y FA(a)p 1970 2731 34 3 v 1974 2772 a(\017)2038 2716 y Fo(\000)g Fq(m)2209 2746 y Fr(x)p Fn(\000)2314 2719 y FA(b)p 2314 2731 28 3 v 2315 2772 a(\017)2351 2746 y Fr(e)2384 2727 y Fi(0)2406 2746 y Fr(;)2436 2719 y FA(a)p 2436 2731 34 3 v 2440 2772 a(\017)p 1664 2787 820 4 v 1990 2870 a Fp(2)p Fq(b=\017)2493 2734 y Fk(\003)2531 2707 y(i)3409 2722 y Fu(\(4.10\))118 3079 y Fh(and)k Fq(J)35 b Fh(varies)24 b(in)h Fo(f)p Fq(J)35 b Fo(2)25 b Fq(C)7 b Fp(\()p Fs(T)1106 3046 y Fr(d)1148 3079 y Fp(\))41 b(:)f Fo(k)p Fq(J)9 b Fo(k)1438 3093 y Fn(1)1539 3079 y Fo(\024)25 b Fp(1)p Fo(g)p Fh(.)118 3283 y Fu(4.1.)45 b FE(Some)22 b(technical)g(tools)f(to)h(bound)h(the)e (spectrum.)44 b Fu(Before)20 b(we)h(turn)f(to)h(the)h(proof)e(of)i (propo-)118 3391 y(sition)i(4.1,)e(let)g(us)g(introduce)g(some)g(tools) h(to)f(deal)g(with)i(the)e(eigenvalue)f(problem)g(posed)h(in)h (\(4.9\).)118 3499 y(W)-8 b(e)26 b(begin)f(by)g(recalling)e(a)j(useful) f(subadditivity)j(property)d(of)h(the)f(supremum)f(of)i(the)g(spectrum) e(of)118 3606 y(a)h(selfadjoint)i(operator)e(and)g(explain)f(its)j (role)d(in)i(the)f(so\226called)g Fh(localization)f(technique)p Fu(.)118 3744 y(Given)h(a)g(\002nite)h(family)f Fo(f)p Fq(X)1067 3758 y Fr(i)1096 3744 y Fo(g)1141 3758 y Fr(i)p Fn(2)p Fr(I)1278 3744 y Fu(of)g(self)n(-adjoint)i(operators)e(on)g Fq(L)2436 3711 y Fl(2)2475 3744 y Fp(\()p Fq(\026)2565 3758 y Fr(\017)2598 3744 y Fp(\))p Fu(,)939 3929 y Fp(sup)14 b Fq(spec)1261 3949 y Fr(L)1309 3930 y Fg(2)1344 3949 y Fl(\()p Fr(\026)1413 3957 y FA(\017)1444 3949 y Fl(\))1476 3929 y Fo(f)1521 3842 y Fk(X)1533 4039 y Fr(i)p Fn(2)p Fr(I)1668 3929 y Fq(X)1743 3943 y Fr(i)1771 3929 y Fo(g)26 b(\024)1938 3842 y Fk(X)1950 4039 y Fr(i)p Fn(2)p Fr(I)2084 3929 y Fp(sup)14 b Fq(spec)2406 3949 y Fr(L)2454 3930 y Fg(2)2489 3949 y Fl(\()p Fr(\026)2558 3957 y FA(\017)2589 3949 y Fl(\))2621 3929 y Fo(f)p Fq(X)2741 3943 y Fr(i)2769 3929 y Fo(g)p Fq(;)570 b Fu(\(4.11\))118 4188 y(and)39 b(similarly)h(with)903 4120 y Fk(P)999 4215 y Fr(i)p Fn(2)p Fr(I)1149 4188 y Fu(replaced)e(by)h Fp(Av)1777 4202 y Fr(i)p Fn(2)p Fr(I)1888 4188 y Fu(.)73 b(The)40 b(subadditivity)i(property)d(allows)g(one)f(to)118 4296 y(exploits)22 b(the)f Fh(localization)f Fu(method)i(which)g(is)g(best)g (explained)e(by)h(means)g(of)h(an)f(example,)g(although)118 4404 y(the)26 b(underlying)e(idea)i(has)g(a)g(much)f(wider)h (application.)32 b(Let)26 b Fq(\017)g(>)f Fp(0)p Fu(,)i Fq(`)e(<)2718 4368 y Fl(1)p 2718 4383 36 4 v 2721 4435 a Fr(\017)2789 4404 y Fu(and)g(for)h(any)f Fq(x)h Fo(2)f Fs(T)3493 4371 y Fr(d)3493 4426 y(\017)3561 4404 y Fu(let)118 4512 y Fq(f)163 4526 y Fr(x)235 4512 y Fu(be)k(a)g(local)f(function)g (with)i(support)f(in)g Fp(\003)1713 4527 y Fr(x;`)1806 4512 y Fu(.)42 b(R)m(ecall)27 b(the)i(de\002nition)g(of)g Fo(M)p Fp(\(\003)3019 4527 y Fr(x;`)3112 4512 y Fp(\))h Fu(as)f(the)g(set)g(of)118 4620 y(canonical)24 b(Gibbs)i(measures)f(on) g Fp(\003)1354 4635 y Fr(x;`)1446 4620 y Fu(.)32 b(Then)711 4798 y Fp(sup)o Fq(spec)1018 4818 y Fr(L)1066 4799 y Fg(2)1101 4818 y Fl(\()p Fr(\026)1170 4826 y FA(\017)1201 4818 y Fl(\))1232 4724 y Fk(\010)1285 4798 y Fp(Av)1403 4820 y Fr(x)p Fn(2)p Fm(T)1540 4801 y FA(d)1540 4837 y(\017)1574 4798 y Fq(f)1619 4812 y Fr(x)1683 4798 y Fp(+)20 b Fq(\017)1811 4760 y Fr(d)p Fn(\000)p Fl(2)1941 4798 y Fo(L)2004 4812 y Fr(\017)2037 4724 y Fk(\011)873 4952 y Fo(\024)25 b Fp(Av)1087 4975 y Fr(x)p Fn(2)p Fm(T)1224 4956 y FA(d)1224 4991 y(\017)1273 4952 y Fp(sup)14 b Fq(spec)1595 4972 y Fr(L)1643 4953 y Fg(2)1678 4972 y Fl(\()p Fr(\026)1747 4980 y FA(\017)1778 4972 y Fl(\))1810 4879 y Fk(\010)1863 4952 y Fq(f)1908 4966 y Fr(x)1971 4952 y Fp(+)20 b Fq(c)15 b(\017)2153 4915 y Fn(\000)p Fl(2)2248 4952 y Fp(Av)2365 4967 y Fr(b)p Fn(2)p Fl(\003)2491 4979 y FA(x;`)2579 4952 y Fo(L)2642 4967 y Fr(b)2676 4879 y Fk(\011)873 5107 y Fo(\024)25 b Fp(Av)1087 5129 y Fr(x)p Fn(2)p Fm(T)1224 5110 y FA(d)1224 5146 y(\017)1384 5107 y Fp(sup)1273 5189 y Fr(\027)t Fn(2M)p Fl(\(\003)1519 5201 y FA(x;`)1603 5189 y Fl(\))1646 5107 y Fp(sup)14 b Fq(spec)1968 5126 y Fr(L)2016 5108 y Fg(2)2051 5126 y Fl(\()p Fr(\027)t Fl(\))2149 5033 y Fk(\010)2202 5107 y Fq(f)2247 5121 y Fr(x)2310 5107 y Fp(+)20 b Fq(c)15 b(\017)2492 5069 y Fn(\000)p Fl(2)2587 5107 y Fp(Av)2704 5122 y Fr(b)p Fn(2)p Fl(\003)2830 5134 y FA(x;`)2918 5107 y Fo(L)2981 5122 y Fr(b)3015 5033 y Fk(\011)3409 4990 y Fu(\(4.12\))118 5366 y(where)33 b(the)h(former)f(inequality)h (follows)g(from)f Fq(\017)1867 5333 y Fr(d)1907 5366 y Fo(L)1970 5380 y Fr(\017)2042 5366 y Fo(\024)39 b Fq(c)15 b Fp(Av)2324 5388 y Fr(x)p Fn(2)p Fm(T)2461 5369 y FA(d)2461 5405 y(\017)2496 5366 y Fp(Av)2613 5381 y Fr(b)p Fn(2)p Fl(\003)2739 5393 y FA(x;`)2827 5366 y Fo(L)2890 5381 y Fr(b)2958 5366 y Fu(together)33 b(with)i(the)118 5480 y(subadditivity)28 b(property)-12 b(,)25 b(while)h(the)f(latter)g(can)g (be)g(easily)g(checked.)p eop %%Page: 13 13 13 12 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(13)218 366 y Fu(Next)32 b(we)h(state)h(a)f(very)g(general)e(result)h(on)h Fp(sup)14 b Fq(spec)2127 386 y Fr(L)2175 367 y Fg(2)2209 386 y Fl(\()p Fr(\027)t Fl(\))2308 366 y Fo(f)p Ff(L)25 b Fp(+)g Fq(\014)5 b(V)21 b Fo(g)p Fu(,)35 b(where)e Ff(L)f Fu(is)i(an)f(ergodic)118 478 y(reversible)26 b(Markov)g (generator)g(on)g(a)i(\002nite)f(set)g Fq(E)33 b Fu(with)28 b(invariant)f(measure)f Fq(\026)p Fu(,)i(and)f(whose)g(proof)118 586 y(is)f(based)g(on)f(perturbation)g(theory)g(for)g(selfadjoint)i (operators)e(\(see)g(e.g.)31 b([23)o(]\).)118 723 y FE(Proposition)37 b(4.2.)47 b Fh(Let)34 b Fp(gap)q(\()p Ff(L)p Fq(;)15 b(\026)p Fp(\))35 b Fh(be)f(the)h(spectral)f(gap)g(of)g Ff(L)g Fh(in)g Fq(L)2547 690 y Fl(2)2587 723 y Fp(\()p Fq(\026)p Fp(\))h Fh(and)e(let,)k(for)d Fq(\014)47 b(>)40 b Fp(0)c Fh(and)118 830 y Fq(V)46 b Fp(:)25 b Fq(E)31 b Fo(7!)25 b Fs(R)s Fh(,)1305 957 y Fq(\025)1358 972 y Fr(\014)1430 957 y Fp(:=)g(sup)14 b Fq(spec)1873 977 y Fr(L)1921 958 y Fg(2)1956 977 y Fl(\()p Fr(\026)p Fl(\))2058 957 y Fo(f)p Ff(L)20 b Fp(+)g Fq(\014)5 b(V)20 b Fo(g)p Fq(:)118 1108 y Fh(Assume)25 b(without)g(loss)g(of)f(generality)g Fq(\026)p Fp(\()p Fq(V)c Fp(\))26 b(=)f(0)p Fh(.)32 b(If)1390 1276 y Fp(2)p Fq(\014)k Fp(gap\()p Ff(L)p Fq(;)15 b(\026)p Fp(\))1889 1238 y Fn(\000)p Fl(1)1984 1276 y Fo(k)p Fq(V)21 b Fo(k)2148 1290 y Fn(1)2248 1276 y Fq(<)k Fp(1)118 1441 y Fh(then)848 1606 y Fp(0)g Fo(\024)g Fq(\025)1067 1621 y Fr(\014)1140 1606 y Fo(\024)1693 1544 y Fq(\014)1749 1511 y Fl(2)p 1246 1585 990 4 v 1246 1668 a Fp(1)20 b Fo(\000)g Fp(2)p Fq(\014)36 b Fp(gap)q(\()p Ff(L)p Fq(;)15 b(\026)p Fp(\))1902 1642 y Fn(\000)p Fl(1)1996 1668 y Fo(k)p Fq(V)21 b Fo(k)2160 1682 y Fn(1)2260 1606 y Fq(\026)2315 1505 y Fk(\020)2369 1606 y Fq(V)5 b(;)15 b Fp(\()p Fo(\000)p Ff(L)p Fp(\))2669 1568 y Fn(\000)p Fl(1)2779 1606 y Fq(V)2852 1505 y Fk(\021)2907 1606 y Fq(:)118 1861 y Fu(The)27 b(above)f(proposition)i(suggests)e(that)i(in)f(order)f(to)h(prove)f (proposition)i(4.1)e(we)g(must)h(be)f(able)g(to)118 1969 y(estimate:)268 2105 y(\(1\))41 b(the)26 b(spectral)e(gap)h(of)h(the)f (generator)f Fo(L)1817 2119 y Fl(\003)1895 2105 y Fu(in)h(a)h(generic)d (box)i Fp(\003)p Fu(;)268 2213 y(\(2\))41 b(the)26 b Fq(H)664 2227 y Fn(\000)p Fl(1)783 2213 y Fu(norm)e(appearing)h(above.) 118 2349 y(W)-8 b(e)26 b(begin)e(with)j(the)e(\002rst)h(one.)118 2486 y FE(Proposition)38 b(4.3.)48 b Fu([10)o(])36 b Fh(Let)g Fp(\003)44 b Fo(\032)f Fs(Z)1497 2453 y Fr(d)1569 2486 y Fh(be)35 b(a)h(parallelepiped)e(with)i(longest)e(side)h Fq(`)p Fh(.)64 b(Then,)38 b(for)e(all)118 2594 y(disorder)24 b(con\002gurations)f Fq(\013)j Fh(and)e(all)h Fq(\027)31 b Fo(2)25 b(M)p Fp(\(\003\))p Fh(,)1526 2762 y Fp(gap)q(\()p Fo(L)1766 2776 y Fl(\003)1819 2762 y Fp(;)15 b Fq(\027)6 b Fp(\))26 b Fo(\025)f Fq(c)15 b(`)2159 2724 y Fn(\000)p Fl(2)3409 2762 y Fu(\(4.13\))118 2926 y Fh(In)33 b(particular)-12 b(,)36 b(for)c(all)h(disorder)e(con\002gurations)h(and)g(all)g Fq(\027)44 b Fo(2)38 b(M)p Fp(\(\003\))p Fh(,)f(the)32 b(following)g(P)-5 b(oincar)m(\264)-42 b(e)32 b(in-)118 3034 y(equality)25 b(holds)1439 3162 y Fp(V)-8 b(ar)1580 3176 y Fr(\027)1624 3162 y Fp(\()p Fq(f)10 b Fp(\))25 b Fo(\024)g Fq(c)15 b(`)1962 3124 y Fl(2)2002 3162 y Fo(D)2072 3176 y Fl(\003)2125 3162 y Fp(\()p Fq(f)10 b Fp(;)15 b Fq(\027)6 b Fp(\))1068 b Fu(\(4.14\))118 3307 y FE(R)m(emark)31 b(4.4.)43 b Fh(The)29 b(key)g(aspect)f(of)g(the) g(above)h(result)f(is)h(the)g(uniformity)g(in)f(the)h(disorder)e (con\002gura-)118 3415 y(tion.)43 b(Its)30 b(proof)e(is)h(based)f(on)g (some)g(clever)h(technique)f(developed)f(recently)h(in)h Fu([13)o(])h Fh(to)f(deal)f(with)i(the)118 3523 y(K)n(ac)25 b(model)g(for)g(the)g(Boltzmann)g(equation)f(and)h(extended)e(in)i Fu([12])h Fh(and)e Fu([11)o(])i Fh(to)g(other)f(kind)g(of)f(diffu-)118 3631 y(sions.)33 b(F)l(or)26 b(other)f(models)g(of)g(lattice)h(gas)f (dynamics)g(like)g(the)h(dilute)g(Ising)f(lattice)h(gas)f(in)h(the)g (Grif\002ths)118 3739 y(regime)h(the)g(above)g(uniformity)g(will)h(no)e (longer)h(be)g(available)f(and)h(a)g(more)g(sophisticated)f(analysis)g (is)118 3847 y(required)e(\(see)g Fu([15])h Fh(for)g(a)g(discussion\).) 118 4012 y Fu(Let)i(us)h(now)e(tackle)h(with)h(the)f Fq(H)1275 4026 y Fn(\000)p Fl(1)1396 4012 y Fu(norm.)36 b(Unfortunately)26 b(that)j(will)d(requires)h(the)g(proof)h(of)f(some) 118 4120 y(technical)e(bounds)g(that,)i(on)d(a)i(\002rst)g(reading,)f (can)f(be)h(just)h(skipped.)218 4228 y(F)-6 b(ollowing)29 b(the)i(theory)g(of)g(non)f(disordered)h(non-gradient)f(systems,)j(we)e (introduce)f(the)h(space)118 4336 y Fo(G)g(\032)25 b Fs(G)50 b Fu(de\002ned)25 b(as)421 4500 y Fo(G)30 b Fp(:=)25 b Fo(f)p Fq(g)30 b Fo(2)25 b Fs(G)65 b Fp(:)40 b Fo(9)15 b Fp(\003)25 b Fo(2)g Fs(F)61 b Fu(such)26 b(that,)51 b Fo(8)p Fq(\013)26 b Fu(and)f Fo(8)p Fq(\027)31 b Fo(2)24 b(M)2421 4463 y Fr(\013)2471 4500 y Fp(\(\003\))15 b Fq(;)42 b(\027)6 b Fp(\()p Fq(g)s Fp(\))26 b(=)f(0)15 b Fo(g)p Fq(:)304 b Fu(\(4.15\))118 4665 y(Equivalently)-12 b(,)21 b Fo(G)k Fu(can)19 b(be)g(de\002ned)h(as)g(the)g(set)g(of)g (functions)g Fq(g)29 b Fo(2)c Fs(G)44 b Fu(such)20 b(that)h(there)e (exists)i Fp(\003)k Fo(2)g Fs(F)30 b Fu(and)118 4773 y Fq(h)h Fo(2)f Fs(G)53 b Fu(with)29 b Fq(g)34 b Fp(=)d Fo(L)838 4787 y Fl(\003)890 4773 y Fq(h)p Fu(.)41 b(Since)28 b(the)g(dynamics)h(is)g(reversible)e(w)-11 b(.r)h(.t.)41 b(Gibbs)30 b(measures,)e(this)i(second)118 4881 y(characterization)19 b(assures)i(an)e(integration)h(by)g(parts)g(property)g(that)g(will)g (play)g(an)f(important)i(role)d(in)118 4989 y(the)27 b(sequel.)34 b(More)25 b(precisely)-12 b(,)26 b(if)h Fq(g)j Fp(=)d Fo(L)1517 5003 y Fl(\003)1570 4989 y Fq(h)p Fu(,)g(then,)g(for)f(any)g Fp(\003)2275 4956 y Fn(0)2325 4989 y Fu(containing)g Fp(\003)g Fu(and)g(any)h Fq(\027)32 b Fo(2)27 b(M)p Fp(\(\003)3577 4956 y Fn(0)3601 4989 y Fp(\))p Fu(,)118 5097 y Fq(\027)169 5023 y Fk(\000)210 5097 y Fq(g)s(;)15 b(f)10 b Fp(\))30 b(=)f Fq(\027)6 b Fp(\()p Fq(h;)15 b Fo(L)757 5111 y Fl(\003)811 5097 y Fq(f)10 b Fp(\))p Fu(.)38 b(Moreover)-10 b(,)26 b(if)i Fq(V)48 b Fu(and)28 b Fp(\001)f Fu(are)g(such)g(that)i Fp(\003)2459 5111 y Fr(x)2532 5097 y Fo(\032)g Fq(V)48 b Fu(for)27 b(any)g Fq(x)i Fo(2)g Fp(\001)p Fu(,)f(then)g(for)118 5205 y(any)d Fq(A)h(>)f Fp(0)g Fu(and)h Fq(\027)31 b Fo(2)24 b(M)p Fp(\()p Fq(V)d Fp(\))p Fu(,)403 5312 y Fk(\014)403 5367 y(\014)433 5390 y Fq(\027)484 5316 y Fk(\000)533 5303 y(X)526 5500 y Fr(x)p Fn(2)p Fl(\001)686 5390 y Fq(\034)726 5404 y Fr(x)770 5390 y Fq(g)s(;)15 b(f)911 5316 y Fk(\001)953 5312 y(\014)953 5367 y(\014)1009 5390 y Fo(\024)25 b Fq(c)p Fp(\()p Fq(g)s Fp(\))p Fo(j)p Fp(\001)p Fo(j)1396 5325 y Fg(1)p 1397 5337 31 3 v 1397 5378 a(2)1442 5390 y Fo(D)1512 5404 y Fr(V)1573 5390 y Fp(\()p Fq(f)10 b Fp(;)15 b Fq(\027)6 b Fp(\))1799 5325 y Fg(1)p 1799 5337 V 1799 5378 a(2)1869 5390 y Fo(\024)25 b Fq(A)15 b(c)p Fp(\()p Fq(g)s Fp(\))p Fo(j)p Fp(\001)p Fo(j)22 b Fp(+)e Fq(A)2510 5352 y Fn(\000)p Fl(1)2620 5390 y Fq(c)p Fp(\()p Fq(g)s Fp(\))p Fo(D)2845 5404 y Fr(V)2908 5390 y Fp(\()p Fq(f)10 b Fp(;)15 b Fq(\027)6 b Fp(\))285 b Fu(\(4.16\))p eop %%Page: 14 14 14 13 bop 118 45 a Fj(14)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(where,)25 b(for)g(some)g(suitable)h(constant)f Fq(c)p Fp(\(\003\))p Fu(,)1217 555 y Fq(c)p Fp(\()p Fq(g)s Fp(\))i(:=)e Fq(c)p Fp(\(\003\))15 b(sup)1754 627 y Fr(\013)1955 555 y Fp(sup)1860 638 y Fr(\027)t Fn(2M)2030 619 y FA(\013)2075 638 y Fl(\()p Fr(V)h Fl(\))2186 481 y Fk(\000)2228 555 y Fq(\027)6 b Fp(\()p Fq(h)2366 517 y Fl(2)2406 555 y Fp(\))2441 481 y Fk(\001)2493 477 y Fg(1)p 2493 489 31 3 v 2493 530 a(2)2537 555 y Fq(:)847 b Fu(\(4.17\))118 786 y(A)26 b(\002rst)h(simple)f(consequence)d(of)k(integration)f(by)g(parts)g (\(see)g(chapter)g(7)f(of)i([23)o(])f(and)g(section)h(1.16)118 894 y(of)f([15)o(])f(for)h(a)f(proof\))g(is)h(the)g(following)e(bound.) 118 1025 y FE(Proposition)g(4.5.)37 b Fh(Let)22 b Fq(g)29 b Fo(2)c(G)i Fh(have)22 b(support)g(included)e(in)i Fp(\003)2215 1039 y Fr(s)2252 1025 y Fh(.)30 b(Then)22 b(for)f(any)g(disorder)g (con\002guration)118 1133 y Fq(\013)p Fh(,)26 b(any)f Fq(\015)30 b(>)25 b Fp(0)h Fh(and)e(any)h(family)g(of)f(functions)g Fq(F)38 b Fp(:=)25 b Fo(f)p Fq(f)2023 1147 y Fr(x)2067 1133 y Fo(g)2112 1156 y Fr(x)p Fn(2)p Fm(T)2249 1137 y FA(d)2249 1172 y(\017)2309 1133 y Fh(on)g Fp(\012)2497 1147 y Fr(\017)2529 1133 y Fh(,)176 1311 y Fp(sup)14 b Fq(spec)498 1331 y Fr(L)546 1312 y Fg(2)580 1331 y Fl(\()p Fr(\026)649 1339 y FA(\017)680 1331 y Fl(\))712 1311 y Fo(f)p Fq(\017)794 1274 y Fn(\000)p Fl(1)889 1311 y Fp(Av)1006 1334 y Fr(x)p Fn(2)p Fm(T)1143 1315 y FA(d)1143 1350 y(\017)1177 1311 y Fp(\()p Fq(\034)1252 1325 y Fr(x)1296 1311 y Fq(g)19 b(f)1403 1325 y Fr(x)1447 1311 y Fp(\))h(+)g Fq(\015)5 b(\017)1682 1274 y Fr(d)p Fn(\000)p Fl(2)1813 1311 y Fo(Lg)25 b(\024)712 1505 y Fq(\017)749 1468 y Fn(\000)p Fl(1)843 1505 y Fq(c)p Fp(\()p Fq(g)s(;)15 b Fo(k)p Fq(F)e Fo(k)1164 1519 y Fn(1)1242 1505 y Fp(\))p Fo(kr)p Fq(F)g Fo(k)1514 1519 y Fn(1)1610 1505 y Fp(+)20 b(sup)14 b Fq(spec)2023 1525 y Fr(L)2071 1506 y Fg(2)2105 1525 y Fl(\()p Fr(\026)2174 1533 y FA(\017)2205 1525 y Fl(\))2237 1505 y Fo(f)p Fq(c)p Fp(\()p Fq(g)s Fp(\)Av)2556 1528 y Fr(x)p Fn(2)p Fm(T)2693 1509 y FA(d)2693 1544 y(\017)2728 1505 y Fq(f)2783 1468 y Fl(2)2773 1528 y Fr(x)2842 1505 y Fp(+)2943 1444 y(1)p 2943 1485 46 4 v 2943 1568 a(2)2998 1505 y Fq(\015)5 b(\017)3087 1468 y Fr(d)p Fn(\000)p Fl(2)3218 1505 y Fo(Lg)p Fq(;)3409 1421 y Fu(\(4.18\))118 1691 y Fh(where)25 b Fo(k)p Fq(F)13 b Fo(k)533 1705 y Fn(1)634 1691 y Fp(:=)25 b(sup)892 1714 y Fr(x)p Fn(2)p Fm(T)1029 1695 y FA(d)1029 1730 y(\017)1078 1691 y Fo(k)p Fq(f)1168 1705 y Fr(x)1212 1691 y Fo(k)1257 1705 y Fn(1)1357 1691 y Fh(and)f Fo(kr)p Fq(F)13 b Fo(k)1765 1705 y Fn(1)1866 1691 y Fp(:=)25 b(sup)2124 1714 y Fr(x)p Fn(2)p Fm(T)2261 1695 y FA(d)2261 1730 y(\017)2311 1691 y Fp(sup)2448 1713 y Fr(b)p Fn(\032)p Fl(\003)2582 1721 y FA(x;s)2688 1691 y Fo(kr)2809 1706 y Fr(b)2843 1691 y Fq(f)2888 1705 y Fr(x)2932 1691 y Fo(k)2977 1705 y Fn(1)3052 1691 y Fh(.)218 1830 y Fu(In)j(the)h(space)f Fo(G)34 b Fu(it)c(is)f(also)g(possible)g(to)g(introduce)f(a)g Fq(H)2188 1844 y Fn(\000)p Fl(1)2311 1830 y Fu(norm)f(closely)h (related)g(to)h(that)g(given)118 1938 y(by)c(perturbation)g(theory)h (\(see)f(proposition)h(4.2)f(above\).)118 2046 y(Given)j(positive)h (integers)e Fq(`;)15 b(s)28 b Fu(with)h Fq(s)1450 2013 y Fl(2)1519 2046 y Fo(\024)h Fq(`)e Fu(and)g Fq(f)5 b(;)15 b(g)33 b Fo(2)c(G)34 b Fu(with)29 b Fp(\001)2488 2061 y Fr(f)2533 2046 y Fq(;)15 b Fp(\001)2649 2060 y Fr(g)2719 2046 y Fo(\032)29 b Fp(\003)2882 2060 y Fr(s)2919 2046 y Fu(,)g(for)f(any)g(canonical)118 2154 y(or)d(grand)g(canonical)f (Gibbs)i(measure)e Fq(\026)h Fu(on)g Fp(\003)1763 2169 y Fr(`)1821 2154 y Fu(we)g(de\002ne)819 2334 y Fq(V)872 2349 y Fr(`)905 2334 y Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(;)g Fq(\026)p Fp(\))26 b(:=)g(\(2)p Fq(l)r Fp(\))1498 2297 y Fn(\000)p Fr(d)1594 2334 y Fq(\026)1649 2233 y Fk(\020)1736 2248 y(X)1703 2449 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)1867 2458 y Fg(1)1915 2334 y Fq(\034)1955 2348 y Fr(x)1999 2334 y Fq(f)5 b(;)2089 2261 y Fk(\000)2151 2334 y Fo(\000)19 b(L)2304 2348 y Fl(\003)2353 2360 y FA(`)2387 2261 y Fk(\001)2429 2283 y Fn(\000)p Fl(1)2571 2248 y Fk(X)2538 2449 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)2702 2458 y Fg(1)2751 2334 y Fq(\034)2791 2348 y Fr(x)2834 2334 y Fq(g)2880 2233 y Fk(\021)2935 2334 y Fq(:)449 b Fu(\(4.19\))118 2598 y(If)39 b Fp(\003)278 2613 y Fr(`)350 2598 y Fu(is)g(replaced)e (by)i Fp(\003)1022 2613 y Fr(z)s(;`)1149 2598 y Fu(and)f(the)h(above)f (sums)h(are)f(over)f Fq(x)48 b Fo(2)f Fp(\003)2659 2613 y Fr(z)s(;`)2744 2622 y Fg(1)2820 2598 y Fu(we)38 b(will)g(simply)h (write)118 2706 y Fq(V)171 2721 y Fr(z)s(;`)259 2706 y Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(;)g Fq(\026)p Fp(\))27 b Fu(and)e(if)h Fq(f)35 b Fp(=)25 b Fq(g)j Fu(we)d(write)h Fq(V)1512 2721 y Fr(`)1545 2706 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)p Fp(\))27 b Fu(or)e Fq(V)1945 2721 y Fr(z)s(;`)2033 2706 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)p Fp(\))p Fu(.)118 2814 y(It)26 b(is)g(simple)f(to)h(check)e(that)j Fq(V)1165 2829 y Fr(`)1198 2814 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)p Fp(\))26 b Fu(can)f(be)g(variationally)g(characterized)f(as)i(follows:) 880 2987 y Fq(V)933 3002 y Fr(`)966 2987 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)p Fp(\))27 b(=)e(\(2)p Fq(l)r Fp(\))1444 2950 y Fn(\000)p Fr(d)1555 2987 y Fp(sup)1603 3066 y Fr(h)1692 2914 y Fk(\010)1745 2987 y Fp(2)p Fq(\026)1845 2914 y Fk(\000)1920 2901 y(X)1887 3102 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)2051 3111 y Fg(1)2099 2987 y Fq(\034)2139 3001 y Fr(x)2183 2987 y Fq(g)s(;)15 b(h)2321 2914 y Fk(\001)2384 2987 y Fo(\000)20 b(D)2545 3001 y Fl(\003)2594 3013 y FA(`)2628 2987 y Fp(\()p Fq(h)p Fp(;)15 b Fq(\026)p Fp(\))2845 2914 y Fk(\011)1204 3318 y Fp(=)25 b(\(2)p Fq(l)r Fp(\))1444 3280 y Fn(\000)p Fr(d)1555 3318 y Fp(sup)1603 3397 y Fr(h)1717 3239 y Fq(\026)p Fp(\()1807 3171 y Fk(P)1903 3266 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)2067 3275 y Fg(1)2120 3239 y Fq(\034)2160 3253 y Fr(x)2203 3239 y Fq(g)s(;)15 b(h)p Fp(\))2376 3206 y Fl(2)p 1717 3297 701 4 v 1882 3380 a Fo(D)1952 3394 y Fl(\003)2001 3406 y FA(`)2035 3380 y Fp(\()p Fq(h)p Fp(;)g Fq(\026)p Fp(\))3409 3172 y Fu(\(4.20\))118 3529 y(where)25 b Fp(sup)526 3551 y Fr(h)596 3529 y Fu(is)h(taken)f(among)f(the)i(non)e(constant)h (functions)g(with)i(support)f(contained)f(in)g Fp(\003)3417 3544 y Fr(`)3450 3529 y Fu(.)118 3637 y(The)g(variational)g (characterization)g(allows)f(one)g(to)h(derive)g(some)f(simple)h (bounds)g(on)f Fq(V)3237 3652 y Fr(`)3270 3637 y Fp(\()p Fq(g)s 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4565 y Fr(x)861 4551 y Fq(g)s(;)15 b(h)p Fp(\))1034 4513 y Fl(2)1100 4551 y Fo(\024)25 b Fq(c)1283 4465 y Fk(X)1250 4666 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)1414 4675 y Fg(1)1463 4551 y Fq(\026)1518 4477 y Fk(\000)1574 4551 y Fp(V)-8 b(ar)1716 4565 y Fr(\026)1763 4551 y Fp(\()p Fq(\034)1838 4565 y Fr(x)1882 4551 y Fq(g)19 b Fo(j)c(F)2049 4565 y Fr(x)2093 4551 y Fp(\))2128 4477 y Fk(\001)2218 4465 y(X)2185 4666 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)2349 4675 y Fg(1)2398 4551 y Fq(\026)2453 4477 y Fk(\000)2509 4551 y Fp(V)-8 b(ar)2651 4565 y Fr(\026)2698 4551 y Fp(\()p Fq(h)15 b Fo(j)g(F)2905 4565 y Fr(x)2950 4551 y Fp(\))2985 4477 y Fk(\001)3027 4551 y Fq(:)357 b Fu(\(4.21\))118 4815 y(If)26 b(we)f(appeal)g(now)g(to)h(the)f(P)-5 b(oincar\264)-45 b(e)25 b(inequality)1087 4986 y Fp(V)-8 b(ar)1228 5000 y Fr(\026)1275 4986 y Fp(\()p Fq(h)15 b Fo(j)g(F)1482 5000 y Fr(x)1527 4986 y Fp(\))26 b Fo(\024)f Fq(cs)1766 4948 y Fl(2)1846 4899 y Fk(X)1821 5097 y Fr(b)p Fn(\032)p Fl(\001)1965 5105 y FA(x)2018 4986 y Fq(\026)2073 4912 y Fk(\000)2115 4986 y Fq(c)2154 5001 y Fr(b)2189 4986 y Fp(\()p Fo(r)2300 5001 y Fr(b)2334 4986 y Fq(h)p Fp(\))2421 4948 y Fl(2)2476 4986 y Fo(j)15 b(F)2581 5000 y Fr(x)2626 4912 y Fk(\001)2668 4986 y Fq(;)118 5249 y Fu(the)28 b(last)g(sum)f(in)h(\(4.21\))e(is)i(bounded)f(by)h Fq(c)15 b(s)1699 5216 y Fr(d)p Fl(+2)1829 5249 y Fo(D)1899 5263 y Fl(\003)1948 5275 y FA(`)1983 5249 y Fp(\()p Fq(h)p Fp(;)g Fq(\026)p Fp(\))p Fu(.)39 b(R)m(ecalling)25 b(\(4.20\),)i(for)g (any)h Fq(`)h(>)f(s)3480 5216 y Fl(2)3547 5249 y Fu(we)118 5357 y(\002nally)d(get)1033 5478 y Fq(V)1086 5493 y Fr(`)1119 5478 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)p Fp(\))27 b Fo(\024)e Fq(c)15 b(s)1550 5441 y Fr(d)p Fl(+2)1680 5478 y Fp(Av)1798 5497 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)1962 5506 y Fg(1)1999 5478 y Fq(\026)2054 5405 y Fk(\000)2111 5478 y Fp(V)-8 b(ar)2252 5492 y Fr(\026)2299 5478 y Fp(\()p Fq(\034)2374 5492 y Fr(x)2418 5478 y Fq(g)19 b Fo(j)c(F)2585 5492 y Fr(x)2629 5478 y Fp(\))2679 5405 y Fk(\001)2721 5478 y Fq(:)663 b Fu(\(4.22\))p eop %%Page: 15 15 15 14 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(15)118 366 y Fu(In)25 b(particular)1447 526 y Fq(V)1500 541 y Fr(`)1533 526 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)p Fp(\))27 b Fo(\024)e Fq(c)15 b(s)1964 488 y Fr(d)p Fl(+2)2095 526 y Fo(k)p Fq(g)s Fo(k)2231 488 y Fl(2)2231 548 y Fn(1)2307 526 y Fq(:)1077 b Fu(\(4.23\))118 699 y(In)34 b(order)f(to)h(bene\002t) f(of)h(the)g(ergodicity)f(of)h(the)g(random)f(\002eld,)j(it)e(is)h (natural)e(to)h(de\002ne,)h(for)f(any)118 807 y Fq(m)25 b Fo(2)g Fp(\(0)p Fq(;)15 b Fp(1\))27 b Fu(and)f(any)f Fq(g)k Fo(2)24 b(G)5 b Fu(,)689 1028 y Fq(V)742 1042 y Fr(m)808 1028 y Fp(\()p Fq(g)s Fp(\))27 b(:=)i(lim)1072 1089 y Fr(`)p Fn("1)1207 1028 y Fp(\(2)p Fq(`)p Fp(\))1360 990 y Fn(\000)p Fr(d)1471 1028 y Fs(E)1526 927 y Fk(h)1575 1028 y Fq(\026)1630 990 y Fr(\025)1671 999 y Fg(0)1706 990 y Fl(\()p Fr(m)p Fl(\))1827 954 y Fk(\000)1917 941 y(X)1884 1143 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)2048 1152 y Fg(1)2096 1028 y Fq(\034)2136 1042 y Fr(x)2180 1028 y Fq(g)s(;)15 b Fp(\()p Fo(\000L)2435 1042 y Fl(\003)2484 1054 y FA(`)2519 1028 y Fp(\))2554 990 y Fn(\000)p Fl(1)2696 941 y Fk(X)2663 1143 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)2827 1152 y Fg(1)2876 1028 y Fq(\034)2916 1042 y Fr(x)2959 1028 y Fq(g)3005 954 y Fk(\001)3048 927 y(i)3409 1028 y Fu(\(4.24\))118 1330 y(where,)30 b(we)e(recall,)h Fq(\025)888 1344 y Fl(0)927 1330 y Fp(\()p Fq(m)p Fp(\))g Fu(is)h(the)f(annealed)f (chemical)g(potential)h(corresponding)e(to)j(the)f(particle)118 1438 y(density)f Fq(m)p Fu(.)35 b(If)27 b Fq(m)g Fp(=)g(0)p Fq(;)15 b Fp(1)28 b Fu(we)f(simply)f(set)h Fq(V)1636 1452 y Fr(m)1703 1438 y Fp(\()p Fq(g)s Fp(\))i(=)e(0)g Fu(for)f(any)h Fq(g)k Fo(2)c(G)5 b Fu(.)36 b(In)26 b(section)h(7)f(we)g (will)h(prove,)118 1545 y(among)36 b(other)g(results,)k(that)e(the)f (limit)g(appearing)f(in)h(\(4.24\))e(exists)j(\002nite)f(and)f(that)i (it)g(de\002nes)118 1653 y(a)g(semi\226inner)f(product)h(on)f Fo(G)44 b Fu(\(see)37 b(theorem)g(7.2)g(there\).)69 b(W)n(ith)39 b(this)g(de\002nition)g(we)e(have)h(the)118 1761 y(following)25 b(result.)118 1912 y FE(L)r(emma)i(4.6.)42 b Fh(Let)25 b Fq(g)k Fo(2)c(G)5 b Fh(.)32 b(Then)894 2104 y Fp(lim)15 b(sup)901 2183 y Fr(`)p Fn("1)p Fr(;)c(\017)p Fn(#)p Fl(0)1188 2104 y Fp(Av)1305 2132 y Fn(j)p Fr(x)p Fn(j\024)1450 2105 y Fg(1)p 1449 2117 31 3 v 1451 2158 a FA(\017)1619 2104 y Fp(sup)1509 2187 y Fr(\027)t Fn(2M)p Fl(\(\003)1755 2199 y FA(x;`)1839 2187 y Fl(\))1881 2104 y Fq(V)1934 2119 y Fr(x;`)2027 2031 y Fk(\000)2068 2104 y Fq(g)s Fp(;)k Fq(\027)2205 2031 y Fk(\001)2273 2104 y Fo(\024)76 b Fp(sup)2369 2187 y Fr(m)p Fn(2)p Fl([0)p Fr(;)p Fl(1])2623 2104 y Fq(V)2676 2118 y Fr(m)2743 2104 y Fp(\()p Fq(g)s Fp(\))p Fq(:)525 b Fu(\(4.25\))118 2405 y Fh(Proof)-8 b(.)42 b Fu(As)32 b(in)g([23],)i(chapter)e(7,)i(lemma)d(4.3,)i(we)f (introduce)g(a)g(scale)f(parameter)h Fq(k)s Fu(,)i(with)f Fq(k)40 b Fo(")e(1)118 2531 y Fu(after)31 b Fq(`)36 b Fo(")g(1)p Fu(,)d(and)e(partition)i Fp(\003)1256 2546 y Fr(`)1320 2531 y Fu(in)f(cubes)e(of)i(side)g Fp(2)p Fq(k)c Fp(+)c(1)p Fu(.)50 b(More)30 b(precisely)-12 b(,)32 b(we)g(de\002ne)e Fp(\003)3432 2483 y Fl(\()p Fr(k)r Fl(\))3432 2560 y Fr(`)3565 2531 y Fp(:=)118 2648 y(\003)181 2663 y Fr(`)220 2648 y Fo(\\)6 b Fp(\(2)p Fq(k)j Fp(+)d(1\))p Fs(Z)645 2615 y Fr(d)705 2648 y Fu(and)21 b(write)g Fp(\003)1164 2663 y Fr(`)1223 2648 y Fp(=)k Fq(B)1388 2663 y Fr(k)r(;l)1478 2648 y Fo([)1545 2574 y Fk(\000)1586 2648 y Fo([)1647 2689 y Fr(z)s Fn(2)p Fl(\003)1779 2654 y Fg(\()p FA(k)q Fg(\))1779 2713 y FA(`)1869 2648 y Fp(\003)1932 2663 y Fr(z)s(;k)2030 2574 y Fk(\001)2093 2648 y Fu(where)20 b Fq(B)2428 2663 y Fr(k)r(;`)2544 2648 y Fp(:=)26 b(\003)2729 2663 y Fr(`)2768 2648 y Fo(n)6 b([)2880 2689 y Fr(z)s Fn(2)p Fl(\003)3012 2654 y Fg(\()p FA(k)q Fg(\))3012 2713 y FA(`)3102 2648 y Fp(\003)3165 2663 y Fr(z)s(;k)3263 2648 y Fu(.)30 b(Then,)22 b(by)118 2788 y(proceeding)28 b(as)j(in)f([23)o(])g(and)g(by)f(using)h(the)g(variational)g (characterization)f(\(4.20\))g(together)g(with)118 2896 y(the)d(integration)f(by)g(parts)h(formula)e(\(4.16\),)h(for)g(any)g Fq(\027)31 b Fo(2)25 b(M)p Fp(\(\003)2391 2911 y Fr(`)2424 2896 y Fp(\))h Fu(we)f(get)696 3150 y Fq(V)749 3165 y Fr(`)782 3150 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\027)6 b Fp(\))26 b Fo(\024)f Fp(\(2)p Fq(`)p Fp(\))1264 3112 y Fn(\000)p Fr(d)1376 3150 y Fp(sup)1424 3229 y Fr(h)p 1424 3241 41 3 v 1513 3076 a Fk(\010)1609 3064 y(X)1566 3288 y Fr(z)s Fn(2)p Fl(\003)1698 3252 y Fg(\()p FA(k)q Fg(\))1698 3311 y FA(`)1799 3150 y Fq(F)1857 3164 y Fr(z)1897 3150 y Fp(\()p Fq(h)1984 3164 y Fr(z)2024 3150 y Fp(;)15 b Fq(\027)6 b Fp(\))2150 3076 y Fk(\011)2224 3150 y Fp(+)20 b Fq(c)p Fp(\()p Fq(g)s Fp(\))2470 3019 y Fk(q)p 2563 3019 454 4 v 2563 3150 a Fq(k)s(`)2651 3124 y Fn(\000)p Fl(1)2765 3150 y Fp(+)g Fq(k)2906 3114 y Fn(\000)2971 3087 y Fg(1)p 2971 3099 31 3 v 2971 3140 a(2)3031 3076 y Fk(\011)3409 3150 y 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FA(z)r(;k)3512 3612 y Fu(and)118 3764 y Fo(D)s Fp(\()p Fq(h)278 3778 y Fr(z)318 3764 y Fp(;)15 b Fq(\027)6 b Fp(\))26 b Fo(\024)f Fq(c)p Fp(\()p Fq(g)s Fp(\))p Fq(k)771 3731 y Fr(d)813 3764 y Fu(.)118 3872 y(Actually)32 b(it)j(is)f(simple)f(to)g(check)g(that)h(in)f (\(4.26\))f(we)h(can)g(restrict)g(the)h(supremum)e(to)h(families)h Fq(h)p 3609 3887 V 118 3980 a Fu(that)26 b(satisfy)i(the)d(extra)g (condition)h Fo(k)p Fq(h)p Fo(k)1496 3994 y Fn(1)1597 3980 y Fo(\024)f Fq(c)p Fp(\()p Fq(g)s Fp(\))p Fq(c)1887 3995 y Fr(k)1957 3980 y Fu(for)g(some)g(constant)g Fq(c)2727 3995 y Fr(k)2795 3980 y Fu(depending)g(on)g Fq(k)s Fu(.)118 4088 y(Therefore,)f(if)g Fq(m)g Fu(is)h(the)f(particle)g(density)h (associated)g(to)g(the)f(canonical)f(measure)f Fq(\027)30 b Fu(and)24 b(thanks)h(to)118 4196 y(the)i(equivalence)d(of)i (ensembles)f(\(see)g(lemmas)h(A.4)g(and)g(A.3\),)g(for)g(any)g (disorder)h(con\002guration)e Fq(\013)118 4304 y Fu(we)g(get)419 4438 y Fk(\014)419 4492 y(\014)493 4429 y(X)450 4653 y Fr(z)s Fn(2)p Fl(\003)582 4617 y Fg(\()p FA(k)q Fg(\))582 4676 y FA(`)683 4515 y Fq(F)741 4529 y Fr(z)781 4515 y Fp(\()p Fq(h)868 4529 y Fr(z)908 4515 y Fp(;)15 b Fq(\027)6 b Fp(\))21 b Fo(\000)1189 4429 y Fk(X)1146 4653 y Fr(z)s Fn(2)p Fl(\003)1278 4617 y Fg(\()p FA(k)q Fg(\))1278 4676 y FA(`)1379 4515 y Fq(F)1437 4529 y Fr(z)1477 4515 y Fp(\()p Fq(h)1564 4529 y Fr(z)1604 4515 y Fp(;)15 b Fq(\026)1699 4467 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))1699 4543 y(\003)1748 4555 y FA(`)1862 4515 y Fp(\))1897 4438 y Fk(\014)1897 4492 y(\014)1953 4515 y Fo(\024)25 b Fq(c)p Fp(\()p Fq(g)s Fp(\))p Fq(c)2243 4530 y Fr(k)2288 4515 y Fq(;)419 4741 y Fk(\014)419 4796 y(\014)493 4732 y(X)450 4956 y Fr(z)s Fn(2)p Fl(\003)582 4921 y Fg(\()p FA(k)q Fg(\))582 4980 y FA(`)683 4818 y Fq(F)741 4832 y Fr(z)781 4818 y Fp(\()p Fq(h)868 4832 y Fr(z)908 4818 y Fp(;)15 b Fq(\026)1003 4771 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))1003 4847 y(\003)1052 4859 y FA(`)1166 4818 y Fp(\))21 b Fo(\000)1356 4732 y Fk(X)1313 4956 y Fr(z)s Fn(2)p Fl(\003)1445 4921 y Fg(\()p FA(k)q Fg(\))1445 4980 y FA(`)1545 4818 y Fq(F)1603 4832 y Fr(z)1644 4818 y Fp(\()p Fq(h)1731 4832 y Fr(z)1771 4818 y Fp(;)15 b Fq(\026)1866 4781 y Fr(\025)1907 4790 y Fg(0)1942 4781 y Fl(\()p Fr(m)p Fl(\))2064 4818 y Fp(\))2099 4741 y Fk(\014)2099 4796 y(\014)2154 4818 y Fo(\024)25 b Fq(c)p Fp(\()p Fq(g)s Fp(\))p Fq(c)2444 4833 y Fr(k)2504 4818 y Fq(`)2542 4781 y Fr(d)2598 4741 y Fk(\014)2598 4796 y(\014)2628 4818 y Fq(m)20 b Fo(\000)g Fq(\026)2874 4781 y Fr(\025)2915 4790 y Fg(0)2950 4781 y Fl(\()p Fr(m)p Fl(\))3071 4818 y Fp(\()p Fq(m)3186 4832 y Fl(\003)3235 4844 y FA(`)3269 4818 y Fp(\))3304 4741 y Fk(\014)3304 4796 y(\014)3335 4818 y Fq(:)118 5149 y Fu(Thanks)26 b(to)g(the)f(previous)g(observations)h(we)f(\002nally)g(obtain)517 5400 y Fp(Av)634 5428 y Fn(j)p Fr(x)p Fn(j\024)779 5401 y Fg(1)p 778 5413 31 3 v 780 5454 a FA(\017)948 5400 y Fp(sup)838 5483 y Fr(\027)t Fn(2M)p Fl(\(\003)1084 5495 y FA(x;`)1168 5483 y Fl(\))1210 5400 y Fq(V)1263 5415 y Fr(x;`)1356 5327 y Fk(\000)1397 5400 y Fq(g)s(;)15 b(\027)1534 5327 y Fk(\001)1602 5400 y Fo(\024)25 b Fq(c)1737 5270 y Fk(q)p 1828 5270 454 4 v 130 x Fq(k)s(`)1916 5374 y Fn(\000)p Fl(1)2031 5400 y Fp(+)20 b Fq(k)2172 5365 y Fn(\000)2237 5337 y Fg(1)p 2237 5349 31 3 v 2237 5391 a(2)2302 5400 y Fp(+)g Fq(c)2432 5415 y Fr(k)2475 5400 y Fq(`)2513 5363 y Fn(\000)p Fr(d)2628 5400 y Fp(+)g Fq(c)2758 5415 y Fr(k)2801 5400 y Fp(\002)2872 5414 y Fl(1)2931 5400 y Fp(+)g Fq(c)3061 5415 y Fr(k)r(;`)3153 5400 y Fp(\002)3224 5414 y Fl(2)p eop %%Page: 16 16 16 15 bop 118 47 a Fj(16)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(where)25 b Fq(c)428 381 y Fr(k)r(;`)544 366 y Fu(is)i(a)e(positive)h(constant)g (depending)e(on)h Fq(k)s(;)15 b(`)26 b Fu(such)f(that)i Fp(lim)2634 381 y Fr(k)r Fn("1)p Fr(;`)p Fn("1)2952 366 y Fq(c)2991 381 y Fr(k)r(;`)3107 366 y Fp(=)e(1)h Fu(and)667 561 y Fp(\002)738 575 y Fl(1)802 561 y Fp(:=)f(Av)1041 589 y Fn(j)p Fr(x)p Fn(j\024)1186 562 y Fg(1)p 1184 574 31 3 v 1186 615 a FA(\017)1295 561 y Fp(sup)1244 644 y Fr(m)p Fn(2)p Fl([0)p Fr(;)p Fl(1])1483 484 y Fk(\014)1483 539 y(\014)1513 561 y Fq(m)20 b Fo(\000)g Fq(\026)1759 524 y Fr(\025)1800 533 y Fg(0)1835 524 y Fl(\()p Fr(m)p Fl(\))1956 561 y Fp(\()p Fq(m)2071 575 y Fl(\003)2120 587 y FA(x;`)2208 561 y Fp(\))2243 484 y Fk(\014)2243 539 y(\014)2274 561 y Fq(;)667 809 y Fp(\002)738 823 y Fl(2)802 809 y Fp(:=)25 b(Av)1041 837 y Fn(j)p Fr(x)p Fn(j\024)1186 810 y Fg(1)p 1184 822 V 1186 863 a FA(\017)1295 809 y Fp(sup)1244 892 y Fr(m)p Fn(2)p Fl([0)p Fr(;)p Fl(1])1498 809 y Fq(\034)1538 823 y Fr(x)1582 709 y Fk(\020)1636 809 y Fp(sup)1684 888 y Fr(h)p 1684 900 41 3 v 1773 736 a Fk(\010)1826 809 y Fp(\(2)p Fq(k)s Fp(\))1991 772 y Fn(\000)p Fr(d)2088 809 y Fp(Av)2205 851 y Fr(z)s Fn(2)p Fl(\003)2337 815 y Fg(\()p FA(k)q Fg(\))2337 874 y FA(`)2427 809 y Fq(F)2485 823 y Fr(z)2525 809 y Fp(\()p Fq(h)2612 823 y Fr(z)2653 809 y Fq(;)15 b(\026)2748 772 y Fr(\025)2789 781 y Fg(0)2824 772 y Fl(\()p Fr(m)p Fl(\))2945 809 y Fp(\))2980 736 y Fk(\011)3033 709 y(\021)3088 809 y Fq(;)118 1065 y Fu(and)25 b Fp(sup)429 1087 y Fr(h)p 429 1099 45 3 v 499 1065 a Fu(is)i(as)e(before.)118 1213 y(It)h(is)g(clear)e (that,)j(by)e(considering)g(a)g(\002xed)g(density)h Fq(m)f Fu(in)h(the)f(de\002nition)h(of)g Fp(\002)2860 1227 y Fl(1)2924 1213 y Fu(and)f Fp(\002)3169 1227 y Fl(2)3209 1213 y Fu(,)g(for)g(almost)118 1321 y(all)31 b(disorder)h (con\002gurations)f Fq(\013)p Fu(,)j Fp(\002)1392 1335 y Fl(1)1463 1321 y Fu(is)f(negligible)d(as)i Fq(`)k Fo(")g(1)p Fq(;)15 b(\017)36 b Fo(#)h Fp(0)p Fu(.)51 b(Moreover)-10 b(,)32 b(because)e(of)i(the)118 1429 y(ergodicity)f(of)g(the)g(random)f (\002eld)h Fq(\013)g Fu(and)g(of)g(the)g(variational)g (characterization)f(\(4.20\),)h(it)h(is)f(also)118 1537 y(clear)24 b(that)i(for)g(almost)f(all)g(disorder)g(con\002gurations)g Fq(\013)1303 1724 y Fp(lim)15 b(sup)1315 1803 y Fr(`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)1596 1724 y Fp(\002)1667 1738 y Fl(2)1732 1724 y Fo(\024)24 b Fs(E)1882 1650 y Fk(\000)1930 1724 y Fq(V)1983 1739 y Fr(k)2025 1724 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)2201 1686 y Fr(\025)2242 1695 y Fg(0)2278 1686 y Fl(\()p Fr(m)p Fl(\))2400 1724 y Fp(\))2435 1650 y Fk(\001)118 1973 y Fu(T)-9 b(o)37 b(handle)e(the)i(supremum)d(over)i Fq(m)43 b Fo(2)g Fp([0)p Fq(;)15 b Fp(1])38 b Fu(requires)e(some)f(simple)h(additional)i (observations)118 2081 y(based)26 b(on)f(compactness)f(of)i Fp([0)p Fq(;)15 b Fp(1])27 b Fu(and)e(lemma)f(A.3)h(\(see)g(e.g)g (section)g(1.13)f(in)i([15)o(]\).)448 b Fe(\003)118 2310 y Fu(4.2.)45 b FE(Back)36 b(to)g(the)e(proof)i(of)g(proposition)f(4.1.) 46 b Fu(Given)33 b(the)h(technical)f(tools)h(developed)e(in)i(the)118 2418 y(previous)c(paragraph,)i(let)e(us)h(now)f(complete)f(the)i(proof) f(of)h(proposition)g(4.1)f(modulo)g(some)g(non)118 2526 y(trivial)c(results)f(to)h(be)f(proved)g(later)f(on.)118 2669 y(The)37 b(basic)h(idea)g(would)f(be)f(to)i(bene\002t)e(of)i(the)f (ergodicity)g(of)h(the)f(model)f(by)i(means)e(of)h(the)h Fh(lo-)118 2777 y(calization)e(technique)f Fu(discussed)j(in)f (subsection)f(4.1.)65 b(Unfortunately)-12 b(,)39 b(the)e(function)3283 2754 y Fp(\026)3261 2777 y Fq(J)3320 2733 y Fr(g)3311 2806 y(b;a;`;\017)3536 2777 y Fu(ap-)118 2895 y(pearing)32 b(in)h(\(4.9\))f(cannot)g(be)h(written)g(as)g Fp(Av)1778 2917 y Fr(x)p Fn(2)p Fm(T)1915 2898 y FA(d)1915 2934 y(\017)1949 2895 y Fq(f)1994 2909 y Fr(x)2070 2895 y Fu(\(or)g(as)g(a)g(more)f(complex)f(spatial)j(average\))118 3009 y(for)40 b(suitable)g(functions)g Fq(f)1071 3023 y Fr(x)1154 3009 y Fu(having)g(support)h(independent)e(of)h Fq(\017)p Fu(.)76 b(W)-8 b(e)40 b(will)g(need)f(some)h(subtle)118 3117 y(techniques)29 b(developed)f(for)g(non-gradient)g(systems)h(in)g (order)f(to)h(approximate)3031 3094 y Fp(\026)3009 3117 y Fq(J)3068 3073 y Fr(g)3059 3147 y(b;a;`;\017)3276 3117 y Fu(with)h(such)118 3242 y(a)35 b(spatial)g(average.)58 b(There)34 b(is)h(however)f(one)f(piece)h(of)2174 3219 y Fp(\026)2152 3242 y Fq(J)2211 3198 y Fr(g)2202 3272 y(b;a;`;\017)2390 3242 y Fu(,)j(namely)c(the)i(density)g (\223gradient\224)118 3313 y Fk(\000)160 3387 y Fp(2)p Fq(b=\017)326 3313 y Fk(\001)368 3336 y Fn(\000)p Fl(1)463 3313 y Fk(\002)500 3387 y Fq(m)580 3417 y Fr(x)p Fl(+)685 3390 y FA(b)p 685 3402 28 3 v 686 3443 a(\017)722 3417 y Fr(e)755 3398 y Fi(0)777 3417 y Fr(;)807 3390 y FA(a)p 807 3402 34 3 v 811 3443 a(\017)882 3387 y Fo(\000)28 b Fq(m)1061 3417 y Fr(x)p Fn(\000)1166 3390 y FA(b)p 1165 3402 28 3 v 1166 3443 a(\017)1202 3417 y Fr(e)1235 3398 y Fi(0)1257 3417 y Fr(;)1287 3390 y FA(a)p 1287 3402 34 3 v 1291 3443 a(\017)1335 3313 y Fk(\003)1409 3387 y Fu(which)37 b(can)f(be)g(conveniently)e(written)j(as)g(a)g (suitable)f(spatial)118 3533 y(average.)d(T)-9 b(o)27 b(this)h(aim)e(recall)f(the)h(de\002nition)h(\(2.2\))f(of)g(the)h (spatial)g(average)e Fp(Av)2979 3498 y Fr(`;s)2979 3556 y(z)s(;y)3102 3533 y Fu(and)i(de\002ne)f(for)118 3664 y(any)k(particle)f(con\002guration)g Fq(\021)s Fu(,)j Fq(m)1363 3620 y Fl(1)p Fr(;e)1363 3694 y(`)1454 3664 y Fq(;)f(m)1590 3620 y Fl(2)p Fr(;e)1590 3694 y(`)1711 3664 y Fu(and)f Fq(m)1970 3631 y Fr(e)1970 3692 y(`)2037 3664 y Fu(to)g(be)f(the)i(particle)e(density)i(associated)g(to)f Fq(\021)118 3789 y Fu(in)j(the)h(sets)f Fp(\003)631 3745 y Fl(1)p Fr(;e)631 3818 y(`)723 3789 y Fq(;)15 b Fp(\003)826 3745 y Fl(1)p Fr(;e)826 3818 y(`)951 3789 y Fu(and)34 b Fp(\003)1197 3756 y Fr(e)1197 3817 y(`)1267 3789 y Fu(de\002ned)e(in)i(\(2.1\))e(respectively)-12 b(.)53 b(It)34 b(is)g(then)f(simple)g(to)g(check)f(the)118 3908 y(following)25 b(identity)h(\(which)g(motivates)g(the)g(introduction)f (of)h Fp(Av)2446 3873 y Fr(`;s)2446 3931 y(z)s(;y)2543 3908 y Fu(\):)1155 4175 y Fp(Av)1272 4138 y Fr(`;s)1272 4198 y(z)s(;y)1369 4175 y Fq(\034)1409 4189 y Fr(z)1458 4111 y Fq(m)1538 4066 y Fl(2)p Fr(;e)1538 4140 y(`)1650 4111 y Fo(\000)20 b Fq(m)1821 4066 y Fl(1)p Fr(;e)1821 4140 y(`)p 1458 4155 455 4 v 1667 4238 a Fq(`)1948 4175 y Fp(=)25 b Fq(\034)2084 4189 y Fr(y)2135 4114 y Fq(m)2215 4070 y Fl(2)p Fr(;e)2215 4126 y(s)2327 4114 y Fo(\000)20 b Fq(m)2498 4070 y Fl(1)p Fr(;e)2498 4126 y(s)p 2135 4155 V 2341 4238 a Fq(s)2599 4175 y(:)785 b Fu(\(4.27\))118 4399 y(Let)30 b(now)g Fq(n;)576 4364 y Fr(a)p 576 4379 38 4 v 580 4431 a(\017)623 4399 y Fq(;)674 4364 y Fr(b)p 674 4379 31 4 v 675 4431 a(\017)744 4399 y Fu(be)f(odd)i(integers)e (such)h(that)1819 4364 y Fr(a)p 1802 4379 72 4 v 1802 4431 a(n\017)1916 4399 y Fo(2)j Fs(N)42 b Fu(and)2298 4364 y Fr(b)p 2294 4379 38 4 v 2294 4431 a(a)2374 4399 y Fo(2)33 b Fs(N)7 b Fu(.)51 b(Then,)31 b(it)g(is)f(simple)g(to)g (check)118 4507 y(that)988 4730 y Fp(Av)1115 4646 y Fg(2)p FA(b)p 1115 4658 58 3 v 1127 4699 a(a)1183 4673 y Fn(\000)p Fl(1)1105 4756 y Fr(u)p Fl(=0)1277 4730 y Fq(\034)1317 4744 y Fr(x)1357 4752 y FA(u)1412 4637 y Fq(m)1492 4593 y Fl(2)p Fr(;e)1502 4642 y FA(a)p 1502 4654 34 3 v 1506 4695 a(\017)1604 4637 y Fo(\000)19 b Fq(m)1774 4593 y Fl(1)p Fr(;e)1784 4642 y FA(a)p 1784 4654 V 1788 4695 a(\017)p 1412 4709 455 4 v 1574 4793 a Fq(a=\017)1901 4730 y Fp(=)2007 4638 y Fq(m)2087 4669 y Fr(x)p Fl(+)2192 4642 y FA(b)p 2192 4654 28 3 v 2193 4695 a(\017)2229 4669 y Fr(e;)2292 4642 y FA(a)p 2291 4654 34 3 v 2295 4695 a(\017)2359 4638 y Fo(\000)h Fq(m)2530 4669 y Fr(x)p Fn(\000)2635 4642 y FA(b)p 2634 4654 28 3 v 2635 4695 a(\017)2672 4669 y Fr(e;)2735 4642 y FA(a)p 2734 4654 34 3 v 2738 4695 a(\017)p 2007 4709 775 4 v 2311 4793 a Fp(2)p Fq(b=\017)3409 4730 y Fu(\(4.28\))118 4944 y(where)1133 5111 y Fq(x)1185 5125 y Fr(u)1255 5111 y Fp(:=)25 b Fq(x)c Fp(+)1539 5010 y Fk(\020)1594 5111 y Fq(u)1656 5049 y(a)p 1656 5090 49 4 v 1659 5173 a(")1734 5111 y Fo(\000)1835 5049 y Fq(b)p 1835 5090 40 4 v 1836 5173 a(\017)1904 5111 y Fp(+)2005 5049 y(1)p 2005 5090 46 4 v 2005 5173 a(2)2060 5037 y Fk(\000)2112 5049 y Fq(a)p 2112 5090 49 4 v 2118 5173 a(\017)2190 5111 y Fo(\000)f Fp(1)2326 5037 y Fk(\001)2388 5111 y Fp(+)g(1)2524 5010 y Fk(\021)2579 5111 y Fq(e:)118 5301 y Fu(Therefore,)25 b(if)h(we)f(de\002ne)1343 5478 y Fp(Av)1460 5441 y Fn(\003)1460 5501 y Fr(z)s(;x)1575 5478 y Fq(f)1620 5492 y Fr(z)1684 5478 y Fp(:=)g(Av)1933 5394 y Fg(2)p FA(b)p 1933 5406 58 3 v 1945 5447 a(a)2001 5421 y Fn(\000)p Fl(1)1923 5504 y Fr(u)p Fl(=0)2095 5478 y Fp(Av)2212 5421 y Fr(n;)2285 5394 y FA(a)p 2285 5406 34 3 v 2289 5447 a(\017)2212 5490 y Fr(z)s(;x)2308 5498 y FA(u)2352 5478 y Fq(f)2397 5492 y Fr(z)3409 5478 y Fu(\(4.29\))p eop %%Page: 17 17 17 16 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(17)118 366 y Fu(\(when)32 b(necessary)g(we)h(will)g(also)f(add)i(the)f(versor) f Fq(e)39 b Fo(2)e(E)k Fu(into)33 b(the)g(notation)h(by)f(writing)g Fp(Av)3499 331 y Fn(\003)p Fr(;e)3499 389 y(z)s(;x)3598 366 y Fu(\),)118 477 y(thanks)26 b(to)g(\(4.27\))e(and)h(\(4.28\))f(we) h(obtain:)1066 829 y Fp(Av)1184 792 y Fn(\003)1184 852 y Fr(z)s(;x)1293 768 y Fq(m)1373 724 y Fl(2)p Fr(;e)1373 780 y(n)1485 768 y Fo(\000)20 b Fq(m)1656 724 y Fl(1)p Fr(;e)1656 780 y(n)p 1293 809 455 4 v 1493 892 a Fq(n)1782 829 y Fp(=)1888 738 y Fq(m)1968 768 y Fr(x)p Fl(+)2073 741 y FA(b)p 2073 753 28 3 v 2074 794 a(\017)2110 768 y Fr(e;)2173 741 y FA(a)p 2172 753 34 3 v 2176 794 a(\017)2240 738 y Fo(\000)g Fq(m)2411 768 y Fr(x)p Fn(\000)2516 741 y FA(b)p 2515 753 28 3 v 2516 794 a(\017)2553 768 y Fr(e;)2616 741 y FA(a)p 2615 753 34 3 v 2619 794 a(\017)p 1888 809 775 4 v 2192 892 a Fp(2)p Fq(b=\017)2688 829 y(:)696 b Fu(\(4.30\))118 1187 y(If)33 b(the)g(above)f(conditions)h(on)f Fq(n;)1314 1152 y Fr(a)p 1314 1167 38 4 v 1318 1219 a(\017)1362 1187 y Fq(;)1412 1152 y Fr(b)p 1412 1167 31 4 v 1413 1219 a(\017)1485 1187 y Fu(are)f(not)i(satis\002ed,)j(we)c(extend)g (the)g(de\002nition)h(of)g Fp(Av)3436 1152 y Fn(\003)3436 1210 y Fr(z)s(;x)3567 1187 y Fu(by)118 1313 y(replacing)28 b(in)h(\(4.29\))912 1277 y Fr(a)p 912 1292 38 4 v 916 1344 a(\017)960 1313 y Fu(,)1025 1277 y Fr(b)p 1025 1292 31 4 v 1026 1344 a(\017)1065 1313 y Fu(,)1131 1277 y Fl(2)p Fr(b)p 1131 1292 66 4 v 1145 1344 a(a)1236 1313 y Fu(with)h Fq(r)1483 1327 y Fl(1)1523 1313 y Fu(,)g Fq(r)1619 1327 y Fl(2)1688 1313 y Fu(and)1876 1276 y Fl(2)p Fr(r)1943 1285 y Fg(2)p 1876 1292 102 4 v 1894 1344 a Fr(r)1926 1353 y Fg(1)2017 1313 y Fu(respectively)-12 b(,)30 b(where)e Fq(r)2843 1327 y Fl(1)2912 1313 y Fu(is)i(the)g (smallest)f(odd)118 1444 y(number)23 b(in)h Fq(n)p Fs(Z)c Fu(such)k(that)1088 1408 y Fr(a)p 1088 1423 38 4 v 1092 1475 a(\017)1161 1444 y Fo(\024)h Fq(r)1298 1458 y Fl(1)1362 1444 y Fu(and)f Fq(r)1576 1458 y Fl(2)1640 1444 y Fu(is)h(the)f (smallest)g(odd)h(number)e(in)h Fq(r)2873 1458 y Fl(1)2913 1444 y Fs(Z)c Fu(such)k(that)3394 1408 y Fr(b)p 3394 1423 31 4 v 3395 1475 a(\017)3460 1444 y Fo(\024)h Fq(r)3597 1458 y Fl(2)3636 1444 y Fu(.)118 1657 y FE(W)-8 b(arning.)33 b Fu(In)26 b(the)g(sequel,)f(for)h(the)g(sake)f(of)h(simplicity)h(we)e (will)h(always)f(assume)h Fq(n;)3122 1621 y Fr(a)p 3122 1636 38 4 v 3126 1688 a(\017)3169 1657 y Fq(;)3220 1621 y Fr(b)p 3220 1636 31 4 v 3221 1688 a(\017)3285 1657 y Fu(to)g(be)g(odd)118 1778 y(integers)c(such)g(that)865 1743 y Fr(a)p 848 1758 72 4 v 848 1810 a(n\017)954 1778 y Fo(2)j Fs(N)35 b Fu(and)1313 1743 y Fr(b)p 1309 1758 38 4 v 1309 1810 a(a)1382 1778 y Fo(2)25 b Fs(N)6 b Fu(.)37 b(The)22 b(way)g(to)h(treat)f(the)g(general)e(case)i(is)h(shortly)f (discussed)118 1886 y(right)k(after)f(section)g(4.5.)118 2086 y(It)h(is)g(convenient)e(to)i(introduce)e(also)i Fp(Av)1559 2050 y Fr(?)1559 2108 y(z)s(;x)1684 2086 y Fu(de\002ned)f(as)g(the)h(dual)f(average)f(of)h Fp(Av)3019 2050 y Fn(\003)3019 2108 y Fr(z)s(;x)3118 2086 y Fu(,)g Fh(i.e.)954 2416 y Fp(Av)1072 2439 y Fr(x)p Fn(2)p Fm(T)1209 2420 y FA(d)1209 2455 y(\017)1243 2316 y Fk(\020)1297 2416 y Fq(f)1342 2430 y Fr(x)1386 2343 y Fk(\000)1427 2416 y Fp(Av)1545 2379 y Fn(\003)1545 2439 y Fr(z)s(;x)1644 2416 y Fq(g)1687 2430 y Fr(z)1727 2416 y Fp(\))1762 2316 y Fk(\021)1842 2416 y Fp(=)g(Av)2055 2439 y Fr(x)p Fn(2)p Fm(T)2192 2420 y FA(d)2192 2455 y(\017)2227 2316 y Fk(\020)2281 2416 y Fq(g)2324 2430 y Fr(x)2368 2343 y Fk(\000)2410 2416 y Fp(Av)2527 2379 y Fr(?)2527 2439 y(z)s(;x)2626 2416 y Fq(f)2671 2430 y Fr(z)2711 2416 y Fp(\))2746 2316 y Fk(\021)2800 2416 y Fq(:)584 b Fu(\(4.31\))118 2734 y(The)27 b(explicit)g(formula)f(of)h Fp(Av)1172 2698 y Fr(?)1172 2756 y(z)s(;x)1271 2734 y Fq(f)1316 2748 y Fr(z)1383 2734 y Fu(can)f(be)g(easily)h(computed)f(and)h(it)h(is)g (similar)f(to)g(the)g(formula)f(of)118 2844 y Fp(Av)236 2808 y Fn(\003)236 2867 y Fr(z)s(;x)335 2844 y Fq(f)380 2858 y Fr(z)419 2844 y Fu(.)218 3060 y(W)-8 b(e)27 b(introduce)f(at)h (this)g(point)h(some)e(special)g(functions)g(related)g(to)h(the)g (gradient)f(of)h(the)g(density)118 3168 y(\002eld.)68 b(Given)37 b(two)g(integers)g Fp(0)46 b Fo(\024)f Fq(n)g Fo(\024)g Fq(s)p Fu(,)40 b Fq(e)46 b Fo(2)f(E)g Fu(and)38 b(a)f(grand)g(canonical)f(measure)g Fq(\026)h Fu(on)g(an)118 3276 y(arbitrary)26 b(set)f Fp(\003)h Fu(containing)e Fp(\003)1222 3243 y Fr(e)1222 3298 y(s)1259 3276 y Fu(,)i(we)f(write) 463 3570 y Fq(m)543 3532 y Fl(2)p Fr(;e)543 3592 y(n)655 3570 y Fo(\000)20 b Fq(m)826 3532 y Fl(1)p Fr(;e)826 3592 y(n)943 3570 y Fp(=)k Fq( )1100 3532 y Fr(e)1097 3592 y(n;s)1217 3570 y Fp(+)c Fq(\036)1362 3532 y Fr(e)1362 3592 y(n;s)1477 3570 y Fq(;)131 b Fu(with)i Fq(\036)1996 3532 y Fr(e)1996 3592 y(n;s)2120 3570 y Fp(:=)25 b Fq(\026)2296 3496 y Fk(\002)2349 3570 y Fq(m)2429 3532 y Fl(2)p Fr(;e)2429 3592 y(n)2541 3570 y Fo(\000)20 b Fq(m)2712 3532 y Fl(1)p Fr(;e)2712 3592 y(n)2819 3570 y Fo(j)15 b(F)2933 3532 y Fr(e)2924 3592 y(s)2986 3496 y Fk(\003)3039 3570 y Fq(;)345 b Fu(\(4.32\))118 3863 y(where)39 b Fo(F)477 3830 y Fr(e)468 3886 y(s)554 3863 y Fu(is)i(the)e Fq(\033)s Fu(\226algebra)g(generated)g(by)g Fq(m)1904 3830 y Fr(e)1904 3886 y(s)1941 3863 y Fu(.)75 b(Notice)39 b(that,)45 b(in)40 b(absence)e(of)i(disorder)-10 b(,)44 b(the)118 3971 y(function)34 b Fq(\036)536 3938 y Fr(e)536 3994 y(n;s)670 3971 y Fu(would)h(be)f (identically)h(equal)f(to)h(zero)f(and)h(that)g Fq( )2537 3938 y Fr(e)2534 3994 y(n;s)2675 3971 y Fo(2)40 b(G)h Fu(for)34 b(all)g Fq(n)41 b(<)g(s)p Fu(,)c(since)118 4083 y Fq(\027)6 b Fp(\()p Fq( )266 4050 y Fr(e)263 4106 y(n;s)363 4083 y Fp(\))36 b(=)g(0)c Fu(for)g(all)f Fq(\027)42 b Fo(2)36 b(M)p Fp(\(\003\))d Fu(and)f(all)f Fp(\003)h Fu(containing)f Fp(\003)2270 4050 y Fr(e)2270 4106 y(s)2307 4083 y Fu(.)52 b(Thanks)32 b(to)g(\(4.22\))f(with)i Fp(\001)j(:=)g (\003)3614 4050 y Fr(e)3614 4106 y(n)118 4203 y Fu(and)24 b Fq(s)h Fp(:=)g Fq(n)f Fu(and)g(thanks)h(to)g(the)f(equivalence)e(of)j (ensembles)d(\(see)i(lemma)f(A.5\),)h(given)f Fq(`)j Fo(\025)f Fq(n)3457 4170 y Fl(2)3520 4203 y Fu(it)g(is)118 4311 y(easy)g(to)h(check)e(that)393 4656 y Fq(V)446 4671 y Fr(`)479 4582 y Fk(\000)531 4585 y Fq( )593 4552 y Fr(e)590 4607 y(n;n)p 531 4635 169 4 v 588 4718 a Fq(n)710 4656 y Fp(;)15 b Fq(\027)801 4582 y Fk(\001)868 4656 y Fo(\024)25 b Fq(c)91 b Fo(8)p Fq(\027)30 b Fo(2)25 b(M)p Fp(\(\003)1513 4671 y Fr(`)1547 4656 y Fp(\))p Fq(;)288 b(V)1948 4671 y Fr(`)1981 4582 y Fk(\000)2033 4585 y Fq( )2095 4552 y Fr(e)2092 4607 y(n;n)p 2033 4635 V 2090 4718 a Fq(n)2211 4656 y Fp(;)15 b Fq(\026)2306 4618 y Fr(\025)2347 4627 y Fg(0)2382 4618 y Fl(\()p Fr(m)p Fl(\))2504 4582 y Fk(\001)2571 4656 y Fo(\024)25 b Fq(c)15 b(m)p Fp(\(1)21 b Fo(\000)f Fq(m)p Fp(\))p Fq(:)276 b Fu(\(4.33\))118 4984 y(Using)25 b(decomposition)h(\(4.32\))e(we)h(can)g (now)g(write)2002 4961 y Fp(\026)1980 4984 y Fq(J)2039 4939 y Fr(g)2030 5013 y(b;a;`;\017)2244 4984 y Fu(as)1320 5359 y Fp(\026)1298 5382 y Fq(J)1357 5338 y Fr(g)1348 5412 y(b;a;`;\017)1562 5382 y Fp(=)1706 5269 y Fl(5)1658 5296 y Fk(X)1662 5492 y Fr(j)t Fl(=0)1805 5382 y Fp(Av)1922 5405 y Fr(x)p Fn(2)p Fm(T)2059 5386 y FA(d)2059 5421 y(\017)2093 5382 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))15 b Fq( )2388 5345 y Fl(\()p Fr(j)t Fl(\))2385 5405 y Fr(x)p eop %%Page: 18 18 18 17 bop 118 45 a Fj(18)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(where)25 b(\(we)g(omit)g(in)h(the)f(notation)h(the)g(suf\002x)f Fq(b;)15 b(a;)g(`;)g(\017;)g(g)30 b Fu(\))498 605 y Fq( )560 568 y Fl(\(0\))557 628 y Fr(x)680 605 y Fp(:=)25 b(Av)918 624 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)1231 633 y Fg(1)1269 504 y Fk(h)1312 605 y Fq(j)1349 619 y Fr(y)r(;y)r Fl(+)p Fr(e)1555 605 y Fp(+)20 b Fq(\034)1686 619 y Fr(y)1727 605 y Fo(L)p Fq(g)k Fp(+)1955 519 y Fk(X)1948 717 y Fr(e)1981 698 y Fi(0)2003 717 y Fn(2E)2108 605 y Fq(D)2183 621 y Fr(e;e)2269 602 y Fi(0)2295 605 y Fp(\()p Fq(m)2410 620 y Fr(x;`)2502 605 y Fp(\))p Fq(\034)2577 619 y Fr(y)2629 534 y Fq( )2691 502 y Fr(e)2724 478 y Fi(0)2688 557 y Fr(n;n)p 2629 584 169 4 v 2686 668 a Fq(n)2807 504 y Fk(i)2850 605 y Fq(;)498 921 y( )560 883 y Fl(\(1\))557 943 y Fr(x)680 921 y Fp(:=)808 834 y Fk(X)801 1032 y Fr(e)834 1013 y Fi(0)856 1032 y Fn(2E)961 921 y Fq(D)1036 936 y Fr(e;e)1122 917 y Fi(0)1148 921 y Fp(\()p Fq(m)1263 936 y Fr(x;`)1356 921 y Fp(\))1391 820 y Fk(h)1434 921 y Fq(\034)1474 935 y Fr(x)1527 850 y Fq( )1589 817 y Fr(e)1622 793 y Fi(0)1586 872 y Fr(n;n)p 1527 900 V 1584 983 a Fq(n)1726 921 y Fo(\000)c Fp(Av)1935 939 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)2248 948 y Fg(1)2285 921 y Fq(\034)2325 935 y Fr(y)2376 850 y Fq( )2438 817 y Fr(e)2471 793 y Fi(0)2435 872 y Fr(n;n)p 2376 900 V 2433 983 a Fq(n)2555 820 y Fk(i)498 1236 y Fq( )560 1198 y Fl(\(2\))557 1258 y Fr(x)680 1236 y Fp(:=)808 1149 y Fk(X)801 1348 y Fr(e)834 1329 y Fi(0)856 1348 y Fn(2E)946 1135 y Fk(h)989 1236 y Fq(D)1064 1251 y Fr(e;e)1150 1232 y Fi(0)1176 1236 y Fp(\()p Fq(m)1291 1253 y Fr(x;)1361 1226 y FA(a)p 1360 1238 34 3 v 1364 1279 a(\017)1408 1236 y Fp(\))h Fo(\000)f Fq(D)1630 1251 y Fr(e;e)1716 1232 y Fi(0)1741 1236 y Fp(\()p Fq(m)1856 1251 y Fr(x;`)1949 1236 y Fp(\))1984 1135 y Fk(i)2027 1236 y Fq(\034)2067 1250 y Fr(x)2121 1165 y Fq( )2183 1132 y Fr(e)2216 1109 y Fi(0)2180 1187 y Fr(n;n)p 2121 1215 169 4 v 2178 1298 a Fq(n)498 1551 y( )560 1514 y Fl(\(3\))557 1574 y Fr(x)680 1551 y Fp(:=)808 1465 y Fk(X)801 1663 y Fr(e)834 1644 y Fi(0)856 1663 y Fn(2E)961 1551 y Fq(D)1036 1566 y Fr(e;e)1122 1547 y Fi(0)1148 1551 y Fp(\()p Fq(m)1263 1568 y Fr(x;)1333 1541 y FA(a)p 1333 1553 34 3 v 1337 1594 a(\017)1380 1551 y Fp(\))1415 1450 y Fk(h)1459 1551 y Fp(Av)1576 1514 y Fn(\003)p Fr(;e)1664 1490 y Fi(0)1576 1574 y Fr(z)s(;x)1690 1551 y Fq(\034)1730 1565 y Fr(z)1780 1480 y Fq( )1842 1447 y Fr(e)1875 1424 y Fi(0)1839 1503 y Fr(n;n)p 1780 1530 169 4 v 1837 1613 a Fq(n)1978 1551 y Fo(\000)g Fq(\034)2109 1565 y Fr(x)2163 1480 y Fq( )2225 1447 y Fr(e)2258 1424 y Fi(0)2222 1503 y Fr(n;n)p 2163 1530 V 2220 1613 a Fq(n)2341 1450 y Fk(i)498 1868 y Fq( )560 1831 y Fl(\(4\))557 1891 y Fr(x)680 1868 y Fp(:=)808 1782 y Fk(X)801 1980 y Fr(e)834 1961 y Fi(0)856 1980 y Fn(2E)961 1868 y Fq(D)1036 1884 y Fr(e;e)1122 1865 y Fi(0)1148 1868 y Fp(\()p Fq(m)1263 1886 y Fr(x;)1333 1858 y FA(a)p 1333 1870 34 3 v 1337 1912 a(\017)1380 1868 y Fp(\))1415 1767 y Fk(h)1469 1776 y Fq(m)1549 1807 y Fr(x)p Fl(+)1654 1780 y FA(b)p 1653 1792 28 3 v 1654 1833 a(\017)1690 1807 y Fr(e)1723 1788 y Fi(0)1745 1807 y Fr(;)1775 1780 y FA(a)p 1775 1792 34 3 v 1779 1833 a(\017)1843 1776 y Fo(\000)g Fq(m)2014 1807 y Fr(x)p Fn(\000)2119 1780 y FA(b)p 2118 1792 28 3 v 2119 1833 a(\017)2155 1807 y Fr(e)2188 1788 y Fi(0)2210 1807 y Fr(;)2240 1780 y FA(a)p 2240 1792 34 3 v 2244 1833 a(\017)p 1469 1847 820 4 v 1795 1931 a Fp(2)p Fq(b=\017)2318 1868 y Fo(\000)g Fp(Av)2526 1831 y Fn(\003)p Fr(;e)2614 1807 y Fi(0)2526 1891 y Fr(z)s(;x)2640 1868 y Fq(\034)2680 1882 y Fr(z)2730 1807 y Fq(m)2810 1763 y Fl(2)p Fr(;e)2898 1739 y Fi(0)2810 1819 y Fr(n)2944 1807 y Fo(\000)g Fq(m)3115 1763 y Fl(1)p Fr(;e)3203 1739 y Fi(0)3115 1819 y Fr(n)p 2730 1847 500 4 v 2952 1931 a Fq(n)3239 1767 y Fk(i)498 2183 y Fq( )560 2146 y Fl(\(5\))557 2206 y Fr(x)680 2183 y Fp(:=)808 2097 y Fk(X)801 2295 y Fr(e)834 2276 y Fi(0)856 2295 y Fn(2E)961 2183 y Fq(D)1036 2199 y Fr(e;e)1122 2180 y Fi(0)1148 2183 y Fp(\()p Fq(m)1263 2201 y Fr(x;)1333 2174 y FA(a)p 1333 2186 34 3 v 1337 2227 a(\017)1380 2183 y Fp(\)Av)1533 2146 y Fn(\003)p Fr(;e)1621 2123 y Fi(0)1533 2206 y Fr(z)s(;x)1647 2183 y Fq(\034)1687 2197 y Fr(z)1737 2113 y Fq(\036)1791 2080 y Fr(e)1824 2056 y Fi(0)1791 2135 y Fr(n;n)p 1737 2163 164 4 v 1791 2246 a Fq(n)118 2442 y Fu(and)25 b(we)g(de\002ne)605 2608 y Fp(\012)671 2622 y Fr(j)732 2608 y Fp(:=)g(sup)14 b Fq(spec)1175 2628 y Fr(L)1223 2609 y Fg(2)1258 2628 y Fl(\()p Fr(\026)1327 2636 y FA(\017)1358 2628 y Fl(\))1390 2534 y Fk(\010)1443 2608 y Fq(\017)1480 2570 y Fn(\000)p Fl(1)1574 2608 y Fp(Av)1691 2622 y Fr(x)1735 2608 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))p Fq( )2015 2570 y Fl(\()p Fr(j)t Fl(\))2012 2630 y Fr(x)2128 2608 y Fp(+)20 b Fq(\015)5 b(\017)2308 2570 y Fr(d)p Fn(\000)p Fl(2)2438 2608 y Fo(L)2501 2622 y Fr(\017)2534 2534 y Fk(\011)2678 2608 y Fq(j)31 b Fp(=)25 b(0)p Fq(;)15 b(:)g(:)g(:)i(;)e Fp(5)g Fq(:)118 2778 y Fu(Then,)44 b(thanks)c(to)h(the)f(subadditivity)i(of)e (\223)p Fp(sup)14 b Fq(spec)p Fu(\224,)44 b(proposition)d(4.1)e (follows)h(from)f(the)h(next)118 2886 y(result.)118 3023 y FE(Proposition)27 b(4.7.)42 b Fh(Let)25 b Fq(d)h Fo(\025)e Fp(3)i Fh(and)e Fq(\015)31 b(>)25 b Fp(0)p Fh(.)32 b(Then,)25 b(for)f(almost)h(any)g(disorder)e(con\002guration)h Fq(\013)p Fh(,)1397 3188 y Fp(inf)1385 3249 y Fr(g)r Fn(2)p Fm(G)1606 3188 y Fp(lim)15 b(sup)1534 3267 y Fr(n)p Fn("1)p Fr(;`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)1971 3188 y Fp(sup)2017 3267 y Fr(J)2123 3188 y Fp(\012)2189 3202 y Fl(0)2254 3188 y Fo(\024)25 b Fp(0)1014 b Fu(\(4.34\))118 3422 y Fh(and,)24 b(for)h(any)g Fq(j)31 b Fp(=)25 b(1)p Fq(;)15 b(:)g(:)g(:)i(;)e Fp(5)p Fh(,)1507 3551 y Fp(lim)g(sup)1312 3630 y Fr(n)p Fn("1)p Fr(;b)p Fn(#)p Fl(0)p Fr(;a)p Fn(#)p Fl(0)p Fr(;`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)2022 3551 y Fp(sup)2068 3629 y Fr(J)2199 3551 y Fp(\012)2265 3565 y Fr(j)2326 3551 y Fo(\024)25 b Fp(0)942 b Fu(\(4.35\))118 3778 y Fh(where)25 b Fq(J)34 b Fh(varies)25 b(in)g Fo(f)p Fq(J)34 b Fo(2)25 b Fq(C)7 b Fp(\()p Fs(T)1188 3745 y Fr(d)1230 3778 y Fp(\))41 b(:)g Fo(k)p Fq(J)9 b Fo(k)1521 3792 y Fn(1)1622 3778 y Fo(\024)25 b Fp(1)p Fo(g)p Fh(.)218 3915 y Fu(The)e(proof)h(of)f(proposition)i(4.7)d(is)j(best)e(divided)i (into)f(several)e(pieces)h(according)g(to)g(the)h(value)e(of)118 4023 y(the)k(index)f Fq(j)5 b Fu(.)118 4225 y(4.3.)45 b FE(The)34 b(term)g Fd(\012)816 4239 y Fc(0)860 4225 y FE(.)46 b Fu(Let)33 b(us)g(\002rst)g(prove)f(\(4.34\).)52 b(By)33 b(localizing)e(on)h(cubes)g(of)h(side)g Fp(2)p Fq(`)26 b Fp(+)f(1)33 b Fu(\(see)118 4333 y(\(4.12\)\))24 b(and)h(using)f(the)h(regularity)f(of)h Fq(J)9 b Fp(\()p Fo(\001)p Fp(\))p Fu(,)27 b(it)e(is)h(enough)e(to)h(prove)f(that)i(for) f(almost)g(any)g(disorder)118 4441 y(con\002guration)f Fq(\013)p Fu(,)137 4615 y Fp(inf)125 4675 y Fr(g)r Fn(2)p Fm(G)372 4615 y Fp(lim)15 b(sup)300 4693 y Fr(n)p Fn("1)p Fr(;`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)762 4615 y Fp(Av)880 4637 y Fr(x)p Fn(2)p Fm(T)1017 4618 y FA(d)1017 4653 y(\017)1084 4615 y Fp(sup)1066 4697 y Fn(j)p Fr(\014)s Fn(j\024)p Fl(1)1269 4615 y Fp(sup)1306 4687 y Fr(m)1447 4615 y Fp(sup)f Fq(spec)1421 4698 y Fr(L)1469 4679 y Fg(2)1504 4698 y Fl(\()p Fr(\027)1566 4709 y Fg(\003)1607 4727 y FA(x;`)1691 4709 y(;m)1768 4698 y Fl(\))1796 4541 y Fk(\010)1849 4615 y Fq(\017)1886 4577 y Fn(\000)p Fl(1)1980 4615 y Fq(\014)20 b Fp(Av)2169 4633 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)2482 4642 y Fg(1)2519 4615 y Fq(\034)2559 4629 y Fr(y)2600 4615 y Fq( )2662 4577 y Fl(\()p Fr(n;g)r Fl(\))2659 4637 y Fr(m)2840 4615 y Fp(+)g Fq(c`)3008 4577 y Fn(\000)p Fr(d)3104 4615 y Fq(\017)3141 4577 y Fn(\000)p Fl(2)3235 4615 y Fo(L)3298 4629 y Fl(\003)3347 4641 y FA(x;`)3435 4541 y Fk(\011)3513 4615 y Fo(\024)25 b Fp(0)3409 4819 y Fu(\(4.36\))118 4927 y(where)1120 5111 y Fq( )1182 5073 y Fl(\()p Fr(n;g)r Fl(\))1179 5133 y Fr(m)1366 5111 y Fp(:=)g Fq(j)1524 5125 y Fl(0)p Fr(;e)1636 5111 y Fp(+)20 b Fo(L)p Fq(g)k Fp(+)1955 5025 y Fk(X)1948 5223 y Fr(e)1981 5204 y Fi(0)2003 5223 y Fn(2E)2108 5111 y Fq(D)2183 5126 y Fr(e;e)2269 5107 y Fi(0)2295 5111 y Fp(\()p Fq(m)p Fp(\))2455 5040 y Fq( )2517 5007 y Fr(e)2550 4984 y Fi(0)2514 5063 y Fr(n;n)p 2455 5090 169 4 v 2512 5173 a Fq(n)2634 5111 y(:)118 5352 y Fu(Since)35 b Fq(\017)42 b Fo(#)h Fp(0)36 b Fu(before)f Fq(`)43 b Fo(")g(1)36 b Fu(and)f(since)h(for)f(any)h Fq(`)f Fu(large)g(enough,)i(any)e Fo(j)p Fq(y)c Fo(\000)c Fq(x)p Fo(j)43 b(\024)f Fq(`)3259 5366 y Fl(1)3334 5352 y Fu(and)36 b(any)118 5480 y Fq(\027)31 b Fo(2)25 b(M)p Fp(\(\003)487 5495 y Fr(x;`)580 5480 y Fp(\))p Fu(,)h Fq(\027)717 5407 y Fk(\000)758 5480 y Fq(\034)798 5494 y Fr(y)854 5480 y Fq( )916 5432 y Fl(\()p Fr(n;g)r Fl(\))913 5492 y Fr(m)1074 5407 y Fk(\001)1141 5480 y Fp(=)f(0)p Fu(,)h(we)f(can)f(appeal)h(to)h(perturbation)f(theory)g(\(see)g (proposition)h(4.2\))p eop %%Page: 19 19 19 18 bop 1015 45 a 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Fu(and)32 b Fp(\012)3512 1540 y Fl(3)3582 1526 y Fu(to)118 1634 y(the)25 b(T)-7 b(wo)25 b(Blocks)f(estimate)h(\(see)g(subsection)f (A.7\).)31 b(T)-9 b(o)26 b(this)g(aim,)f(by)f(integrating)h(by)f (parts,)i(we)e(can)118 1741 y(write)750 1912 y Fq(\017)787 1874 y Fn(\000)p Fl(1)881 1912 y Fp(Av)999 1926 y Fr(x)1043 1912 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))p Fq( )1323 1874 y Fl(\()p Fr(j)t Fl(\))1320 1934 y Fr(x)1440 1912 y Fp(=)1543 1826 y Fk(X)1536 2024 y Fr(e)1569 2005 y Fi(0)1591 2024 y Fn(2E)1697 1912 y Fq(\017)1734 1874 y Fn(\000)p Fl(1)1828 1912 y Fp(Av)1945 1926 y Fr(x)1989 1912 y Fq(\034)2029 1926 y Fr(x)2083 1841 y Fq( )2145 1808 y Fr(e)2178 1785 y Fi(0)2142 1864 y Fr(n;n)p 2083 1891 169 4 v 2140 1974 a Fq(n)2282 1912 y Fo(\001)20 b Fq(B)2401 1874 y Fl(\()p Fr(j)t Fl(\))2396 1934 y Fr(x)2583 1912 y Fo(8)15 b Fq(j)30 b Fp(=)25 b(1)p Fq(;)15 b Fp(2)p Fq(;)g Fp(3)118 2138 y Fu(where)798 2290 y Fq(B)872 2252 y Fl(\(1\))867 2312 y Fr(x)991 2290 y Fp(:=)25 b Fq(J)9 b Fp(\()p 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y Fr(d)p 2784 2978 89 4 v 2784 3062 a Fq(a)2832 3035 y Fr(d)2883 2926 y Fk(\001)2924 2999 y Fq(:)118 3181 y Fu(Therefore,)31 b(using)f(proposition)h(4.5,)g (it)g(is)g(enough)d(to)j(prove)e(that)j(for)e(almost)g(any)g(disorder)g (con-)118 3289 y(\002guration)25 b Fq(\013)p Fu(,)h(given)f Fq(\015)30 b(>)25 b Fp(0)p Fu(,)275 3482 y Fp(lim)15 b(sup)164 3561 y Fr(b)p Fn(#)p Fl(0)p Fr(;a)p Fn(#)p Fl(0)p Fr(;`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)680 3482 y Fp(sup)726 3560 y Fr(J)847 3482 y Fp(sup)f Fq(spec)1169 3502 y Fr(L)1217 3483 y Fg(2)1252 3502 y Fl(\()p Fr(\026)1321 3510 y FA(\017)1352 3502 y Fl(\))1384 3408 y Fk(\010)1437 3482 y Fp(Av)1554 3504 y Fr(x)p Fn(2)p Fm(T)1691 3485 y FA(d)1691 3521 y(\017)1726 3408 y Fk(\000)1767 3482 y Fq(B)1841 3444 y Fl(\()p Fr(j)t Fl(\))1836 3504 y Fr(x)1932 3408 y Fk(\001)1974 3431 y Fl(2)2033 3482 y Fp(+)2134 3420 y(1)p 2134 3461 46 4 v 2134 3544 a(2)2190 3482 y Fq(\015)5 b(\017)2279 3444 y Fr(d)p Fn(\000)p Fl(2)2409 3482 y Fo(L)2472 3496 y Fr(\017)2505 3408 y Fk(\011)2583 3482 y Fp(=)25 b(0)182 b Fo(8)p Fq(j)31 b Fp(=)25 b(1)p Fq(;)15 b Fp(2)p Fq(;)g Fp(3)p Fq(:)48 b Fu(\(4.39\))118 3703 y(Since)18 b Fq(D)k Fu(can)d(be)g(approximated)h(by)f(Lipschitz)i (functions)e(and)g Fq(J)29 b Fu(is)20 b(smooth,)i(\(4.39\))c(can)g(be)h (derived)118 3811 y(from)40 b(the)h(T)-7 b(wo)40 b(Blocks)g(estimate)h (\(see)f(subsection)g(A.7\).)77 b(F)-6 b(or)40 b(simplicity)h(of)g (notation,)j(let)c(us)118 3919 y(consider)26 b(the)g(case)g Fq(j)32 b Fp(=)26 b(2)h Fu(\(the)f(case)g Fq(j)32 b Fp(=)26 b(1)h Fu(is)g(simpler)-10 b(,)26 b(while)g Fq(j)32 b Fp(=)27 b(3)f Fu(is)h(a)f(slight)h(variation\))f(and)h Fq(D)118 4045 y Fu(Lipschitz)j(continuous.)42 b(Since)1242 3971 y Fk(\000)1284 4045 y Fq(B)1358 3997 y Fl(\(2\))1353 4056 y Fr(x)1452 3971 y Fk(\001)1493 3994 y Fl(2)1564 4045 y Fo(\024)31 b Fq(c)1720 3967 y Fk(\014)1720 4022 y(\014)1766 4045 y Fq(m)1846 4060 y Fr(x;`)1960 4045 y Fo(\000)22 b Fq(m)2133 4062 y Fr(x;)2203 4035 y FA(a)p 2203 4047 34 3 v 2207 4088 a(\017)2265 4045 y Fo(j)p Fu(,)31 b(by)d(introducing)h(a)f(scale)g(parameter)118 4164 y Fq(k)h Fu(such)c(that)h Fq(k)j Fo(")c(1)g Fu(after)h Fq(a)f Fo(#)h Fp(0)p Fq(;)15 b(`)26 b Fo(")f(1)h Fu(and)f Fq(\017)g Fo(#)h Fp(0)p Fu(,)g(we)f(can)f(estimate)728 4285 y Fk(\000)770 4358 y Fq(B)844 4321 y Fl(\(2\))839 4381 y Fr(x)938 4285 y Fk(\001)980 4307 y Fl(2)1044 4358 y Fo(\024)h Fq(c)15 b Fp(Av)1312 4377 y Fn(j)p Fr(y)r Fn(j\024)p Fr(`)1477 4358 y Fp(Av)1594 4377 y Fn(j)p Fr(z)s Fn(j\024)1735 4350 y FA(a)p 1734 4362 V 1738 4403 a(\017)1781 4358 y Fo(j)p Fq(m)1886 4373 y Fr(x)p Fl(+)p Fr(y)r(;k)2101 4358 y Fo(\000)20 b Fq(m)2272 4373 y Fr(x)p Fl(+)p Fr(z)s(;k)2464 4358 y Fo(j)g Fp(+)g Fq(c)2665 4297 y(k)p 2665 4337 51 4 v 2671 4421 a(`)2745 4358 y Fp(+)g Fq(c)2926 4297 y(k)p 2885 4337 131 4 v 2885 4421 a(a=\017)3026 4358 y(:)118 4563 y Fu(A)n(t)35 b(this)g(point,)i(by)d (the)g(subadditivity)j(\(4.11\))32 b(of)i(\223)p Fp(sup)14 b Fq(spec)p Fu(\224,)36 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5351 y Fl(\(4\))379 5411 y Fr(x)502 5389 y Fp(=)605 5302 y Fk(X)598 5500 y Fr(e)631 5481 y Fi(0)653 5500 y Fn(2E)759 5389 y Fq(D)834 5404 y Fr(e;e)920 5385 y Fi(0)945 5389 y Fp(\()p Fq(m)1060 5403 y Fr(Q)1116 5421 y FA(x;)1180 5401 y(a)p 1180 5409 34 3 v 1184 5442 a(\017)1232 5389 y Fp(\))1267 5288 y Fk(h)1320 5297 y Fq(m)1400 5327 y Fr(x)p Fl(+)1505 5300 y FA(b)p 1504 5312 28 3 v 1505 5353 a(\017)1542 5327 y Fr(e)1575 5308 y Fi(0)1597 5327 y Fr(;)1627 5300 y FA(a)p 1626 5312 34 3 v 1630 5353 a(\017)1694 5297 y Fo(\000)20 b Fq(m)1865 5327 y Fr(x)p Fn(\000)1970 5300 y FA(b)p 1969 5312 28 3 v 1970 5353 a(\017)2007 5327 y Fr(e)2040 5308 y Fi(0)2062 5327 y Fr(;)2092 5300 y FA(a)p 2091 5312 34 3 v 2095 5353 a(\017)p 1320 5368 820 4 v 1646 5451 a Fp(2)p Fq(b=\017)2169 5389 y Fo(\000)2270 5325 y Fq(m)2350 5340 y Fr(x)p Fl(+)p Fr(r)2477 5349 y Fg(2)2511 5340 y Fr(e)2544 5321 y Fi(0)2566 5340 y Fr(;r)2618 5349 y Fg(1)2677 5325 y Fo(\000)f Fq(m)2847 5340 y Fr(x)p Fn(\000)p Fr(r)2974 5349 y Fg(2)3008 5340 y Fr(e)3041 5321 y Fi(0)3063 5340 y Fr(;r)3115 5349 y Fg(1)p 2270 5368 884 4 v 2649 5451 a Fp(2)p Fq(r)2735 5465 y Fl(2)3164 5288 y Fk(i)3409 5389 y Fu(\(4.40\))p eop %%Page: 20 20 20 19 bop 118 45 a Fj(20)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(where)27 b Fq(r)432 380 y Fl(1)471 366 y Fq(;)15 b(r)552 380 y Fl(2)620 366 y Fu(have)27 b(been)g(de\002ned)g(in)g(subsection)h(4.2.) 38 b(By)27 b(the)h(T)-7 b(wo)28 b(Blocks)f(estimate)h(it)g(is)h(simple) 118 474 y(to)d(check)e(that)i(for)g(any)f Fq(\015)30 b(>)25 b Fp(0)h Fu(and)f(for)g(almost)h(any)f(disorder)g (con\002guration)g Fq(\013)479 675 y Fp(lim)429 737 y Fr(a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)671 675 y Fp(sup)14 b Fq(spec)993 695 y Fr(L)1041 676 y Fg(2)1075 695 y Fl(\()p Fr(\026)1144 703 y FA(\017)1176 695 y Fl(\))1207 602 y Fk(\010)1260 675 y Fp(Av)1378 698 y Fr(x)p Fn(2)p Fm(T)1515 679 y FA(d)1515 714 y(\017)1549 598 y Fk(\014)1549 653 y(\014)1579 675 y Fq(m)1659 693 y Fr(x;)1729 665 y FA(a)p 1728 677 34 3 v 1732 719 a(\017)1796 675 y Fo(\000)20 b Fq(m)1967 689 y Fr(x;r)2059 698 y Fg(1)2097 675 y Fo(j)g Fp(+)g Fq(\015)5 b(\017)2322 638 y Fr(d)p Fn(\000)p Fl(2)2453 675 y Fo(L)2516 689 y Fr(\017)2548 602 y Fk(\011)2627 675 y Fp(=)25 b(0)641 b Fu(\(4.41\))479 878 y Fp(lim)429 939 y Fr(a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)720 878 y Fp(sup)671 960 y Fn(j)p Fr(w)r Fn(j\024)p Fl(2)864 933 y FA(a)p 863 945 V 867 986 a(\017)921 878 y Fp(sup)14 b Fq(spec)1243 897 y Fr(L)1291 878 y Fg(2)1326 897 y Fl(\()p Fr(\026)1395 905 y FA(\017)1426 897 y Fl(\))1458 804 y Fk(\010)1511 878 y Fp(Av)1628 900 y Fr(x)p Fn(2)p Fm(T)1765 881 y FA(d)1765 916 y(\017)1800 800 y Fk(\014)1800 855 y(\014)1830 878 y Fq(m)1910 895 y Fr(x;)1980 868 y FA(a)p 1979 880 V 1983 921 a(\017)2047 878 y Fo(\000)20 b Fq(m)2218 895 y Fr(x)p Fl(+)p Fr(w)r(;)2396 868 y FA(a)p 2394 880 V 2398 921 a(\017)2442 878 y Fo(j)g Fp(+)g Fq(\015)5 b(\017)2667 840 y Fr(d)p Fn(\000)p Fl(2)2798 878 y Fo(L)2861 892 y Fr(\017)2893 804 y Fk(\011)2972 878 y Fp(=)25 b(0)296 b Fu(\(4.42\))118 1160 y(\(hint:)103 b(introduce)60 b(the)h(scale)f (parameter)g Fq(k)j Fu(with)f Fq(a)85 b Fo(#)g Fp(0)p Fq(;)15 b(k)89 b Fo(")84 b(1)p Fq(;)15 b(\017)85 b Fo(#)g Fp(0)61 b Fu(and)g(write)118 1268 y Fq(m)198 1282 y Fr(x;s)319 1268 y Fp(=)25 b(Av)533 1282 y Fr(y)r Fn(2)p Fl(\003)666 1290 y FA(x;s)757 1268 y Fq(m)837 1283 y Fr(y)r(;k)957 1268 y Fp(+)19 b Fq(O)s Fp(\()p Fq(k)s(=s)p Fp(\))27 b Fu(for)e Fq(s)g Fp(=)1666 1232 y Fr(a)p 1666 1247 38 4 v 1670 1299 a(\017)1713 1268 y Fq(;)15 b(r)1794 1282 y Fl(1)1834 1268 y Fu(\).)118 1376 y(In)24 b(\(4.40\))e(we)i(can)e (substitute)j Fq(r)1250 1390 y Fl(1)1313 1376 y Fu(by)1441 1340 y Fr(a)p 1441 1355 V 1445 1407 a(\017)1512 1376 y Fu(\(thanks)f(to)g(\(4.41\)\))f(and)g(after)h(that)h(in)e(the)h (numerators)f(we)118 1497 y(can)28 b(substitute)i Fq(r)741 1511 y Fl(2)809 1497 y Fu(by)941 1461 y Fr(b)p 941 1476 31 4 v 942 1529 a(\017)1010 1497 y Fu(\(thanks)f(to)g(\(4.42\)\).)40 b(In)29 b(order)f(to)h(conclude)e(is)i(enough)f(to)g(observe)g(that)118 1615 y Fq(\017)155 1582 y Fn(\000)p Fl(1)249 1538 y Fk(\014)249 1593 y(\014)319 1580 y Fl(1)p 290 1595 95 4 v 290 1648 a Fr(b=\017)414 1615 y Fo(\000)530 1580 y Fl(1)p 515 1595 67 4 v 515 1647 a Fr(r)547 1656 y Fg(2)591 1538 y Fk(\014)591 1593 y(\014)647 1615 y Fo(\024)d Fq(c)821 1580 y Fr(a)p 807 1595 65 4 v 807 1649 a(b)837 1630 y Fg(2)907 1615 y Fu(which)h(goes)f(to)g Fp(0)p Fu(.)118 1897 y(4.6.)45 b FE(The)34 b(term)f Fd(\012)815 1911 y Fc(5)860 1897 y FE(.)45 b Fu(The)33 b(proof)g(of)f(\(4.35\))f(for)i Fq(j)43 b Fp(=)37 b(5)32 b Fu(is)i(based)f(on)f(the)g(key)h(results)f (of)g(section)118 2004 y(5)f(and)g(it)i(is)f(one)e(place)h(where)f(the) i(restriction)g(on)f(the)g(dimension)h Fq(d)k Fo(\025)f Fp(3)d Fu(is)g(crucial)e(for)h(us.)50 b(W)-8 b(e)118 2112 y(refer)33 b(the)i(reader)e(to)i(the)f(beginning)f(of)h(section)h (5)e(for)i(an)f(heuristic)g(justi\002cation)i(of)e(the)h(above)118 2220 y(condition.)g(Here)25 b(it)i(is)g(enough)e(to)h(say)h(that)g(the) f(main)g(contribution)g(to)h(the)f(term)f Fp(\012)3133 2234 y Fl(5)3199 2220 y Fu(comes)g(from)118 2328 y(the)h (\003uctuations)f(in)g(the)h(density)g(\002eld)f(induced)g(by)g(the)h (\003uctuations)f(of)g(the)h Fh(disorder)e(\002eld)p Fu(.)118 2481 y(By)g(the)g(subadditivity)j(of)e(\224)p Fp(sup)13 b Fq(spec)p Fu(\224)24 b(we)g(only)f(need)g(to)i(prove)e (that)i(for)f(almost)g(all)g Fq(\013)p Fu(,)h(given)e Fq(e;)15 b(e)3551 2448 y Fn(0)3601 2481 y Fo(2)118 2589 y(E)33 b Fu(and)26 b Fq(\015)k(>)25 b Fp(0)p Fu(,)294 2858 y Fp(lim)15 b(sup)158 2937 y Fr(n)p Fn("1)p Fr(;)c(b)p Fn(#)p Fl(0)p Fr(;)h(a)p Fn(#)p Fl(0)p Fr(;)g(\017)p Fn(#)p Fl(0)724 2858 y Fp(sup)770 2936 y Fr(J)901 2858 y Fp(sup)i Fq(spec)1223 2878 y Fr(L)1271 2859 y Fg(2)1306 2878 y Fl(\()p Fr(\026)1375 2886 y FA(\017)1406 2878 y Fl(\))1437 2785 y Fk(\010)1490 2858 y Fq(\017)1527 2821 y Fn(\000)p Fl(1)1622 2858 y Fp(Av)1739 2872 y Fr(x)1783 2858 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))p Fq(D)2076 2874 y Fr(e;e)2162 2855 y Fi(0)2188 2858 y Fp(\()p Fq(m)2303 2876 y Fr(x;)2373 2848 y FA(a)p 2373 2860 34 3 v 2377 2902 a(\017)2421 2858 y Fp(\)Av)2573 2821 y Fn(\003)p Fr(;e)2661 2797 y Fi(0)2573 2881 y Fr(z)s(;x)2688 2858 y Fq(\034)2728 2872 y Fr(z)2777 2787 y Fq(\036)2831 2754 y Fr(e)2864 2731 y Fi(0)2831 2810 y Fr(n;n)p 2777 2837 164 4 v 2832 2921 a Fq(n)2971 2858 y Fp(+)20 b Fq(\015)5 b(\017)3151 2821 y Fr(d)p Fn(\000)p Fl(2)3281 2858 y Fo(L)3344 2872 y Fr(\017)3377 2785 y Fk(\011)3455 2858 y Fo(\024)25 b Fp(0)p Fq(:)3409 3032 y Fu(\(4.43\))118 3140 y(R)m(ecall)g(the)j(de\002nition)g(of)f Fp(Av)1169 3104 y Fn(\003)p Fr(;e)1257 3081 y Fi(0)1169 3163 y Fr(z)s(;x)1310 3140 y Fu(and)h Fq(x)1539 3154 y Fr(u)1611 3140 y Fu(given)e(in)i (\(4.29\).)36 b(Then,)28 b(thanks)g(again)f(to)h(the)f(subaddi-)118 3251 y(tivity)g(of)e(\224)p Fp(sup)14 b Fq(spec)p Fu(\224,)25 b(the)h(\224)p Fp(sup)13 b Fq(spec)p Fu(\224)25 b(in)h(the)f(l.h.s.)33 b(of)25 b(\(4.43\))f(is)i(bounded)f(from)g(above)g(by)321 3523 y Fp(Av)449 3439 y Fg(2)p FA(b)p 449 3451 58 3 v 461 3492 a(a)516 3466 y Fn(\000)p Fl(1)439 3550 y Fr(u)p Fl(=0)626 3523 y Fp(sup)14 b Fq(spec)948 3543 y Fr(L)996 3524 y Fg(2)1030 3543 y Fl(\()p Fr(\026)1099 3551 y FA(\017)1130 3543 y Fl(\))1162 3450 y Fk(\010)1215 3523 y Fq(\017)1252 3486 y Fn(\000)p Fl(1)1346 3523 y Fp(Av)1464 3537 y Fr(x)1508 3523 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))p Fq(D)1801 3539 y Fr(e;e)1887 3520 y Fi(0)1913 3523 y Fp(\()p Fq(m)2028 3541 y Fr(x;)2098 3514 y FA(a)p 2098 3526 34 3 v 2102 3567 a(\017)2145 3523 y Fp(\)Av)2298 3466 y Fr(n;)2371 3439 y FA(a)p 2371 3451 V 2375 3492 a(\017)2298 3535 y Fr(z)s(;x)2394 3543 y FA(u)2438 3523 y Fq(\034)2478 3537 y Fr(z)2528 3453 y Fq(\036)2582 3420 y Fr(e)2615 3396 y Fi(0)2582 3475 y Fr(n;n)p 2528 3503 164 4 v 2582 3586 a Fq(n)2721 3523 y Fp(+)20 b Fq(\015)5 b(\017)2901 3486 y Fr(d)p Fn(\000)p Fl(2)3032 3523 y Fo(L)3095 3537 y Fr(\017)3127 3450 y Fk(\011)3180 3523 y Fq(:)204 b Fu(\(4.44\))118 3809 y(Observe)23 b(that)h Fp(Av)761 3752 y Fr(n;)834 3725 y FA(a)p 833 3737 34 3 v 837 3778 a(\017)761 3821 y Fr(z)s(;x)857 3829 y FA(u)901 3809 y Fq(\034)941 3823 y Fr(z)990 3755 y(\036)1032 3732 y FA(e)1061 3711 y Fi(0)1032 3772 y FA(n;n)p 990 3789 142 4 v 1040 3841 a Fr(n)1165 3809 y Fu(has)g(support)f(inside)h Fp(\003)1968 3776 y Fr(e)2001 3753 y Fi(0)1968 3840 y Fr(x)2008 3848 y FA(u)2049 3840 y Fr(;)2079 3812 y FA(a)p 2079 3824 34 3 v 2083 3866 a(\017)2126 3809 y Fu(.)31 b(W)-8 b(e)24 b(would)f(like)g(at)g(this)i(point)e(to)h(localize)118 3951 y(on)h(boxes)g(of)g(side)h(length)e(of)h(order)f Fq(O)1470 3878 y Fk(\000)1522 3915 y Fr(a)p 1522 3930 38 4 v 1526 3983 a(\017)1569 3878 y Fk(\001)1636 3951 y Fu(in)h(such)g(a)g(way)h(that)g Fq(D)2459 3967 y Fr(e;e)2545 3948 y Fi(0)2571 3951 y Fp(\()p Fq(m)2686 3969 y Fr(x;)2756 3941 y FA(a)p 2755 3953 34 3 v 2759 3995 a(\017)2803 3951 y Fp(\))f Fu(becomes)f(a)h(constant.)118 4085 y(T)-9 b(o)26 b(this)h(aim,)e(given)g Fq(u)g Fo(2)g(f)p Fp(0)p Fq(;)15 b(:)g(:)g(:)i(;)1311 4049 y Fl(2)p Fr(b)p 1311 4064 66 4 v 1325 4116 a(a)1406 4085 y Fo(\000)j Fp(1)p Fo(g)27 b Fu(and)e Fq(x)g Fo(2)g Fs(T)2010 4052 y Fr(d)2010 4107 y(\017)2052 4085 y Fu(,)h(we)f(set)1093 4372 y Fp(\001)1169 4386 y Fr(x;u)1298 4372 y Fp(:=)1420 4217 y Fk(\()1493 4306 y Fq(Q)1565 4324 y Fr(x;)p Fl(10)1705 4296 y FA(a)p 1705 4308 34 3 v 1709 4350 a(\017)1843 4306 y Fu(if)h Fq(Q)1995 4324 y Fr(x;)2065 4296 y FA(a)p 2064 4308 V 2068 4350 a(\017)2132 4306 y Fo(\\)20 b Fp(\003)2276 4324 y Fr(x)2316 4332 y FA(u)2357 4324 y Fr(;)p Fl(2)2422 4296 y FA(a)p 2422 4308 V 2426 4350 a(\017)2495 4306 y Fo(6)p Fp(=)25 b Fo(;)1493 4440 y Fq(Q)1565 4458 y Fr(x;)1635 4430 y FA(a)p 1634 4442 V 1638 4484 a(\017)1843 4440 y Fu(otherwise)2661 4372 y Fq(:)118 4650 y Fu(and)f(we)g(observe)g (that)h(either)f Fp(\001)1271 4664 y Fr(x;u)1399 4650 y Fu(is)h(disjoint)h(from)e(or)g(completely)e(contains)j Fp(\003)3000 4668 y Fr(x)3040 4676 y FA(u)3080 4668 y Fr(;)p Fl(2)3145 4641 y FA(a)p 3145 4653 V 3149 4694 a(\017)3193 4650 y Fu(.)31 b(Therefore,)118 4770 y(if)f(in)g(\(4.44\))e (we)i(could)f(replace)f(the)i(term)f Fq(D)1747 4786 y Fr(e;e)1833 4767 y Fi(0)1858 4770 y Fp(\()p Fq(m)1973 4788 y Fr(x;)2043 4760 y FA(a)p 2043 4772 V 2047 4814 a(\017)2091 4770 y Fp(\))h Fu(by)f(the)h(new)f(term)g Fq(D)2922 4786 y Fr(e;e)3008 4767 y Fi(0)3034 4770 y Fp(\()p Fq(m)3149 4784 y Fl(\001)3208 4792 y FA(x;u)3307 4770 y Fp(\))p Fu(,)i(then)f(it)118 4889 y(would)g(be)g(simple)g(to)h (check)e(\(by)i(localizing)e(on)h(boxes)g Fp(\003)2195 4906 y Fr(x)2235 4914 y FA(u)2275 4906 y Fr(;)p Fl(2)2340 4879 y FA(a)p 2340 4891 V 2344 4933 a(\017)2388 4889 y Fu(\))g(that)i(all)d(what)i(is)h(needed)d(is)i(that)118 5008 y(for)25 b Fq(d)h Fo(\025)f Fp(3)p Fu(,)h(for)f(all)g Fq(T)38 b Fo(2)24 b Fs(N)38 b Fu(and)26 b(for)f(almost)g(all)g Fq(\013)p Fu(,)176 5276 y Fp(lim)15 b(sup)118 5355 y Fr(n)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)528 5276 y Fp(Av)646 5298 y Fr(x)p Fn(2)p Fm(T)783 5279 y FA(d)783 5315 y(\017)858 5276 y Fp(sup)832 5359 y Fn(j)p Fr(\014)s Fn(j\024)p Fr(T)1175 5276 y Fp(sup)1035 5359 y Fr(\027)t Fn(2M)p Fl(\(\003)1281 5377 y FA(x;)p Fg(2)1376 5357 y FA(a)p 1376 5365 V 1380 5398 a(\017)1424 5359 y Fl(\))1491 5276 y Fp(sup)f Fq(spec)1813 5296 y Fr(L)1861 5277 y Fg(2)1896 5296 y Fl(\()p Fr(\027)t Fl(\))1994 5202 y Fk(\010)2047 5276 y Fq(\017)2084 5238 y Fn(\000)p Fl(1)2178 5276 y Fq(\014)5 b Fp(Av)2352 5219 y Fr(n;)2425 5191 y FA(a)p 2424 5203 V 2428 5245 a(\017)2352 5288 y Fr(z)s(;x)2472 5276 y Fq(\034)2512 5290 y Fr(z)2562 5205 y Fq(\036)2616 5172 y Fr(e)2649 5149 y Fi(0)2616 5228 y Fr(n;n)p 2562 5255 164 4 v 2616 5338 a Fq(n)2735 5276 y Fp(+)p Fq(\017)2843 5238 y Fn(\000)p Fl(2)2937 5276 y Fp(Av)3055 5291 y Fr(b)p Fn(2)p Fl(\003)3181 5305 y FA(x;)p Fg(2)p FA(a=\017)3365 5276 y Fo(L)3428 5291 y Fr(b)3462 5202 y Fk(\011)3540 5276 y Fo(\024)25 b Fp(0)3409 5480 y Fu(\(4.45\))p eop %%Page: 21 21 21 20 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(21)118 366 y Fu(Section)25 b(5)f(is)i(devoted)g(to)g(the)f(proof)g(of)h (\(4.45\))e(\(see)h(theorem)f(5.3)h(there\).)118 474 y(Therefore,)g(it)h(remains)f(to)g(prove)g(that)h(for)f Fq(d)h Fo(\025)f Fp(3)p Fu(,)h(for)f(almost)h(all)e Fq(\013)i Fu(and)f(for)g(any)g Fq(\015)31 b(>)25 b Fp(0)460 670 y(lim)15 b(sup)342 749 y Fr(n)p Fn("1)p Fr(;b)p Fn(#)p Fl(0)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)872 670 y Fp(sup)918 748 y Fr(J)1040 670 y Fp(Av)1167 585 y Fg(2)p FA(b)p 1167 597 58 3 v 1179 639 a(a)1235 613 y Fn(\000)p Fl(1)1157 696 y Fr(u)p Fl(=0)1344 670 y Fp(sup)f Fq(spec)1666 690 y Fr(L)1714 671 y Fg(2)1749 690 y Fl(\()p Fr(\026)1818 698 y FA(\017)1849 690 y Fl(\))1880 596 y Fk(\010)1933 670 y Fq(\017)1970 632 y Fn(\000)p Fl(1)2065 670 y Fp(Av)2182 692 y Fr(x)p Fn(2)p Fm(T)2319 673 y FA(d)2319 709 y(\017)2353 670 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))771 961 y Fo(\002)857 887 y Fk(\002)895 961 y Fq(D)970 976 y Fr(e;e)1056 957 y Fi(0)1082 961 y Fp(\()p Fq(m)1197 978 y Fr(x;)1267 951 y FA(a)p 1266 963 34 3 v 1270 1004 a(\017)1314 961 y Fp(\))21 b Fo(\000)f Fq(D)1536 976 y Fr(e;e)1622 957 y Fi(0)1648 961 y Fp(\()p Fq(m)1763 975 y Fl(\001)1822 983 y FA(x;u)1920 961 y Fp(\))1955 887 y Fk(\003)1993 961 y Fp(Av)2111 904 y Fr(n;)2184 877 y FA(a)p 2183 889 V 2187 930 a(\017)2111 973 y Fr(z)s(;x)2207 981 y FA(u)2251 961 y Fq(\034)2291 975 y Fr(z)2340 890 y Fq(\036)2394 857 y Fr(e)2427 834 y Fi(0)2394 913 y Fr(n;n)p 2340 940 164 4 v 2395 1023 a Fq(n)2534 961 y Fp(+)g Fq(\015)5 b(\017)2714 924 y Fr(d)p Fn(\000)p Fl(2)2845 961 y Fo(L)2908 975 y Fr(\017)2940 887 y Fk(\011)3018 961 y Fo(\024)25 b Fp(0)p Fq(:)3409 810 y Fu(\(4.46\))118 1199 y(Notice)33 b(that)h(the)f(only)f(values)g(of)h Fq(u)g Fu(which)g(contribute)f(to)h(the)g Fp(Av)2548 1114 y Fg(2)p FA(b)p 2548 1126 58 3 v 2560 1168 a(a)2615 1141 y Fn(\000)p Fl(1)2538 1225 y Fr(u)p Fl(=0)2743 1199 y Fu(above,)h(in)f(what)h(follows)118 1312 y(called)26 b(\223bad)h(values\224,)g(are)g(those)h(for)f(which)h Fq(Q)1798 1330 y Fr(x;)1868 1302 y FA(a)p 1867 1314 34 3 v 1871 1356 a(\017)1943 1312 y Fo(6)p Fp(=)g(\001)2118 1326 y Fr(x;u)2249 1312 y Fu(for)f(some)g Fq(x)h Fo(2)g Fs(T)2851 1279 y Fr(d)2851 1335 y(\017)2894 1312 y Fu(.)37 b(It)28 b(is)g(easy)f(to)g(check)118 1443 y(that)i(the)f(cardinality)g (of)g(the)g(bad)g(values)f(of)g Fq(u)h Fu(is)h(of)e(order)h Fq(O)s Fp(\(1\))g Fu(for)g(any)f(\002xed)h Fq(x)h Fo(2)g Fs(T)3220 1410 y Fr(d)3220 1465 y(\017)3262 1443 y Fu(.)38 b(Thus)29 b(we)118 1556 y(only)e(need)g(to)h(bound)f(the)g(\224)p Fp(sup)14 b Fq(spec)p Fu(\224)27 b(appearing)g(in)h(\(4.46\))e(by)h Fq(o)p Fp(\()2523 1520 y Fr(b)p 2519 1535 38 4 v 2519 1587 a(a)2567 1556 y Fp(\))p Fu(,)i(uniformly)d(in)i Fq(u)g Fu(in)f(the)h(bad)118 1664 y(set.)k(Thanks)26 b(to)g(\(4.30\))e(and)h(\(4.32\))f(we)h(can)f(write)845 1888 y Fp(Av)963 1830 y Fr(n;)1036 1803 y FA(a)p 1035 1815 34 3 v 1039 1856 a(\017)963 1899 y Fr(z)s(;x)1059 1907 y FA(u)1103 1888 y Fq(\034)1143 1902 y Fr(z)1192 1817 y Fq(\036)1246 1784 y Fr(e)1279 1760 y Fi(0)1246 1839 y Fr(n;n)p 1192 1867 164 4 v 1247 1950 a Fq(n)1391 1888 y Fp(=)1497 1796 y Fq(m)1577 1826 y Fr(x)1617 1834 y FA(u)1657 1826 y Fl(+)1722 1799 y FA(b)p 1722 1811 28 3 v 1723 1852 a(\017)1759 1826 y Fr(;)1789 1799 y FA(a)p 1789 1811 34 3 v 1793 1852 a(\017)1857 1796 y Fo(\000)c Fq(m)2028 1826 y Fr(x)2068 1834 y FA(u)2108 1826 y Fn(\000)2173 1799 y FA(b)p 2173 1811 28 3 v 2174 1852 a(\017)2210 1826 y Fr(;)2240 1799 y FA(a)p 2240 1811 34 3 v 2244 1852 a(\017)p 1497 1867 791 4 v 1809 1950 a Fp(2)p Fq(b=\017)2318 1888 y Fo(\000)g Fp(Av)2526 1830 y Fr(n;)2599 1803 y FA(a)p 2599 1815 34 3 v 2603 1856 a(\017)2526 1899 y Fr(z)s(;x)2622 1907 y FA(u)2666 1888 y Fq(\034)2706 1902 y Fr(z)2756 1817 y Fq( )2818 1784 y Fr(e)2851 1760 y Fi(0)2815 1839 y Fr(n;n)p 2756 1867 169 4 v 2813 1950 a Fq(n)3409 1888 y Fu(\(4.47\))118 2096 y(Then,)27 b(the)f(contribution)g(in)h(\(4.46\))e(coming)g(from)h (the)g(\002rst)h(addendum)f(in)g(the)h(r)-10 b(.h.s.)35 b(of)27 b(\(4.47\))e(is)118 2204 y(not)g(larger)f(than)i Fq(O)s Fp(\()855 2168 y Fl(1)p 855 2183 36 4 v 858 2236 a Fr(b)900 2204 y Fp(\))g Fu(and)f(therefore)g(negligible.)118 2312 y(Let)j(us)g(consider)g(the)g(contribution)g(of)g(the)g(second)g (addendum.)39 b(An)28 b(integration)g(by)g(parts)g(shows)118 2420 y(that)391 2632 y Fp(Av)509 2654 y Fr(x)p Fn(2)p Fm(T)646 2635 y FA(d)646 2671 y(\017)680 2632 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))898 2558 y Fk(\000)941 2632 y Fq(D)1016 2647 y Fr(e;e)1102 2628 y Fi(0)1127 2632 y Fp(\()p Fq(m)1242 2649 y Fr(x;)1312 2622 y FA(a)p 1312 2634 34 3 v 1316 2675 a(\017)1360 2632 y Fp(\))20 b Fo(\000)g Fq(D)1581 2647 y Fr(e;e)1667 2628 y Fi(0)1693 2632 y Fp(\()p Fq(m)1808 2646 y Fl(\001)1867 2654 y FA(x;u)1966 2558 y Fk(\001)2007 2632 y Fp(Av)2125 2575 y Fr(n;)2198 2548 y FA(a)p 2197 2560 V 2201 2601 a(\017)2125 2644 y Fr(z)s(;x)2221 2652 y FA(u)2265 2632 y Fq(\034)2305 2646 y Fr(z)2354 2561 y Fq( )2416 2528 y Fr(e)2449 2505 y Fi(0)2413 2584 y Fr(n;n)p 2354 2611 169 4 v 2411 2694 a Fq(n)2558 2632 y Fp(=)25 b(Av)2771 2654 y Fr(x)p Fn(2)p Fm(T)2908 2635 y FA(d)2908 2671 y(\017)2943 2632 y Fq(\034)2983 2646 y Fr(x)3036 2561 y Fq( )3098 2528 y Fr(e)3131 2505 y Fi(0)3095 2584 y Fr(n;n)p 3036 2611 V 3093 2694 a Fq(n)3215 2632 y(B)3284 2646 y Fr(x;u)118 2819 y Fu(where)g(the)g(functions)g Fq(B)1000 2833 y Fr(x;u)1130 2819 y Fu(satisfy)i Fo(k)p Fq(B)1518 2833 y Fr(x;u)1623 2819 y Fo(k)f(\024)f Fq(c)g Fu(together)g(with)1044 3042 y Fo(jr)1145 3057 y Fr(b)1179 3042 y Fq(B)1248 3056 y Fr(x;u)1353 3042 y Fo(j)g(\024)g Fq(c)1548 2981 y(n\017)p 1548 3021 92 4 v 1570 3105 a(a)1650 3042 y(O)s(sc)p Fp(\()p Fq(D)1914 3058 y Fr(e;e)2000 3039 y Fi(0)2026 3042 y Fq(;)15 b(c)2121 2981 y(\017)2158 2948 y Fr(d)p 2115 3021 89 4 v 2115 3105 a Fq(a)2163 3078 y Fr(d)2214 3042 y Fp(\))91 b Fo(8)p Fq(b)25 b Fo(2)g Fp(\003)2604 3005 y Fr(e)2637 2981 y Fi(0)2604 3065 y Fr(x;n)2710 3042 y Fq(:)118 3267 y Fu(Moreover)-10 b(,)42 b Fq(B)626 3281 y Fr(x;u)770 3267 y Fu(is)f(a)f(particular)f(spatial)i (average)d(\(dual)i(to)g Fp(Av)2520 3210 y Fr(n;)2593 3182 y FA(a)p 2593 3194 34 3 v 2597 3236 a(\017)2520 3279 y Fr(z)s(;x)2616 3287 y FA(u)2660 3267 y Fu(\))g(of)g Fq(J)9 b Fp(\()p Fq(\017z)t Fp(\))3068 3193 y Fk(\000)3110 3267 y Fq(D)3185 3282 y Fr(e;e)3271 3263 y Fi(0)3297 3267 y Fp(\()p Fq(m)3412 3284 y Fr(z)s(;)3478 3257 y FA(a)p 3478 3269 V 3482 3310 a(\017)3525 3267 y Fp(\))31 b Fo(\000)118 3396 y Fq(D)193 3411 y Fr(e;e)279 3392 y Fi(0)305 3396 y Fp(\()p Fq(m)420 3410 y Fl(\001)479 3418 y FA(z)r(;u)575 3322 y Fk(\001)616 3396 y Fu(.)42 b(Therefore,)28 b(by)h(proposition)h(4.5)e(and)g(the)h(T)-7 b(wo)29 b(Blocks)f(estimate)h(\(see)f(subsection)118 3504 y(A.7\),)h(the)g(contribution)g(of)f(the)h(second)f(addendum)g(is) i(also)e(negligible)f(\(see)h(also)g(the)h(discussion)118 3612 y(at)d(the)f(end)g(of)h(subsection)f(4.4\).)118 3794 y(4.7.)45 b FE(Proof)33 b(of)e(the)g(energy)f(estimate.)44 b Fu(In)30 b(this)h(subsection)e(we)h(prove)f(lemma)f(3.1.)44 b(It)30 b(is)h(simple)118 3902 y(to)26 b(check)e(that)1048 4020 y(spatial\226time)j(average)c(in)j Fp(\(3)p Fq(:)p Fp(4\))h(=)57 b(sup)2300 4098 y Fr(H)5 b Fn(2H)2470 4110 y FA(b)2517 4020 y Fq(H)2600 3983 y Fr(?)2593 4043 y(b;a;\017)3409 4020 y Fu(\(4.48\))118 4236 y(where)25 b Fo(H)466 4251 y Fr(b)525 4236 y Fp(:=)g Fo(f)p Fq(H)33 b Fu(smooth)26 b(on)f Fp([0)p Fq(;)15 b(T)e Fp(])21 b Fo(\002)f Fs(T)1616 4203 y Fr(d)1658 4236 y Fq(;)41 b Fo(k)p Fq(H)7 b Fo(k)1897 4250 y Fn(1)1998 4236 y Fo(\024)2104 4200 y Fl(1)p 2104 4215 36 4 v 2107 4267 a Fr(b)2149 4236 y Fo(g)25 b Fu(and)463 4483 y Fq(H)546 4446 y Fr(?)539 4506 y(b;a;\017)703 4483 y Fp(:=)g(Av)942 4506 y Fr(x)p Fn(2)p Fm(T)1079 4487 y FA(d)1079 4522 y(\017)1128 4360 y Fk(Z)1219 4386 y Fr(T)1179 4566 y Fl(0)1274 4382 y Fk(\020)1328 4483 y Fp(2)p Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))1698 4382 y Fk(h)1752 4391 y Fq(m)1832 4422 y Fr(x)p Fl(+)1937 4395 y FA(b)p 1937 4407 28 3 v 1938 4448 a(\017)1974 4422 y Fr(e;)2037 4395 y FA(a)p 2036 4407 34 3 v 2040 4448 a(\017)2084 4391 y Fp(\()p Fq(s)p Fp(\))20 b Fo(\000)g Fq(m)2388 4409 y Fr(x;)2458 4382 y FA(a)p 2457 4394 V 2461 4435 a(\017)2505 4391 y Fp(\()p Fq(s)p Fp(\))p 1752 4462 867 4 v 2166 4546 a Fq(b)2628 4382 y Fk(i)2692 4483 y Fo(\000)f Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))3107 4446 y Fl(2)3147 4382 y Fk(\021)3202 4483 y Fq(ds:)118 4689 y Fu(In)39 b(what)h(follows)f(let)g Fq(H)46 b Fu(belong)38 b(to)h Fo(H)1563 4704 y Fr(b)1597 4689 y Fu(.)74 b(By)39 b(the)g(entropy)g(inequality)g(and)g(the)h(F)-6 b(eynman-Kac)118 4797 y(formula)24 b(\(see)h(\(4.6\))g(and)g(\(4.7\))14 b(\),)26 b(for)f(any)g Fq(\015)30 b(>)25 b Fp(0)p Fu(,)121 5016 y Fs(E)176 4979 y Fr(\026)219 4955 y FA(\017)259 4943 y Fk(\000)301 5016 y Fq(H)384 4979 y Fr(?)377 5040 y Fl(2)p Fr(b;a;\017)551 4943 y Fk(\001)618 5016 y Fo(\024)724 4955 y Fq(\024)p 724 4996 53 4 v 724 5079 a(\015)806 5016 y Fo(\000)20 b Fq(\015)5 b Fp(Av)1067 5039 y Fr(x)p Fn(2)p Fm(T)1204 5020 y FA(d)1204 5055 y(\017)1268 4893 y Fk(Z)1359 4919 y Fr(T)1319 5099 y Fl(0)1430 5016 y Fq(dsH)i Fp(\()p Fq(s;)15 b(\017x)p Fp(\))1845 4979 y Fl(2)613 5294 y Fp(+)704 5171 y Fk(Z)795 5197 y Fr(T)754 5377 y Fl(0)865 5294 y Fq(ds)g Fp(sup)f Fq(spec)1026 5378 y Fr(L)1074 5359 y Fg(2)1108 5378 y Fl(\()p Fr(\026)1177 5386 y FA(\017)1209 5378 y Fl(\))1292 5221 y Fk(\010)1345 5294 y Fq(\017)1382 5257 y Fn(\000)p Fl(1)1476 5294 y Fq(\015)5 b Fp(Av)1646 5316 y Fr(x)p Fn(2)p Fm(T)1783 5298 y FA(d)1783 5333 y(\017)1832 5294 y Fp(2)p Fq(H)i Fp(\()p Fq(s;)15 b(\017x)p Fp(\))2202 5193 y Fk(h)2256 5202 y Fq(m)2336 5233 y Fr(x)p Fl(+)2441 5206 y Fg(2)p FA(b)p 2441 5218 58 3 v 2456 5259 a(\017)2508 5233 y Fr(e;)2571 5206 y FA(a)p 2571 5218 34 3 v 2575 5259 a(\017)2618 5202 y Fp(\()p Fq(s)p Fp(\))21 b Fo(\000)f Fq(m)2923 5220 y Fr(x;)2993 5192 y FA(a)p 2992 5204 V 2996 5246 a(\017)3040 5202 y Fp(\()p Fq(s)p Fp(\))p 2256 5273 897 4 v 2621 5357 a(2)p Fq(b=\017)3163 5193 y Fk(i)3226 5294 y Fp(+)g Fq(\017)3354 5257 y Fr(d)p Fn(\000)p Fl(2)3484 5294 y Fo(L)3547 5308 y Fr(\017)3580 5221 y Fk(\011)3633 5294 y Fq(:)3409 5477 y Fu(\(4.49\))p eop %%Page: 22 22 22 21 bop 118 45 a Fj(22)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(It)31 b(is)g(convenient)e(to)h(introduce)g(a)g(free)f(scale)h(parameter)f Fq(n)p Fu(,)i(with)g Fq(n)i Fo(")h(1)c Fu(after)g Fq(a)k Fo(#)g Fp(0)c Fu(and)h Fq(\017)i Fo(#)h Fp(0)p Fu(,)118 495 y(and)28 b(write)g(the)f(gradient)h(of)f(masses)h(appearing)f(in)h (\(4.49\))e(as)i Fp(Av)2487 460 y Fn(\003)2487 518 y Fr(z)s(;x)2586 495 y Fq(\034)2626 509 y Fr(z)2665 422 y Fk(\000)2717 441 y Fr( )2765 418 y FA(e)2763 458 y(n;n)p 2717 475 146 4 v 2768 527 a Fr(n)2894 495 y Fp(+)2997 441 y Fr(\036)3039 418 y FA(e)3039 458 y(n;n)p 2997 475 142 4 v 3046 527 a Fr(n)3148 422 y Fk(\001)3218 495 y Fu(\(see)f(\(4.30\))118 606 y(and)e(\(4.32\)\).)118 714 y(By)g(the)h(de\002nition)g(of)f Fp(Av)1023 679 y Fn(\003)1023 737 y Fr(z)s(;x)1122 714 y Fu(,)g(the)h(subadditivity)i(of)e Fp(sup)14 b Fq(spec)25 b Fu(and)g(theorem)g(5.3,)439 943 y Fp(lim)15 b(sup)381 1022 y Fr(n)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)791 820 y Fk(Z)882 846 y Fr(T)841 1026 y Fl(0)952 943 y Fq(ds)25 b Fp(sup)14 b Fq(spec)1123 1027 y Fr(L)1171 1008 y Fg(2)1206 1027 y Fl(\()p Fr(\026)1275 1035 y FA(\017)1306 1027 y Fl(\))1389 870 y Fk(\010)1442 943 y Fq(\017)1479 906 y Fn(\000)p Fl(1)1573 943 y Fq(\015)5 b Fp(Av)1743 966 y Fr(x)p Fn(2)p Fm(T)1880 947 y FA(d)1880 982 y(\017)1929 943 y Fp(2)p Fq(H)i Fp(\()p Fq(s;)15 b(\017x)p Fp(\)Av)2418 906 y Fn(\003)2418 966 y Fr(z)s(;x)2517 943 y Fq(\034)2557 957 y Fr(z)2606 872 y Fq(\036)2660 839 y Fr(e)2660 895 y(n;n)p 2606 922 164 4 v 2661 1006 a Fq(n)2800 943 y Fp(+)20 b Fq(\017)2928 906 y Fr(d)p Fn(\000)p Fl(2)3059 943 y Fo(L)3122 957 y Fr(\017)3154 870 y Fk(\011)3232 943 y Fo(\024)25 b Fp(0)p Fq(:)118 1171 y Fu(Let)h(us)f(consider)-10 b(,)25 b(for)g(\002xed)h Fq(b;)15 b(n;)g(a)p Fu(,)698 1373 y Fp(sup)f Fq(spec)1020 1393 y Fr(L)1068 1374 y Fg(2)1102 1393 y Fl(\()p Fr(\026)1171 1401 y FA(\017)1202 1393 y Fl(\))1234 1300 y Fk(\010)1287 1373 y Fq(\017)1324 1336 y Fn(\000)p Fl(1)1418 1373 y Fq(\015)5 b Fp(Av)1588 1395 y Fr(x)p Fn(2)p Fm(T)1725 1377 y FA(d)1725 1412 y(\017)1774 1373 y Fp(2)p Fq(H)i Fp(\()p Fq(s;)15 b(\017x)p Fp(\)Av)2263 1336 y Fn(\003)2263 1396 y Fr(z)s(;x)2362 1373 y Fq(\034)2402 1387 y Fr(z)2451 1302 y Fq( )2513 1269 y Fr(e)2510 1325 y(n;n)p 2451 1352 169 4 v 2508 1436 a Fq(n)2650 1373 y Fp(+)20 b Fq(\017)2778 1336 y Fr(d)p Fn(\000)p Fl(2)2908 1373 y Fo(L)2971 1387 y Fr(\017)3004 1300 y Fk(\011)3057 1373 y Fq(:)327 b Fu(\(4.50\))118 1554 y(Thanks)26 b(to)g(the)f(de\002nition)h(of)g(the)f(dual)g(average) f Fp(Av)2000 1518 y Fr(?)2000 1576 y(z)s(;x)2124 1554 y Fu(we)h(can)g(write)915 1774 y Fp(Av)1033 1796 y Fr(x)p Fn(2)p Fm(T)1170 1777 y FA(d)1170 1813 y(\017)1219 1774 y Fp(2)p Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\)Av)1708 1736 y Fn(\003)1708 1796 y Fr(z)s(;x)1807 1774 y Fq(\034)1847 1788 y Fr(z)1896 1703 y Fq( )1958 1670 y Fr(e)1955 1725 y(n;n)p 1896 1753 V 1953 1836 a Fq(n)2100 1774 y Fp(=)25 b(Av)2314 1796 y Fr(x)p Fn(2)p Fm(T)2451 1777 y FA(d)2451 1813 y(\017)2500 1774 y Fq(a)2548 1788 y Fr(x)2592 1774 y Fq(\034)2632 1788 y Fr(x)2686 1703 y Fq( )2748 1670 y Fr(e)2745 1725 y(n;n)p 2686 1753 V 2743 1836 a Fq(n)118 1955 y Fu(where)36 b Fq(a)448 1969 y Fr(x)536 1955 y Fp(:=)45 b(Av)794 1919 y Fr(?)794 1977 y(z)s(;x)893 1955 y Fp(2)p Fq(H)7 b Fp(\()p Fq(s;)15 b(\017z)t Fp(\))p Fu(.)67 b(Since)36 b Fp(Av)1716 1919 y Fr(?)1716 1977 y(z)s(;x)1852 1955 y Fu(is)i(translationally)e(invariant)h(w)-11 b(.r)h(.t.)67 b Fq(x)36 b Fu(and)h Fq(H)44 b Fu(is)118 2099 y(regular)-10 b(,)27 b(we)g(can)g(proceed)f(as)i(at)g(the)g(very)f (beginning)f(of)i(this)h(section)f(and)g(safely)f(replace)f Fq(\034)3452 2113 y Fr(x)3506 2045 y( )3554 2021 y FA(e)3552 2061 y(n;n)p 3506 2078 146 4 v 3557 2131 a Fr(n)118 2240 y Fu(by)f(a)h(local)e(average)g Fp(Av)972 2258 y Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)1228 2267 y Fg(1)1266 2240 y Fq(\034)1306 2254 y Fr(y)1357 2185 y( )1405 2162 y FA(e)1403 2202 y(n;n)p 1357 2219 V 1408 2271 a Fr(n)1513 2240 y Fu(,)h Fq(`)g Fo(\035)h Fq(n)p Fu(,)f(to)g(get)201 2459 y Fp(\(4)p Fq(:)p Fp(50\))i Fo(\024)e Fp(sup)14 b Fq(spec)876 2479 y Fr(L)924 2460 y Fg(2)959 2479 y Fl(\()p Fr(\026)1028 2487 y FA(\017)1059 2479 y Fl(\))1091 2386 y Fk(\010)1144 2459 y Fq(\017)1181 2422 y Fn(\000)p Fl(1)1275 2459 y Fq(\015)5 b Fp(Av)1444 2482 y Fr(x)p Fn(2)p Fm(T)1581 2463 y FA(d)1581 2498 y(\017)1631 2459 y Fq(a)1679 2473 y Fr(x)1723 2459 y Fp(Av)1840 2478 y Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)2096 2487 y Fg(1)2134 2459 y Fq(\034)2174 2473 y Fr(y)2225 2389 y Fq( )2287 2356 y Fr(e)2284 2411 y(n;n)p 2225 2438 169 4 v 2282 2522 a Fq(n)2424 2459 y Fp(+)20 b Fq(\017)2552 2422 y Fr(d)p Fn(\000)p Fl(2)2682 2459 y Fo(L)2745 2473 y Fr(\017)2777 2386 y Fk(\011)2851 2459 y Fp(+)g Fq(c)p Fp(\()p Fq(H)7 b Fp(\))p Fq(\015)e(\017`)3261 2422 y Fl(2)3301 2459 y Fq(:)83 b Fu(\(4.51\))118 2640 y(By)24 b(the)f(usual)g(trick)h(of)g(localizing)e(on)h(boxes)g Fp(\003)1768 2655 y Fr(x;`)1884 2640 y Fu(and)g(proposition)i(4.2,)f (if)g Fq(\017)f Fu(is)h(small)f(enough)g(then)118 2748 y(the)j(\002rst)g(term)e(in)i(the)f(r)-10 b(.h.s.)32 b(of)26 b(\(4.51\))e(is)i(bounded)f(from)g(above)g(by)1174 2973 y Fq(c)15 b(\015)1280 2936 y Fl(2)1320 2973 y Fp(Av)1437 2996 y Fr(x)p Fn(2)p Fm(T)1574 2977 y FA(d)1574 3012 y(\017)1624 2973 y Fq(a)1672 2936 y Fl(2)1672 2996 y Fr(x)1838 2973 y Fp(sup)1731 3056 y Fr(\027)t Fn(2M)p Fl(\(\003)1977 3068 y FA(x;l)2055 3056 y Fl(\))2098 2973 y Fq(V)2151 2988 y Fr(x;`)2243 2900 y Fk(\000)2295 2903 y Fq( )2357 2870 y Fr(e)2390 2846 y Fi(0)2354 2925 y Fr(n;n)p 2295 2953 V 2352 3036 a Fq(n)2473 2973 y Fp(;)g Fq(\027)2564 2900 y Fk(\001)118 3205 y Fu(which)26 b(in)f(turn,)g (thanks)h(to)g(\(4.33\),)e(is)i(bounded)f(from)g(above)g(by)1131 3356 y Fq(c)15 b(\015)1237 3318 y Fl(2)1277 3356 y Fp(Av)1394 3378 y Fr(x)p Fn(2)p Fm(T)1531 3359 y FA(d)1531 3395 y(\017)1581 3356 y Fq(a)1629 3318 y Fl(2)1629 3378 y Fr(x)1698 3356 y Fo(\024)25 b Fq(c)1833 3318 y Fn(\003)1888 3356 y Fq(\015)1940 3318 y Fl(2)1980 3356 y Fp(Av)2097 3378 y Fr(x)p Fn(2)p Fm(T)2234 3359 y FA(d)2234 3395 y(\017)2284 3356 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))2609 3318 y Fl(2)118 3521 y Fu(for)31 b(some)g(suitable)h (positive)g(constant)g Fq(c)1585 3488 y Fn(\003)1625 3521 y Fu(.)49 b(Let)32 b(us)f(now)g(choose)g Fq(\015)37 b Fu(so)32 b(small)e(that)j Fq(c)3147 3488 y Fn(\003)3186 3521 y Fq(\015)3238 3488 y Fl(2)3302 3521 y Fo(\000)24 b Fq(\015)41 b(<)35 b Fp(0)p Fu(.)118 3629 y(Then,)25 b(by)h(the)f(previous)g(estimates,)i(if)f Fq(\017)f Fu(is)h(small)f (enough,)440 3843 y Fp(lim)15 b(sup)382 3922 y Fr(n)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p 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b(+)55 b(sup)2026 4361 y Fr(H)5 b Fn(2H)2196 4338 y Fi(\003)2196 4385 y FA(b)2248 4282 y Fq(H)2331 4245 y Fr(?)2324 4305 y(b;a;\017)118 4493 y Fu(so)26 b(that)443 4645 y Fp(lim)15 b(sup)384 4724 y Fr(n)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)795 4645 y Fs(E)849 4608 y Fr(\026)892 4584 y FA(\017)932 4571 y Fk(\000)1006 4645 y Fp(sup)974 4723 y Fr(H)5 b Fn(2H)1144 4735 y FA(b)1190 4645 y Fq(H)1273 4608 y Fr(?)1266 4668 y(b;a;\017)1405 4571 y Fk(\001)1472 4645 y Fo(\024)25 b Fp(1)c(+)78 b(lim)15 b(sup)1725 4724 y Fr(n)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)2135 4645 y Fs(E)2190 4608 y Fr(\026)2232 4584 y FA(\017)2273 4571 y Fk(\000)2349 4645 y Fp(sup)2314 4724 y Fr(H)5 b Fn(2H)2484 4701 y Fi(\003)2484 4748 y FA(b)2536 4645 y Fq(H)2619 4608 y Fr(?)2612 4668 y(b;a;\017)2751 4571 y Fk(\001)2818 4645 y Fo(\024)25 b Fp(1)20 b(+)3080 4584 y Fq(\024)p 3080 4624 V 3080 4707 a(\015)3409 4645 y Fu(\(4.53\))118 4875 y(thus)26 b(allowing)e(to)i(conclude)d(the)j(proof)f(of)h (\(3.4\).)118 4983 y(Let)39 b(us)g(now)f(sketch)h(the)g(proof)g(of)g (\(3.5\).)71 b(Since)37 b Fq(C)2059 4950 y Fl(1)2098 4983 y Fp(\([0)p Fq(;)15 b(T)e Fp(])31 b Fo(\002)d Fs(T)2523 4950 y Fr(d)2566 4983 y Fp(\))39 b Fu(has)g(a)g(countable)e(base,)42 b(by)118 5091 y(Beppo\226Levi)28 b(theorem)g(it)g(is)h(enough)e(to)h (prove)f(that)i(there)e(exists)i(a)f(constant)g Fq(c)2960 5105 y Fl(0)3028 5091 y Fu(such)f(that,)j(given)118 5198 y Fq(H)194 5212 y Fl(1)233 5198 y Fq(;)15 b(:)g(:)g(:)i(;)e(H)511 5212 y Fr(n)583 5198 y Fu(in)25 b Fq(C)757 5165 y Fl(1)796 5198 y Fp(\([0)p Fq(;)15 b(T)e Fp(])22 b Fo(\002)e Fs(T)1204 5165 y Fr(d)1246 5198 y Fp(\))p Fu(,)26 b(then)357 5291 y Fk(Z)463 5415 y Fq(dQ)p Fp(\()p Fq(m)p Fp(\))732 5314 y Fk(h)850 5415 y Fp(sup)791 5492 y Fr(i)p Fl(=1)p Fr(;:::)o(;n)1061 5291 y Fk(Z)1152 5317 y Fr(T)1112 5497 y Fl(0)1222 5291 y Fk(Z)1273 5497 y Fm(T)1323 5478 y FA(d)1357 5341 y Fk(\000)1399 5415 y Fp(2)15 b Fq(m)p Fp(\()p Fq(s;)g(\022)s Fp(\))1788 5353 y Fq(@)p 1748 5394 133 4 v 1748 5477 a(@)5 b(\022)1844 5491 y Fr(e)1891 5415 y Fq(H)1967 5429 y Fr(i)1995 5415 y Fp(\()p Fq(s;)15 b(\022)s Fp(\))20 b Fo(\000)g Fq(H)2381 5429 y Fr(i)2409 5415 y Fp(\()p Fq(s;)15 b(\022)s Fp(\))2608 5377 y Fl(2)2647 5341 y Fk(\001)2689 5415 y Fq(d\022)i(ds)2901 5314 y Fk(i)2970 5415 y Fo(\024)25 b Fq(c)3105 5429 y Fl(0)3144 5415 y Fq(:)240 b Fu(\(4.54\))p eop %%Page: 23 23 23 22 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(23)118 366 y Fu(By)31 b(the)g(Lebesgue)f(density)h(theorem)f(and)h(the)h (dominated)f(convergence)c(theorem,)32 b(the)f(l.h.s.)49 b(of)118 478 y Fp(\(4)p Fq(:)p Fp(54\))28 b Fu(is)e(equal)e(to)i Fp(lim)935 493 y Fr(a)p Fn(#)p Fl(0)1063 478 y Fs(E)1117 492 y Fr(Q)1183 404 y Fk(\000)1225 478 y Fp(\002)1296 445 y Fl(\()p Fr(a)p Fl(\))1392 404 y Fk(\001)1459 478 y Fu(where,)f(for)g(any)g Fq(\027)31 b Fo(2)25 b Fq(D)s Fp(\([0)p Fq(;)15 b(T)e Fp(])p Fq(;)i Fo(M)2686 492 y Fl(1)2727 478 y Fp(\))p Fu(,)462 699 y Fp(\002)533 662 y Fl(\()p Fr(a)p Fl(\))629 699 y Fp(\()p Fq(\027)6 b Fp(\))26 b(:=)84 b(sup)897 776 y Fr(i)p Fl(=1)p Fr(;:::)o(;n)1168 575 y Fk(Z)1258 602 y Fr(T)1218 782 y Fl(0)1329 575 y Fk(Z)1379 782 y Fm(T)1429 763 y FA(d)1464 626 y Fk(\000)1505 699 y Fp(2)15 b Fq(\027)1616 662 y Fl(\()p Fr(a)p Fl(\))1713 699 y Fp(\()p Fq(s;)g(\022)s Fp(\))1962 638 y Fq(@)p 1922 678 133 4 v 1922 762 a(@)5 b(\022)2018 776 y Fr(e)2065 699 y Fq(H)2141 713 y Fr(i)2169 699 y Fp(\()p Fq(s;)15 b(\022)s Fp(\))20 b Fo(\000)g Fq(H)2562 662 y Fl(2)2555 722 y Fr(i)2601 699 y Fp(\()p Fq(s;)15 b(\022)s Fp(\))2800 626 y Fk(\001)2842 699 y Fq(d\022)i(ds;)345 b Fu(\(4.55\))118 912 y(with)844 1042 y Fq(\027)895 1005 y Fl(\()p Fr(a)p Fl(\))991 1042 y Fp(\()p Fq(s;)15 b(\022)s Fp(\))26 b(:=)1426 981 y(1)p 1347 1021 205 4 v 1347 1104 a(\(2)p Fq(a)p Fp(\))1510 1078 y Fr(d)1561 1042 y Fq(\027)1606 1056 y Fr(s)1643 968 y Fk(\000)1685 1042 y Fo(f)p Fq(\022)1776 1005 y Fn(0)1824 1042 y Fo(2)f Fs(T)1969 1005 y Fr(d)2036 1042 y Fp(:)81 b(sup)2087 1121 y Fr(i)p Fl(=1)p Fr(;:::)n(;d)2350 1042 y Fo(j)p Fq(\022)2421 1005 y Fn(0)2418 1065 y Fr(i)2467 1042 y Fo(\000)20 b Fq(\022)2601 1056 y Fr(i)2628 1042 y Fo(j)26 b(\024)f Fq(a)p Fo(g)2868 968 y Fk(\001)2910 1042 y Fq(:)118 1238 y Fu(It)h(is)g(simple)f(to)h(prove)f(\(see)f ([15],)i(section)f(1.18\))f(that)752 1429 y Fp(lim)761 1490 y Fr(a)p Fn(#)p Fl(0)894 1305 y Fk(Z)1000 1429 y Fq(dQ)p Fp(\()p Fq(m)p Fp(\))1269 1355 y Fk(\000)1311 1429 y Fp(\002)1382 1391 y Fl(\()p Fr(a)p Fl(\))1478 1429 y Fp(\()p Fq(m)p Fp(\))1628 1355 y Fk(\001)1696 1429 y Fo(\024)h Fp(lim)15 b(sup)1817 1508 y Fr(a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)2085 1305 y Fk(Z)2191 1429 y Fq(dQ)2310 1391 y Fr(\013;)d(\026)2429 1368 y FA(\017)2464 1429 y Fp(\()p Fq(\027)6 b Fp(\))2585 1355 y Fk(\000)2627 1429 y Fp(\002)2698 1391 y Fl(\()p Fr(a)p Fl(\))2794 1429 y Fp(\()p Fq(\027)g Fp(\))2915 1355 y Fk(\001)2982 1429 y Fp(=)153 1708 y(lim)15 b(sup)118 1787 y Fr(b)p Fn(#)p Fl(0)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)480 1708 y Fs(E)535 1671 y Fr(\013;\026)642 1647 y FA(\017)683 1607 y Fk(\020)811 1708 y Fp(sup)752 1785 y Fr(i)p Fl(=1)p Fr(;:::)o(;n)1022 1584 y Fk(Z)1113 1611 y Fr(T)1073 1790 y Fl(0)1183 1708 y Fp(Av)1301 1730 y Fr(x)p Fn(2)p Fm(T)1438 1711 y FA(d)1438 1747 y(\017)1472 1634 y Fk(\000)1529 1708 y Fp(2)p Fq(m)1654 1725 y Fr(x;)1724 1698 y FA(a)p 1724 1710 34 3 v 1728 1751 a(\017)1771 1708 y Fp(\()p Fq(s;)g(\017x)p Fp(\))2013 1634 y Fk(\002)2062 1647 y Fq(H)2138 1661 y Fr(i)2166 1647 y Fp(\()p Fq(s;)g(\017x)20 b Fp(+)g Fq(be)p Fp(\))h Fo(\000)f Fq(H)2788 1661 y Fr(i)2815 1647 y Fp(\()p Fq(s;)15 b(\017x)p Fp(\))p 2062 1687 997 4 v 2540 1770 a Fq(b)3068 1634 y Fk(\003)3126 1708 y Fo(\000)20 b Fq(H)3300 1671 y Fl(2)3293 1731 y Fr(i)3339 1708 y Fp(\()p Fq(s;)15 b(\017x)p Fp(\))3596 1634 y Fk(\001)3654 1607 y(\021)3708 1708 y Fq(:)118 1923 y Fu(By)25 b(integrating)g(by)g (parts)h(and)g(observing)e(that)1250 2070 y Fp(sup)1218 2148 y Fr(H)5 b Fn(2H)1388 2160 y FA(b)1435 2070 y Fq(H)1518 2032 y Fr(?)1511 2093 y(b;a;\017)1675 2070 y Fp(=)224 b(sup)1771 2156 y Fr(H)5 b Fn(2)p Fr(C)1936 2137 y Fg(1)1971 2156 y Fl(\([0)p Fr(;T)10 b Fl(])p Fn(\002)p Fm(T)2248 2137 y FA(d)2279 2156 y Fl(\))2321 2070 y Fq(H)2404 2032 y Fr(?)2397 2093 y(a;b;\017)2536 2070 y Fq(;)118 2296 y Fu(the)26 b(thesis)g(follows)g(from)e(\(4.53\).)118 2459 y(4.8.)45 b FE(Hydrodynamic)30 b(limit)d(without)g(regularity)h (of)h(the)f(diffusion)g(matrix.)46 b Fu(In)27 b(this)i(last)e(para-)118 2567 y(graph)36 b(we)h(shortly)g(discuss)h(the)e(hydrodynamic)h(limit)g (when)f(the)h(regularity)f(condition)h(on)f(the)118 2675 y(diffusion)26 b(matrix)e(is)h(replaced)e(by)h(the)h(two)g(conditions)g (at)g(the)f(end)g(of)h(theorem)e(2.4,)h(in)h(the)f(sequel)118 2783 y(referred)g(to)i(as)g(assumptions)h Fq(A)p Fp(\()p Fq(\032)p Fp(\))p Fu(.)33 b(The)26 b(main)f(idea)h(here)f(is)i(to)f (prove)e(that)j(one)e(can)g(safely)g(intro-)118 2891 y(duce)k(a)i(density)g(cutoff)f(near)f(the)h(edges)g(of)g(the)h (interval)e Fp(\(0)p Fq(;)15 b Fp(1\))p Fu(,)34 b(and)c(for)g(this)h (purpose)f(the)h(main)118 2999 y(technical)25 b(tool)g(is)h(the)f (following)g(result.)118 3126 y FE(L)r(emma)j(4.8.)43 b Fh(Assume)26 b(that)h(the)g(sequence)d(of)i(initial)h(probability)g (measures)e Fq(\026)2897 3093 y Fr(\017)2956 3126 y Fh(satisfy)h Fq(A)p Fp(\()p Fq(\032)p Fp(\))p Fh(.)37 b(Then)118 3234 y(there)25 b(exists)g(a)g(constant)f Fp(0)h Fq(<)34 b Fp(\026)-54 b Fq(\032)25 b Fo(\024)g Fq(\032)h Fh(such)f(that,)h(for)e (any)h Fq(T)38 b(>)25 b Fp(0)h Fh(and)e(any)h(disorder)e (con\002guration)h Fq(\013)p Fh(,)705 3444 y Fp(lim)642 3505 y Fr(`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)910 3444 y Fs(E)965 3406 y Fr(\026)1007 3383 y FA(\017)1048 3343 y Fk(\020)1117 3320 y(Z)1208 3346 y Fr(T)1168 3526 y Fl(0)1278 3444 y Fq(ds)15 b Fp(Av)1501 3466 y Fr(x)p Fn(2)p Fm(T)1638 3447 y FA(d)1638 3483 y(\017)1687 3370 y Fk(\000)1729 3444 y Fp(1)-25 b(I)1782 3462 y Fn(f)p Fr(m)1879 3474 y FA(x;`)1963 3462 y Fl(\()p Fr(s)p Fl(\))p Fr(<)6 b Fl(\026)-41 b Fr(\032)q Fn(g)2201 3444 y Fp(+)20 b(1)-25 b(I)2345 3462 y Fn(f)p Fr(m)2442 3474 y FA(x;`)2527 3462 y Fl(\()p Fr(s)p Fl(\))p Fr(>)p Fl(1)p Fn(\000)6 b Fl(\026)-41 b Fr(\032)p Fn(g)2835 3370 y Fk(\001)2892 3343 y(\021)2971 3444 y Fp(=)25 b(0)p Fq(:)272 b Fu(\(4.56\))118 3654 y Fh(Proof)-8 b(.)42 b Fu(F)-6 b(or)21 b(simplicity)-12 b(,)23 b(we)e(consider)h(in)f(\(4.56\))f(only)h(the)h(contribution)f (coming)g(from)g Fp(1)-25 b(I)3237 3672 y Fn(f)p Fr(m)3334 3684 y FA(x;`)3418 3672 y Fl(\()p Fr(s)p Fl(\))p Fr(<)6 b Fl(\026)-41 b Fr(\032)q Fn(g)3636 3654 y Fu(,)118 3769 y(the)31 b(other)f(one)g(being)f(similar)-10 b(.)48 b(Given)30 b(two)g(probability)i(measures)d Fq(\026)2628 3783 y Fl(1)2667 3769 y Fq(;)i(\026)2778 3783 y Fl(2)2848 3769 y Fu(on)f Fp(\012)3045 3783 y Fr(\017)3077 3769 y Fu(,)i(we)e(will)g (write)118 3877 y Fq(\026)173 3891 y Fl(1)241 3877 y Fo(\024)f Fq(\026)396 3891 y Fl(2)462 3877 y Fu(if)g Fq(\026)600 3891 y Fl(1)639 3877 y Fp(\()p Fq(f)10 b Fp(\))29 b Fo(\024)f Fq(\026)947 3891 y Fl(2)986 3877 y Fp(\()p Fq(f)10 b Fp(\))28 b Fu(for)f(any)g(function)g Fq(f)37 b Fu(which)28 b(is)g(increasing)f(w)-11 b(.r)h(.t.)39 b(the)28 b(partial)f(order)g(in)118 3987 y Fp(\012)184 4001 y Fr(\017)245 3987 y Fu(given)g(by)h Fq(\021)34 b Fo(\024)c Fq(\021)837 3954 y Fn(0)891 3987 y Fo(,)g Fq(\021)s Fp(\()p Fq(x)p Fp(\))h Fo(\024)f Fq(\021)1362 3954 y Fn(0)1385 3987 y Fp(\()p Fq(x)p Fp(\))h Fo(8)15 b Fq(x)30 b Fo(2)g Fs(T)1836 3954 y Fr(d)1836 4010 y(\017)1878 3987 y Fu(.)40 b(It)29 b(is)g(then)f(simple)g(to)h(check)e(that)i(our)f (model)g(is)118 4095 y Fh(attractive)f Fu([26])g(in)g(the)g(sense)f (that)i Fq(\026)1451 4109 y Fl(1)1519 4095 y Fo(\024)f Fq(\026)1672 4109 y Fl(2)1738 4095 y Fu(implies)h(that)g Fs(P)2290 4051 y Fr(\026)2332 4060 y Fg(1)2290 4119 y Fr(t)2399 4095 y Fo(\024)g Fs(P)2553 4051 y Fr(\026)2595 4060 y Fg(2)2553 4119 y Fr(t)2662 4095 y Fu(for)e(any)h Fq(t)h(>)g Fp(0)f Fu(and)g(for)g(any)118 4203 y(disorder)k (con\002guration)e Fq(\013)p Fu(.)46 b(Therefore,)31 b(condition)f Fq(A)p Fp(\()p Fq(\032)p Fp(\))h Fu(implies)f(that)h (there)f(exists)h Fq(\025)i(<)g Fp(0)e Fu(such)118 4328 y(that)e Fq(\026)359 4295 y Fr(\025)359 4350 y(\017)434 4328 y Fo(\024)h Fq(\026)590 4295 y Fr(\017)651 4328 y Fu(for)e(any)g Fq(\017)p Fu(.)40 b(Let)28 b(now)36 b Fp(\026)-53 b Fq(\032)30 b Fp(:=)1627 4292 y Fl(1)p 1627 4307 36 4 v 1627 4359 a(2)1688 4328 y Fp(min)1839 4254 y Fk(\000)1936 4292 y Fr(e)1969 4269 y FA(\025)p Fi(\000)p FA(B)p 1891 4307 261 4 v 1891 4364 a Fl(1+)p Fr(e)2014 4345 y FA(\025)p Fi(\000)p FA(B)2161 4328 y Fq(;)15 b(\032)2248 4254 y Fk(\001)2290 4328 y Fu(.)40 b(Then,)29 b(given)e Fq(\014)36 b(>)29 b Fp(0)g Fu(and)f(thanks)h(to) 118 4457 y(attractivity)-12 b(,)27 b(the)f(entropy)e(inequality)i (\(4.6\))e(and)i(the)f(identity)i Fq(H)7 b Fp([)p Fq(\026)2534 4424 y Fr(\017)2567 4457 y Fo(j)p Fq(\026)2647 4424 y Fr(\017)2647 4480 y Fn(\003)2686 4457 y Fp(])26 b(=)e Fq(H)7 b Fp([)p Fs(P)2995 4424 y Fr(\026)3037 4401 y FA(\017)3073 4457 y Fo(j)p Fs(P)3153 4424 y Fr(\026)3195 4401 y FA(\017)3195 4441 y Fi(\003)3236 4457 y Fp(])p Fu(,)1171 4670 y Fs(E)1226 4632 y Fr(\026)1268 4609 y FA(\017)1309 4596 y Fk(\000)1365 4546 y(Z)1456 4572 y Fr(T)1416 4752 y Fl(0)1527 4670 y Fq(ds)15 b Fp(Av)1749 4692 y Fr(x)p Fn(2)p Fm(T)1886 4673 y FA(d)1886 4708 y(\017)1920 4670 y Fp(1)-25 b(I)1973 4688 y Fr(m)2035 4700 y FA(x;`)2120 4688 y Fl(\()p Fr(s)p Fl(\))p Fr(<)6 b Fl(\026)-41 b Fr(\032)2317 4596 y Fk(\001)2384 4670 y Fo(\024)745 4869 y Fp(1)p 739 4909 57 4 v 739 4992 a Fq(\014)806 4930 y(H)889 4856 y Fk(\002)926 4930 y Fq(\026)981 4893 y Fr(\017)1014 4930 y Fo(j)p Fq(\026)1094 4893 y Fr(\017)1094 4953 y Fn(\003)1133 4856 y Fk(\003)1191 4930 y Fp(+)1298 4869 y(1)p 1292 4909 V 1292 4992 a Fq(\014)1374 4930 y Fp(ln)1449 4829 y Fk(\020)1519 4930 y Fq(\026)1574 4893 y Fr(\025)1574 4953 y(\017)1619 4856 y Fk(\000)1661 4930 y Fp(exp)1800 4856 y Fk(\010)1853 4806 y(Z)1943 4833 y Fr(T)1903 5013 y Fl(0)2014 4930 y Fq(ds)15 b(\014)20 b Fp(Av)2308 4952 y Fr(x)p Fn(2)p Fm(T)2445 4933 y FA(d)2445 4969 y(\017)2479 4930 y Fp(1)-25 b(I)2532 4949 y Fr(m)2594 4961 y FA(x;`)2678 4949 y Fl(\()p Fr(s)p Fl(\))p Fr(<)6 b Fl(\026)-41 b Fr(\032)2861 4856 y Fk(\011\001)2971 4829 y(\021)3025 4930 y Fq(:)3409 4793 y Fu(\(4.57\))118 5139 y(Thanks)22 b(to)f(the)g(Jensen's)g(inequality)h(and)f(the)g (reversibility)g(of)g Fo(L)2400 5153 y Fr(\017)2454 5139 y Fu(w)-11 b(.r)h(.t.)31 b Fq(\026)2729 5106 y Fr(\025)2729 5161 y(\017)2795 5139 y Fu(the)21 b(second)g(addendum)118 5246 y(in)k(the)h(r)-10 b(.h.s.)32 b(of)26 b(\(4.57\))e(can)g(be)h (bounded)g(by)1178 5372 y Fp(1)p 1172 5413 V 1172 5496 a Fq(\014)1254 5433 y Fp(ln)1330 5360 y Fk(\000)1386 5433 y Fq(\026)1441 5396 y Fr(\025)1441 5456 y(\017)1486 5360 y Fk(\000)1528 5433 y Fp(exp)p Fo(f)p Fq(T)j(\014)21 b Fp(Av)1982 5456 y Fr(x)p Fn(2)p Fm(T)2119 5437 y FA(d)2119 5472 y(\017)2154 5433 y Fp(1)-25 b(I)2207 5447 y Fr(m)2269 5459 y FA(x;`)2353 5447 y Fr(<)6 b Fl(\026)-41 b Fr(\032)2448 5433 y Fo(g)2493 5360 y Fk(\001)15 b(\001)2592 5433 y Fq(:)792 b Fu(\(4.58\))p eop %%Page: 24 24 24 23 bop 118 45 a Fj(24)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 377 y Fu(Let)26 b(us)h(call)e Fq(\027)596 344 y Fr(\025)667 377 y Fu(the)h(product)g (measure)f(on)h Fp(\012)1707 391 y Fr(\017)1765 377 y Fu(such)g(that)h Fq(\027)2206 344 y Fr(\025)2251 377 y Fp(\()p Fq(\021)2331 391 y Fr(x)2376 377 y Fp(\))g(=)2590 341 y Fr(e)2623 318 y FA(\025)p Fi(\000)p FA(B)p 2545 356 261 4 v 2545 414 a Fl(1+)p Fr(e)2668 395 y FA(\025)p Fi(\000)p FA(B)2815 377 y Fu(.)34 b(Then,)26 b(it)h(is)g(simple)f(to) 118 508 y(check)e(that)j Fq(\027)602 475 y Fr(\025)672 508 y Fo(\024)e Fq(\026)823 475 y Fr(\025)823 530 y(\017)868 508 y Fu(,)h(which)g(implies)f(that)988 700 y Fp(\(4)p Fq(:)p Fp(58\))i Fo(\024)1357 638 y Fp(1)p 1351 679 57 4 v 1351 762 a Fq(\014)1433 700 y Fp(ln)1508 626 y Fk(\000)1565 700 y Fq(\027)1616 662 y Fr(\025)1661 626 y Fk(\000)1703 700 y Fp(exp)p Fo(f)p Fq(T)h(\014)20 b Fp(Av)2157 722 y Fr(x)p Fn(2)p Fm(T)2294 703 y FA(d)2294 739 y(\017)2328 700 y Fp(1)-25 b(I)2381 714 y Fr(m)2443 726 y FA(x;`)2527 714 y Fr(<)6 b Fl(\026)-41 b Fr(\032)2623 700 y Fo(g)2668 626 y Fk(\001)15 b(\001)2766 700 y Fq(:)118 897 y Fu(A)n(t)34 b(this)g(point,)i(let)d(us)g(recall)e(a)i(general)e(result)h(based)i (on)e(the)i(Herbst's)g(argument)d(and)i(the)g(log-)118 1005 y(arithmic)g(Sobolev)e(inequality)i(\(see)f([1)o(])h(for)f(a)h (complete)e(discussion\):)46 b(for)32 b(any)g Fq(\015)43 b(>)36 b Fp(0)d Fu(and)g(any)118 1113 y(function)25 b Fq(f)34 b Fu(on)25 b Fp(\012)744 1127 y Fr(\017)1445 1237 y Fq(\027)1496 1200 y Fr(\025)1541 1237 y Fp(\()p Fq(e)1618 1200 y Fr(\015)t(f)1704 1237 y Fp(\))g Fo(\024)g Fq(e)1902 1200 y Fr(c)1933 1212 y FA(f)1985 1200 y Fr(\015)2025 1176 y Fg(2)2060 1200 y Fl(+)p Fr(\015)t(\027)2194 1176 y FA(\025)2235 1200 y Fl(\()p Fr(f)7 b Fl(\))118 1368 y Fu(where)23 b Fq(c)426 1383 y Fr(f)497 1368 y Fp(:=)i Fq(c)688 1300 y Fk(P)784 1395 y Fr(x)p Fn(2)p Fm(T)921 1376 y FA(d)921 1411 y(\017)970 1368 y Fo(kr)1091 1382 y Fr(x)1135 1368 y Fq(f)10 b Fo(k)1235 1335 y Fl(2)1235 1390 y Fn(1)1334 1368 y Fu(and)23 b Fq(c)j Fp(=)f Fq(c)p Fp(\()p Fq(B)5 b(;)15 b(\025)p Fp(\))25 b Fu(is)f(a)g(suitable)g (constant)f(independent)g(of)h Fq(\017)g Fu(\()p Fq(c)g Fu(is)118 1499 y(related)h(to)g(the)h(logarithmic)f(Sobolev)f(constant) h(of)h(the)f(Bernoulli)f(measure)g Fq(\027)2909 1466 y Fr(\025)2954 1499 y Fu(\).)118 1607 y(Thus)315 1715 y Fp(1)p 309 1756 V 309 1839 a Fq(\014)391 1777 y Fp(ln)466 1703 y Fk(\000)523 1777 y Fq(\027)574 1739 y Fr(\025)619 1703 y Fk(\000)661 1777 y Fp(exp)p Fo(f)p Fq(T)k(\014)21 b Fp(Av)1115 1799 y Fr(x)p Fn(2)p Fm(T)1252 1780 y FA(d)1252 1816 y(\017)1286 1777 y Fp(1)-25 b(I)1339 1791 y Fr(m)1401 1803 y FA(x;`)1485 1791 y Fr(<)6 b Fl(\026)-41 b Fr(\032)1581 1777 y Fo(g)1626 1703 y Fk(\001)15 b(\001)1750 1777 y Fo(\024)25 b Fq(c)15 b(T)1966 1739 y Fl(2)2021 1777 y Fq(\014)20 b(\017)2129 1739 y Fr(d)2169 1777 y Fq(`)2207 1739 y Fr(d)2268 1777 y Fp(+)g Fq(T)28 b Fp(Av)2557 1799 y Fr(x)p Fn(2)p Fm(T)2694 1780 y FA(d)2694 1816 y(\017)2728 1777 y Fq(\027)2779 1739 y Fr(\025)2825 1703 y Fk(\000)2866 1777 y Fp(1)-25 b(I)2919 1791 y Fr(m)2981 1803 y FA(x;`)3065 1791 y Fr(<)6 b Fl(\026)-41 b Fr(\032)3160 1703 y Fk(\001)3202 1777 y Fq(:)182 b Fu(\(4.59\))118 1985 y(Since)39 b Fp(\026)-54 b Fq(\032)35 b(<)g(\027)601 1952 y Fr(\025)646 1985 y Fp(\()p Fq(\021)726 1999 y Fl(0)766 1985 y Fp(\))p Fu(,)f(by)d (choosing)g Fq(\014)1423 1952 y Fl(2)1498 1985 y Fp(:=)k Fq(H)7 b Fp([)p Fq(\026)1792 1952 y Fr(\017)1825 1985 y Fo(j)p Fq(\026)1905 1952 y Fr(\017)1905 2008 y Fn(\003)1944 1985 y Fp(])p Fq(=)p Fp(\()p Fq(T)2115 1952 y Fl(2)2171 1985 y Fq(\017)2208 1952 y Fr(d)2248 1985 y Fq(`)2286 1952 y Fr(d)2326 1985 y Fp(\))32 b Fu(the)f(r)-10 b(.h.s.)51 b(of)31 b(\(4.59\))f(is)i(negligible)118 2096 y(as)i Fq(`)39 b Fo(")h(1)p Fq(;)15 b(\017)39 b Fo(#)h Fp(0)p Fu(.)57 b(Since)32 b Fq(H)7 b Fp([)p Fq(\026)1224 2063 y Fr(\017)1257 2096 y Fo(j)p Fq(\026)1337 2063 y Fr(\017)1337 2118 y Fn(\003)1376 2096 y Fp(])40 b(=)f Fq(o)p Fp(\()p Fq(\017)1667 2063 y Fn(\000)p Fr(d)1762 2096 y Fp(\))p Fu(,)d(the)e(thesis)h(follows)f(by)f(collecting)f(all)h(the)h(above)118 2204 y(estimates.)3078 b Fe(\003)218 2366 y Fu(Using)33 b(the)h(above)f(result)g(we)g(are)g(in)h(position)h(to)e(discuss)i(our) e(density)h(cutoof)m(.)56 b(Let)34 b(us)f(recall)118 2474 y(\002rst)25 b(that,)i(given)c(a)i(generic)e(continuous)h (extensions)2052 2451 y Fp(\026)2031 2474 y Fq(D)j Fu(of)e Fq(D)i Fu(outside)f(the)f(interval)f Fp([)p Fq(\032;)15 b Fp(1)k Fo(\000)f Fq(\032)p Fp(])p Fu(,)26 b(any)118 2582 y(weak)g(solution)g Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))26 b Fu(of)g(the)g(Cauchy)g(problem)e(\(2.7\),)i(where)f Fq(D)k Fu(has)d(been)f(replaced)g(by)3430 2559 y Fp(\026)3409 2582 y Fq(D)j Fu(and)118 2692 y Fq(\032)e Fo(\024)g Fq(m)368 2706 y Fl(0)407 2692 y Fp(\()p Fq(\022)s Fp(\))h Fo(\024)e Fp(1)d Fo(\000)e Fq(\032)26 b Fu(for)f(any)h Fq(\022)i Fo(2)e Fs(T)1401 2659 y Fr(d)1443 2692 y Fu(,)g(satis\002es)i Fq(\032)e Fo(\024)g Fq(m)p Fp(\()p Fq(t;)15 b(\022)s Fp(\))26 b Fo(\024)g Fp(1)21 b Fo(\000)g Fq(\032)k Fu(for)h(any)g Fp(0)g Fo(\024)g Fq(t)g Fo(\024)g Fq(T)39 b Fu(and)26 b(any)118 2802 y Fq(\022)i Fo(2)c Fs(T)333 2769 y Fr(d)375 2802 y Fu(.)32 b(Let)601 2779 y Fp(\026)580 2802 y Fq(D)c Fu(be)d(de\002ned)g(as)1106 3049 y Fp(\026)1085 3072 y Fq(D)r Fp(\()p Fq(m)p Fp(\))h(:=)1459 2858 y Fk(8)1459 2940 y(>)1459 2967 y(<)1459 3131 y(>)1459 3158 y(:)1540 2946 y Fq(D)s Fp(\()8 b(\026)-53 b Fq(\032)p Fp(\))273 b Fu(if)26 b Fp(0)g Fo(\024)f Fq(m)g Fo(\024)33 b Fp(\026)-53 b Fq(\032)1540 3075 y(D)s Fp(\()p Fq(m)p Fp(\))240 b Fu(if)35 b Fp(\026)-54 b Fq(\032)26 b Fo(\024)f Fq(m)g Fo(\024)g Fp(1)20 b Fo(\000)29 b Fp(\026)-54 b Fq(\032)1540 3205 y(D)s Fp(\(1)21 b Fo(\000)28 b Fp(\026)-53 b Fq(\032)p Fp(\))116 b Fu(if)26 b Fp(1)21 b Fo(\000)28 b Fp(\026)-53 b Fq(\032)25 b Fo(\024)g Fq(m)g Fo(\024)g Fp(1)p Fq(:)118 3360 y Fu(Let)31 b(us)g(explain)e(next)h(how)h(one)f(should)h(modify)g (the)g(proof)g(of)g(theorem)e(3.2)h(in)h(order)f(to)h(get)f(the)118 3468 y(same)37 b(result)g(but)h(with)h Fq(D)h Fu(replaced)c(by)1647 3445 y Fp(\026)1626 3468 y Fq(D)k Fu(in)e(the)g(de\002nition)g(of)2579 3445 y Fp(\026)2555 3468 y Fq(H)2631 3483 y Fr(b;a;\017)2808 3468 y Fu(\(in)f(what)h(follows)g(this)118 3576 y(replacement)31 b(will)i(be)g(understood)h(without)g(further)f(notice\).)56 b(T)-9 b(o)34 b(this)g(aim)g(it)g(is)g(convenient)e(to)118 3684 y(introduce)25 b(the)g(following)g(shorter)g(notation)1334 3833 y Fq(\037)1391 3848 y Fr(x;`)1509 3833 y Fp(:=)g(1)-25 b(I)1683 3847 y Fr(m)1745 3859 y FA(x;`)1829 3847 y Fr(<)6 b Fl(\026)-41 b Fr(\032)1945 3833 y Fp(+)19 b(1)-25 b(I)2088 3847 y Fr(m)2150 3859 y FA(x;`)2235 3847 y Fr(>)p Fl(1)p Fn(\000)6 b Fl(\026)-41 b Fr(\032)2420 3833 y Fq(:)118 3985 y Fu(Then,)25 b(thanks)h(to)g(lemma)e(4.8,)h(equation)g(\(4.5\))g (can)f(be)h(substituted)i(by)293 4197 y Fp(inf)281 4258 y Fr(g)r Fn(2)p Fm(G)437 4197 y Fp(inf)431 4257 y Fr(r)r Fn(\025)p Fl(0)678 4197 y Fp(lim)15 b(sup)570 4276 y Fr(b)p Fn(#)p Fl(0)p Fr(;a)p Fn(#)p Fl(0)p Fr(;l)q Fn("1)p Fr(;\017)p Fn(#)p Fl(0)1079 4197 y Fs(E)1134 4160 y Fr(\026)1176 4136 y FA(\017)1217 4096 y Fk(\020)1286 4120 y(\014)1286 4175 y(\014)1316 4074 y(Z)1407 4100 y Fr(T)1367 4280 y Fl(0)1478 4197 y Fq(\017)1515 4160 y Fn(\000)p Fl(1)1609 4197 y Fp(Av)1726 4219 y Fr(x)p Fn(2)p Fm(T)1863 4201 y FA(d)1863 4236 y(\017)1898 4197 y Fo(r)1974 4160 y Fr(\017)1974 4220 y(e)2010 4197 y Fq(H)7 b Fp(\()p Fq(s;)15 b(\017x)p Fp(\))2335 4096 y Fk(h)2379 4197 y Fp(Av)2496 4216 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)2809 4225 y Fg(1)2846 4197 y Fp(\()p Fq(j)2918 4211 y Fr(y)r(;y)r Fl(+)p Fr(e)3125 4197 y Fp(+)20 b Fq(\034)3256 4211 y Fr(y)3297 4197 y Fo(L)p Fq(g)s Fp(\))575 4478 y(+)673 4392 y Fk(X)666 4590 y Fr(e)699 4571 y Fi(0)721 4590 y Fn(2E)847 4456 y Fp(\026)826 4478 y Fq(D)901 4494 y Fr(e;e)987 4475 y Fi(0)1013 4478 y Fp(\()p Fq(m)1128 4496 y Fr(x;)1198 4469 y FA(a)p 1197 4481 34 3 v 1201 4522 a(\017)1245 4478 y Fp(\))1280 4405 y Fk(\002)1328 4387 y Fq(m)1408 4417 y Fr(x)p Fl(+)1513 4390 y FA(b)p 1513 4402 28 3 v 1514 4443 a(\017)1550 4417 y Fr(e)1583 4398 y Fi(0)1605 4417 y Fr(;)1635 4390 y FA(a)p 1635 4402 34 3 v 1639 4443 a(\017)1702 4387 y Fo(\000)g Fq(m)1873 4417 y Fr(x)p Fn(\000)1978 4390 y FA(b)p 1978 4402 28 3 v 1979 4443 a(\017)2015 4417 y Fr(e)2048 4398 y Fi(0)2070 4417 y Fr(;)2100 4390 y FA(a)p 2100 4402 34 3 v 2104 4443 a(\017)p 1328 4458 820 4 v 1654 4541 a Fp(2)p Fq(b=\017)2157 4405 y Fk(\003)2195 4378 y(i)2274 4478 y Fo(\000)f Fq(r)2423 4355 y Fk(Z)2514 4381 y Fr(T)2474 4561 y Fl(0)2584 4478 y Fq(ds)c Fp(Av)2807 4501 y Fr(x)p Fn(2)p Fm(T)2944 4482 y FA(d)2944 4517 y(\017)2978 4478 y Fq(\037)3035 4493 y Fr(x;`)3142 4478 y Fq(ds)3247 4401 y Fk(\014)3247 4456 y(\014)3278 4378 y(\021)3357 4478 y Fp(=)25 b(0)118 4728 y Fu(and)g(the)h(main)f(issue)h(is)g(to)g(prove)e(proposition)j(4.1)d (with)2182 4705 y Fp(\026)2160 4728 y Fq(J)2219 4683 y Fr(g)2210 4757 y(b;a;`;\017)2423 4728 y Fu(replaced)g(by)628 4939 y Fp(\026)606 4962 y Fq(J)665 4918 y Fr(g)r(;r)656 4992 y(b;a;`;\017)870 4962 y Fp(:=)991 4861 y Fk(n)1051 4962 y Fu(r)-10 b(.h.s.)33 b(of)25 b Fp(\(4)p Fq(:)p Fp(10\))j Fu(with)e Fq(D)i Fo(!)2075 4939 y Fp(\026)2054 4962 y Fq(D)2146 4861 y Fk(o)2227 4962 y Fo(\000)20 b Fq(\017)15 b(r)2429 4838 y Fk(Z)2520 4865 y Fr(T)2479 5045 y Fl(0)2590 4962 y Fq(ds)g Fp(Av)2813 4984 y Fr(x)p Fn(2)p Fm(T)2950 4965 y FA(d)2950 5001 y(\017)2984 4962 y Fq(\037)3041 4977 y Fr(x;`)3148 4962 y Fq(:)118 5161 y Fu(In)30 b(turn)f(the)h(proof)g(of)g(the)g(modi\002ed)g(version)f(of) h(proposition)h(4.1)e(is)i(splitted)g(into)f(several)e(steps,)118 5287 y(one)d(for)g(each)g(term)f Fp(\012)912 5240 y Fl(\()p Fr(r)r Fl(\))912 5315 y Fr(j)1005 5287 y Fu(,)h Fq(j)31 b Fp(=)25 b(0)p Fq(;)15 b Fp(1)p Fq(;)g(:)g(:)g(:)j Fp(5)p Fu(,)26 b(where)f(now)576 5475 y Fp(\012)642 5427 y Fl(\()p Fr(r)r Fl(\))642 5501 y(0)760 5475 y Fp(:=)g(sup)14 b Fq(spec)1203 5494 y Fr(L)1251 5476 y Fg(2)1286 5494 y Fl(\()p Fr(\026)1355 5502 y FA(\017)1386 5494 y Fl(\))1418 5401 y Fk(\010)1471 5475 y Fq(\017)1508 5437 y Fn(\000)p Fl(1)1602 5475 y Fp(Av)1719 5489 y Fr(x)1763 5475 y Fq(J)9 b Fp(\()p Fq(\017x)p Fp(\))p Fq( )2043 5437 y Fl(\(0\))2040 5497 y Fr(x)2159 5475 y Fo(\000)20 b Fq(r)s Fp(Av)2411 5497 y Fr(x)p Fn(2)p Fm(T)2548 5478 y FA(d)2548 5514 y(\017)2582 5475 y Fq(\037)2639 5490 y Fr(x;`)2752 5475 y Fp(+)f Fq(\015)5 b(\017)2931 5437 y Fr(d)p Fn(\000)p Fl(2)3062 5475 y Fo(L)3125 5401 y Fk(\011)3178 5475 y Fq(:)p eop %%Page: 25 25 25 24 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(25)118 366 y Fu(and)37 b(all)f(the)h(other)g Fp(\012)914 380 y Fr(j)987 366 y Fu(are)f(unchanged.)64 b(It)38 b(thus)f(remains)f(to)h (explain)f(how)h(the)g(discussion)h(in)118 499 y(subsection)26 b(4.3)f(has)i(to)f(be)g(modi\002ed)g(in)g(order)g(to)g(apply)g(to)g Fp(\012)2335 451 y Fl(\()p Fr(r)r Fl(\))2335 525 y(0)2428 499 y Fu(.)33 b(Because)25 b(of)h(the)g(new)g(de\002nition)118 607 y(of)g Fp(\012)288 621 y Fl(0)327 607 y Fu(,)f(\(4.36\))f(has)i(to) g(be)f(replaced)f(by)341 764 y Fp(inf)329 825 y Fr(g)r Fn(2)p Fm(G)511 764 y Fp(inf)504 823 y Fr(r)r Fn(\025)p Fl(0)715 764 y Fp(lim)15 b(sup)643 843 y Fr(n)p Fn("1)p Fr(;`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)1105 764 y Fp(Av)1222 786 y Fr(x)p Fn(2)p Fm(T)1359 767 y FA(d)1359 803 y(\017)1435 764 y Fp(sup)1409 846 y Fn(j)p Fr(\014)s Fn(j\024)p Fr(T)1678 764 y Fp(sup)1627 846 y Fr(m)p Fn(2)p Fl([0)p Fr(;)p Fl(1])314 911 y Fk(h)383 1012 y Fp(sup)f Fq(spec)357 1096 y Fr(L)405 1077 y Fg(2)439 1096 y Fl(\()p Fr(\027)501 1107 y Fg(\003)542 1124 y FA(x;`)626 1107 y(;m)704 1096 y Fl(\))731 938 y Fk(\010)784 1012 y Fq(\017)821 974 y Fn(\000)p Fl(1)916 1012 y Fq(\014)20 b Fp(Av)1104 1030 y Fr(y)r Fl(:)p Fn(j)p Fr(y)r Fn(\000)p Fr(x)p Fn(j\024)p Fr(`)1417 1039 y Fg(1)1455 1012 y Fq(\034)1495 1026 y Fr(y)1536 1012 y Fq( )1598 974 y Fl(\()p Fr(n;g)r Fl(\))1595 1034 y Fr(m)1776 1012 y Fp(+)g Fq(\017)1904 974 y Fr(d)p Fn(\000)p Fl(2)2034 1012 y Fo(L)2097 1026 y Fr(\017)2130 938 y Fk(\011)2203 1012 y Fo(\000)g Fq(r)s Fp(1)-25 b(I)2391 1026 y Fr(m<)6 b Fl(\026)-41 b Fr(\032)2568 1012 y Fo(\000)20 b Fq(r)s Fp(1)-25 b(I)2756 1026 y Fr(m>)p Fl(1)p Fn(\000)6 b Fl(\026)-41 b Fr(\032)3004 911 y Fk(i)3072 1012 y Fo(\024)25 b Fp(0)3409 945 y Fu(\(4.60\))118 1314 y(where)g Fq(D)j Fo(!)629 1291 y Fp(\026)608 1314 y Fq(D)g Fu(in)d(the)h(de\002nition)g (of)f Fq( )1537 1267 y Fl(\()p Fr(n;g)r Fl(\))1534 1326 y Fr(m)1695 1314 y Fu(.)118 1422 y(W)-8 b(e)31 b(observe)e(that,)k (provided)e Fq(\017`)1275 1389 y Fr(d)p Fl(+2)1438 1422 y Fo(\034)j Fp(1)p Fu(,)e(the)e Fp(sup)14 b Fq(spec)30 b Fu(inside)i(the)e(square)g(bracket)f(in)i(\(4.60\))e(is)118 1530 y(bounded)j(by)g Fq(c)664 1544 y Fr(g)720 1530 y Fq(T)786 1497 y Fl(2)825 1530 y Fu(,)j(for)d(a)g(suitable)h(constant)f Fq(c)1857 1544 y Fr(g)1930 1530 y Fu(depending)g(on)g Fq(g)s Fu(.)53 b(That)33 b(follows)g(immediately)118 1638 y(from)j(perturbation)g(theory)g(\(see)f(proposition)j(4.2\))d (and)h(the)h(estimate)f(\(4.33\).)63 b(Therefore,)38 b(by)118 1746 y(choosing)30 b Fq(r)i Fu(large)d(enough,)h(we)g(only)f (need)g(to)h(prove)g(\(4.60\))e(with)k Fq(m)g Fo(2)h Fp([)8 b(\026)-53 b Fq(\032;)15 b Fp(1)25 b Fo(\000)31 b Fp(\026)-53 b Fq(\032)p Fp(])30 b Fu(where)f Fq(D)s Fp(\()p Fq(m)p Fp(\))118 1854 y Fu(and)311 1831 y Fp(\026)290 1854 y Fq(D)s Fp(\()p Fq(m)p Fp(\))23 b Fu(coincide.)31 b(Similarly)22 b(one)g(shows)i(that)g(the)g(two)f(\223)p Fp(sup)2415 1876 y Fr(m)p Fn(2)p Fl([0)p Fr(;)p Fl(1])2658 1854 y Fu(\224)f(appearing)h(in)g(\(4.37\))e(and)118 1968 y(\(4.38\))j(can)h(be)g(sa\003ey)g(replaced)f(by)h(\223)p Fp(sup)1584 1990 y Fr(m)p Fn(2)p Fl([)6 b(\026)-41 b Fr(\032)q(;)p Fl(1)p Fn(\000)6 b Fl(\026)-41 b Fr(\032)o Fl(])1919 1968 y Fu(\224.)288 2230 y(5.)48 b(D)t Ft(I)t(S)t(O)t(R)t(D)t (E)t(R)33 b(I)t(N)t(D)t(U)t(C)t(E)t(D)h(FL)t(U)t(C)t(T)t(U)r(A)l(T)t(I) t(O)t(N)t(S)i(I)t(N)30 b(T)t(H)t(E)h(AV)t(E)t(R)t(A)t(G)t(E)t(D)j(G)t (R)t(A)t(D)t(I)t(E)t(N)t(T)h(D)t(E)t(N)t(S)t(I)t(T)t(Y)e(FI)t(E)t(L)t (D)218 2392 y Fu(In)j(this)i(section)e(we)g(analyze)f(a)i(key)f(term)f (that,)41 b(as)36 b(we)g(seen)g(in)g(section)h(4,)i(arises)e(naturally) 118 2500 y(when)f(one)g(tries)h(to)f(approximate)g(spatial)i(averages)d (of)i(the)f(current)f(with)i(spatial)h(averages)d(of)118 2608 y Fh(gradients)21 b(of)h(the)h(density)e(pro\002le)p Fu(.)31 b(Since)21 b(the)i(currents)f Fq(j)2094 2622 y Fr(x;x)p Fl(+)p Fr(e)2308 2608 y Fu(have,)h(by)g(construction,)g (zero)f(canon-)118 2716 y(ical)31 b(expectation)g(with)i(respect)e(to)h Fh(any)f(canonical)e(measure)i Fu(on)g(any)g(set)h Fp(\003)k Fo(3)f Fq(x;)15 b(x)25 b Fp(+)f Fq(e)p Fu(,)33 b(in)f(order)118 2824 y(to)e(approximate)g Fp(Av)873 2838 y Fr(x)932 2824 y Fq(j)969 2838 y Fr(x;x)p Fl(+)p Fr(e)1190 2824 y Fu(with)h(suitable)f (averages)f(of)h(gradients)f(of)h(the)g(density)h(\002eld,)g(one)e(is) 118 2932 y(forced)g(to)h(subtract)g(from)g(these)g(gradients)g (appropriate)g(canonical)e(expectations.)45 b(Therefore,)30 b(a)118 3040 y(key)h(point)h(in)f(order)g(to)h(establish)g(the)g (hydrodynamical)f(limit,)j(is)e(to)f(prove)g(that)i(these)e (\223counter)118 3147 y(terms\224)25 b(vanish)h(as)g Fq(\017)f Fo(#)h Fp(0)p Fu(.)33 b(These)25 b(kind)h(of)g(terms)f(arise) g(also)h(in)f(the)h(hydrodynamical)f(limit)h(of)g(non\226)118 3255 y(disordered)i(lattice)g(gases)g(\(see)f([37)o(],)i(section)f Fp(7)p Fu(\))g(with)g(short)h(range)d(interaction.)38 b(In)28 b(our)f(context)118 3363 y(however)h(their)h(nature)f(is)i (quite)f(different)g(and,)h(as)f(we)g(will)f(show)i(next,)f(they)g(are) f(basically)h(pro-)118 3471 y(duced)c(by)g(\003uctuations)g(in)h(the)f (disorder)h(\002eld.)118 3619 y(In)34 b(order)f(to)g(be)g(more)g (precise)g(recall)f(\002rst,)k(for)d(any)h(given)e Fq(e)39 b Fo(2)g(E)8 b Fu(,)36 b(the)e(notation)g Fp(\003)3192 3575 y Fl(1)p Fr(;e)3192 3631 y(n)3284 3619 y Fq(;)15 b Fp(\003)3387 3575 y Fl(2)p Fr(;e)3387 3631 y(n)3512 3619 y Fu(and)118 3735 y Fp(\003)181 3702 y Fr(e)181 3757 y(n)259 3735 y Fp(:=)31 b(\003)449 3690 y Fl(1)p Fr(;e)449 3747 y(n)564 3735 y Fo([)22 b Fp(\003)710 3690 y Fl(2)p Fr(;e)710 3747 y(n)830 3735 y Fu(described)29 b(in)g(section)g(2.1,)g(together)f(with)i(the)f(associated)h(densities) g Fq(m)3443 3690 y Fl(1)p Fr(;e)3443 3747 y(n)3565 3735 y Fp(:=)118 3853 y Fq(m)198 3886 y Fl(\003)247 3853 y Fg(1)p FA(;e)247 3896 y(n)334 3853 y Fq(;)g(m)469 3808 y Fl(2)p Fr(;e)469 3864 y(n)586 3853 y Fp(:=)25 b Fq(m)787 3886 y Fl(\003)836 3853 y Fg(2)p FA(;e)836 3896 y(n)923 3853 y Fq(;)31 b(m)1059 3820 y Fr(e)1059 3875 y(n)1131 3853 y Fp(:=)25 b Fq(m)1332 3867 y Fl(\003)1381 3848 y FA(e)1381 3883 y(n)1427 3853 y Fu(.)118 4005 y(Using)e(the)g(above)g (notation)g(and)g(given)f(two)i(integers)e Fq(n)j Fo(\024)g Fq(s)d Fu(and)h(a)g(vector)f Fq(e)k Fo(2)e(E)8 b Fu(,)24 b(the)f(basic)h(object)118 4113 y(of)i(our)e(investigation)i(is)h (de\002ned)e(as)g(\(see)g(\(4.32\)\):)1374 4286 y Fq(\036)1428 4300 y Fr(n;s)1553 4286 y Fp(:=)g Fq(\026)p Fp([)p Fq(m)1834 4249 y Fl(2)p Fr(;e)1834 4309 y(n)1946 4286 y Fo(\000)19 b Fq(m)2116 4249 y Fl(1)p Fr(;e)2116 4309 y(n)2223 4286 y Fo(j)c Fq(m)2343 4249 y Fr(e)2343 4309 y(s)2380 4286 y Fp(])1054 b Fu(\(5.1\))118 4474 y(Notice)20 b(that)i(if)f(the)g (disorder)g(con\002guration)e Fq(\013)i Fu(was)g(identical)g(in)g(the)f (two)h(cubes)f Fp(\003)3026 4429 y Fl(1)p Fr(;e)3026 4485 y(n)3139 4474 y Fu(and)g Fp(\003)3371 4429 y Fl(2)p Fr(;e)3371 4485 y(n)3484 4474 y Fu(then)118 4582 y Fq(\036)172 4596 y Fr(n;n)311 4582 y Fu(would)30 b(be)f(identically)g(equal)g(to)h (zero.)44 b(Moreover)28 b Fs(E)2214 4508 y Fk(\000)2261 4582 y Fq(\036)2315 4596 y Fr(n;s)2415 4508 y Fk(\001)2489 4582 y Fp(=)k(0)e Fu(and)g Fs(E)2900 4508 y Fk(\000)2948 4582 y Fp([)p Fq(\036)3027 4596 y Fr(n;s)3127 4582 y Fp(])3152 4549 y Fl(2)3191 4508 y Fk(\001)3266 4582 y Fp(=)i Fq(O)s Fp(\()p Fq(n)3531 4549 y Fn(\000)p Fr(d)3626 4582 y Fp(\))118 4689 y Fu(uniformly)24 b(in)i Fq(s)f Fo(\025)g Fq(n)p Fu(.)118 4830 y FE(R)m(emark)f(5.1.)38 b Fh(The)22 b(fact)g(that)h Fq(\036)1222 4844 y Fr(n;s)1344 4830 y Fh(is)f(small)g(\(on)g(some)g(average)f(sense\))f(with)j Fq(n)f Fh(and)g(not)g(with)g Fq(s)g Fh(is)h(one)118 4938 y(of)k(the)i(main)e(differences)e(with)j(non)f(disordered)f(lattice)i (gases)e(where,)j(instead,)e(the)h(analogous)e(term)118 5046 y(goes)e(very)h(fast)g(to)g(zero)g(as)f Fq(s)h Fo(")h(1)f Fh(\(see)f Fu([37)o(])p Fh(,)i(section)e Fp(10)p Fh(\).)218 5186 y Fu(The)h(main)g(result)g(of)g(this)h(section)g(is)g(the)f(proof) g(that)h(the)g(contribution)f(to)g(the)h(hydrodynamical)118 5307 y(limit)g(of)f(suitable)h(spatial)g(averages)f(of)1534 5261 y Fr(\036)1576 5269 y FA(n;n)p 1534 5286 142 4 v 1583 5338 a Fr(n)1711 5307 y Fu(is)h(negligible)d(as)j Fq(\017)f Fo(#)h Fp(0)g Fu(at)f(least)h(in)f(dimension)h Fq(d)f Fo(\025)g Fp(3)p Fu(.)118 5480 y(In)g(order)g(to)h(be)f(more)f (precise)h(let)g(us)g(introduce)g(the)g(following)g(equivalence)e (relation.)p eop %%Page: 26 26 26 25 bop 118 47 a Fj(26)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y FE(De\002nition)42 b(5.2.)50 b Fh(Given)41 b(two)f(families)g(of)g(functions)f Fq(f)2115 380 y Fr(x;n;a;\017)2326 366 y Fp(\()p Fq(\013;)15 b(\021)s Fp(\))43 b Fh(and)d Fq(g)2815 380 y Fr(x;n;a;\017)3026 366 y Fp(\()p Fq(\013;)15 b(\021)s Fp(\))43 b Fh(with)f Fq(x)51 b Fo(2)118 480 y Fs(T)177 447 y Fr(d)177 503 y(\017)219 480 y Fq(;)15 b(n)39 b Fo(2)g Fs(N)6 b Fq(;)16 b(a)45 b(>)38 b Fp(0)p Fq(;)15 b(\017)40 b(>)f Fp(0)33 b Fh(we)g(will)h(write)f Fq(f)1691 494 y Fr(x)1773 480 y Fo(\031)39 b Fq(g)1926 494 y Fr(x)2004 480 y Fh(if)-8 b(,)35 b(for)e(any)g(given)g Fq(T)51 b(>)39 b Fp(0)34 b Fh(and)e(for)h(almost)g(all)118 588 y(disorder)24 b(con\002gurations) f Fq(\013)p Fh(,)357 737 y Fp(lim)15 b(sup)299 816 y Fr(n)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)709 737 y Fp(Av)827 759 y Fr(x)p Fn(2)p Fm(T)964 741 y FA(d)964 776 y(\017)1039 737 y Fp(sup)1013 820 y Fn(j)p Fr(\014)s Fn(j\024)p Fr(T)1216 737 y Fp(sup)1265 810 y Fr(\027)1393 737 y Fp(sup)f Fq(spec)1715 757 y Fr(L)1763 738 y Fg(2)1798 757 y Fl(\()p Fr(\027)t Fl(\))1896 664 y Fk(\010)1949 737 y Fq(\017)1986 700 y Fn(\000)p Fl(1)2080 737 y Fq(\014)5 b Fp(\()p Fq(f)2216 751 y Fr(x)2281 737 y Fo(\000)19 b Fq(g)2414 751 y Fr(x)2459 737 y Fp(\))h(+)g Fq(\017)2642 700 y Fn(\000)p Fl(2)2736 737 y Fp(Av)2854 752 y Fr(b)p Fn(2)p Fl(\003)2980 766 y FA(x;)p Fg(2)p FA(a=\017)3164 737 y Fo(L)3227 752 y Fr(b)3261 664 y Fk(\011)3339 737 y Fo(\024)25 b Fp(0)118 962 y Fh(where)33 b Fp(sup)517 984 y Fr(\027)594 962 y Fh(is)h(the)g(supremum)g(over)g Fq(\027)39 b Fh(in)34 b(the)g(set)f Fo(M)p Fp(\(\003)2166 980 y Fr(x;)p Fl(2)2271 953 y FA(a)p 2272 965 34 3 v 2276 1006 a(\017)2319 962 y Fp(\))i Fh(of)e(all)h(the)g(canonical)e (measures)g(on)118 1076 y Fp(\003)181 1093 y Fr(x;)p Fl(2)286 1066 y FA(a)p 286 1078 V 290 1119 a(\017)334 1076 y Fh(.)218 1215 y Fu(W)-8 b(e)28 b(are)f(now)h(in)g(a)f(position)i (to)g(state)f(our)f(main)h(result.)38 b(Assume)28 b(that)h(a)f(given)f (direction)h Fq(e)g Fu(has)118 1323 y(been)h(\002xed)h(once)e(and)i (for)g(all)f(and,)j(given)d(two)h(integers)f Fq(`)k Fo(\024)g Fq(s)d Fu(with)2665 1287 y Fr(s)p 2665 1302 33 4 v 2667 1354 a(`)2740 1323 y Fo(2)j Fs(N)42 b Fu(and)30 b Fq(x)j Fo(2)g Fs(T)3346 1290 y Fr(d)3346 1346 y(\017)3388 1323 y Fu(,)e(recall)118 1445 y(the)26 b(de\002nition)g(of)f(the)h(spatial)g (average)e Fp(Av)1668 1409 y Fr(`;s)1668 1467 y(z)s(;x)1793 1445 y Fu(given)g(in)i(\(2.2\).)118 1576 y FE(Theorem)h(5.3.)41 b Fh(F)l(or)26 b(any)e Fq(d)i Fo(\025)f Fp(3)1536 1732 y(Av)1653 1675 y Fr(n;)1726 1648 y FA(a)p 1726 1660 34 3 v 1730 1701 a(\017)1653 1744 y Fr(z)s(;x)1773 1732 y Fq(\034)1813 1746 y Fr(z)1863 1670 y Fq(\036)1917 1684 y Fr(n;n)p 1863 1711 164 4 v 1917 1794 a Fq(n)2062 1732 y Fo(\031)g Fp(0)15 b Fq(:)218 1891 y Fu(Before)37 b(discussing)j(the)f (plan)f(of)g(the)h(proof)g(of)g(the)f(theorem)g(we)h(would)f(like)g(to) h(justify)h(the)118 1999 y(restriction)25 b Fq(d)h Fo(\025)f Fp(3)p Fu(.)31 b(If)26 b(we)e(pretend)g(that)i(the)f(particle)f (density)h(is)h(constant)f(everywhere,)e(say)i(equal)118 2107 y(to)h Fq(m)p Fu(,)f(then)810 2297 y Fp(sup)14 b Fq(spec)1132 2316 y Fr(L)1180 2297 y Fg(2)1215 2316 y Fl(\()p Fr(\027)t Fl(\))1313 2223 y Fk(\010)1366 2297 y Fq(\017)1403 2259 y Fn(\000)p Fl(1)1497 2297 y Fq(\014)5 b Fp(Av)1671 2239 y Fr(n;)1744 2212 y FA(a)p 1743 2224 34 3 v 1747 2265 a(\017)1671 2308 y Fr(z)s(;x)1791 2297 y Fq(\034)1831 2311 y Fr(z)1880 2235 y Fq(\036)1934 2249 y Fr(n;n)p 1880 2276 164 4 v 1935 2359 a Fq(n)2074 2297 y Fp(+)20 b Fq(\017)2202 2259 y Fn(\000)p Fl(2)2296 2297 y Fp(Av)2413 2312 y Fr(b)p Fn(2)p Fl(\003)2539 2326 y FA(x;)p Fg(2)p FA(a=\017)2724 2297 y Fo(L)2787 2312 y Fr(b)2821 2223 y Fk(\011)2899 2297 y Fo(\024)1471 2524 y Fq(\017)1508 2487 y Fn(\000)p Fl(1)1602 2524 y Fq(\014)5 b Fp(Av)1775 2467 y Fr(n;)1848 2440 y FA(a)p 1848 2452 34 3 v 1852 2493 a(\017)1775 2536 y Fr(z)s(;x)1895 2524 y Fq(\034)1935 2538 y Fr(z)1985 2462 y Fq(\036)2039 2476 y Fr(n;n)2149 2462 y Fp(\()p Fq(m)p Fp(\))p 1985 2504 315 4 v 2115 2587 a Fq(n)118 2703 y Fu(Since)24 b(the)i(typical)f (\003uctuations)g(\(in)h Fq(\013)p Fu(\))f(of)h(the)f(quantity)1499 2902 y Fq(\017)1536 2864 y Fn(\000)p Fl(1)1630 2902 y Fp(Av)1747 2844 y Fr(n;)1820 2817 y FA(a)p 1820 2829 34 3 v 1824 2870 a(\017)1747 2913 y Fr(z)s(;x)1867 2902 y Fq(\034)1907 2916 y Fr(z)1957 2839 y Fq(\036)2011 2853 y Fr(n;n)2120 2839 y Fp(\()p Fq(m)p Fp(\))p 1957 2881 315 4 v 2087 2964 a Fq(n)118 3111 y Fu(are)h(of)g(the)g(order)g(of)g Fq(\017)918 3047 y FA(d)p Fi(\000)p Fg(2)p 918 3060 111 3 v 958 3101 a(2)1043 3111 y Fq(C)7 b Fp(\()p Fq(a;)15 b(n)p Fp(\))p Fu(,)26 b(necessarily)g(we)g(must)g(assume)g Fq(d)h Fo(\025)f Fp(3)h Fu(since)e Fq(\017)i Fo(#)g Fp(0)g Fu(before)e Fq(a)i Fo(#)g Fp(0)118 3219 y Fu(and)e Fq(n)g Fo(")h(1)p Fu(.)118 3494 y(5.1.)45 b FE(Plan)31 b(of)f(the)g(proof)h (of)g(theorem)f(5.3.)46 b Fu(The)29 b(main)f(dif\002culty)h(in)g (proving)g(theorem)f(5.3)g(lyes)118 3602 y(in)e(the)g(fact)g(that)h (\002rst)g Fq(\017)f Fo(#)g Fp(0)h Fu(and)f(only)f(afterward)h Fq(n)g Fo(")g(1)p Fu(.)33 b(In)26 b(particular)g(there)f(is)i(no)e (hope)h(to)g(beat)118 3728 y(the)31 b(diverging)f(factor)h Fq(\017)977 3695 y Fn(\000)p Fl(1)1102 3728 y Fu(appearing)f(in)h (de\002nition)h(5.2)e(with)i(the)f(typical)g(smallness)f Fq(O)s Fp(\()p Fq(n)3438 3693 y Fn(\000)3503 3665 y FA(d)p Fg(+2)p 3503 3678 110 3 v 3542 3719 a(2)3626 3728 y Fp(\))118 3852 y Fu(of)232 3806 y Fr(\036)274 3814 y FA(n;n)p 232 3831 142 4 v 281 3883 a Fr(n)383 3852 y Fu(.)i(The)25 b(main)g(idea)h(is)g(therefore)f(\002rst)h(to)f(try)h(to)f(prove)g (that)1344 4057 y Fp(Av)1461 3999 y Fr(n;)1534 3972 y FA(a)p 1534 3984 34 3 v 1538 4025 a(\017)1461 4069 y Fr(z)s(;x)1581 4057 y Fq(\034)1621 4071 y Fr(z)1671 3995 y Fq(\036)1725 4009 y Fr(n;n)p 1671 4036 164 4 v 1725 4119 a Fq(n)1870 4057 y Fo(\031)g Fp(Av)2083 3999 y Fr(s;)2146 3972 y FA(a)p 2145 3984 34 3 v 2149 4025 a(\017)2083 4069 y Fr(z)s(;x)2193 4057 y Fq(\034)2233 4071 y Fr(z)2282 3995 y Fq(\036)2336 4009 y Fr(s;s)p 2282 4036 144 4 v 2333 4119 a Fq(s)3459 4057 y Fu(\(5.2\))118 4236 y(where)30 b(the)g(new)g(mesoscopic)f(scale)h Fq(s)j Fp(=)g Fq(s)p Fp(\()p Fq(\017)p Fp(\))e Fu(diverges)e(suf\002ciently)h(fast)i(as)e Fq(\017)k Fo(#)g Fp(0)p Fu(.)46 b(By)30 b(standard)118 4344 y(large)25 b(deviations)j(estimates)g(\(see)e(lemma)f(6.7\))h (it's)i(simple)e(to)h(verify)g(that,)h(given)e Fp(0)i Fq(<)f(\016)k Fo(\034)c Fp(1)g Fu(and)118 4452 y Fp(0)f Fq(<)f(\015)30 b(<)25 b Fp(1)p Fu(,)h(for)f(almost)h(any)f(disorder)h (con\002guration)e Fq(\013)h Fu(and)h Fq(s)f Fp(=)g Fq(O)s Fp(\()p Fq(\017)2624 4419 y Fn(\000)p Fr(\015)2723 4452 y Fp(\))1450 4622 y(sup)1435 4709 y Fr(x)p Fn(2)p Fm(T)1572 4690 y FA(d)1572 4725 y(\017)1617 4622 y Fo(j)p Fq(\034)1682 4636 y Fr(x)1741 4622 y Fq(\036)1795 4636 y Fr(s;s)1884 4622 y Fo(j)h(\024)f Fq(C)7 b(s)2146 4585 y Fn(\000)2211 4558 y FA(d)p 2210 4570 33 3 v 2211 4611 a Fg(2)2252 4585 y Fl(+)p Fr(\016)3459 4622 y Fu(\(5.3\))118 4850 y(for)25 b(any)g Fq(\017)g Fu(small)g(enough.)30 b(In)c(particular)-10 b(,)25 b(by)g(a)g(trivial)h Fq(L)2117 4817 y Fn(1)2217 4850 y Fu(estimate,)1217 5044 y Fp(Av)1334 4987 y Fr(s;)1397 4959 y FA(a)p 1396 4971 34 3 v 1400 5013 a(\017)1334 5056 y Fr(z)s(;x)1444 5044 y Fq(\034)1484 5058 y Fr(z)1534 4982 y Fq(\036)1588 4996 y Fr(s;s)p 1534 5023 144 4 v 1584 5106 a Fq(s)1712 5044 y Fo(\031)f Fp(0)117 b Fu(if)f Fq(\015)31 b(>)2403 4982 y Fp(2)p 2324 5023 204 4 v 2324 5106 a Fq(d)20 b Fp(+)g(2)2538 5044 y Fq(:)896 b Fu(\(5.4\))118 5264 y(The)22 b(above)f(simple)h(reasoning)e(suggests)i(to)g(de\002ne)f (a)h(\002rst)g(mesoscopic)f(critical)h(scale)f Fq(s)3216 5278 y Fn(1)3315 5264 y Fp(:=)26 b Fq(\017)3474 5222 y Fn(\000)3578 5195 y Fg(2)p 3538 5207 110 3 v 3538 5248 a FA(d)p Fg(+2)118 5372 y Fu(above)k(which)i(things)f(become)e (trivial.)49 b(It)31 b(is)h(important)f(to)g(outline)f(that)i(we)e (will)h Fh(not)f Fu(be)g(able)g(to)118 5480 y(prove)23 b(\(5.2\))g(with)h Fq(s)h Fo(\035)g Fq(s)1018 5494 y Fn(1)1116 5480 y Fu(but)f(only)e(with)j Fq(s)g Fp(=)j(\026)-48 b Fq(s)23 b Fu(where)j Fp(\026)-48 b Fq(s)25 b Fp(:=)g Fq(\017)2391 5447 y Fr(\016)2429 5480 y Fq(s)2472 5494 y Fn(1)2569 5480 y Fu(and)f Fp(0)i Fq(<)f(\016)k Fo(\034)c Fp(1)f Fu(can)f(be)g(taken)p eop %%Page: 27 27 27 26 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(27)118 366 y Fu(arbitrarily)26 b(small.)118 474 y(Once)f(we)f(have)i(reached)e (scale)k Fp(\026)-48 b Fq(s)24 b Fu(we)h(cannot)g(simply)g(use)g Fq(L)2262 441 y Fn(1)2362 474 y Fu(bounds)g(but)g(we)g(need)g(to)g (appeal)g(to)118 582 y(an)j(improved)f(version)h(of)g(the)g(well)f (known)g(T)-7 b(wo)29 b(Blocks)e(Estimate)i(\(see)f(proposition)h(5.9)e (below\))118 717 y(in)e(order)g(to)h(conclude)d(that)k Fp(Av)1249 659 y Fl(\026)-38 b Fr(s;)1309 632 y FA(a)p 1308 644 34 3 v 1312 685 a(\017)1246 729 y Fr(z)s(;x)1356 717 y Fq(\034)1396 731 y Fr(z)1445 671 y(\036)1490 679 y Fg(\026)-33 b FA(s;)s Fg(\026)g FA(s)p 1445 696 124 4 v 1493 748 a Fl(\026)c Fr(s)1604 717 y Fo(\031)25 b Fp(0)p Fu(.)118 856 y(W)-8 b(e)23 b(now)g(explain)f(the)h(main)g(steps) g(in)g(the)g(proof)g(of)g(\(5.2\))f(with)i Fq(s)h Fp(=)j(\026)-48 b Fq(s)o Fu(.)31 b(As)23 b(discussed)h(in)f(subsection)118 964 y(4.1,)31 b(a)f(main)f(tool)h(for)g(estimating)g(eigenvalues)e(is)j (given)e(by)h(localization)f(together)g(with)i(pertur-)118 1072 y(bation)25 b(theory)-12 b(.)31 b(However)-10 b(,)23 b(because)h(of)g(proposition)h(4.2,)f(it)h(turns)f(out)h(that)g(this)h (technique)d(can)h(be)118 1180 y(applied)i(to)f(prove)g(\(5.2\))f(only) h(if)984 1397 y Fq(\017s)1064 1360 y Fr(d)p Fl(+2)1194 1397 y Fo(k)p Fp(Av)1357 1340 y Fr(n;)1430 1313 y FA(a)p 1429 1325 34 3 v 1433 1366 a(\017)1357 1409 y Fr(z)s(;x)1477 1397 y Fq(\034)1517 1411 y Fr(z)1567 1335 y Fq(\036)1621 1349 y Fr(n;n)p 1567 1376 164 4 v 1621 1460 a Fq(n)1760 1397 y Fo(\000)20 b Fp(Av)1969 1340 y Fr(s;)2032 1313 y FA(a)p 2031 1325 34 3 v 2035 1366 a(\017)1969 1409 y Fr(z)s(;x)2079 1397 y Fq(\034)2119 1411 y Fr(z)2168 1335 y Fq(\036)2222 1349 y Fr(s;s)p 2168 1376 144 4 v 2218 1460 a Fq(s)2321 1397 y Fo(k)2366 1411 y Fn(1)2467 1397 y Fo(\024)25 b Fp(const)p Fq(;)118 1611 y Fu(that)34 b(is)g(if)f Fq(\017s)571 1578 y Fr(d)p Fl(+2)739 1611 y Fo(\024)38 b Fp(const)p Fu(.)55 b(In)33 b(particular)f(we)g(see)h (immediately)g(that)h(this)g(approach)f(cannot)f(be)118 1719 y(used)38 b(directly)f(to)h(prove)e(\(5.2\))h(for)g Fq(s)46 b Fp(=)j(\026)-49 b Fq(s)o Fu(,)41 b(but)d(only)e(up)i(to)g(a)f (new)g(critical)g(mesoscopic)g(scale)118 1849 y Fq(s)161 1863 y Fl(0)225 1849 y Fp(:=)26 b Fq(\017)384 1806 y Fn(\000)488 1779 y Fg(1)p 448 1791 110 3 v 448 1832 a FA(d)p Fg(+2)572 1849 y Fu(.)118 1977 y(Assuming)g(that)i(we)e(have)g (been)f(able)h(to)g(replace)f Fp(Av)2011 1920 y Fr(n;)2084 1893 y FA(a)p 2083 1905 34 3 v 2087 1946 a(\017)2011 1989 y Fr(z)s(;x)2131 1977 y Fq(\034)2171 1991 y Fr(z)2220 1931 y(\036)2262 1939 y FA(n;n)p 2220 1956 142 4 v 2270 2008 a Fr(n)2398 1977 y Fu(with)j Fp(Av)2719 1920 y Fr(s)2752 1929 y Fg(0)2786 1920 y Fr(;)2816 1893 y FA(a)p 2816 1905 34 3 v 2820 1946 a(\017)2719 1989 y Fr(z)s(;x)2864 1977 y Fq(\034)2904 1991 y Fr(z)2953 1926 y(\036)2995 1934 y FA(s)3024 1949 y Fg(0)3059 1934 y FA(;s)3107 1949 y Fg(0)p 2953 1956 193 4 v 3016 2008 a Fr(s)3049 2017 y Fg(0)3155 1977 y Fu(,)f(we)f(face)g(the)118 2093 y(problem)e(to)i (increase)e(the)i(mesoscopic)f(scale)f(from)h Fq(s)2029 2107 y Fl(0)2093 2093 y Fu(to)k Fp(\026)-48 b Fq(s)o Fu(.)118 2228 y(The)32 b(main)f(observation)h(now)g(is)g(that)h(the)f Fq(L)1730 2195 y Fn(1)1836 2228 y Fu(norm)f(of)h(the)f(new)g(quantity)i Fp(Av)3023 2170 y Fr(s)3056 2179 y Fg(0)3091 2170 y Fr(;)3121 2143 y FA(a)p 3120 2155 34 3 v 3124 2196 a(\017)3023 2239 y Fr(z)s(;x)3168 2228 y Fq(\034)3208 2242 y Fr(z)3257 2176 y(\036)3299 2184 y FA(s)3328 2199 y Fg(0)3363 2184 y FA(;s)3411 2199 y Fg(0)p 3257 2207 193 4 v 3320 2259 a Fr(s)3353 2268 y Fg(0)3491 2228 y Fu(is)g(at)118 2396 y(least)d(smaller)e(than)i Fq(s)902 2339 y Fn(\000)967 2311 y FA(d)p Fg(+2)p 966 2324 110 3 v 1006 2365 a(2)902 2422 y Fl(0)1119 2396 y Fu(\(see)f(\(5.3\)\))g(almost)g(surely)g (\(here)g(and)g(in)h(what)g(follows)g(we)f(deliber-)118 2510 y(ately)24 b(neglect)e(the)j(correction)e Fq(s)1262 2477 y Fr(\016)1323 2510 y Fu(appearing)h(in)g(\(5.3\)\).)31 b(This)25 b(means)f(that)h(the)g(limit)f(scale)g(beyond)118 2618 y(which)i(perturbation)g(theory)g(cannot)f(be)g(applied,)i (previously)e(equal)g(to)h Fq(s)2769 2632 y Fl(0)2808 2618 y Fu(,)g(is)h(now)e(pushed)h(up)g(to)118 2726 y(a)f(new)g(scale)g Fq(s)643 2740 y Fl(1)707 2726 y Fu(given)f(by)1148 2938 y Fq(\017s)1228 2899 y Fr(d)p Fl(+2)1228 2964 y(1)1373 2938 y Fq(s)1416 2880 y Fn(\000)1481 2852 y FA(d)p Fg(+2)p 1481 2865 V 1520 2906 a(2)1416 2964 y Fl(0)1629 2938 y Fo(\024)h Fp(const)51 b Fo(\))g Fq(s)2168 2952 y Fl(1)2232 2938 y Fp(=)25 b Fq(\017)2365 2889 y Fn(\000)2508 2862 y Fg(3)p 2430 2874 188 3 v 2430 2915 a(2\()p FA(d)p Fg(+2\))118 3109 y Fu(The)35 b(above)g(remark)f(clearly)f(suggests)j(an)e (inductive)i(scheme)e(on)h(a)g(sequence)e(of)j(length)e(scales)118 3217 y Fo(f)p Fq(s)206 3232 y Fr(k)249 3217 y Fo(g)294 3232 y Fr(k)r Fn(\025)p Fl(0)452 3217 y Fu(given)25 b(by)1190 3379 y Fq(s)1233 3393 y Fl(0)1298 3379 y Fp(:=)g Fq(\017)1456 3337 y Fn(\000)1560 3309 y Fg(1)p 1521 3321 110 3 v 1521 3363 a FA(d)p Fg(+2)1659 3379 y Fp(;)197 b Fq(s)1924 3394 y Fr(k)r Fl(+1)2082 3379 y Fp(:=)25 b Fq(\017)2240 3337 y Fn(\000)2344 3309 y Fg(1)p 2305 3321 V 2305 3363 a FA(d)p Fg(+2)2428 3317 y Fo(p)p 2504 3317 86 4 v 62 x Fq(s)2547 3394 y Fr(k)118 3532 y Fu(in)f(which)g(one)f(proves)g (recursively)-12 b(,)22 b(by)i(means)f(of)g(localization)g(on)g(scale)g Fq(s)2761 3547 y Fr(k)r Fl(+1)2917 3532 y Fu(combined)g(together)118 3640 y(with)k(perturbation)e(theory)-12 b(,)25 b(that)1038 3863 y Fp(Av)1156 3805 y Fr(s)1189 3817 y FA(k)1227 3805 y Fr(;)1257 3778 y FA(a)p 1256 3790 34 3 v 1260 3831 a(\017)1156 3875 y Fr(z)s(;x)1304 3863 y Fq(\034)1344 3877 y Fr(z)1393 3801 y Fq(\036)1447 3815 y Fr(s)1480 3827 y FA(k)1518 3815 y Fr(;s)1571 3827 y FA(k)p 1393 3842 220 4 v 1460 3925 a Fq(s)1503 3940 y Fr(k)1643 3863 y Fo(\000)20 b Fp(Av)1851 3805 y Fr(s)1884 3817 y FA(k)q Fg(+1)1999 3805 y Fr(;)2029 3778 y FA(a)p 2029 3790 34 3 v 2033 3831 a(\017)1851 3875 y Fr(z)s(;x)2076 3863 y Fq(\034)2116 3877 y Fr(z)2166 3795 y Fq(\036)2220 3809 y Fr(s)2253 3821 y FA(k)q Fg(+1)2368 3809 y Fr(;s)2421 3821 y FA(k)q Fg(+1)p 2166 3842 374 4 v 2265 3925 a Fq(s)2308 3940 y Fr(k)r Fl(+1)2574 3863 y Fo(\031)25 b Fp(0)p Fq(:)118 4121 y Fu(Notice)30 b(that)i Fp(lim)718 4136 y Fr(k)r Fn(!1)917 4121 y Fq(s)960 4136 y Fr(k)1036 4121 y Fp(=)h Fq(s)1183 4135 y Fn(1)1288 4121 y Fu(where)c Fq(s)1606 4135 y Fn(1)1714 4121 y Fp(=)k Fq(\017)1855 4078 y Fn(\000)1960 4051 y Fg(2)p 1920 4063 110 3 v 1920 4104 a FA(d)p Fg(+2)2074 4121 y Fu(represents)c(the)i(limiting)f(scale)g(introduced)118 4229 y(at)c(the)f(beginning)f(of)i(this)g(section.)118 4337 y(A)f(large)f(but)i(\002nite)f(number)f(of)h(steps)h(of)g(the)f (inductive)h(scheme)e(proves)h(that)1266 4554 y Fp(Av)1383 4496 y Fr(n;)1456 4469 y FA(a)p 1455 4481 34 3 v 1459 4523 a(\017)1383 4566 y Fr(z)s(;x)1503 4554 y Fq(\034)1543 4568 y Fr(z)1593 4492 y Fq(\036)1647 4506 y Fr(n;n)p 1593 4533 164 4 v 1647 4616 a Fq(n)1786 4554 y Fo(\000)20 b Fp(Av)1997 4496 y Fl(\026)-37 b Fr(s)o(;)2057 4469 y FA(a)p 2057 4481 34 3 v 2061 4523 a(\017)1995 4566 y Fr(z)s(;x)2105 4554 y Fq(\034)2145 4568 y Fr(z)2194 4492 y Fq(\036)2251 4506 y Fl(\026)f Fr(s;)s Fl(\026)g Fr(s)p 2194 4533 144 4 v 2248 4616 a Fp(\026)-49 b Fq(s)2373 4554 y Fo(\031)24 b Fp(0)118 4767 y Fu(where,)h(as)h(before,)j Fp(\026)-49 b Fq(s)26 b Fp(=)f Fq(\017)1026 4734 y Fr(\016)1064 4767 y Fq(s)1107 4781 y Fn(1)1181 4767 y Fu(.)33 b(W)-8 b(e)26 b(remark)e(that)j(for)e(this)i(part)f(of)g(the)g(proof)f(we)h (only)e(need)h Fq(d)h Fo(\025)g Fp(2)p Fu(,)118 4875 y(while)21 b(we)f(will)h(assume)f Fq(d)26 b Fo(\025)f Fp(3)c Fu(when)f(proving)g(the)h(improved)f(version)h(of)g(the)g(T)-7 b(wo)21 b(Blocks)f(estimate)118 4983 y(\(see)25 b(proposition)h(5.9\).) 118 5197 y(5.2.)45 b FE(Preliminary)28 b(tools.)46 b Fu(In)27 b(this)h(section)f(we)g(collect)e(some)h(general)f(techniques) i(that)h(are)e(com-)118 5305 y(mon)d(to)h(all)f(the)h(steps)g(of)g(the) g(proof)f(of)h(theorem)f(5.3.)30 b(W)-8 b(e)24 b(recall)e(that)j Fp(\003)2671 5272 y Fr(e)2671 5332 y(z)s(;`)2783 5305 y Fu(denotes)e(the)h(translated)118 5421 y(by)h Fq(z)30 b Fu(of)25 b(the)h(box)f Fp(\003)796 5388 y Fr(e)796 5448 y(`)833 5421 y Fu(.)p eop %%Page: 28 28 28 27 bop 118 45 a Fj(28)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 367 y FE(L)r(emma)38 b(5.4.)48 b Fh(Let)36 b Fq(`)860 381 y Fl(0)942 367 y Fq(<)43 b(`)1094 381 y Fl(1)1177 367 y Fq(<)g(`)1329 381 y Fl(2)1404 367 y Fh(be)36 b(odd)f(integers)f(such)i(that)2445 330 y Fr(`)2474 339 y Fg(2)p 2445 346 64 4 v 2445 398 a Fr(`)2474 407 y Fg(0)2562 367 y Fo(2)43 b Fs(N)6 b Fh(.)70 b(Let)36 b Fq(\027)42 b Fh(be)36 b(an)f(arbitrary)118 483 y(canonical)23 b(measure)i(on)f(the)h(cube)g Fp(\003)1377 498 y Fr(`)1406 507 y Fg(2)1470 483 y Fh(and)f(let)h Fq(f)35 b Fh(be)25 b(a)g(function)f(with)h(support)g(in)g Fp(\003)3040 450 y Fr(e)3040 510 y(`)3069 519 y Fg(1)3108 483 y Fh(.)32 b(Then)502 658 y Fp(sup)14 b Fq(spec)824 678 y Fr(L)872 659 y Fg(2)907 678 y Fl(\()p Fr(\027)t Fl(\))1005 658 y Fo(f)p Fp(Av)1168 614 y Fr(`)1197 623 y Fg(0)1231 614 y Fr(;`)1280 623 y Fg(2)1168 684 y Fr(z)s(;)p Fl(0)1318 658 y Fq(\034)1358 672 y Fr(z)1398 658 y Fq(f)29 b Fp(+)20 b(Av)1680 673 y Fr(b)p Fn(2)p Fl(\003)1806 685 y FA(`)1832 700 y Fg(2)1875 658 y Fo(L)1938 673 y Fr(b)1972 658 y Fo(g)26 b(\024)1452 825 y Fp(Av)1569 781 y Fr(`)1598 790 y Fg(0)1633 781 y Fr(;`)1682 790 y Fg(2)1569 851 y Fr(z)s(;)p Fl(0)1735 825 y Fp(sup)1773 904 y Fr(\027)1812 885 y Fi(0)1902 825 y Fp(sup)14 b Fq(spec)2224 845 y Fr(L)2272 826 y Fg(2)2307 845 y Fl(\()p Fr(\027)2373 826 y Fi(0)2396 845 y Fl(\))2427 825 y Fo(f)p Fq(\034)2512 839 y Fr(z)2552 825 y Fq(f)30 b Fp(+)20 b Fq(c)p Fp(Av)2874 840 y Fr(b)p Fn(2)p Fl(\003)3000 817 y FA(e)3000 863 y(z)r(;`)3077 878 y Fg(1)3135 825 y Fo(L)3198 840 y Fr(b)3232 825 y Fo(g)118 1031 y Fh(where)25 b Fq(\027)423 998 y Fn(0)471 1031 y Fh(varies)g(in)f Fo(M)930 1045 y Fr(\013)980 1031 y Fp(\(\003)1078 998 y Fr(e)1078 1059 y(z)s(;`)1163 1068 y Fg(1)1201 1031 y Fp(\))i Fh(and)e Fq(c)i Fh(is)f(a)g(suitable)g(constant.)118 1199 y(Proof)-8 b(.)42 b Fu(It)26 b(is)g(suf\002cient)f(to)h(observe)e (that)1184 1358 y Fp(Av)1301 1373 y Fr(b)p Fn(2)p Fl(\003)1427 1385 y FA(`)1453 1400 y Fg(2)1496 1358 y Fo(L)1559 1373 y Fr(b)1619 1358 y Fo(\024)h Fq(c)15 b Fp(Av)1887 1313 y Fr(`)1916 1322 y Fg(0)1950 1313 y Fr(;`)1999 1322 y Fg(2)1887 1384 y Fr(z)s(;)p Fl(0)2037 1284 y Fk(\000)2079 1358 y Fp(Av)2196 1373 y Fr(b)p Fn(2)p Fl(\003)2322 1350 y FA(e)2322 1396 y(z)r(;`)2399 1411 y Fg(1)2457 1358 y Fo(L)2520 1373 y Fr(b)2554 1284 y Fk(\001)118 1526 y Fu(and)25 b(localize)f(in)i(the)f(box)g Fp(\003)1103 1493 y Fr(e)1103 1554 y(z)s(;`)1188 1563 y Fg(1)1226 1526 y Fu(.)2340 b Fe(\003)218 1709 y Fu(A)n(t)26 b(this)g(point,)g(it) g(is)g(convenient)e(to)h(observe)g(the)g(factorization)h(property)e(of) i(the)f(average)f Fp(Av)3562 1673 y Fr(`;s)3562 1732 y(z)s(;x)118 1840 y Fu(de\002ned)h(in)h(\(2.2\):)k(given)24 b(odd)i(integers)f Fq(`;)15 b(`)1677 1807 y Fn(0)1701 1840 y Fq(;)g(L)25 b Fu(such)g(that)2227 1804 y Fr(`)2256 1780 y Fi(0)p 2227 1819 52 4 v 2238 1871 a Fr(`)2288 1840 y Fq(;)2340 1804 y Fr(L)p 2338 1819 V 2338 1871 a(`)2367 1852 y Fi(0)2425 1840 y Fo(2)f Fs(N)7 b Fu(,)32 b(then)1338 2007 y Fp(Av)1456 1970 y Fr(`;L)1456 2030 y(z)s(;x)1556 2007 y Fq(f)1601 2021 y Fr(z)1666 2007 y Fp(=)25 b(Av)1879 1970 y Fr(`)1908 1946 y Fi(0)1930 1970 y Fr(;L)1879 2030 y(z)s(;x)2002 1934 y Fk(\000)2044 2007 y Fp(Av)2161 1970 y Fr(`;`)2239 1946 y Fi(0)2161 2030 y Fr(w)r(;z)2273 2007 y Fq(f)2318 2021 y Fr(w)2374 1934 y Fk(\001)2416 2007 y Fq(:)1018 b Fu(\(5.5\))118 2169 y FE(Proposition)39 b(5.5.)47 b Fh(Let)37 b Fq(d)44 b Fo(\025)f Fp(2)p Fh(,)d Fp(0)k Fq(<)g(\015)49 b Fo(\024)43 b Fq(\015)1780 2136 y Fn(0)1847 2169 y Fq(<)h Fp(1)36 b Fh(and)g Fq(\015)2278 2136 y Fn(0)2345 2169 y Fq(<)2515 2133 y Fl(1)p 2469 2148 127 4 v 2469 2201 a Fr(d)p Fl(+2)2633 2169 y Fp(+)2741 2128 y Fr(\015)p 2741 2148 41 4 v 2743 2201 a Fl(2)2792 2169 y Fh(.)64 b(If)37 b(either)f Fq(`)43 b Fp(=)h Fq(n)36 b Fh(and)118 2318 y Fq(s)25 b Fp(=)g Fq(O)354 2244 y Fk(\000)395 2318 y Fq(\017)432 2275 y Fn(\000)537 2248 y Fg(1)p 497 2260 110 3 v 497 2301 a FA(d)p Fg(+2)620 2244 y Fk(\001)687 2318 y Fh(or)g Fq(`)h Fp(=)e Fq(O)s Fp(\()p Fq(\017)1097 2285 y Fn(\000)p Fr(\015)1197 2318 y Fp(\))i Fh(and)e Fq(s)g Fp(=)h Fq(O)s Fp(\()p Fq(\017)1736 2285 y Fn(\000)p Fr(\015)1831 2261 y Fi(0)1858 2318 y Fp(\))p Fh(,)h(then)1333 2524 y Fp(Av)1450 2466 y Fr(`;)1509 2439 y FA(a)p 1509 2451 34 3 v 1513 2492 a(\017)1450 2536 y Fr(z)s(;x)1556 2524 y Fq(\034)1596 2538 y Fr(z)1646 2461 y Fq(\036)1700 2476 y Fr(`;s)p 1646 2503 140 4 v 1696 2586 a Fq(`)1820 2524 y Fo(\031)f Fp(Av)2034 2466 y Fr(s;)2097 2439 y FA(a)p 2096 2451 34 3 v 2100 2492 a(\017)2034 2536 y Fr(z)s(;x)2144 2524 y Fq(\034)2184 2538 y Fr(z)2233 2462 y Fq(\036)2287 2476 y Fr(s;)p Fl(2)p Fr(s)p 2233 2503 179 4 v 2301 2586 a Fq(s)2422 2524 y(:)118 2714 y Fh(Proof)-8 b(.)42 b Fu(By)25 b(the)g(factorization)h(property)f(\(5.5\),)g(we)g(have)684 2908 y Fp(Av)801 2851 y Fr(`;)860 2824 y FA(a)p 859 2836 34 3 v 863 2877 a(\017)801 2920 y Fr(z)s(;x)907 2908 y Fq(\034)947 2922 y Fr(z)997 2845 y Fq(\036)1051 2860 y Fr(`;s)p 997 2887 140 4 v 1047 2970 a Fq(`)1166 2908 y Fo(\000)20 b Fp(Av)1375 2851 y Fr(s;)1438 2824 y FA(a)p 1437 2836 34 3 v 1441 2877 a(\017)1375 2920 y Fr(z)s(;x)1484 2908 y Fq(\034)1524 2922 y Fr(z)1574 2846 y Fq(\036)1628 2860 y Fr(s;)p Fl(2)p Fr(s)p 1574 2887 179 4 v 1642 2970 a Fq(s)1788 2908 y Fp(=)25 b(Av)2001 2851 y Fr(s;)2064 2824 y FA(a)p 2063 2836 34 3 v 2067 2877 a(\017)2001 2920 y Fr(z)s(;x)2111 2807 y Fk(h)2154 2908 y Fp(Av)2271 2870 y Fr(`;s)2271 2931 y(w)r(;z)2383 2908 y Fq(\034)2423 2922 y Fr(w)2489 2845 y Fq(\036)2543 2860 y Fr(`;s)p 2489 2887 140 4 v 2540 2970 a Fq(`)2659 2908 y Fo(\000)20 b Fq(\034)2790 2922 y Fr(z)2839 2846 y Fq(\036)2893 2860 y Fr(s;)p Fl(2)p Fr(s)p 2839 2887 179 4 v 2907 2970 a Fq(s)3028 2908 y(:)3053 2807 y Fk(i)118 3085 y Fu(Therefore,)39 b(by)e(lemma)e(5.4,)k(it)e(is)h(enough)d(to)i(prove)f(that)i(for)f(any) f Fq(T)57 b(>)44 b Fp(0)37 b Fu(and)g(for)f(almost)h(all)118 3193 y(disorder)26 b(con\002guration)e Fq(\013)1098 3341 y Fp(lim)15 b(sup)1039 3420 y Fr(n)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)1449 3341 y Fp(Av)1567 3363 y Fr(x)p Fn(2)p Fm(T)1704 3344 y FA(d)1704 3380 y(\017)1779 3341 y Fp(sup)1753 3423 y Fn(j)p Fr(\014)s Fn(j\024)p Fr(T)2116 3341 y Fp(sup)1976 3423 y Fr(\027)t Fn(2M)2146 3431 y FA(\013)2190 3423 y Fl(\(\003)2266 3401 y FA(e)2266 3445 y(x;)p Fg(2)p FA(s)2384 3423 y Fl(\))2427 3341 y Fq(f)2472 3355 y Fr(x;\027)2599 3341 y Fo(\024)25 b Fp(0)719 b Fu(\(5.6\))118 3579 y(where)498 3753 y Fq(f)543 3767 y Fr(x;\027)671 3753 y Fp(:=)25 b(sup)14 b Fq(spec)1114 3773 y Fr(L)1162 3754 y Fg(2)1196 3773 y Fl(\()p Fr(\027)t Fl(\))1294 3653 y Fk(n)1355 3753 y Fq(\017)1392 3716 y Fn(\000)p Fl(1)1486 3753 y Fq(\014)1542 3653 y Fk(h)1585 3753 y Fp(Av)1703 3716 y Fr(`;s)1703 3776 y(z)s(;x)1802 3753 y Fq(\034)1842 3767 y Fr(z)1891 3690 y Fq(\036)1945 3705 y Fr(`;s)p 1891 3733 140 4 v 1942 3816 a Fq(`)2061 3753 y Fo(\000)20 b Fq(\034)2192 3767 y Fr(x)2245 3691 y Fq(\036)2299 3705 y Fr(s;)p Fl(2)p Fr(s)p 2245 3733 179 4 v 2313 3816 a Fq(s)2434 3653 y Fk(i)2497 3753 y Fp(+)g Fq(c\017)2664 3716 y Fn(\000)p Fl(2)2758 3753 y Fp(Av)2876 3768 y Fr(b)p Fn(2)p Fl(\003)3002 3745 y FA(e)3002 3789 y(x;)p Fg(2)p FA(s)3124 3753 y Fo(L)3187 3768 y Fr(b)3221 3653 y Fk(o)118 3959 y Fu(for)25 b(a)h(suitable)f(constant)g Fq(c)p Fu(.)32 b(Notice)25 b(that)i Fq(\034)1623 3973 y Fr(x)1676 3912 y(\036)1718 3921 y FA(s;)p Fg(2)p FA(s)p 1676 3938 154 4 v 1737 3991 a Fr(s)1865 3959 y Fp(=)e Fq(\027)2012 3886 y Fk(\000)2054 3959 y Fp(Av)2171 3923 y Fr(`;s)2171 3982 y(z)s(;x)2270 3959 y Fq(\034)2310 3973 y Fr(z)2360 3910 y(\036)2402 3922 y FA(`;s)p 2360 3938 120 4 v 2405 3991 a Fr(`)2490 3886 y Fk(\001)2556 3959 y Fq(\027)31 b Fu(a.s..)118 4175 y(Because)24 b(of)i(lemma)e(6.7,)h(given)f Fp(0)i Fq(<)f(\016)k Fo(\034)c Fp(1)p Fu(,)h(for)f(almost)g(all)g Fq(\013)h Fu(and)f Fq(\017)g Fu(small)g(enough)974 4402 y Fp(sup)959 4488 y Fr(x)p Fn(2)p Fm(T)1096 4470 y FA(d)1096 4505 y(\017)1126 4325 y Fk(\015)1126 4379 y(\015)1177 4402 y Fq(\034)1217 4416 y Fr(x)1270 4339 y Fq(\036)1324 4354 y Fr(`;s)p 1270 4381 140 4 v 1321 4465 a Fq(`)1420 4325 y Fk(\015)1420 4379 y(\015)1470 4438 y Fn(1)1570 4402 y Fo(\024)1666 4247 y Fk(\()1739 4341 y Fq(`)1777 4308 y Fn(\000)p Fl(1)2293 4341 y Fu(if)h Fq(`)f Fp(=)g Fq(n)1739 4470 y(`)1777 4437 y Fn(\000)p Fl(\()p Fr(d)p Fl(+2\))p Fr(=)p Fl(2+)p Fr(\016)2293 4470 y Fu(if)h Fq(`)f Fp(=)g Fq(O)s Fp(\()p Fq(\017)2676 4437 y Fn(\000)p Fr(\015)2775 4470 y Fp(\))118 4642 y Fu(Thanks)30 b(to)g(the)f(above)g (bound)g(and)h(to)g(the)f(choice)g Fq(\015)2019 4609 y Fn(0)2075 4642 y Fq(<)2233 4606 y Fl(1)p 2187 4621 127 4 v 2187 4674 a Fr(d)p Fl(+2)2347 4642 y Fp(+)2450 4601 y Fr(\015)p 2450 4621 41 4 v 2452 4674 a Fl(2)2500 4642 y Fu(,)i(for)e(almost)h(all)f Fq(\013)g Fu(and)h Fq(\017)f Fu(small)118 4756 y(enough,)24 b(we)h(can)g(apply)g (proposition)i(4.2)d(together)h(with)i(lemma)c(A.5)j(to)f(get)1207 4908 y Fp(sup)1057 4990 y Fr(\027)t Fn(2M)1227 4998 y FA(\013)1271 4990 y Fl(\(\003)1347 4967 y FA(e)1347 5011 y(x;)p Fg(2)p FA(s)1465 4990 y Fl(\))1508 4908 y Fq(f)1553 4922 y Fr(x;\027)1680 4908 y Fo(\024)g Fq(c)15 b(T)1896 4870 y Fl(2)1936 4908 y Fq(`)1974 4870 y Fn(\000)p Fl(2)2068 4908 y Fq(s)2111 4870 y Fr(d)p Fl(+2)2256 4908 y Fp(sup)2294 4980 y Fr(m)2408 4908 y Fq(F)e Fp(\()p Fq(x;)i(m)p Fp(\))738 b Fu(\(5.7\))118 5145 y(where)25 b Fq(m)g Fu(varies)g(among)f(all)h (possible)h(particle)f(densities)h(in)g Fp(\003)2372 5112 y Fr(e)2372 5170 y(x;)p Fl(2)p Fr(s)2528 5145 y Fu(and)1207 5314 y Fq(F)13 b Fp(\()p Fq(x;)i(m)p Fp(\))27 b(:=)e(V)-8 b(ar)1809 5340 y Fr(\026)1851 5320 y FA(\025)1888 5328 y(x)1927 5320 y Fg(\()p FA(m)p Fg(\))2038 5240 y Fk(\000)2080 5314 y Fp(Av)2197 5276 y Fr(`;s)2197 5337 y(z)s(;x)2312 5314 y Fq(\034)2352 5328 y Fr(z)2391 5314 y Fq(\036)2445 5329 y Fr(`;s)2531 5240 y Fk(\001)118 5469 y Fu(and)25 b Fq(\025)345 5483 y Fr(x)389 5469 y Fp(\()p Fq(m)p Fp(\))h(:=)f Fq(\025)739 5483 y Fl(\003)788 5460 y FA(e)788 5504 y(x;)p Fg(2)p FA(s)910 5469 y Fp(\()p Fq(m)p Fp(\))p Fu(.)p eop %%Page: 29 29 29 28 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(29)118 366 y Fu(W)-8 b(e)26 b(claim)e(that)j(for)e(almost)g(all)g Fq(\013)h Fu(and)f Fq(\017)g Fu(small)g(enough)1344 557 y Fp(sup)1329 644 y Fr(x)p Fn(2)p Fm(T)1466 625 y FA(d)1466 660 y(\017)1512 557 y Fp(sup)1549 629 y Fr(m)1664 557 y Fq(F)13 b Fp(\()p Fq(x;)i(m)p Fp(\))26 b Fo(\024)f Fq(c)15 b(s)2196 520 y Fn(\000)p Fl(2)p Fr(d)p Fl(+2)p Fr(\016)3459 557 y Fu(\(5.8\))118 824 y(thus)24 b(proving)f(the)h (proposition)h(since)e Fq(d)i Fo(\025)g Fp(2)p Fu(.)32 b(The)23 b(proof)h(of)f(\(5.8\))g(follows)g(exactly)g(the)h(same)f (lines)118 931 y(of)28 b(the)f(proof)g(of)h(proposition)g(6.5)f(with)h (the)g(main)f(difference)f(that)i(it)g(is)g(necessery)e(to)i(use)f (lemma)118 1039 y(6.4)e(in)g(order)g(to)g(control)g(the)g(empirical)g (chemical)f(potetials)i(\(see)f(also)g(section)h(4.7)e(in)i([15)o(]\)) 3591 1147 y Fe(\003)118 1359 y FE(Proposition)34 b(5.6.)46 b Fh(Let)32 b Fq(d)37 b Fo(\025)f Fp(2)p Fh(,)1339 1323 y Fl(1)p 1294 1338 127 4 v 1294 1390 a Fr(d)p Fl(+2)1467 1359 y Fo(\024)g Fq(\015)42 b Fo(\024)36 b Fq(\015)1822 1326 y Fn(0)1882 1359 y Fq(<)h Fp(1)32 b Fh(and)f Fq(\015)2297 1326 y Fn(0)2357 1359 y Fq(<)2520 1323 y Fl(1)p 2474 1338 V 2474 1390 a Fr(d)p Fl(+2)2636 1359 y Fp(+)2741 1318 y Fr(\015)p 2741 1338 41 4 v 2743 1390 a Fl(2)2791 1359 y Fh(.)52 b(Set)32 b Fq(s)k Fp(=)h Fq(O)s Fp(\()p Fq(\017)3349 1326 y Fn(\000)p Fr(\015)3448 1359 y Fp(\))c Fh(and)118 1490 y Fq(s)161 1457 y Fn(0)209 1490 y Fp(=)25 b Fq(O)s Fp(\()p Fq(\017)449 1457 y Fn(\000)p Fr(\015)544 1433 y Fi(0)571 1490 y Fp(\))p Fh(.)32 b(Then)1335 1688 y Fp(Av)1453 1631 y Fr(s;)1516 1604 y FA(a)p 1515 1616 34 3 v 1519 1657 a(\017)1453 1700 y Fr(z)s(;x)1563 1688 y Fq(\034)1603 1702 y Fr(z)1652 1626 y Fq(\036)1706 1640 y Fr(s;s)p 1652 1668 144 4 v 1702 1751 a Fq(s)1831 1688 y Fo(\031)25 b Fp(Av)2044 1631 y Fr(s;)2107 1604 y FA(a)p 2106 1616 34 3 v 2110 1657 a(\017)2044 1700 y Fr(z)s(;x)2154 1688 y Fq(\034)2194 1702 y Fr(z)2243 1625 y Fq(\036)2297 1640 y Fr(s;s)2383 1621 y Fi(0)p 2243 1668 166 4 v 2305 1751 a Fq(s)2419 1688 y(:)118 1930 y Fh(Proof)-8 b(.)42 b Fu(By)26 b(lemma)g Fp(5)p Fq(:)p Fp(4)i Fu(it)g(is)g(enough)d(to)j (prove)e(that)i(for)f(any)f Fq(T)41 b(>)28 b Fp(0)f Fu(and)g(for)g (almost)g(any)g(disorder)118 2038 y(con\002guration)d Fq(\013)1141 2187 y Fp(lim)15 b(sup)1166 2266 y Fr(a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)1434 2187 y Fp(Av)1552 2209 y Fr(x)p Fn(2)p Fm(T)1689 2190 y FA(d)1689 2226 y(\017)1764 2187 y Fp(sup)1738 2270 y Fn(j)p Fr(\014)s Fn(j\024)p Fr(T)2086 2187 y Fp(sup)1941 2270 y Fr(\027)t Fn(2M)2111 2278 y FA(\013)2155 2270 y Fl(\(\003)2231 2247 y FA(e)2231 2301 y(x;s)2314 2287 y Fi(0)2341 2270 y Fl(\))2384 2187 y Fq(f)2429 2201 y Fr(x)2497 2187 y Fo(\024)25 b Fp(0)118 2455 y Fu(where)684 2632 y Fq(f)729 2646 y Fr(x)798 2632 y Fp(:=)g(sup)14 b Fq(spec)1241 2651 y Fr(L)1289 2632 y Fg(2)1324 2651 y Fl(\()p Fr(\027)t Fl(\))1422 2531 y Fk(n)1482 2632 y Fq(\017)1519 2594 y Fn(\000)p Fl(1)1614 2632 y Fq(\014)5 b(\034)1710 2646 y Fr(x)1753 2531 y Fk(h)1806 2570 y Fq(\036)1860 2584 y Fr(s;s)p 1806 2611 144 4 v 1857 2694 a Fq(s)1980 2632 y Fo(\000)2081 2568 y Fq(\036)2135 2583 y Fr(s;s)2221 2564 y Fi(0)p 2081 2611 166 4 v 2142 2694 a Fq(s)2256 2531 y Fk(i)2319 2632 y Fp(+)20 b Fq(c\017)2486 2594 y Fn(\000)p Fl(2)2581 2632 y Fp(Av)2698 2647 y Fr(b)p Fn(2)p Fl(\003)2824 2624 y FA(e)2824 2677 y(x;s)2907 2663 y Fi(0)2938 2632 y Fo(L)3001 2647 y Fr(b)3035 2531 y Fk(o)118 2830 y Fu(for)25 b(a)h(suitable)f(constant)g Fq(c)p Fu(.)32 b(Notice)25 b(that)i Fq(\027)6 b Fp(\()p Fq(\036)1723 2844 y Fr(s;s)1812 2830 y Fp(\))26 b(=)f Fq(\036)2023 2846 y Fr(s;s)2109 2827 y Fi(0)2159 2830 y Fq(\027)31 b Fu(a.s..)118 3046 y(Because)24 b(of)i(lemma)e(6.7,)h (given)f Fp(0)i Fq(<)f(\016)k Fo(\034)c Fp(1)p Fu(,)h(for)f(almost)g (all)g Fq(\013)h Fu(and)f Fq(\017)g Fu(enough)f(small)1316 3278 y Fp(sup)1301 3365 y Fr(x)p Fn(2)p Fm(T)1438 3346 y FA(d)1438 3381 y(\017)1468 3201 y Fk(\015)1468 3256 y(\015)1518 3278 y Fq(\034)1558 3292 y Fr(x)1612 3216 y Fq(\036)1666 3230 y Fr(s;s)p 1612 3257 144 4 v 1662 3341 a Fq(s)1765 3201 y Fk(\015)1765 3256 y(\015)1816 3314 y Fn(1)1916 3278 y Fo(\024)g Fq(s)2054 3241 y Fn(\000)p Fl(\()p Fr(d)p Fl(+2\))p Fr(=)p Fl(2+)p Fr(\016)2454 3278 y Fq(:)118 3556 y Fu(Thanks)k(to)f(the)h(above)f(bound)g(and)g(to) g(the)h(choice)f Fq(\015)2000 3523 y Fn(0)2051 3556 y Fq(<)2206 3520 y Fl(1)p 2161 3535 127 4 v 2161 3587 a Fr(d)p Fl(+2)2318 3556 y Fp(+)2421 3515 y Fr(\015)p 2421 3535 41 4 v 2423 3587 a Fl(2)2471 3556 y Fu(,)h(for)f(almost)g(any)g Fq(\013)h Fu(and)f Fq(\017)g Fu(small)118 3670 y(enough,)d(we)h(can)g (apply)g(proposition)i(4.2)d(together)h(with)i(lemma)c(A.5)j(to)f(get) 1201 3861 y Fp(sup)1056 3943 y Fr(\027)t Fn(2M)1226 3951 y FA(\013)1270 3943 y Fl(\(\003)1346 3920 y FA(e)1346 3974 y(x;s)1429 3960 y Fi(0)1456 3943 y Fl(\))1498 3861 y Fq(f)1543 3875 y Fr(x)1612 3861 y Fo(\024)g Fq(T)1774 3823 y Fl(2)1813 3861 y Fq(s)1856 3823 y Fn(\000)p Fl(2)1950 3861 y Fp(\()p Fq(s)2028 3823 y Fn(0)2051 3861 y Fp(\))2086 3823 y Fr(d)p Fl(+2)2232 3861 y Fp(sup)2270 3933 y Fr(m)2385 3861 y Fq(F)13 b Fp(\()p Fq(x;)i(m)p Fp(\))p Fq(;)736 b Fu(\(5.9\))118 4147 y(where)25 b Fq(m)g Fu(varies)g(among)f(all)h (possible)h(particle)f(densities)h(in)g Fp(\003)2372 4114 y Fr(e)2372 4175 y(x;s)2465 4157 y Fi(0)2490 4147 y Fu(,)1322 4353 y Fq(F)13 b Fp(\()p Fq(x;)i(m)p Fp(\))27 b(:=)e(V)-8 b(ar)1925 4378 y Fr(\026)1967 4359 y FA(\025)2004 4367 y(x)2043 4359 y Fg(\()p FA(m)p Fg(\))2153 4279 y Fk(\000)2195 4353 y Fq(\034)2235 4367 y Fr(x)2279 4353 y Fq(\036)2333 4367 y Fr(s;s)2422 4353 y Fp(\))118 4546 y Fu(and)25 b(now)g Fq(\025)541 4560 y Fr(x)585 4546 y Fp(\()p Fq(m)p Fp(\))h(=)f Fq(\025)910 4560 y Fl(\003)959 4538 y FA(e)959 4592 y(x;s)1042 4578 y Fi(0)1073 4546 y Fp(\()p Fq(m)p Fp(\))p Fu(.)118 4677 y(W)-8 b(e)26 b(claim)e(that)j(for)e(almost)g(all)g Fq(\013)h Fu(and)f Fq(\017)g Fu(small)g(enough)1344 4868 y Fp(sup)1329 4954 y Fr(x)p Fn(2)p Fm(T)1466 4935 y FA(d)1466 4970 y(\017)1512 4868 y Fp(sup)1549 4940 y Fr(m)1664 4868 y Fq(F)13 b Fp(\()p Fq(x;)i(m)p Fp(\))26 b Fo(\024)f Fq(c)15 b(s)2196 4830 y Fn(\000)p Fl(2)p Fr(d)p Fl(+2)p Fr(\016)3409 4868 y Fu(\(5.10\))118 5136 y(thus)39 b(proving)f(the)g(proposition)i (because)d(of)i(the)f(constraint)h(on)e Fq(\015)5 b(;)15 b(\015)2671 5103 y Fn(0)2695 5136 y Fq(;)g(d)p Fu(.)72 b(The)38 b(proof)g(of)h(\(5.8\),)118 5244 y(requiring)32 b Fq(d)38 b Fo(\025)g Fp(2)p Fu(,)e(follows)c(exactly)g(the)h(same)g (lines)g(of)g(the)g(proof)g(of)g(proposition)h(6.5)e(with)i(the)118 5352 y(main)i(difference)f(that)i(it)f(is)h(necessery)d(to)j(use)e (lemma)g(6.4)g(in)h(order)f(to)h(control)f(the)h(empirical)118 5460 y(chemical)24 b(potetials)i(\(see)f(also)h(section)f(4.6)g(in)g ([15)o(]\))1590 b Fe(\003)p eop %%Page: 30 30 30 29 bop 118 45 a Fj(30)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(5.3.)45 b FE(F)m(rom)25 b(scale)f Fq(n)g FE(to)h(scale)f Fq(s)1256 380 y Fl(0)1295 366 y FE(.)46 b Fu(Here)22 b(we)h(show)h(how)g(to)g (replace)d(the)j(starting)g(scale)f Fq(n)f Fu(with)j(our)118 497 y(\002rst)h(mesoscopic)f(scale)f(increasing)h(with)h Fq(\017)p Fu(,)g Fq(s)1752 511 y Fl(0)1816 497 y Fp(=)f Fq(O)1984 423 y Fk(\000)2026 497 y Fq(\017)2063 454 y Fn(\000)2167 427 y Fg(1)p 2127 439 110 3 v 2127 480 a FA(d)p Fg(+2)2250 423 y Fk(\001)2292 497 y Fu(.)118 628 y FE(Proposition)i(5.7.)42 b Fh(Let)25 b Fq(d)h Fo(\025)e Fp(3)p Fh(.)32 b(Then)1341 826 y Fp(Av)1459 769 y Fr(n;)1532 742 y FA(a)p 1531 754 34 3 v 1535 795 a(\017)1459 838 y Fr(z)s(;x)1579 826 y Fq(\034)1619 840 y Fr(z)1658 725 y Fk(h)1711 764 y Fq(\036)1765 778 y Fr(n;n)p 1711 805 164 4 v 1766 888 a Fq(n)1905 826 y Fo(\000)2006 764 y Fq(\036)2060 778 y Fr(n;s)2156 787 y Fg(0)p 2006 805 188 4 v 2072 888 a Fq(n)2204 725 y Fk(i)2272 826 y Fo(\031)25 b Fp(0)p Fq(:)971 b Fu(\(5.11\))118 1051 y Fh(Proof)-8 b(.)42 b Fu(W)n(ithout)27 b(loss)e(of)h(generality)-12 b(,)24 b(we)h(assume)g(that)2087 1014 y Fr(s)2120 1023 y Fg(0)p 2087 1030 68 4 v 2099 1082 a Fr(n)2190 1051 y Fo(2)g Fs(N)38 b Fu(and)25 b(similarly)g(for)3055 1006 y Fr(a=\017)p 3055 1030 102 4 v 3085 1082 a(n)3167 1051 y Fu(.)118 1197 y(By)37 b(the)g(de\002nition)g(of)g Fp(Av)1068 1139 y Fr(n;)1141 1112 y FA(a)p 1141 1124 34 3 v 1145 1165 a(\017)1068 1209 y Fr(z)s(;x)1225 1197 y Fu(and)g(setting)g Fq(B)49 b Fp(=)44 b Fq(Q)2019 1215 y Fr(a=\017)2152 1197 y Fo(\\)28 b Fq(n)15 b Fs(Z)2376 1164 y Fr(d)2412 1197 y Fu(,)40 b(in)c(order)g(to)i(prove)e(\(5.11\))f(it)j(is)118 1307 y(enough)24 b(to)i(show)g(that)192 1462 y Fp(lim)15 b(sup)134 1540 y Fr(n)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)544 1462 y Fp(Av)661 1484 y Fr(x)p Fn(2)p Fm(T)798 1465 y FA(d)798 1500 y(\017)873 1462 y Fp(sup)848 1544 y Fn(j)p Fr(\014)s Fn(j\024)p Fr(T)1051 1462 y Fp(sup)1100 1534 y Fr(\027)1218 1462 y Fp(sup)f Fq(spec)1540 1481 y Fr(L)1588 1462 y Fg(2)1623 1481 y Fl(\()p Fr(\027)t Fl(\))1721 1388 y Fk(\010)1774 1462 y Fq(\017)1811 1424 y Fn(\000)p Fl(1)1905 1462 y Fq(\014)5 b Fp(Av)2079 1476 y Fr(z)s Fn(2)p Fr(B)s Fl(+)p Fr(x)2317 1462 y Fq(\034)2357 1476 y Fr(z)2396 1462 y Fq(f)2441 1476 y Fr(n;s)2537 1485 y Fg(0)2595 1462 y Fp(+)20 b Fq(c)15 b(\017)2777 1424 y Fn(\000)p Fl(2)2871 1462 y Fp(Av)2989 1477 y Fr(b)p Fn(2)p Fl(\003)3115 1491 y FA(x;)p Fg(3)p FA(a=)p Fg(2)p FA(\017)3329 1462 y Fo(L)3392 1477 y Fr(b)3426 1388 y Fk(\011)3505 1462 y Fo(\024)24 b Fp(0)3409 1643 y Fu(\(5.12\))118 1761 y(where)29 b Fq(f)438 1775 y Fr(n;s)534 1784 y Fg(0)603 1761 y Fp(:=)741 1715 y Fr(\036)783 1723 y FA(n;n)p 741 1740 142 4 v 791 1793 a Fr(n)916 1761 y Fo(\000)1019 1710 y Fr(\036)1061 1718 y FA(n;s)1147 1733 y Fg(0)p 1019 1740 168 4 v 1082 1793 a Fr(n)1226 1761 y Fu(and)g Fq(\027)35 b Fu(varies)30 b(in)f Fo(M)p Fp(\(\003)2059 1780 y Fr(x;)p Fl(3)p Fr(a=)p Fl(2)p Fr(\017)2295 1761 y Fp(\))p Fu(.)44 b(The)30 b(proof)f(is)h (nothing)g(more)e(than)118 1911 y(a)g(careful)f(writing)h(of)g(the)g (spatial)h(average)e Fp(Av)1808 1853 y Fr(n;)1881 1826 y FA(a)p 1880 1838 34 3 v 1884 1879 a(\017)1808 1923 y Fr(z)s(;x)1956 1911 y Fu(together)h(with)h(the)f(subadditivity)j (property)c(of)118 2019 y Fp(sup)14 b Fq(spec)p Fu(.)118 2127 y(Setting)25 b Fq(B)498 2094 y Fn(0)546 2127 y Fp(=)g Fq(Q)714 2141 y Fr(s)747 2150 y Fg(0)805 2127 y Fo(\\)20 b Fq(n)15 b Fs(Z)1021 2094 y Fr(d)1057 2127 y Fu(,)26 b Fq(Y)45 b Fp(=)25 b Fq(Q)1374 2145 y Fr(a=\017)1499 2127 y Fo(\\)20 b Fq(s)1623 2141 y Fl(0)1662 2127 y Fs(Z)1727 2094 y Fr(d)1788 2127 y Fu(we)25 b(can)g(write)h Fq(B)j Fp(=)c Fo([)2578 2141 y Fr(y)r Fn(2)p Fr(Y)2723 2127 y Fp(\()p Fq(B)2832 2094 y Fn(0)2875 2127 y Fp(+)20 b Fq(y)s Fp(\))26 b Fu(so)f(that)1027 2280 y Fp(Av)1145 2294 y Fr(z)s Fn(2)p Fr(B)s Fl(+)p Fr(x)1383 2280 y Fq(\034)1423 2294 y Fr(z)1462 2280 y Fq(f)1507 2294 y Fr(n;s)1603 2303 y Fg(0)1666 2280 y Fp(=)g(Av)1879 2294 y Fr(y)r Fn(2)p Fr(Y)16 b Fl(+)p Fr(x)2119 2280 y Fp(Av)2236 2295 y Fr(z)s Fn(2)p Fr(B)2375 2276 y Fi(0)2398 2295 y Fl(+)p Fr(y)2494 2280 y Fq(\034)2534 2294 y Fr(z)2574 2280 y Fq(f)2619 2294 y Fr(n;s)2715 2303 y Fg(0)118 2433 y Fu(and)1078 2549 y Fp(Av)1195 2564 y Fr(b)p Fn(2)p Fl(\003)1321 2578 y FA(x;)p Fg(3)p FA(a=)p Fg(2)p FA(\017)1536 2549 y Fo(L)1599 2564 y Fr(b)1658 2549 y Fo(\024)25 b Fq(c)15 b Fp(Av)1926 2563 y Fr(y)r Fn(2)p Fr(Y)h Fl(+)p Fr(x)2166 2549 y Fp(Av)2283 2564 y Fr(b)p Fn(2)p Fl(\003)2409 2573 y FA(y)r(;)p Fg(2)p FA(s)2521 2588 y Fg(0)2564 2549 y Fo(L)2627 2564 y Fr(b)2676 2549 y Fq(:)118 2693 y Fu(By)25 b(the)h(subadditivity)i(property)d(of)g Fp(sup)14 b 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b(among)f Fo(M)p Fp(\(\003)1225 3138 y Fr(x;)p Fl(2)p Fr(s)1353 3147 y Fg(0)1392 3124 y Fp(\))i Fu(and)f Fp(Av)1744 3076 y Fl(\()p Fr(n)p Fl(\))1744 3152 y Fr(y)r Fn(2)p Fl(\003)1907 3124 y Fp(:=)g(Av)2146 3146 y Fr(y)r Fn(2)p Fl(\003)p Fn(\\)12 b Fr(n)p Fm(Z)2431 3127 y FA(d)2465 3124 y Fu(.)118 3264 y(A)n(t)76 b(this)g(point)f(we)f(can)g(apply)h(perturbation)g(theory)f (\(see)g(proposition)i(4.2\):)130 b(since)118 3376 y Fp(lim)244 3391 y Fr(n)p Fn("1)412 3376 y Fp(sup)549 3398 y Fr(\017>)p Fl(0)687 3376 y Fq(\017s)767 3338 y Fr(d)p Fl(+2)767 3402 y(0)897 3376 y Fo(k)p Fq(f)987 3390 y Fr(n;s)1083 3399 y Fg(0)1121 3376 y Fo(k)1166 3390 y Fn(1)1267 3376 y Fp(=)25 b(0)p Fu(,)h(it)g(is)g(enough)e(to)h (prove)g(that)h(for)g(almost)f(all)g(disorder)g Fq(\013)730 3578 y Fp(lim)15 b(sup)736 3657 y Fr(n)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)1058 3516 y Fp(1)p 1034 3557 95 4 v 1034 3640 a Fq(n)1089 3614 y Fl(2)1138 3578 y Fp(Av)1255 3600 y Fr(x)p Fn(2)p Fm(T)1392 3581 y FA(d)1392 3617 y(\017)1442 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3915 y Fp(Av)2519 3867 y Fl(\()p Fr(n)p Fl(\))2519 3944 y Fr(y)r Fn(2)p Fr(Q)2659 3952 y FA(s)2688 3967 y Fg(0)2731 3915 y Fq(\034)2771 3929 y Fr(y)2812 3915 y Fq(f)2867 3842 y Fk(\001)2908 3915 y Fq(:)118 4069 y Fu(In)37 b(order)g(to)h(prove)e(\(5.13\))g(it)i(is)g(clearly)e (suf\002cient)h(to)h(prove)e(it)j(with)f Fq(\036)2785 4083 y Fr(n;n)2932 4069 y Fu(replaced)e(by)h Fq(\036)3492 4089 y Fr(n;n)3598 4070 y Fg(4)3636 4069 y Fu(,)118 4177 y(provided)26 b(one)e(is)j(able)d(to)i(show)g(that)g(for)f(almost)h (any)f(disorder)h Fq(\013)713 4371 y Fp(lim)15 b(sup)718 4450 y Fr(n)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)1040 4310 y Fp(1)p 1016 4350 V 1016 4434 a Fq(n)1071 4407 y Fl(2)1120 4371 y Fp(Av)1237 4393 y Fr(x)p Fn(2)p Fm(T)1374 4375 y FA(d)1374 4410 y(\017)1424 4371 y Fp(sup)1561 4393 y Fr(\027)t Fn(2M)p Fl(\(\003)1807 4402 y FA(x;)p Fg(2)p FA(s)1920 4417 y Fg(0)1959 4393 y Fl(\))1991 4371 y Fp(\011)2062 4334 y Fl(\()p Fr(\027)t Fl(\))2062 4394 y Fr(s)2095 4403 y Fg(0)2160 4298 y Fk(\000)2201 4371 y Fq(\034)2241 4385 y Fr(x)2285 4298 y Fk(\002)2323 4371 y Fq(\036)2377 4385 y Fr(n;n)2506 4371 y Fo(\000)20 b Fq(\036)2651 4391 y Fr(n;n)2757 4372 y Fg(4)2795 4298 y Fk(\003\001)2900 4371 y Fp(=)25 b(0)p Fq(:)343 b Fu(\(5.14\))118 4593 y(W)-8 b(e)32 b(will)e(concentrate)g(only)g(on)g(the)i(\002rst)f(step)h (and)f(refer)f(the)h(reader)f(to)h(section)g(4.5)g(in)g([15)o(])h(for) 118 4701 y(the)26 b(details)g(of)f(the)h(proof)f(of)h(\(5.14\).)118 4827 y(Given)i Fq(\027)37 b Fo(2)30 b(M)p Fp(\(\003)756 4841 y Fl(2)p Fr(s)824 4850 y Fg(0)863 4827 y Fp(\))f Fu(we)f(\002rst)h(estimate)h Fp(\011)1687 4779 y Fl(\()p Fr(\027)t Fl(\))1687 4839 y Fr(s)1720 4848 y Fg(0)1784 4754 y Fk(\000)1826 4827 y Fq(\036)1880 4847 y Fr(n;n)1986 4828 y Fg(4)2046 4827 y Fo(\000)23 b Fq(\036)2194 4841 y Fr(n;s)2290 4850 y Fg(0)2327 4754 y Fk(\001)2398 4827 y Fu(as)29 b(follows)f(\(a)h(similar)f(bound)h(will)118 4935 y(then)c(be)g(applied)h(to)f(any)g(translation)h(by)f Fq(x)p Fu(\).)118 5043 y(Assume,)i(without)g(loss)g(of)f(generality)-12 b(,)25 b(that)i Fq(s)1748 5057 y Fl(0)1814 5043 y Fp(=)g Fq(N)1995 5010 y Fl(4)2060 5043 y Fu(for)f(some)g Fq(N)37 b Fo(2)26 b Fs(N)39 b Fu(and)26 b(set)h Fq(`)3072 5058 y Fr(k)3141 5043 y Fp(:=)g Fq(k)3314 5010 y Fl(4)3380 5043 y Fu(for)f(any)118 5151 y Fq(k)j Fo(2)24 b Fs(N)7 b Fu(.)38 b(Then,)25 b(given)f Fp(0)i Fq(<)f(\032)g Fo(\034)g Fp(1)p Fu(,)h(by)g(Schwarz)e(inequality)-12 b(,)817 5393 y Fp(\011)888 5356 y Fl(\()p Fr(\027)t Fl(\))888 5416 y Fr(s)921 5425 y Fg(0)986 5393 y Fp(\()p Fq(\036)1075 5413 y Fr(n;n)1181 5394 y Fg(4)1240 5393 y Fo(\000)20 b Fq(\036)1385 5407 y Fr(n;s)1481 5416 y Fg(0)1519 5393 y Fp(\))25 b Fo(\024)g Fq(c)1714 5407 y Fr(\032)1770 5280 y(N)7 b Fn(\000)p Fl(1)1781 5307 y Fk(X)1778 5504 y Fr(k)r Fl(=)p Fr(n)1938 5393 y Fq(k)1988 5356 y Fl(1+)p Fr(\032)2134 5393 y Fp(\011)2205 5356 y Fl(\()p Fr(\027)t Fl(\))2205 5416 y Fr(s)2238 5425 y Fg(0)2303 5393 y Fp(\()p Fq(\036)2392 5408 y Fr(n;`)2484 5420 y FA(k)2546 5393 y Fo(\000)20 b Fq(\036)2691 5408 y Fr(n;`)2783 5420 y FA(k)q Fg(+1)2902 5393 y Fp(\))p Fq(:)p eop %%Page: 31 31 31 30 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(31)118 381 y Fu(In)32 b(order)f(to)h(estimate)h Fp(\011)1026 333 y Fl(\()p Fr(\027)t Fl(\))1026 393 y Fr(s)1059 402 y Fg(0)1123 381 y Fp(\()p Fq(\036)1212 396 y Fr(n;`)1304 408 y FA(k)1371 381 y Fo(\000)24 b Fq(\036)1520 396 y Fr(n;`)1612 408 y FA(k)q Fg(+1)1731 381 y Fp(\))32 b Fu(we)g(divide)h Fq(Q)2292 395 y Fr(s)2325 404 y Fg(0)2395 381 y Fu(in)e(cubes)h Fo(f)p Fq(Q)2875 396 y Fr(i;k)2961 381 y Fo(g)3006 395 y Fr(i)p Fn(2)p Fr(I)3108 407 y FA(k)3182 381 y Fu(with)h(side)g Fq(`)3619 396 y Fr(k)118 491 y Fu(where,)g(without)f(loss)g(of)g(generality)-12 b(,)31 b(we)g(assume)h(that)g Fq(s)2193 505 y Fl(0)2232 491 y Fq(=`)2315 506 y Fr(k)2394 491 y Fo(2)j Fs(N)44 b Fu(and)31 b(similarly)g(for)h Fq(`)3322 506 y Fr(k)3364 491 y Fq(=n)p Fu(.)50 b(Let)139 576 y Fp(\026)118 599 y Fq(Q)190 614 y Fr(i;k)302 599 y Fu(be)24 b(the)i(cube)e(of)i(side)g Fp(10)p Fq(`)1198 614 y Fr(k)1266 599 y Fu(concentric)e(to)h Fq(Q)1877 614 y Fr(i;k)1964 599 y Fu(.)31 b(Then)25 b(by)g(lemma)f(6.1) h(with)408 771 y Fq(I)33 b Fp(:=)25 b Fq(I)642 786 y Fr(k)r Fl(+1)775 771 y Fq(;)106 b Fp(\003)25 b(:=)h(\003)1179 785 y Fl(2)p Fr(s)1247 794 y Fg(0)1286 771 y Fq(;)106 b Fp(\003)1480 785 y Fr(i)1533 771 y Fp(:=)1675 748 y(\026)1655 771 y Fq(Q)1727 786 y Fr(i;k)r Fl(+1)1903 771 y Fq(;)g(f)2079 785 y Fr(i)2132 771 y Fp(:=)25 b(Av)2371 723 y Fl(\()p Fr(n)p Fl(\))2371 800 y Fr(x)p Fn(2)p Fr(Q)2514 812 y FA(i;k)q Fg(+1)2673 771 y Fq(\034)2713 785 y Fr(x)2757 771 y Fp([)p Fq(\036)2836 786 y Fr(n;`)2928 798 y FA(k)2990 771 y Fo(\000)20 b Fq(\036)3135 786 y Fr(n;`)3227 798 y FA(k)q Fg(+1)3346 771 y Fp(])118 935 y Fu(we)25 b(obtain)h(\(thanks)g (also)f(to)h(lemma)e(A.5\))292 1099 y Fp(\011)363 1061 y Fl(\()p Fr(\027)t Fl(\))363 1121 y Fr(s)396 1130 y Fg(0)461 1099 y Fp(\()p Fq(\036)550 1114 y Fr(n;`)642 1126 y FA(k)704 1099 y Fo(\000)c Fq(\036)849 1114 y Fr(n;`)941 1126 y FA(k)q Fg(+1)1059 1099 y Fp(\))26 b Fo(\024)f Fq(c)15 b(`)1308 1061 y Fr(d)p Fl(+2)1308 1128 y Fr(k)r Fl(+1)1441 1099 y Fp(Av)1559 1113 y Fr(i)p Fn(2)p Fr(I)1661 1125 y FA(k)q Fg(+1)1780 1099 y Fp(V)-8 b(ar)1921 1121 y Fr(\026)1963 1102 y FA(\025)2009 998 y Fk(\020)2063 1099 y Fp(Av)2180 1051 y Fl(\()p Fr(n)p Fl(\))2180 1128 y Fr(x)p Fn(2)p Fr(Q)2323 1140 y FA(i;k)q Fg(+1)2483 1099 y Fq(\034)2523 1113 y Fr(x)2567 1099 y Fp([)p Fq(\036)2646 1114 y Fr(n;`)2738 1126 y FA(k)2800 1099 y Fo(\000)20 b Fq(\036)2945 1114 y Fr(n;`)3037 1126 y FA(k)q Fg(+1)3155 1099 y Fp(])3180 998 y Fk(\021)3409 1099 y Fu(\(5.15\))118 1284 y(where)25 b Fq(\026)444 1251 y Fr(\025)514 1284 y Fu(is)h(the)g(grand)e(canonical)g(measure)g(corresponding)g(to)i Fq(\027)6 b Fu(.)118 1392 y(Let)27 b(now)f Fq(J)35 b Fu(be)26 b(the)h(set)g(of)f(possible)h(densities)h(on)e Fp(\003)1970 1406 y Fl(2)p Fr(s)2038 1415 y Fg(0)2077 1392 y Fu(.)34 b(Then,)27 b(thanks)g(to)g(\(5.15\),)f(it)h(is)g(enough) e(to)118 1500 y(prove)g(that,)h(for)g Fq(\032)f Fu(small)f(enough)h (and)g(for)g(almost)g(any)g(disorder)h Fq(\013)p Fu(,)283 1735 y Fp(lim)212 1796 y Fr(n)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)495 1735 y Fp(Av)612 1757 y Fr(x)p Fn(2)p Fm(T)749 1738 y FA(d)749 1773 y(\017)818 1673 y Fp(1)p 794 1714 95 4 v 794 1797 a Fq(n)849 1771 y Fl(2)994 1621 y Fr(N)960 1648 y Fk(X)913 1846 y Fr(k)r Fl(=)p Fr(n)p Fl(+1)1154 1735 y Fq(k)1204 1697 y Fl(1+)p Fr(\032)1335 1735 y Fq(`)1373 1696 y Fr(d)p Fl(+2)1373 1764 y Fr(k)1503 1735 y Fp(Av)1621 1749 y Fr(i)p Fn(2)p Fr(I)1723 1761 y FA(k)1789 1735 y Fp(sup)1780 1813 y Fr(m)p Fn(2)p Fr(J)1949 1735 y Fp(V)-8 b(ar)2091 1776 y Fr(\026)2133 1741 y FA(\025)p Fg(\()p FA(m)p Fg(\))2133 1799 y(\003)2174 1814 y FA(x;)p Fg(2)p FA(s)2287 1829 y Fg(0)2335 1634 y Fk(\020)2389 1735 y Fp(Av)2507 1687 y Fl(\()p Fr(n)p Fl(\))2507 1763 y Fr(y)r Fn(2)p Fr(x)p Fl(+)p Fr(Q)2742 1775 y FA(i;k)2825 1735 y Fq(\034)2865 1749 y Fr(y)2906 1735 y Fq(\036)2960 1750 y Fr(n;`)3052 1762 y FA(k)3094 1634 y Fk(\021)3173 1735 y Fp(=)25 b(0)95 b Fu(\(5.16\))118 1976 y(and)25 b(similarly)h(with)g Fq(\036)915 1991 y Fr(n;`)1007 2003 y FA(k)1074 1976 y Fu(replaced)e(by)h Fq(\036)1610 1991 y Fr(n;`)1702 2003 y FA(k)q Fi(\000)p Fg(1)1822 1976 y Fu(.)118 2102 y(Given)c Fq(\015)30 b(>)25 b Fp(0)d Fu(we)f(set)h Fq(J)928 2117 y Fr(k)996 2102 y Fp(=)j Fo(f)p Fq(`)1175 2058 y Fn(\000)p Fr(\015)1175 2131 y(k)1275 2102 y Fq(;)15 b Fp(2)p Fq(`)1398 2058 y Fn(\000)p Fr(\015)1398 2131 y(k)1498 2102 y Fq(;)g(:)g(:)g(:)h(;)f Fp(1)7 b Fo(\000)g Fq(`)1867 2058 y Fn(\000)p Fr(\015)1867 2131 y(k)1967 2102 y Fo(g)p Fu(.)30 b(Then,)22 b(using)f(\(A.1\),)h(the)g(variance)e(in)i(\(5.16\)) 118 2210 y(can)j(be)g(bounded)g(from)f(above)h(by)1121 2381 y Fp(V)-8 b(ar)1263 2423 y Fr(\026)1305 2387 y FA(\025)p Fg(\()12 b(\026)-42 b FA(m)p Fg(\))1305 2453 y FA(x)p Fg(+)1400 2440 y(\026)1387 2453 y FA(Q)1435 2471 y(i;k)1522 2280 y Fk(\020)1576 2381 y Fp(Av)1694 2333 y Fl(\()p Fr(n)p Fl(\))1694 2410 y Fr(y)r Fn(2)p Fr(x)p Fl(+)p Fr(Q)1929 2422 y FA(i;k)2012 2381 y Fq(\034)2052 2395 y Fr(y)2093 2381 y Fq(\036)2147 2396 y Fr(n;`)2239 2408 y FA(k)2280 2280 y Fk(\021)2355 2381 y Fp(+)20 b Fq(c`)2523 2337 y Fr(d)p Fn(\000)p Fr(\015)2523 2411 y(k)118 2611 y Fu(provided)26 b(that)43 b Fp(\026)-62 b Fq(m)25 b Fo(2)g Fq(J)919 2626 y Fr(k)988 2611 y Fu(satis\002es)i Fo(j)17 b Fp(\026)-62 b Fq(m)20 b Fo(\000)g Fq(m)p Fo(j)25 b(\024)g Fq(`)1801 2567 y Fn(\000)p Fr(\015)1801 2641 y(k)1900 2611 y Fu(.)118 2747 y(Therefore,)h(by)g(choosing)g Fq(\015)32 b Fu(large)25 b(enough,)g(we)h(can)g(replace)f(in)h (\(5.16\))f Fq(\026)2761 2699 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))2761 2775 y(\003)2810 2784 y FA(x;)p Fg(2)p FA(s)2923 2799 y Fg(0)2992 2747 y Fu(by)h Fq(\026)3167 2699 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))3167 2785 y Fr(x)p Fl(+)3278 2768 y(\026)3262 2785 y Fr(Q)3318 2797 y FA(i;k)3427 2747 y Fu(and)g Fq(J)118 2882 y Fu(by)f Fq(J)287 2897 y Fr(k)330 2882 y Fu(.)32 b(W)-8 b(e)26 b(can)e(at)i(this)h(apply)e(proposition)h (6.6)f(to)h(get)e(that)381 3053 y Fp(sup)357 3131 y Fr(m)p Fn(2)p Fr(J)505 3143 y FA(k)558 3053 y Fp(V)-8 b(ar)700 3094 y Fr(\026)742 3059 y FA(\025)p Fg(\()p FA(m)p Fg(\))756 3112 y(\026)742 3125 y FA(Q)790 3142 y(i;k)869 3125 y Fg(+)p FA(x)959 2952 y Fk(\020)1013 3053 y Fp(Av)1131 3005 y Fl(\()p Fr(n)p Fl(\))1131 3082 y Fr(y)r Fn(2)p Fr(x)p Fl(+)p Fr(Q)1366 3094 y FA(i;k)1449 3053 y Fq(\034)1489 3067 y Fr(y)1530 3053 y Fq(\036)1584 3068 y Fr(n;`)1676 3080 y FA(k)1717 2952 y Fk(\021)1797 3053 y Fo(\024)25 b Fq(c)p Fp(1)-25 b(I)1985 3067 y Fn(A)2041 3044 y FA(c)2041 3091 y(x;i;k)2179 3053 y Fp(\()p Fq(\013)p Fp(\))p Fq(`)2345 3068 y Fr(k)2388 3016 y Fn(\000)p Fl(2)p Fr(d)p Fl(+2)p Fr(\016)2663 3053 y Fp(+)20 b(1)-25 b(I)2807 3067 y Fn(A)2863 3079 y FA(x;i;k)3001 3053 y Fp(\()p Fq(\013)p Fp(\))15 b Fq(;)240 b Fu(\(5.17\))118 3290 y(where)25 b Fo(A)462 3305 y Fr(x;i;k)632 3290 y 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Fq(n)1685 3907 y Fn(\000)p Fl(2)1875 3830 y Fr(N)1841 3858 y Fk(X)1794 4055 y Fr(k)r Fl(=)p Fr(n)p Fl(+1)2035 3944 y Fq(k)2085 3907 y Fl(1+)p Fr(\032)2216 3944 y Fq(`)2254 3906 y Fl(2)p Fn(\000)p Fr(d)p Fl(+2)p Fr(\016)2254 3973 y(k)2508 3944 y Fq(:)876 b Fu(\(5.18\))118 4185 y(The)24 b(second)f(addendum)g(in)h (the)f(r)-10 b(.h.s.)32 b(of)24 b(\(5.18\))e(is)i(zero)f(because)f(of)i (the)g(de\002nition)g(of)g Fq(`)3297 4200 y Fr(k)3363 4185 y Fu(and)f(the)118 4293 y(condition)j Fq(d)f Fo(\025)g Fp(3)p Fu(.)118 4401 y(Let)31 b(us)g(consider)f(the)h(\002rst)g (addendum)f(in)h(the)f(r)-10 b(.h.s.)49 b(of)30 b(\(5.18\).)46 b(By)31 b(Chebyschev)f(inequality)-12 b(,)33 b(for)118 4509 y(any)25 b Fq(q)j(>)d Fp(0)h Fu(and)f(any)g Fq(x;)15 b(k)840 4660 y Fs(P)895 4587 y Fk(\000)952 4660 y Fp(Av)1070 4674 y Fr(i)p Fn(2)p Fr(I)1172 4686 y FA(k)1214 4660 y Fp(1)-25 b(I)1267 4674 y Fn(A)1323 4686 y FA(x;i;k)1485 4660 y Fo(\025)25 b Fq(`)1619 4616 y Fn(\000)p Fr(q)1619 4690 y(k)1727 4587 y Fk(\001)1794 4660 y Fo(\024)g Fs(P)1945 4587 y Fk(\000)2003 4660 y Fo(9)p Fq(i)g Fo(2)g Fq(I)2236 4675 y Fr(k)2319 4660 y Fp(:)40 b(1)-25 b(I)2437 4674 y Fn(A)2493 4686 y FA(x;i;k)2656 4660 y Fo(\025)25 b Fq(`)2790 4616 y Fn(\000)p Fr(q)2790 4690 y(k)2898 4587 y Fk(\001)1794 4828 y Fo(\024)g Fq(s)1933 4790 y Fr(d)1933 4850 y Fl(0)1988 4828 y Fq(`)2038 4783 y Fr(q)r Fn(\000)p Fr(d)2026 4857 y(k)2182 4828 y Fq(e)2224 4790 y Fn(\000)p Fr(c)12 b(`)2351 4767 y FA(\016)2351 4813 y(k)2393 4828 y Fq(:)991 b Fu(\(5.19\))118 4988 y(Moreover)-10 b(,)24 b(by)h(setting)954 4965 y Fp(\026)950 4988 y(1)-25 b(I)1003 5002 y Fn(A)1059 5014 y FA(x;i;k)1222 4988 y Fp(=)25 b(1)-25 b(I)1371 5002 y Fn(A)1427 5014 y FA(x;i;k)1584 4988 y Fo(\000)20 b Fs(P)1730 4914 y Fk(\000)1772 4988 y Fo(A)1845 5003 y Fr(x;i;k)1991 4914 y Fk(\001)2032 4988 y Fu(,)26 b(we)f(have)g(for)g(any)g Fq(r)j Fo(2)d Fs(N)38 b Fu(and)25 b(any)g Fq(x;)15 b(k)407 5164 y Fs(P)462 5090 y Fk(\000)519 5164 y Fp(Av)636 5178 y Fr(i)p Fn(2)p 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5478 y Fp(1)-25 b(I)1471 5492 y Fn(A)1527 5504 y FA(x;i;k)1689 5478 y Fo(\025)25 b Fq(l)1814 5434 y Fn(\000)p Fr(q)1812 5507 y(k)1922 5404 y Fk(\001)1989 5478 y Fo(\024)g Fq(c)p Fp(\()p Fq(q)s(;)15 b(r)s Fp(\))p Fq(s)2365 5430 y Fn(\000)p Fr(d)p Fl(\()p Fr(r)r Fn(\000)p Fl(1\))p Fr(=)p Fl(2)2365 5504 y(0)2710 5478 y Fq(:)p eop %%Page: 32 32 32 31 bop 118 45 a Fj(32)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(It)26 b(is)g(enough)e(at)i(this)h(point)f(to)f(choose)g Fq(q)j Fu(and)e Fq(r)i Fu(large)c(enough,)g(de\002ne)749 543 y Fp(\002)820 557 y Fr(\017)877 543 y Fp(:=)i Fo(f)f(9)p Fq(x)g Fo(2)g Fs(T)1342 505 y Fr(d)1342 565 y(\017)1409 543 y Fp(:)h(Av)1577 557 y Fr(i)p Fn(2)p Fr(I)1679 569 y FA(k)1721 543 y Fp(1)-25 b(I)1774 557 y Fn(A)1830 569 y FA(x;i;k)1993 543 y Fo(\025)25 b Fq(`)2127 499 y Fn(\000)p Fr(q)2127 572 y(k)2310 543 y Fu(for)h(some)f Fq(k)j Fo(\024)d Fq(N)35 b Fo(g)p Fq(;)118 717 y Fu(and)27 b(apply)h(Borel-Cantelli)e (lemma)g(to)h(get)g(that)h(also)g(the)f(\002rst)h(addendum)f(in)g(the)h (r)-10 b(.h.s.)38 b(of)28 b(\(5.18\))118 825 y(is)e(negligible.)2972 b Fe(\003)118 1039 y Fu(5.4.)45 b FE(F)m(rom)28 b(scale)g Fq(s)841 1054 y Fr(k)910 1039 y FE(to)g(scale)f Fq(s)1302 1054 y Fr(k)r Fl(+1)1434 1039 y FE(.)46 b Fu(Here)26 b(we)g(de\002ne)f(precisely)h(the)g(sequence)f(of)h(length)g(scales)118 1147 y Fq(s)161 1162 y Fr(k)229 1147 y Fu(and)f(discuss)h(the)g (details)g(of)g(the)f(inductive)h(step)f Fq(s)2019 1162 y Fr(k)2087 1147 y Fo(!)g Fq(s)2246 1162 y Fr(k)r Fl(+1)2404 1147 y Fu(described)g(section)g(5.1.)118 1255 y(Let)h Fo(f)p Fq(a)359 1270 y Fr(k)402 1255 y Fo(g)447 1270 y Fr(k)r Fn(\025)p Fl(0)605 1255 y Fu(be)f(de\002ned)g(inductively)g (by)1068 1473 y Fq(a)1116 1487 y Fl(0)1181 1473 y Fp(=)g(1)117 b Fu(and)f Fq(a)1752 1488 y Fr(k)r Fl(+1)1910 1473 y Fp(=)25 b(1)c(+)2163 1399 y Fk(\000)2214 1411 y Fp(1)p 2214 1452 46 4 v 2214 1535 a(2)2290 1473 y Fo(\000)2457 1411 y Fp(1)p 2391 1452 179 4 v 2391 1535 a(2)2436 1509 y Fr(k)r Fl(+1)2579 1399 y Fk(\001)2621 1473 y Fq(a)2669 1488 y Fr(k)118 1675 y Fu(It)31 b(is)g(easy)f(to)h(verify)f(that)h(the) g(sequence)d Fo(f)p Fq(a)1696 1690 y Fr(k)1740 1675 y Fo(g)1785 1690 y Fr(k)r Fn(\025)p Fl(0)1948 1675 y Fu(is)j(increasing)e (with)j Fp(lim)2808 1690 y Fr(k)r Fn(!1)3007 1675 y Fq(a)3055 1690 y Fr(k)3131 1675 y Fp(=)h(2)p Fu(.)47 b(Let)30 b(also)118 1812 y Fq(s)161 1827 y Fr(k)229 1812 y Fp(:=)25 b Fq(\017)387 1770 y Fn(\000)470 1731 y FA(a)504 1748 y(k)p 452 1754 110 3 v 452 1796 a(d)p Fg(+2)575 1812 y Fu(.)118 1952 y FE(Proposition)i(5.8.)42 b Fh(Let)25 b Fq(d)h Fo(\025)e Fp(2)p Fh(.)32 b(Then)892 2174 y Fp(Av)1010 2117 y Fr(s)1043 2129 y FA(k)1080 2117 y Fr(;)1110 2090 y FA(a)p 1110 2102 34 3 v 1114 2143 a(\017)1010 2186 y Fr(z)s(;x)1158 2174 y Fq(\034)1198 2188 y Fr(z)1247 2112 y Fq(\036)1301 2126 y Fr(s)1334 2138 y FA(k)1372 2126 y Fr(;s)1425 2138 y FA(k)p 1247 2154 220 4 v 1314 2237 a Fq(s)1357 2252 y Fr(k)1502 2174 y Fo(\031)25 b Fp(Av)1715 2117 y Fr(s)1748 2129 y FA(k)q Fg(+1)1863 2117 y Fr(;)1893 2090 y FA(a)p 1892 2102 34 3 v 1896 2143 a(\017)1715 2186 y Fr(z)s(;x)1940 2174 y Fq(\034)1980 2188 y Fr(z)2030 2107 y Fq(\036)2084 2121 y Fr(s)2117 2133 y FA(k)q Fg(+1)2232 2121 y Fr(;s)2285 2133 y FA(k)q Fg(+1)p 2030 2154 374 4 v 2129 2237 a Fq(s)2172 2252 y Fr(k)r Fl(+1)2595 2174 y Fo(8)p Fq(k)j Fo(\025)d Fp(0)p Fq(:)522 b Fu(\(5.21\))118 2419 y Fh(Proof)-8 b(.)42 b Fu(In)27 b(order)g(to)h(prove)e(\(5.21\))h(observe)f(that,)k (by)d(construction,)h(the)f(two)h(exponents)3374 2379 y Fr(a)3411 2391 y FA(k)p 3348 2398 127 4 v 3348 2451 a Fr(d)p Fl(+2)3512 2419 y Fu(and)128 2494 y Fr(a)165 2506 y FA(k)q Fg(+1)p 128 2520 153 4 v 141 2573 a Fr(d)p Fl(+2)328 2541 y Fu(satisfy)39 b(the)e(conditions)h(of)g(propositions)h (5.5)d(and)h(5.6)g(with)h Fq(\015)51 b Fp(:=)2867 2501 y Fr(a)2904 2513 y FA(k)p 2842 2520 127 4 v 2842 2573 a Fr(d)p Fl(+2)3015 2541 y Fu(and)38 b Fq(\015)3254 2508 y Fn(0)3322 2541 y Fp(:=)3474 2494 y Fr(a)3511 2506 y FA(k)q Fg(+1)p 3474 2520 153 4 v 3487 2573 a Fr(d)p Fl(+2)3636 2541 y Fu(.)118 2655 y(Therefore)24 b(we)h(have)h(the)f(following)g (chain)g(of)g(equivalences:)720 2878 y Fp(Av)838 2820 y Fr(s)871 2832 y FA(k)909 2820 y Fr(;)939 2793 y FA(a)p 938 2805 34 3 v 942 2847 a(\017)838 2890 y Fr(z)s(;x)986 2878 y Fq(\034)1026 2892 y Fr(z)1075 2816 y Fq(\036)1129 2830 y Fr(s)1162 2842 y FA(k)1200 2830 y Fr(;s)1253 2842 y FA(k)p 1075 2857 220 4 v 1142 2940 a Fq(s)1185 2955 y Fr(k)1330 2878 y Fo(\031)g Fp(Av)1543 2820 y Fr(s)1576 2832 y FA(k)1614 2820 y Fr(;)1644 2793 y FA(a)p 1644 2805 34 3 v 1648 2847 a(\017)1543 2890 y Fr(z)s(;x)1691 2878 y Fq(\034)1731 2892 y Fr(z)1781 2810 y Fq(\036)1835 2824 y Fr(s)1868 2836 y FA(k)1906 2824 y Fr(;s)1959 2836 y FA(k)q Fg(+1)p 1781 2857 297 4 v 1886 2940 a Fq(s)1929 2955 y Fr(k)2113 2878 y Fo(\031)f Fp(Av)2326 2820 y Fr(s)2359 2832 y FA(k)q Fg(+1)2474 2820 y Fr(;)2504 2793 y FA(a)p 2503 2805 34 3 v 2507 2847 a(\017)2326 2890 y Fr(z)s(;x)2551 2878 y Fq(\034)2591 2892 y Fr(z)2640 2810 y Fq(\036)2694 2824 y Fr(s)2727 2836 y FA(k)q Fg(+1)2842 2824 y Fr(;)p Fl(2)p Fr(s)2930 2836 y FA(k)q Fg(+1)p 2640 2857 409 4 v 2757 2940 a Fq(s)2800 2955 y Fr(k)r Fl(+1)118 3104 y Fu(Finally)-12 b(,)25 b(using)g(again)g(proposition)h(5.6)f(with)h Fq(s)f Fp(=)g Fq(s)1942 3119 y Fr(k)r Fl(+1)2100 3104 y Fu(and)g Fq(s)2317 3071 y Fn(0)2365 3104 y Fp(=)g(2)p Fq(s)p Fu(,)h(we)f(obtain)h(\(5.21\).)295 b Fe(\003)118 3332 y Fu(5.5.)45 b FE(Analysis)27 b(of)811 3285 y Fr(\036)856 3293 y Fg(\026)-33 b FA(s;)s Fg(\026)g FA(s)p 811 3311 124 4 v 859 3363 a Fl(\026)-38 b Fr(s)972 3332 y FE(via)27 b(an)h(improved)f(T)-7 b(wo)28 b(Blocks)g(Estimate.)45 b Fu(Here)25 b(we)h(describe)g(the)h(\002nal)118 3440 y(step)f(in)f(the)h(proof)f(of)g(theorem)g(5.3,)g(namely)f(we)h(show)h (that)1571 3657 y Fp(Av)1691 3599 y Fl(\026)-37 b Fr(s)o(;)1751 3572 y FA(a)p 1751 3584 34 3 v 1755 3625 a(\017)1689 3669 y Fr(z)s(;x)1799 3657 y Fq(\034)1839 3671 y Fr(z)1888 3595 y Fq(\036)1945 3609 y Fl(\026)f Fr(s;)s Fl(\026)g Fr(s)p 1888 3636 144 4 v 1942 3719 a Fp(\026)-48 b Fq(s)2067 3657 y Fo(\031)25 b Fp(0)118 3893 y Fu(where)41 b Fp(\026)-49 b Fq(s)47 b Fp(=)f Fq(\017)645 3860 y Fr(\016)683 3893 y Fq(s)726 3907 y Fn(1)838 3893 y Fu(and)38 b Fq(s)1068 3907 y Fn(1)1189 3893 y Fp(=)46 b Fq(\017)1343 3850 y Fn(\000)1448 3823 y Fg(2)p 1408 3835 110 3 v 1408 3876 a FA(d)p Fg(+2)1569 3893 y Fu(\(see)38 b(section)g(5.1\).)69 b(The)38 b(basic)h(tool)f(is)h(represented)d(by)118 4001 y(the)30 b(following)f(improved)h(version)f(of)h(the)g(T)-7 b(wo)31 b(Blocks)e(Estimate)i(\(see)e(e.g.)45 b([23)o(]\),)31 b(whose)f(proof)118 4109 y(mainly)23 b(relies)f(on)h(the)h(same)f (techniques)g(used)g(for)g(proving)g(proposition)h(A.9)f(\(see)g (section)g(4.10)f(in)118 4217 y([15)o(]\).)118 4356 y FE(Proposition)27 b(5.9.)42 b Fh(\(Improved)25 b(T)-8 b(wo)24 b(Blocks)h(Estimate\))118 4469 y(Let)31 b Fq(d)j Fo(\025)g Fp(3)p Fh(,)e Fp(0)j Fq(<)e(\015)39 b(<)34 b(\015)978 4436 y Fn(0)1036 4469 y Fq(<)f Fp(1)e Fh(and)f(set)f Fq(s)34 b Fp(=)g Fq(\017)1744 4436 y Fn(\000)p Fr(\015)1843 4469 y Fh(,)e Fq(`)i Fp(=)g Fq(\017)2114 4436 y Fn(\000)p Fr(\015)2209 4412 y Fi(0)2235 4469 y Fh(.)47 b(Then,)32 b(for)e(any)g Fq(r)j Fh(such)d(that)h Fp(0)j Fq(<)g(r)j(<)118 4593 y Fp(min)270 4519 y Fk(\000)321 4548 y Fl(2\(1)p Fn(\000)p Fr(\015)513 4525 y Fi(0)537 4548 y Fl(\))p 321 4572 243 4 v 380 4624 a Fr(d)p Fl(+4)574 4593 y Fq(;)625 4552 y Fr(\015)p 625 4572 41 4 v 627 4624 a Fl(2)675 4519 y Fk(\001)742 4593 y Fh(and)24 b(for)g(almost)h(any)g(disorder)e (con\002guration)h Fq(\013)195 4809 y Fp(lim)15 b(sup)221 4888 y Fr(a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)488 4809 y Fp(Av)606 4831 y Fr(x)p Fn(2)p Fm(T)743 4812 y FA(d)743 4848 y(\017)792 4809 y Fp(sup)841 4881 y Fr(\027)944 4809 y Fp(sup)f Fq(spec)1266 4829 y Fr(L)1314 4810 y Fg(2)1349 4829 y Fl(\()p Fr(\027)t Fl(\))1447 4736 y Fk(\010)1500 4809 y Fq(\017)1537 4772 y Fn(\000)p Fr(r)1630 4809 y Fp(Av)1747 4752 y Fr(`;)1806 4725 y FA(a)p 1806 4737 34 3 v 1810 4778 a(\017)1747 4821 y Fr(w)r(;x)1878 4809 y Fq(Av)1993 4772 y Fr(s;`)1990 4832 y(z)s(;w)2118 4809 y Fo(j)p Fq(m)2223 4772 y Fr(e)2223 4832 y(z)s(;s)2335 4809 y Fo(\000)20 b Fq(m)2506 4772 y Fr(e)2506 4832 y(w)r(;`)2611 4809 y Fo(j)g Fp(+)g Fq(\017)2784 4772 y Fn(\000)p Fl(2)2878 4809 y Fp(Av)2996 4824 y Fr(b)p Fn(2)p Fl(\003)3122 4842 y FA(x;)p Fg(2)3216 4823 y FA(a)p 3216 4831 V 3220 4864 a(\017)3268 4809 y Fo(L)3331 4824 y Fr(b)3365 4736 y Fk(\011)3443 4809 y Fo(\024)25 b Fp(0)118 5050 y Fh(where)g Fq(\027)31 b Fh(varies)24 b(among)g Fo(M)p Fp(\(\003)1190 5068 y Fr(x;)p Fl(2)1295 5040 y FA(a)p 1295 5052 V 1299 5094 a(\017)1343 5050 y Fp(\))p Fh(.)118 5201 y FE(Corollary)k(5.10.)41 b Fh(Let)25 b Fq(d)h Fo(\025)f Fp(3)h Fh(and)e Fp(0)h Fq(<)g(\016)30 b Fo(\034)25 b Fp(1)p Fh(.)32 b(Then)1559 5418 y Fp(Av)1679 5361 y Fl(\026)-38 b Fr(s;)1739 5333 y FA(a)p 1738 5345 V 1742 5387 a(\017)1676 5430 y Fr(z)s(;x)1786 5418 y Fq(\034)1826 5432 y Fr(z)1876 5356 y Fq(\036)1933 5370 y Fl(\026)g Fr(s)o(;)s Fl(\026)g Fr(s)p 1876 5397 144 4 v 1930 5480 a Fp(\026)-49 b Fq(s)2054 5418 y Fo(\031)25 b Fp(0)p Fq(:)1189 b Fu(\(5.22\))p eop %%Page: 33 33 33 32 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(33)118 370 y Fh(Proof)-8 b(.)42 b Fu(F)-6 b(or)27 b(simplicity)j(of)e (notation)h(we)f(omit)g(the)h(bar)f(in)k Fp(\026)-49 b Fq(s)28 b Fu(and)g(we)g(set)h Fp(\001)p Fq(m)h Fp(:=)g Fq(m)3106 326 y Fl(2)p Fr(;e)3106 382 y(s)3220 370 y Fo(\000)22 b Fq(m)3393 326 y Fl(1)p Fr(;e)3393 382 y(s)3512 370 y Fu(and)118 478 y Fq(N)35 b Fp(:=)26 b Fq(N)421 492 y Fl(\003)470 474 y FA(e)470 509 y(s)507 478 y Fu(.)118 615 y(Let)291 591 y Fp(^)279 615 y Fq(\036)333 629 y Fr(s;s)422 615 y Fp(\()p Fq(\021)s Fp(\))48 b(=)f Fq(\026)761 567 y Fr(\025)p Fl(\()p Fr(m)891 544 y FA(e)891 584 y(s)925 567 y Fl(\()p Fr(\021)r Fl(\)\))761 644 y(\003)810 625 y FA(e)810 660 y(s)1050 615 y Fp(\(\001)p Fq(m)p Fp(\))p Fu(.)71 b(Then,)41 b(by)e(the)f(equivalence)f(of)h(ensembles)f(\(see)h (proposition)118 756 y(A.4\),)32 b(it)f(is)g(enough)e(to)i(prove)e (\(5.22\))h(with)h Fq(\036)1759 770 y Fr(s;s)1879 756 y Fu(replaced)e(by)2383 732 y Fp(^)2371 756 y Fq(\036)2425 770 y Fr(s;s)2514 756 y Fu(.)47 b(Let)30 b Fq(m)g Fu(be)g(a)h(particle) f(density)118 869 y(on)f Fp(\003)311 836 y Fr(e)311 892 y(s)378 869 y Fu(that,)i(without)g(loss)f(of)g(generality)-12 b(,)29 b(we)g(can)g(suppose)g(in)h Fp(\(0)p Fq(;)2550 833 y Fl(1)p 2550 848 36 4 v 2550 901 a(2)2595 869 y Fp(\))g Fu(and)g(set)f Fq(\025)k Fp(:=)f Fq(\025)3246 883 y Fl(\003)3295 864 y FA(e)3295 900 y(s)3332 869 y Fp(\()p Fq(m)p Fp(\))e Fu(and)118 977 y Fq(\025)171 991 y Fl(0)236 977 y Fp(:=)25 b Fq(\025)410 991 y Fl(0)450 977 y Fp(\()p Fq(m)p Fp(\))p Fu(.)32 b(Then,)25 b(by)g(T)-9 b(aylor)25 b(expansion,)361 1139 y Fq(\026)416 1102 y Fr(\025)461 1139 y Fp(\(\001)p Fq(m)p Fp(\))h(=)f Fq(\026)864 1102 y Fr(\025)905 1111 y Fg(0)943 1139 y Fp(\(\001)p Fq(m)p Fp(\))c(+)f Fq(\026)1336 1102 y Fr(\025)1377 1111 y Fg(0)1415 1139 y Fp(\(\001)p Fq(m)p Fp(;)15 b Fq(N)10 b Fp(\)\()p Fq(\025)22 b Fo(\000)e Fq(\025)2018 1153 y Fl(0)2057 1139 y Fp(\))h(+)f Fq(\026)2259 1102 y Fr(\025)2300 1078 y Fi(0)2326 1139 y Fp(\(\001)p Fq(m)p Fp(;)15 b Fq(N)10 b Fp(;)15 b Fq(N)10 b Fp(\)\()p Fq(\025)22 b Fo(\000)d Fq(\025)3051 1153 y Fl(0)3091 1139 y Fp(\))3126 1102 y Fl(2)3409 1139 y Fu(\(5.23\))118 1290 y(where)25 b Fq(\025)442 1257 y Fn(0)490 1290 y Fu(is)h(between)f Fq(\025)g Fu(and)g Fq(\025)1243 1304 y Fl(0)1283 1290 y Fu(.)118 1398 y(Let)h(us)f(observe)g(that)h Fo(j)p Fq(\026)971 1365 y Fr(\025)1012 1341 y Fi(0)1039 1398 y Fp(\(\001)p Fq(m)p Fp(;)15 b Fq(N)10 b Fp(;)15 b Fq(N)10 b Fp(\))p Fo(j)26 b(\024)f Fq(c)p Fu(,)h(while)f(by)h(lemma)e(A.3)1361 1612 y Fo(j)p Fq(\025)d Fo(\000)f Fq(\025)1604 1626 y Fl(0)1643 1612 y Fo(j)26 b(\024)f Fq(c)15 b Fo(j)p Fp(1)21 b Fo(\000)2036 1551 y Fq(\026)2091 1518 y Fr(\025)2132 1527 y Fg(0)2170 1551 y Fp(\()p Fq(m)2285 1518 y Fr(e)2285 1573 y(s)2323 1551 y Fp(\))p 2036 1591 323 4 v 2157 1674 a Fq(m)2368 1612 y Fo(j)p Fq(:)118 1804 y Fu(Moreover)-10 b(,)39 b Fs(E)608 1731 y Fk(\002)652 1804 y Fq(\026)707 1771 y Fr(\025)748 1780 y Fg(0)787 1804 y Fp(\()p Fq(m)902 1771 y Fr(e)902 1827 y(s)939 1804 y Fp(\))974 1731 y Fk(\003)1057 1804 y Fp(=)45 b Fq(m)37 b Fu(and)g Fs(E)1530 1731 y Fk(\002)1574 1804 y Fq(\026)1629 1771 y Fr(\025)1670 1780 y Fg(0)1709 1804 y Fp(\(\001)p Fq(m)1900 1818 y Fr(s)1937 1804 y Fp(;)15 b Fq(N)10 b Fp(\))2095 1731 y Fk(\003)2178 1804 y Fp(=)45 b(0)p Fu(.)67 b(Therefore,)39 b(thanks)f(to)f(the)h(large)118 1947 y(deviations)g(estimate)g(of)f (lemma)e(A.1)i(applied)g(to)h(the)f(function)f Fq(f)10 b Fp(\()p Fq(\013)p Fp(\))45 b(:=)2879 1903 y Fr(\026)2921 1879 y FA(\025)2958 1894 y Fg(0)2997 1903 y Fl(\()p Fr(\021)3059 1912 y Fg(0)3094 1903 y Fl(\))p 2879 1927 243 4 v 2969 1979 a Fr(m)3160 1947 y Fo(\000)27 b Fp(1)p Fu(,)41 b(for)c(any)118 2055 y Fq(\014)31 b Fo(2)24 b Fp(\(0)p Fq(;)15 b Fp(1\))28 b Fu(and)d Fq(\017)g Fu(small)g(enough)706 2249 y Fs(P)p Fp(\()p Fo(j)p Fq(\025)d Fo(\000)e Fq(\025)1040 2263 y Fl(0)1079 2249 y Fo(j)26 b(\025)f Fq(s)1269 2212 y Fn(\000)1334 2184 y FA(d)p 1333 2196 33 3 v 1334 2238 a Fg(2)1375 2212 y Fl(+)1440 2181 y FA(\014)p 1440 2196 38 3 v 1443 2238 a Fg(2)1491 2249 y Fp(\))h Fo(\024)f Fs(P)p Fp(\()p Fo(j)p Fp(Av)1881 2263 y Fr(x)p Fn(2)p Fl(\003)2017 2244 y FA(e)2017 2280 y(s)2055 2249 y Fq(\034)2095 2263 y Fr(x)2154 2249 y Fq(f)10 b Fo(j)24 b(\025)2364 2188 y Fp(1)p 2364 2228 46 4 v 2367 2311 a Fq(c)2420 2249 y(s)2463 2212 y Fn(\000)2528 2184 y FA(d)p 2527 2196 33 3 v 2528 2238 a Fg(2)2569 2212 y Fl(+)2634 2181 y FA(\014)p 2634 2196 38 3 v 2637 2238 a Fg(2)2685 2249 y Fp(\))i Fo(\024)f Fq(e)2884 2212 y Fn(\000)p Fr(cs)3003 2188 y FA(\014)3048 2249 y Fq(:)118 2441 y Fu(A)33 b(similar)g (reasoning)f(applies)i(to)f(the)g(term)g Fq(\026)1790 2408 y Fr(\025)1831 2417 y Fg(0)1869 2441 y Fp(\(\001)p Fq(m)p Fp(;)15 b Fq(N)10 b Fp(\))34 b Fu(if)f(we)g(consider)g(instead)h (the)f(function)118 2552 y Fq(f)10 b Fp(\()p Fq(\013)p Fp(\))40 b(:=)f Fq(\026)531 2519 y Fr(\025)572 2528 y Fg(0)611 2552 y Fp(\()p Fq(\021)691 2566 y Fl(0)730 2552 y Fp(;)15 b Fq(\021)815 2566 y Fl(0)855 2552 y Fp(\))27 b Fo(\000)e Fs(E)1068 2478 y Fk(\000)1115 2552 y Fq(\026)1170 2519 y Fr(\025)1211 2528 y Fg(0)1250 2552 y Fp(\()p Fq(\021)1330 2566 y Fl(0)1370 2552 y Fp(;)15 b Fq(\021)1455 2566 y Fl(0)1495 2552 y Fp(\))1530 2478 y Fk(\001)1572 2552 y Fu(.)56 b(The)34 b(above)f(bounds)h(together)f(with)i(the)f(fact)f (that)i(the)118 2659 y(number)e(of)h(possible)h(choices)g(of)f Fq(m)g Fu(is)i(polynomially)d(bounded)h(in)g Fq(s)g Fu(and)h(together)f (with)i(Borel)118 2767 y(Cantelli)25 b(lemma,)f(implies)h(in)g (particular)f(that)i(for)e(almost)h(all)f(disorder)i(con\002guration)e Fq(\013)h Fu(and)g(for)g Fq(\017)118 2875 y Fu(small)g(enough)1064 2996 y Fp(sup)1049 3082 y Fr(x)p Fn(2)p Fm(T)1186 3063 y FA(d)1186 3099 y(\017)1232 2996 y Fo(k)p Fq(\034)1317 3010 y Fr(x)1361 2996 y Fp(\()1423 2972 y(^)1411 2996 y Fq(\036)1465 3010 y Fr(s;s)1575 2996 y Fo(\000)19 b Fq(\026)1720 2958 y Fr(\025)1761 2967 y Fg(0)1796 2958 y Fl(\()p Fr(m)1885 2935 y FA(e)1885 2975 y(s)1919 2958 y Fl(\))1951 2996 y Fp(\(\001)p Fq(m)p Fp(\))c(\))p Fo(k)2272 3010 y Fn(1)2373 2996 y Fo(\024)25 b Fq(s)2512 2958 y Fn(\000)p Fr(d)p Fl(+)p Fr(\014)2705 2996 y Fq(:)118 3207 y Fu(Thanks)55 b(to)g(the)g(above)f(estimate)h(it)g(is)g(enough)e (to)i(prove)f(\(5.22\))f(with)j Fq(\036)3032 3221 y Fr(s;s)3175 3207 y Fu(replaced)d(by)118 3325 y Fq(\026)173 3292 y Fr(\025)214 3301 y Fg(0)249 3292 y Fl(\()p Fr(m)338 3268 y FA(e)338 3308 y(s)372 3292 y Fl(\))404 3325 y Fp(\(\001)p Fq(m)p Fp(\))p Fu(,)25 b(that)i(is)1181 3509 y Fp(Av)1298 3452 y Fr(s;)1361 3425 y FA(a)p 1361 3437 34 3 v 1365 3478 a(\017)1298 3521 y Fr(z)s(;x)1408 3509 y Fq(\026)1463 3472 y Fr(\025)1504 3481 y Fg(0)1539 3472 y Fl(\()p Fr(m)1628 3448 y FA(e)1628 3488 y(z)r(;s)1713 3472 y Fl(\))1744 3408 y Fk(\020)1799 3509 y Fq(\034)1839 3523 y Fr(z)1888 3448 y Fq(m)1968 3404 y Fl(2)p Fr(;e)1968 3460 y(s)2080 3448 y Fo(\000)20 b Fq(m)2251 3404 y Fl(1)p Fr(;e)2251 3460 y(s)p 1888 3488 455 4 v 2094 3572 a Fq(s)2352 3408 y Fk(\021)2432 3509 y Fo(\031)25 b Fp(0)p Fq(:)811 b Fu(\(5.24\))118 3671 y(W)-8 b(e)26 b(assert)g(that)g(we)f(only)g(need)f (to)i(show)g(that)942 3894 y(l.h.s.)32 b(of)26 b Fp(\(5)p Fq(:)p Fp(24\))h Fo(\031)e Fp(Av)1737 3836 y Fr(`;)1796 3809 y FA(a)p 1795 3821 34 3 v 1799 3862 a(\017)1737 3906 y Fr(z)s(;x)1843 3894 y Fq(\026)1898 3851 y Fr(\025)1939 3860 y Fg(0)1973 3851 y Fl(\()p Fr(m)2062 3828 y FA(e)2062 3874 y(z)r(;`)2144 3851 y Fl(\))2175 3793 y Fk(\020)2230 3894 y Fq(\034)2270 3908 y Fr(z)2319 3829 y Fq(m)2399 3785 y Fl(2)p Fr(;e)2399 3858 y(`)2511 3829 y Fo(\000)20 b Fq(m)2682 3785 y Fl(1)p Fr(;e)2682 3858 y(`)p 2319 3873 455 4 v 2527 3956 a Fq(`)2783 3793 y Fk(\021)3409 3894 y Fu(\(5.25\))118 4083 y(where)39 b Fq(`)48 b Fp(=)g Fq(\017)645 4050 y Fl(1)p Fn(\000)p Fr(\032)814 4083 y Fu(is)40 b(a)f(new)g(mesoscopic)g(scale)f(with)i Fp(0)49 b Fq(<)f(\032)h(<)f Fp(1)39 b Fu(so)h(small)e(that)i Fq(s)48 b(<)g(`)39 b Fu(and)118 4210 y Fq(\017)155 4177 y Fn(\000)p Fl(1)249 4210 y Fq(`)287 4174 y Fn(\000)352 4146 y FA(d)p Fg(+2)p 352 4159 110 3 v 391 4200 a(2)500 4210 y Fo(#)26 b Fp(0)c Fu(as)h Fq(\017)i Fo(#)g Fp(0)p Fu(.)31 b(In)22 b(fact,)h(thanks)g(to)f(lemma)e(A.1)i(applied)g(with)h Fq(f)10 b Fp(\()p Fq(\013)p Fp(\))26 b(:=)f Fq(\026)3005 4167 y Fr(\025)3046 4176 y Fg(0)3080 4167 y Fl(\()p Fr(m)3169 4144 y FA(e)3169 4190 y(z)r(;`)3251 4167 y Fl(\))3282 4210 y Fp(\()p Fq(\021)3362 4224 y Fl(0)3411 4210 y Fo(\000)8 b Fq(\021)3535 4225 y Fr(`e)3601 4210 y Fp(\))p Fu(,)118 4317 y(given)36 b Fp(0)44 b Fq(<)f(\014)49 b Fo(\034)44 b Fp(1)37 b Fu(for)f(almost)h(any)f(disorder)h(con\002guration)e Fq(\013)i Fu(and)f(for)h Fq(")f Fu(small)g(enough)f(the)118 4444 y(r)-10 b(.h.s.)44 b(of)30 b(\(5.25\))d(is)j(bounded)f(by)g Fq(`)1368 4408 y Fn(\000)1433 4380 y FA(d)p Fg(+2)p 1433 4393 V 1473 4435 a(2)1552 4408 y Fl(+)p Fr(\014)1654 4444 y Fu(.)44 b(Because)28 b(of)h(our)g(choice)f(of)i Fq(`)p Fu(,)g(the)f(r)-10 b(.h.s.)44 b(of)30 b(\(5.25\))e(is)118 4552 y(equivalent)d(to)g Fp(0)p Fu(.)118 4660 y(Let)h(us)f(prove)g (\(5.25\).)30 b(T)-9 b(o)26 b(this)g(aim,)g(we)f(observe)g(that)h (thanks)g(to)g(\(5.5\))e(and)h(\(4.27\))937 4855 y(l.h.s.)33 b(of)25 b Fp(\(5)p Fq(:)p Fp(24\))j(=)d(Av)1732 4797 y Fr(`;)1791 4770 y FA(a)p 1791 4782 34 3 v 1795 4823 a(\017)1732 4867 y Fr(w)r(;x)1863 4855 y Fp(Av)1981 4817 y Fr(s;`)1981 4877 y(z)s(;w)2108 4855 y Fq(\026)2163 4817 y Fr(\025)2204 4826 y Fg(0)2238 4817 y Fl(\()p Fr(m)2327 4794 y FA(e)2327 4834 y(z)r(;s)2412 4817 y Fl(\))2443 4754 y Fk(\020)2498 4855 y Fq(\034)2538 4869 y Fr(z)2587 4793 y Fp(\001)p Fq(m)p 2587 4834 156 4 v 2644 4917 a(s)2753 4754 y Fk(\021)2807 4855 y Fq(;)937 5076 y Fu(r)-10 b(.h.s.)33 b(of)25 b Fp(\(5)p Fq(:)p Fp(25\))j(=)d(Av)1731 5019 y Fr(`;)1790 4992 y FA(a)p 1790 5004 34 3 v 1794 5045 a(\017)1731 5088 y Fr(w)r(;x)1862 5076 y Fp(Av)1980 5039 y Fr(s;`)1980 5099 y(z)s(;w)2107 5076 y Fq(\026)2162 5033 y Fr(\025)2203 5042 y Fg(0)2237 5033 y Fl(\()p Fr(m)2326 5010 y FA(e)2326 5057 y(w)q(;`)2422 5033 y Fl(\))2453 4975 y Fk(\020)2508 5076 y Fq(\034)2548 5090 y Fr(z)2597 5015 y Fp(\001)p Fq(m)p 2597 5055 156 4 v 2654 5138 a(s)2763 4975 y Fk(\021)2817 5076 y Fq(:)118 5257 y Fu(Therefore,)g(we)g(only)f (need)h(to)g(prove)g(that)765 5451 y Fp(Av)883 5394 y Fr(`;)942 5367 y FA(a)p 941 5379 34 3 v 945 5420 a(\017)883 5463 y Fr(w)r(;x)1014 5451 y Fp(Av)1131 5414 y Fr(s;`)1131 5474 y(z)s(;w)1243 5350 y Fk(\020)1297 5451 y Fq(\026)1352 5414 y Fr(\025)1393 5423 y Fg(0)1428 5414 y Fl(\()p Fr(m)1517 5390 y FA(e)1517 5430 y(z)r(;s)1602 5414 y Fl(\))1633 5378 y Fk(\000)1675 5451 y Fq(\034)1715 5465 y Fr(z)1764 5390 y Fp(\001)p Fq(m)p 1764 5430 156 4 v 1821 5514 a(s)1930 5378 y Fk(\001)1992 5451 y Fo(\000)20 b Fq(\026)2138 5409 y Fr(\025)2179 5418 y Fg(0)2213 5409 y Fl(\()p Fr(m)2302 5385 y FA(e)2302 5432 y(w)q(;`)2398 5409 y Fl(\))2429 5378 y Fk(\000)2471 5451 y Fq(\034)2511 5465 y Fr(z)2561 5390 y Fp(\001)p Fq(m)p 2561 5430 V 2617 5514 a(s)2726 5378 y Fk(\001)2768 5350 y(\021)2847 5451 y Fo(\031)25 b Fp(0)p Fq(:)p eop %%Page: 34 34 34 33 bop 118 45 a Fj(34)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)218 366 y Fu(Let)26 b(us)f(assume)h(for)f(the)h(moment)f(that,)i(given)e Fp(0)h Fq(<)g(\014)31 b Fo(\034)26 b Fp(1)p Fu(,)g(for)g(almost)g(all)f (disorder)h(con\002gura-)118 474 y(tion)g Fq(\013)g Fu(and)f Fq(\017)g Fu(small)g(enough)227 651 y Fp(sup)211 737 y Fr(x)p Fn(2)p Fm(T)348 718 y FA(d)348 753 y(\017)394 651 y Fo(j)p Fq(\026)474 613 y Fr(\025)515 622 y Fg(0)549 613 y Fl(\()p Fr(m)p Fl(\))671 577 y Fk(\000)713 651 y Fq(\034)753 665 y Fr(x)796 651 y Fp(\001)p Fq(m)952 577 y Fk(\001)1014 651 y Fo(\000)20 b Fq(\026)1160 613 y Fr(\025)1201 622 y Fg(0)1235 613 y Fl(\()p Fr(m)1324 590 y Fi(0)1347 613 y Fl(\))1379 577 y Fk(\000)1421 651 y Fq(\034)1461 665 y Fr(x)1504 651 y Fp(\001)p Fq(m)1660 577 y Fk(\001)1701 651 y Fo(j)26 b(\024)f Fq(cs)1930 613 y Fn(\000)1995 586 y FA(d)p 1995 598 33 3 v 1996 639 a Fg(2)2037 613 y Fl(+)p Fr(\014)2139 651 y Fo(j)p Fq(m)20 b Fo(\000)g Fq(m)2435 613 y Fn(0)2458 651 y Fo(j)h Fp(+)e Fq(cs)2676 613 y Fn(\000)2741 586 y FA(d)p 2741 598 V 2742 639 a Fg(2)2783 613 y Fn(\000)p Fr(\014)2976 651 y Fo(8)p Fq(m;)c(m)3227 613 y Fn(0)3275 651 y Fo(2)25 b Fp([0)p Fq(;)15 b Fp(1])p Fq(:)3409 836 y Fu(\(5.26\))118 944 y(Then)23 b(it)g(is)g(simple)g(to)g(deduce)f(\(5.25\))f(from)h (\(5.26\))f(and)i(proposition)h(5.9)e(with)i Fq(\015)30 b Fp(=)3181 908 y Fl(2)p 3136 923 127 4 v 3136 975 a Fr(d)p Fl(+2)3283 944 y Fo(\000)11 b Fq(\016)19 b(;)41 b(\015)3542 911 y Fn(0)3591 944 y Fp(=)118 1072 y(1)21 b Fo(\000)f Fq(\032)25 b Fu(and)g Fq(r)j Fp(=)d Fo(\000)p Fq(\016)s(\014)h Fp(+)978 1036 y Fr(d)p Fl(+2)p 978 1051 V 1024 1104 a(2)1114 1072 y Fq(\016)e Fp(+)1325 1036 y Fl(2)p 1279 1051 V 1279 1104 a Fr(d)p Fl(+2)1416 1072 y Fq(\014)30 b Fu(by)25 b(choosing)h(suitable)f Fp(0)h Fq(<)f(\014)30 b Fo(\034)25 b Fq(\016)k Fo(\034)d Fq(\032)f Fo(\034)g Fp(1)p Fu(.)118 1186 y(It)k(remains)e(to)h(prove)f(\(5.26\).) 38 b(F)-6 b(or)27 b(simplicity)i(of)f(notation,)h(let)f(us)g(consider)f (only)g(the)h(case)f Fq(x)j Fp(=)f(0)118 1294 y Fu(\(the)h(general)d (case)i(is)h(a)f(simple)g(variation\).)45 b(By)29 b(continuity)-12 b(,)30 b(we)f(may)g(assume)h Fp(0)i Fq(<)g(m)f(<)h(m)3458 1261 y Fn(0)3513 1294 y Fq(<)g Fp(1)118 1402 y Fu(and)25 b(by)h(T)-9 b(aylor)25 b(expansion,)616 1567 y Fo(j)p Fq(\026)696 1530 y Fr(\025)737 1539 y Fg(0)772 1530 y Fl(\()p Fr(m)861 1506 y Fi(0)884 1530 y Fl(\))916 1494 y Fk(\000)958 1567 y Fp(\001)p Fq(m)1114 1494 y Fk(\001)1175 1567 y Fo(\000)20 b Fq(\026)1321 1530 y Fr(\025)1362 1539 y Fg(0)1396 1530 y Fl(\()p Fr(m)p Fl(\))1518 1494 y Fk(\000)1560 1567 y Fp(\001)p Fq(m)1716 1494 y Fk(\001)1757 1567 y Fo(j)25 b Fp(=)g Fo(j)p Fq(\026)1983 1530 y Fr(\025)2024 1539 y Fg(0)2059 1530 y Fl(\()14 b(\026)-49 b Fr(m)q Fl(\))2181 1567 y Fp(\(\001)p Fq(m)p Fp(;)15 b Fq(N)10 b Fp(\))p Fq(\025)2583 1530 y Fn(0)2583 1590 y Fl(0)2623 1567 y Fp(\()17 b(\026)-62 b Fq(m)p Fp(\)\()p Fq(m)2888 1530 y Fn(0)2932 1567 y Fo(\000)20 b Fq(m)p Fp(\))p Fo(j)1807 1783 y(\024)25 b Fq(c)15 b Fo(j)1992 1722 y Fq(\026)2047 1689 y Fr(\025)2088 1698 y Fg(0)2124 1689 y Fl(\()f(\026)-49 b Fr(m)p Fl(\))2245 1722 y Fp(\(\001)p Fq(m)p Fp(;)15 b Fq(N)10 b Fp(\))p 1992 1763 602 4 v 2271 1846 a(\026)-62 b Fq(m)2605 1783 y Fo(j)p Fp(\()p Fq(m)2745 1746 y Fn(0)2789 1783 y Fo(\000)19 b Fq(m)p Fp(\))118 1982 y Fu(where)30 b Fq(m)k(<)51 b Fp(\026)-62 b Fq(m)34 b(<)g(m)912 1949 y Fn(0)935 1982 y Fu(.)48 b(If)31 b(we)f(could)g(restrict)h(the)g (possible)g(values)f(of)48 b Fp(\026)-62 b Fq(m)30 b Fu(to)h Fo(f)p Fq(s)3000 1949 y Fn(\000)p Fr(d)3096 1982 y Fq(;)15 b Fp(2)p Fq(s)3224 1949 y Fn(\000)p Fr(d)3319 1982 y Fq(;)g(:)g(:)g(:)i(;)e Fp(1)25 b Fo(\000)118 2092 y Fq(s)161 2059 y Fn(\000)p Fr(d)256 2092 y Fo(g)p Fu(,)42 b(then,)e(by)e(means)f(of)h(large)e(deviations)j(estimate)f(as)g(in)g (the)g(\002rst)g(part)g(of)g(the)g(proof)m(,)j(we)118 2220 y(would)d(obtain)718 2185 y Fl(1)p 704 2200 63 4 v 718 2252 a(\026)-49 b Fr(m)777 2220 y Fo(j)p Fq(\026)857 2187 y Fr(\025)898 2196 y Fg(0)933 2187 y Fl(\()14 b(\026)-49 b Fr(m)p Fl(\))1054 2220 y Fp(\(\001)p Fq(m)p Fp(;)15 b Fq(N)10 b Fp(\))p Fo(j)48 b(\024)f Fq(c)15 b(s)1691 2185 y Fn(\000)1756 2157 y FA(d)p 1756 2169 33 3 v 1757 2211 a Fg(2)1798 2185 y Fl(+)p Fr(\014)1939 2220 y Fu(for)38 b(almost)h(any)f(disorder)h Fq(\013)g Fu(and)f(for)h Fq(\017)f Fu(small)118 2328 y(enough,)20 b(thus)h(implying)f(\(5.26\).) 28 b(The)21 b(complete)e(proof)h(requires)f(some)h(addional)h (straightforward)118 2436 y(computations)26 b(\(see)e(also)i(section)f Fp(4)p Fq(:)p Fp(10)i Fu(in)e([15]\).)1706 b Fe(\003)873 2650 y Fu(6.)48 b(S)t Ft(O)t(M)t(E)30 b(T)t(E)t(C)t(H)t(N)t(I)t(C)s(A)t (L)k(R)t(E)t(S)t(U)t(L)n(T)t(S)g(N)t(E)t(E)t(D)t(E)t(D)e(I)t(N)e(S)t(E) t(C)t(T)t(I)t(O)t(N)j Fu(5)218 2812 y(In)k(this)h(section)f(we)g (collect)e(some)i(technical)f(results,)k(mostly)d(based)g(on)g (estimates)h(of)f(large)118 2920 y(deviations)30 b(in)e(the)g(disorder) h(\002eld)f Fq(\013)p Fu(,)i(that)f(are)f(used)g(in)g(the)h(proof)f(of) g(theorem)g(5.3.)39 b(Our)28 b(bounds)118 3028 y(mainly)22 b(concern)d(canonical)i(or)h(grand)f(canonical)f(variances)i(of)g (suitable)g(spatial)h(averages)e(of)i(local)118 3136 y(functions.)39 b(Such)26 b(variances)h(arise)h(naturally)f(from)g (eigenvalue)e(estimates)k(via)f(perturbation)f(the-)118 3244 y(ory)-12 b(.)30 b(W)-8 b(e)24 b(have)f(seen)f(in)h(fact)g(that,)i (when)e(perturbation)f(theory)h(applies)h(\(see)e(proposition)i(4.2\),) f(the)118 3352 y(maximal)e(eigenvalue)f(is)i(bounded)f(by)h(an)f (expression)h(containing)f(an)g(H)2687 3366 y Fn(\000)p Fl(1)2803 3352 y Fu(norm)g(that,)j(in)d(general,)118 3460 y(can)k(be)g(bounded)g(from)f(above)h(by:)1072 3618 y Fq(\027)1123 3544 y Fk(\000)1165 3618 y Fq(f)5 b(;)15 b Fo(\000L)1389 3579 y Fn(\000)p Fl(1)1389 3646 y(\003)1482 3618 y Fq(f)1537 3544 y Fk(\001)1603 3618 y Fo(\024)25 b Fq(c)15 b(`)1791 3580 y Fl(2)1831 3618 y Fp(V)-8 b(ar)1973 3632 y Fr(\027)2016 3618 y Fp(\()p Fq(f)10 b Fp(\))25 b Fo(\024)g Fq(c)15 b(`)2354 3580 y Fl(2)2394 3618 y Fp(V)-8 b(ar)2536 3632 y Fr(\026)2582 3618 y Fp(\()p Fq(f)10 b Fp(\))752 b Fu(\(6.1\))118 3774 y(where)32 b Fq(\027)38 b Fu(is)33 b(a)f(canonical)f(measure)g(on)h(the)g(cube)f Fp(\003)i Fu(of)f(side)h Fq(`)f Fu(with)i(disorder)f Fq(\013)p Fu(,)h Fq(\026)e Fu(is)h(the)g(corre-)118 3882 y(sponding)e(grand)f(canonical)f(measure)g(\(with)j(suitable)f (empirical)f(chemical)f(potential\))i(and)g Fq(f)40 b Fu(is)118 3990 y(a)30 b(\(mean)f(zero)g(w)-11 b(.r)h(.t.)47 b Fq(\027)6 b Fu(\))30 b(function.)46 b(Above)30 b(we)f(used)i(the)f (spectral)g(gap)g(bound)f Fp(gap)q(\()p Fo(L)3264 4004 y Fl(\003)3317 3990 y Fp(\))34 b Fo(\025)f Fq(c`)3567 3957 y Fn(\000)p Fl(2)118 4098 y Fu(together)25 b(with)h(lemma)e(A.5.) 218 4206 y(When)j(the)g(function)f Fq(f)37 b Fu(is)27 b(the)g(spatial)i(average)c(of)i(local)f(functions)h Fo(f)p Fq(f)2734 4220 y Fr(i)2762 4206 y Fo(g)2807 4220 y Fr(i)p Fn(2)p Fr(I)2945 4206 y Fu(each)g(with)h(support)118 4314 y(much)d(smaller)f(than)h Fp(\003)h Fu(it)g(is)g(possible)g(to)f (do)h(better)f(than)g(\(6.1\).)31 b(W)-8 b(e)26 b(have)f(in)g(fact:)118 4446 y FE(L)r(emma)37 b(6.1.)46 b Fh(Let)35 b Fp(\003)f Fh(be)h(a)f(box)g(in)g Fs(Z)1462 4413 y Fr(d)1533 4446 y Fh(and)f Fo(f)p Fp(\003)1821 4460 y Fr(i)1850 4446 y Fo(g)1895 4460 y Fr(i)p Fn(2)p Fr(I)2041 4446 y Fh(be)h(a)g(family)g (of)g(cubes)g Fp(\003)2928 4460 y Fr(i)2997 4446 y Fo(\032)40 b Fp(\003)35 b Fh(with)f(side)g Fq(R)118 4554 y Fh(satisfying)1166 4599 y Fk(\014)1166 4654 y(\014)1211 4676 y Fo(f)p Fq(i)26 b Fo(2)f Fq(I)48 b Fp(:)40 b Fq(x)26 b Fo(2)e Fp(\003)1778 4690 y Fr(i)1807 4676 y Fo(g)1867 4599 y Fk(\014)1867 4654 y(\014)1923 4676 y Fo(\024)h Fp(10)2109 4639 y Fl(10)p Fr(d)2312 4676 y Fo(8)p Fq(x)f Fo(2)h Fp(\003)p Fq(:)118 4814 y Fh(Let)j Fq(f)38 b Fp(=)29 b(Av)561 4828 y Fr(i)p Fn(2)p Fr(I)672 4814 y Fq(f)717 4828 y Fr(i)772 4814 y Fh(where,)e(for)g(any)h Fq(i)h Fo(2)g Fq(I)34 b Fh(and)27 b(for)g(all)g Fq(\013)p Fh(,)i Fq(f)2175 4828 y Fr(i)2230 4814 y Fh(has)e(support)h(in)f Fp(\003)2874 4828 y Fr(i)2930 4814 y Fh(and)f(has)i(zero)f(mean)118 4922 y(w)-5 b(.r)-12 b(.t.)45 b(any)29 b(canonical)f(measure)g(on)h Fp(\003)1455 4936 y Fr(i)1483 4922 y Fh(.)45 b(Then,)30 b(for)f(any)g(canonical)e (measure)i Fq(\027)35 b Fh(on)29 b Fp(\003)g Fh(with)h(disorder)118 5030 y(con\002guration)24 b Fq(\013)p Fh(,)951 5150 y Fq(\027)1002 5076 y Fk(\000)1043 5150 y Fq(f)5 b(;)15 b Fo(\000L)1267 5112 y Fn(\000)p Fl(1)1267 5178 y(\003)1361 5150 y Fq(f)1416 5076 y Fk(\001)1482 5150 y Fo(\024)25 b Fq(c)15 b(R)1702 5112 y Fl(2)1741 5150 y Fo(j)p Fq(I)7 b Fo(j)1838 5112 y Fn(\000)p Fl(1)1933 5150 y Fp(Av)2051 5164 y Fr(i)p Fn(2)p Fr(I)2162 5150 y Fq(\027)2213 5076 y Fk(\000)2269 5150 y Fp(V)-8 b(ar)2411 5164 y Fr(\027)2454 5150 y Fp(\()p Fq(f)2534 5164 y Fr(i)2577 5150 y Fo(j)15 b(F)2682 5164 y Fr(i)2711 5150 y Fp(\))2761 5076 y Fk(\001)2803 5150 y Fq(:)118 5322 y Fh(Proof)-8 b(.)42 b Fu(Let)25 b Fo(F)593 5336 y Fr(i)647 5322 y Fp(:=)g Fq(\033)s Fp(\()p Fq(m)938 5336 y Fl(\003)987 5346 y FA(i)1018 5322 y Fq(;)15 b(\021)1103 5336 y Fr(x)1173 5322 y Fu(with)26 b Fq(x)f Fo(62)g Fp(\003)1601 5336 y Fr(i)1629 5322 y Fp(\))h Fu(and)f(observe)g(that)1305 5480 y Fq(\027)6 b Fp(\()p Fq(f)1436 5494 y Fr(i)1464 5480 y Fq(;)15 b(g)s Fp(\))26 b(=)f Fq(\027)1758 5407 y Fk(\000)1800 5480 y Fq(\027)6 b Fp(\()p Fq(f)1931 5494 y Fr(i)1958 5480 y Fp(;)15 b Fq(g)k Fo(j)c(F)2165 5494 y Fr(i)2194 5480 y Fp(\))2229 5407 y Fk(\001)2362 5480 y Fo(8)g Fq(g)p eop %%Page: 35 35 35 34 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(35)118 366 y Fu(Thus,)26 b(by)g(Schwarz)e(and)h(P)-5 b(oincar\264)-45 b(e)24 b(inequalities)i(and)g(the)f(diffusive)i(scaling)d(of)i(the)f (spectral)g(gap)696 536 y Fo(j)p Fq(\027)6 b Fp(\()p Fq(f)f(;)15 b(g)s Fp(\))p Fo(j)27 b(\024)e Fq(c)15 b(R)h Fp(Av)1383 550 y Fr(i)p Fn(2)p Fr(I)1509 536 y Fq(\027)1560 462 y Fk(\010)f(\002)1665 536 y Fp(V)-8 b(ar)1807 550 y Fr(\027)1850 536 y Fp(\()p Fq(f)1930 550 y Fr(i)1973 536 y Fo(j)15 b(F)2078 550 y Fr(i)2107 536 y Fp(\))p Fo(D)2212 550 y Fl(\003)2261 560 y FA(i)2292 462 y Fk(\000)2334 536 y Fq(g)s Fp(;)g Fq(\027)6 b Fp(\()p Fo(\001jF)2621 550 y Fr(i)2651 536 y Fp(\))2686 462 y Fk(\001\003)2766 485 y Fl(1)p Fr(=)p Fl(2)2891 462 y Fk(\011)1030 736 y Fo(\024)25 b Fq(c)15 b(R)h Fo(j)p Fq(I)7 b Fo(j)1362 698 y Fn(\000)p Fl(1)p Fr(=)p Fl(2)1528 635 y Fk(\020)1582 736 y Fp(Av)1699 750 y Fr(i)p Fn(2)p Fr(I)1825 736 y Fq(\027)1876 662 y Fk(\000)1933 736 y Fp(V)-8 b(ar)2075 750 y Fr(\027)2118 736 y Fp(\()p Fq(f)2198 750 y Fr(i)2241 736 y Fo(j)15 b(F)2346 750 y Fr(i)2375 736 y Fp(\))2410 662 y Fk(\001)2452 635 y(\021)2506 658 y Fl(1)p Fr(=)p Fl(2)2616 736 y Fo(D)2686 750 y Fl(\003)2739 736 y Fp(\()p Fq(g)s Fp(;)g Fq(\027)6 b Fp(\))2946 698 y Fl(1)p Fr(=)p Fl(2)3058 736 y Fq(:)118 924 y Fu(It)26 b(is)g(enough)e(now)h(to)h (take)f Fq(g)k Fp(=)c Fo(\000L)1407 886 y Fn(\000)p Fl(1)1407 953 y(\003)1501 924 y Fq(f)10 b Fu(.)2010 b Fe(\003)118 1091 y Fu(6.1.)45 b FE(V)-10 b(ariance)35 b(bounds.)46 b Fu(One)33 b(of)g(the)g(key)f(issues)i(is)g(to)f(provide)g(sharp)h (enough)d(upper)h(bounds)118 1199 y(\(see)25 b(proposition)h(6.5)f (below\))g(on)f(the)i(variance)1399 1347 y Fp(V)-8 b(ar)1541 1375 y Fr(\026)1583 1353 y FA(\025)1620 1368 y Fg(0)1654 1353 y(\()p FA(m)p Fg(\))1765 1274 y Fk(\000)1807 1347 y Fp(Av)1924 1361 y Fr(x)p Fn(2)p Fl(\003)2060 1373 y FA(k)2102 1347 y Fq(\034)2142 1361 y Fr(x)2186 1347 y Fq(\036)2240 1361 y Fr(n;s)2339 1274 y Fk(\001)3459 1347 y Fu(\(6.2\))118 1515 y(where)28 b Fq(n;)15 b(s;)g(k)32 b Fu(are)d(positive)h(integers)e(satisfying)i Fq(n)h Fo(\024)g Fq(s)g Fo(\024)g Fq(k)h Fu(and)c Fq(m)j Fo(2)g Fp(\(0)p Fq(;)2839 1480 y Fl(1)p 2839 1495 36 4 v 2839 1547 a(2)2885 1515 y Fp(\))e Fu(and)g Fq(\036)3181 1529 y Fr(n;s)3309 1515 y Fu(has)h(been)118 1623 y(de\002ned)c(in)g (\(5.1\).)33 b(Actually)25 b(the)i(method)f(developed)g(below)f(is)i (very)e(general)f(and)i(it)h(can)f(be)f(used)118 1731 y(to)f(estimate)g(also)f(other)g(similar)h(variances,)f(like)h(for)f (example)e(\(6.2\))i(with)h Fq(\025)2843 1745 y Fl(0)2883 1731 y Fp(\()p Fq(m)p Fp(\))g Fu(replaced)e(by)h(the)118 1839 y(empirical)i(chemical)f(potential)h Fq(\025)1320 1853 y Fl(\003)1369 1865 y FA(k)1412 1839 y Fp(\()p Fq(\013;)15 b(m)p Fp(\))p Fu(.)218 1947 y(It)21 b(is)f(convenient)f(to)h(de\002ne)g (\002rst)g(some)g(additional)h(convenient)e(notation)h(besides)h(those) g(already)118 2055 y(de\002ned)k(at)h(the)f(beginning)f(of)i(section)f (5:)1184 2197 y Fp(^)1172 2221 y Fq(\036)1226 2235 y Fr(n;s)1325 2221 y Fp(\()p Fq(\021)s Fp(\))i(:=)e Fq(\026)1646 2173 y Fr(\025)p Fl(\()p Fr(m)1776 2150 y FA(e)1776 2190 y(s)1810 2173 y Fl(\))1646 2250 y(\003)1695 2231 y FA(e)1695 2267 y(s)1842 2221 y Fp(\()p Fq(m)1957 2184 y Fl(2)p Fr(;e)1957 2244 y(n)2069 2221 y Fo(\000)20 b Fq(m)2240 2184 y Fl(1)p Fr(;e)2240 2244 y(n)2332 2221 y Fp(\))1215 2389 y Fq(\030)1255 2403 y Fl(0)1294 2389 y Fp(\()p Fq(m)p Fp(\))26 b(:=)f Fq(\026)1646 2352 y Fr(\025)1687 2361 y Fg(0)1722 2352 y Fl(\()p Fr(m)p Fl(\))1843 2389 y Fp(\()15 b Fq(m)1973 2352 y Fl(2)p Fr(;e)1973 2412 y(n)2085 2389 y Fo(\000)20 b Fq(m)2256 2352 y Fl(1)p Fr(;e)2256 2412 y(n)2363 2389 y Fp(;)15 b Fq(N)2476 2403 y Fl(\003)2525 2384 y FA(e)2525 2420 y(n)2572 2389 y Fp(\))1250 2559 y Fq(\030)t Fp(\()p Fq(m)p Fp(\))26 b(:=)f Fq(\026)1646 2511 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))1646 2587 y(\003)1695 2569 y FA(e)1695 2604 y(s)1808 2559 y Fp(\()15 b Fq(m)1938 2521 y Fl(2)p Fr(;e)1938 2581 y(n)2051 2559 y Fo(\000)20 b Fq(m)2222 2521 y Fl(1)p Fr(;e)2222 2581 y(n)2328 2559 y Fp(;)15 b Fq(N)2441 2573 y Fl(\003)2490 2554 y FA(e)2490 2589 y(s)2528 2559 y Fp(\))1199 2727 y Fq(\033)1254 2689 y Fl(2)1251 2749 y(0)1294 2727 y Fp(\()p Fq(m)p Fp(\))26 b(:=)f Fq(\026)1646 2689 y Fr(\025)1687 2698 y Fg(0)1722 2689 y Fl(\()p Fr(m)p Fl(\))1843 2727 y Fp(\()15 b Fq(m)1973 2689 y Fr(e)1973 2749 y(s)2010 2727 y Fp(;)g Fq(N)2123 2741 y Fl(\003)2172 2722 y FA(e)2172 2757 y(s)2210 2727 y Fp(\))1199 2896 y Fq(\033)1254 2859 y Fl(2)1294 2896 y Fp(\()p Fq(m)p Fp(\))26 b(:=)f Fq(\026)1646 2848 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))1646 2925 y(\003)1695 2906 y FA(e)1695 2941 y(s)1808 2896 y Fp(\()15 b Fq(m)1938 2859 y Fr(e)1938 2919 y(s)1976 2896 y Fp(;)g Fq(N)2089 2910 y Fl(\003)2138 2891 y FA(e)2138 2927 y(s)2176 2896 y Fp(\))p Fq(;)1223 b Fu(\(6.3\))118 3057 y(where)33 b Fq(N)470 3071 y Fl(\003)519 3052 y FA(e)519 3088 y(n)566 3057 y Fu(,)j Fq(N)700 3071 y Fl(\003)749 3052 y FA(e)749 3088 y(s)820 3057 y Fu(denote)e(the)g(particle)f(number)f(respectively) h(in)h(the)g(box)f Fp(\003)2971 3024 y Fr(e)2971 3080 y(n)3052 3057 y Fu(and)h Fp(\003)3298 3024 y Fr(e)3298 3080 y(s)3335 3057 y Fu(.)57 b(Let)34 b(us)118 3179 y(recall)24 b(the)h(de\002nition)h(of)g(static)g(compressibility)g Fq(\037)p Fp(\()p Fq(m)p Fp(\))g(=)f Fs(E)2260 3105 y Fk(\000)2308 3179 y Fq(\026)2363 3146 y Fr(\025)2404 3155 y Fg(0)2439 3146 y Fl(\()p Fr(m)p Fl(\))2560 3179 y Fp(\()15 b Fq(\021)2655 3193 y Fl(0)2695 3179 y Fp(;)g Fq(\021)2780 3193 y Fl(0)2835 3179 y Fp(\))2870 3105 y Fk(\001)2912 3179 y Fu(.)118 3287 y(Moreover)-10 b(,)24 b(given)g Fp(0)i Fq(<)f(\016)k Fo(\034)c Fp(1)h Fu(and)f(a)h(site)g Fq(x)p Fu(,)f(we)g(de\002ne)g(the)h(events:)994 3455 y Fo(M)1103 3469 y Fr(x)1147 3455 y Fp(\()p Fq(m)p Fp(\))f(:=)h Fo(fj)p Fq(m)1594 3417 y Fr(e)1594 3477 y(x;s)1690 3455 y Fp(\()p Fq(\021)s Fp(\))c Fo(\000)d Fq(m)p Fo(j)26 b(\025)2147 3385 y(p)p 2223 3385 80 4 v 70 x Fq(m)14 b(s)2360 3417 y Fn(\000)2425 3390 y FA(d)p 2425 3402 33 3 v 2426 3443 a Fg(2)2467 3417 y Fl(+)2532 3390 y FA(\016)p 2532 3402 31 3 v 2532 3443 a Fg(2)2591 3455 y Fo(g)980 3647 y(A)1053 3609 y Fl(\(1\))1053 3669 y Fr(x)1147 3647 y Fp(\()p Fq(m)p Fp(\))25 b(:=)1444 3573 y Fk(\010)1539 3585 y Fp(1)p 1522 3626 80 4 v 1522 3709 a Fq(m)1612 3647 y Fo(j)p Fq(m)20 b Fo(\000)g Fq(\026)1883 3609 y Fr(\025)1924 3618 y Fg(0)1958 3609 y Fl(\()p Fr(m)p Fl(\))2080 3647 y Fp(\()p Fq(m)2195 3609 y Fr(e)2195 3669 y(x;s)2291 3647 y Fp(\))p Fo(j)26 b(\025)f Fq(s)2516 3609 y Fn(\000)2581 3582 y FA(d)p 2580 3594 33 3 v 2581 3635 a Fg(2)2622 3609 y Fl(+)2687 3582 y FA(\016)p 2687 3594 31 3 v 2687 3635 a Fg(2)2747 3573 y Fk(\011)980 3882 y Fo(A)1053 3844 y Fl(\(2\))1053 3904 y Fr(x)1147 3882 y Fp(\()p Fq(m)p Fp(\))g(:=)1444 3808 y Fk(\010)1512 3804 y(\014)1512 3859 y(\014)1542 3882 y Fq(\034)1582 3896 y Fr(x)1636 3820 y Fq(\033)1691 3787 y Fl(2)1688 3845 y(0)1730 3820 y Fp(\()p Fq(m)p Fp(\))p 1636 3861 246 4 v 1655 3944 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1911 3882 y Fo(\000)20 b Fp(1)2047 3804 y Fk(\014)2047 3859 y(\014)2103 3882 y Fo(\025)25 b Fq(s)2242 3844 y Fn(\000)2307 3817 y FA(d)p 2306 3829 33 3 v 2307 3870 a Fg(2)2348 3844 y Fl(+)2413 3817 y FA(\016)p 2413 3829 31 3 v 2413 3870 a Fg(2)2473 3808 y Fk(\011)3459 3882 y Fu(\(6.4\))118 4083 y FE(R)m(emark)i(6.2.)42 b Fh(Notice)25 b(that)g(the)h(\002rst)f (event)g(is)g(an)g(event)f(for)h(the)g(particles)g(con\002guration)f Fq(\021)29 b Fh(while)c(all)118 4191 y(the)g(others)g(are)g(events)f (for)g(the)i(disorder)d(\002eld.)118 4319 y FE(L)r(emma)38 b(6.3.)47 b Fh(There)35 b(exists)g Fq(s)1205 4333 y Fl(0)1244 4319 y Fp(\()p Fq(\016)s Fp(\))i Fh(such)f(that)f(the)h(following)d (holds)i(for)f(any)h Fq(s)42 b Fo(\025)g Fq(s)3135 4333 y Fl(0)3174 4319 y Fp(\()p Fq(\016)s Fp(\))p Fh(.)64 b(Assume)118 4448 y Fq(n)25 b Fo(\024)g Fq(s)p Fh(,)g Fp(4)p Fq(s)475 4415 y Fn(\000)p Fr(d)p Fl(+)p Fr(\016)684 4448 y Fo(\024)g Fq(m)g Fo(\024)g Fp(1)p Fq(=)p Fp(2)p Fh(,)i Fq(\021)i Fo(62)c(M)1437 4462 y Fr(x)1481 4448 y Fp(\()p Fq(m)p Fp(\))h Fh(and)e Fq(\013)h Fo(62)g(A)2070 4400 y Fl(\(1\))2070 4459 y Fr(x)2164 4448 y Fp(\()p Fq(m)p Fp(\))c Fo([)f(A)2489 4400 y Fl(\(2\))2489 4459 y Fr(x)2583 4448 y Fp(\()p Fq(m)p Fp(\))p Fh(.)32 b(Then,)25 b(for)f(any)h(site)g Fq(y)s Fh(,)604 4555 y Fk(\014)604 4609 y(\014)604 4664 y(\014)634 4659 y Fo(r)710 4673 y Fr(y)752 4659 y Fp([)p Fq(\034)817 4673 y Fr(x)872 4635 y Fp(^)860 4659 y Fq(\036)914 4673 y Fr(n;s)1014 4659 y Fp(]\()p Fq(\021)s Fp(\))c Fo(\000)1279 4597 y Fp(\(1)g Fo(\000)f Fp(2)p Fq(\021)1561 4611 y Fr(y)1603 4597 y Fp(\))p 1279 4638 360 4 v 1395 4722 a(2)p Fq(s)1483 4696 y Fr(d)1648 4659 y Fq(\034)1688 4673 y Fr(x)1742 4598 y Fq(\030)1782 4612 y Fl(0)1821 4598 y Fp(\()p Fq(m)p Fp(\))p 1742 4638 230 4 v 1753 4722 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1982 4555 y Fk(\014)1982 4609 y(\014)1982 4664 y(\014)2037 4659 y Fo(\024)25 b Fq(c)15 b(s)2230 4622 y Fn(\000)p Fr(d)2326 4558 y Fk(n)2396 4598 y Fq(s)2439 4565 y Fn(\000)p Fr(d)p 2396 4638 138 4 v 2425 4722 a Fq(m)2564 4659 y Fp(+)2720 4598 y(1)p 2665 4638 156 4 v 2665 4657 a Fo(p)p 2741 4657 80 4 v 65 x Fq(m)2831 4659 y(s)2874 4622 y Fn(\000)2939 4595 y FA(d)p 2938 4607 33 3 v 2939 4648 a Fg(2)2980 4622 y Fl(+)3045 4595 y FA(\016)p 3045 4607 31 3 v 3045 4648 a Fg(2)3090 4558 y Fk(o)3150 4659 y Fq(:)284 b Fu(\(6.5\))118 4873 y Fh(Proof)-8 b(.)42 b Fu(By)25 b(Lagrange)f(theorem)g(we)h(can)g(write)1091 5089 y Fo(r)1167 5103 y Fr(y)1208 5089 y Fp([)p Fq(\034)1273 5103 y Fr(x)1329 5065 y Fp(^)1317 5089 y Fq(\036)1371 5103 y Fr(n;s)1470 5089 y Fp(]\()p Fq(\021)s Fp(\))i(=)1736 4965 y Fk(Z)1827 4991 y Fr(m)1889 4968 y FA(e)1889 5008 y(x;s)1976 4991 y Fl(\()p Fr(\021)2040 4968 y FA(y)2079 4991 y Fl(\))1786 5171 y Fr(m)1848 5152 y FA(e)1848 5188 y(x;s)1936 5171 y Fl(\()p Fr(\021)r Fl(\))2126 5089 y Fq(\034)2166 5103 y Fr(x)2245 5027 y Fq(\030)t Fp(\()p Fq(m)2404 4994 y Fn(0)2427 5027 y Fp(\))p 2219 5068 269 4 v 2219 5151 a Fq(\033)2274 5125 y Fl(2)2314 5151 y Fp(\()p Fq(m)2429 5125 y Fn(0)2452 5151 y Fp(\))2513 5089 y Fq(dm)2640 5051 y Fn(0)2663 5089 y Fq(:)771 b Fu(\(6.6\))118 5316 y(Assume)25 b Fq(m)527 5283 y Fn(0)576 5316 y Fu(in)g(the)g(interval)g(with)i(end-points)e Fq(m)1885 5283 y Fr(e)1885 5339 y(x;s)1981 5316 y Fp(\()p Fq(\021)s Fp(\))i Fu(and)e Fq(m)2380 5283 y Fr(e)2380 5339 y(x;s)2476 5316 y Fp(\()p Fq(\021)2559 5283 y Fr(y)2601 5316 y Fp(\))p Fu(.)32 b(Then,)25 b(by)h(lemma)e(A.2,)310 5480 y Fq(\030)350 5494 y Fl(0)389 5480 y Fp(\()p Fq(m)504 5443 y Fn(0)528 5480 y Fp(\))h Fo(\024)g Fq(c)15 b(m)818 5443 y Fn(0)842 5480 y Fq(;)106 b(\030)t Fp(\()p Fq(m)1132 5443 y Fn(0)1155 5480 y Fp(\))26 b Fo(\024)f Fq(c)15 b(m)1446 5443 y Fn(0)1470 5480 y Fq(;)106 b(\033)1656 5443 y Fl(2)1653 5503 y(0)1696 5480 y Fp(\()p Fq(m)1811 5443 y Fn(0)1834 5480 y Fp(\))26 b Fo(\025)f Fq(c)15 b(m)2125 5443 y Fn(0)2148 5480 y Fq(;)107 b(\033)2335 5443 y Fl(2)2374 5480 y Fp(\()p Fq(m)2489 5443 y Fn(0)2513 5480 y Fp(\))25 b Fo(\025)g Fq(c)15 b(m)2803 5443 y Fn(0)2827 5480 y Fq(;)106 b(\037)p Fp(\()p Fq(m)3130 5443 y Fn(0)3153 5480 y Fp(\))26 b Fo(\025)f Fq(c)15 b(m:)p eop %%Page: 36 36 36 35 bop 118 47 a Fj(36)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(Moreover)-10 b(,)25 b(since)h Fq(\021)k Fo(62)c(M)1036 380 y Fr(x)1080 366 y Fp(\()p Fq(m)p Fp(\))p Fu(,)h Fq(m)1362 333 y Fn(0)1412 366 y Fo(\025)g Fq(c)15 b(m)26 b Fu(if)h Fq(s)f Fu(is)h(large)d(enough) h(depending)h(on)g Fq(\016)s Fu(.)35 b(Therefore,)25 b(by)118 474 y(lemma)f Fq(A:)p Fp(3)583 663 y Fk(\014)583 717 y(\014)583 772 y(\014)614 767 y Fq(\034)654 781 y Fr(x)733 706 y Fq(\030)t Fp(\()p Fq(m)892 673 y Fn(0)915 706 y Fp(\))p 707 746 269 4 v 707 829 a Fq(\033)762 803 y Fl(2)802 829 y Fp(\()p Fq(m)917 803 y Fn(0)940 829 y Fp(\))1006 767 y Fo(\000)c Fq(\034)1137 781 y Fr(x)1198 706 y Fq(\030)1238 720 y Fl(0)1277 706 y Fp(\()p Fq(m)1392 673 y Fn(0)1416 706 y Fp(\))p 1190 746 V 1190 831 a Fq(\033)1245 800 y Fl(2)1242 857 y(0)1285 831 y Fp(\()p Fq(m)1400 805 y Fn(0)1424 831 y Fp(\))1469 663 y Fk(\014)1469 717 y(\014)1469 772 y(\014)1524 767 y Fo(\024)1651 706 y Fq(c)p 1630 746 80 4 v 1630 829 a(m)1720 767 y Fo(j)p Fq(m)1825 730 y Fn(0)1869 767 y Fo(\000)g Fq(\026)2015 730 y Fr(\025)2056 739 y Fg(0)2090 730 y Fl(\()p Fr(m)2179 706 y Fi(0)2202 730 y Fl(\))2234 767 y Fp(\()p Fq(m)2349 730 y Fr(e)2349 790 y(x;s)2445 767 y Fp(\))p Fo(j)p Fq(;)929 b Fu(\(6.7\))607 919 y Fk(\014)607 974 y(\014)607 1028 y(\014)637 1024 y Fq(\034)677 1038 y Fr(x)738 962 y Fq(\030)778 976 y Fl(0)817 962 y Fp(\()p Fq(m)932 929 y Fn(0)956 962 y Fp(\))p 731 1003 269 4 v 731 1088 a Fq(\033)786 1057 y Fl(2)783 1114 y(0)825 1088 y Fp(\()p Fq(m)940 1062 y Fn(0)964 1088 y Fp(\))1029 1024 y Fo(\000)20 b Fq(\034)1160 1038 y Fr(x)1221 962 y Fq(\030)1261 976 y Fl(0)1301 962 y Fp(\()p Fq(m)p Fp(\))p 1214 1003 246 4 v 1214 1088 a Fq(\033)1269 1057 y Fl(2)1266 1114 y(0)1308 1088 y Fp(\()p Fq(m)p Fp(\))1469 919 y Fk(\014)1469 974 y(\014)1469 1028 y(\014)1524 1024 y Fo(\024)1651 962 y Fq(c)p 1630 1003 80 4 v 1630 1086 a(m)1720 1024 y Fo(j)p Fq(m)1825 986 y Fn(0)1869 1024 y Fo(\000)g Fq(m)p Fo(j)25 b(\024)2216 962 y Fq(c)p 2196 1003 V 2196 1086 a(m)2286 1024 y Fo(j)p Fq(m)2391 986 y Fr(e)2391 1046 y(x;s)2487 1024 y Fp(\()p Fq(\021)s Fp(\))c Fo(\000)f Fq(m)p Fo(j)g Fp(+)2963 962 y Fq(c)p 2943 1003 V 2943 1086 a(m)3033 1024 y(s)3076 986 y Fn(\000)p Fr(d)3171 1024 y Fq(;)263 b Fu(\(6.8\))645 1182 y Fk(\014)645 1237 y(\014)645 1291 y(\014)676 1287 y Fq(\034)716 1301 y Fr(x)777 1225 y Fq(\030)817 1239 y Fl(0)856 1225 y Fp(\()p Fq(m)p Fp(\))p 769 1266 246 4 v 769 1351 a Fq(\033)824 1319 y Fl(2)821 1377 y(0)864 1351 y Fp(\()p Fq(m)p Fp(\))1045 1287 y Fo(\000)20 b Fq(\034)1176 1301 y Fr(x)1229 1225 y Fq(\030)1269 1239 y Fl(0)1308 1225 y Fp(\()p Fq(m)p Fp(\))p 1229 1266 230 4 v 1240 1349 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1469 1182 y Fk(\014)1469 1237 y(\014)1469 1291 y(\014)1524 1287 y Fo(\024)25 b Fq(c)1659 1182 y Fk(\014)1659 1237 y(\014)1659 1291 y(\014)1690 1287 y Fq(\034)1730 1301 y Fr(x)1784 1225 y Fq(\033)1839 1192 y Fl(2)1836 1249 y(0)1878 1225 y Fp(\()p Fq(m)p Fp(\))p 1784 1266 246 4 v 1802 1349 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))2059 1287 y Fo(\000)20 b Fp(1)2195 1182 y Fk(\014)2195 1237 y(\014)2195 1291 y(\014)2226 1287 y Fq(:)1208 b Fu(\(6.9\))118 1528 y(By)29 b(lemma)f(A.3)h(and)g(the)g(assumption)h Fq(\013)i Fo(62)f(A)1780 1480 y Fl(\(1\))1780 1540 y Fr(x)1874 1528 y Fp(\()p Fq(m)p Fp(\))p Fu(,)g(the)e(r)-10 b(.h.s.)44 b(of)29 b(\(6.7\))f(can)g(be)h(bounded)g(from)118 1636 y(above)c(by)876 1818 y Fq(c)p 856 1858 80 4 v 856 1942 a(m)945 1879 y Fo(j)p Fq(m)c Fo(\000)f Fq(\026)1217 1842 y Fr(\025)1258 1851 y Fg(0)1292 1842 y Fl(\()p Fr(m)p Fl(\))1414 1879 y Fp(\()p Fq(m)1529 1842 y Fr(e)1529 1902 y(x;s)1625 1879 y Fp(\))p Fo(j)h Fp(+)1827 1818 y Fq(c)p 1807 1858 V 1807 1942 a(m)1896 1879 y(s)1939 1842 y Fn(\000)p Fr(d)2060 1879 y Fo(\024)k Fq(c)2195 1806 y Fk(\002)2248 1879 y Fq(s)2291 1842 y Fn(\000)2356 1815 y FA(d)p 2355 1827 33 3 v 2356 1868 a Fg(2)2397 1842 y Fl(+)2462 1815 y FA(\016)p 2462 1827 31 3 v 2462 1868 a Fg(2)2527 1879 y Fp(+)2645 1818 y(1)p 2628 1858 80 4 v 2628 1942 a Fq(m)2718 1879 y(s)2761 1842 y Fn(\000)p Fr(d)2871 1806 y Fk(\003)2909 1879 y Fq(:)475 b Fu(\(6.10\))118 2043 y(Similarly)-12 b(,)28 b(the)g(contribution)g(of)h(the)f(r)-10 b(.h.s.)41 b(of)28 b(\(6.8\))f(together)h(with)h(\(6.9\))e(can)g(be)h (bounded)g(from)118 2151 y(above)d(by)1475 2328 y Fq(c)1514 2227 y Fk(h)1568 2266 y Fq(s)1611 2233 y Fn(\000)p Fr(d)p 1568 2307 138 4 v 1597 2390 a Fq(m)1736 2328 y Fp(+)1892 2266 y(1)p 1837 2307 156 4 v 1837 2325 a Fo(p)p 1912 2325 80 4 v 1912 2390 a Fq(m)2002 2328 y(s)2045 2290 y Fn(\000)2110 2263 y FA(d)p 2110 2275 33 3 v 2111 2316 a Fg(2)2152 2290 y Fl(+)2217 2263 y FA(\016)p 2217 2275 31 3 v 2217 2316 a Fg(2)2261 2227 y Fk(i)3409 2328 y Fu(\(6.11\))118 2536 y(The)h(thesis)g(follows)f(immediately)h(from)e (\(6.6\))h(together)g(with)h(\(6.10\),)e(\(6.11\).)636 b Fe(\003)118 2704 y FE(L)r(emma)36 b(6.4.)46 b Fh(There)34 b(exists)g Fq(s)1200 2718 y Fl(0)1239 2704 y Fp(\()p Fq(\016)s Fp(\))h Fh(such)f(that)g(the)g(following)e(holds)h(for)g(any) g Fq(s)40 b Fo(\025)f Fq(s)3112 2718 y Fl(0)3151 2704 y Fp(\()p Fq(\016)s Fp(\))p Fh(.)59 b(Let)34 b Fq(n)40 b Fo(\024)118 2813 y Fq(s)p Fh(,)d Fq(m)k Fo(2)g Fp(\(0)p Fq(;)577 2778 y Fl(1)p 577 2793 36 4 v 577 2845 a(2)622 2813 y Fp(\))p Fh(,)d(and)c(let)h Fq(\025)p Fp(\()p Fq(\013)p Fp(\))h Fh(be)e(a)h(bounded)e(measurable)g(function)h(such)g(that)i (for)e(any)g(disorder)118 2921 y(con\002guration)24 b Fq(\013)1394 3056 y Fo(j)p Fq(\025)p Fp(\()p Fq(\013)p Fp(\))d Fo(\000)f Fq(\025)1765 3070 y Fl(0)1805 3056 y Fp(\()p Fq(m)p Fp(\))p Fo(j)26 b(\024)f Fq(s)2145 3019 y Fn(\000)2210 2992 y FA(d)p 2209 3004 33 3 v 2210 3045 a Fg(2)2251 3019 y Fl(+)2316 2992 y FA(\016)p 2316 3004 31 3 v 2316 3045 a Fg(4)2361 3056 y Fq(:)1023 b Fu(\(6.12\))118 3193 y Fh(Then,)25 b(for)g(any)g Fq(s)f Fo(\025)h Fq(s)866 3207 y Fl(0)905 3193 y Fp(\()p Fq(\016)s Fp(\))i Fh(and)d(any)h (\002nite)g(set)g Fp(\001)g Fo(\032)g Fs(Z)1996 3160 y Fr(d)2032 3193 y Fh(,)946 3373 y Fs(P)1001 3300 y Fk(\000)1058 3373 y Fq(\026)1113 3336 y Fr(\025)p Fl(\()p Fr(\013)p Fl(\))1258 3300 y Fk(\000)1300 3373 y Fo([)1361 3387 y Fr(x)p Fn(2)p Fl(\001)1510 3373 y Fo(M)1619 3387 y Fr(x)1663 3373 y Fp(\()p Fq(m)p Fp(\))1813 3300 y Fk(\001)1881 3373 y Fo(\025)g(j)p Fp(\001)p Fo(j)p Fq(e)2145 3336 y Fn(\000)p Fr(s)2233 3312 y FA(\016)r(=)p Fg(2)2348 3272 y Fk(\021)2428 3373 y Fo(\024)g Fq(c)15 b(e)2620 3336 y Fn(\000)p Fr(s)2708 3312 y FA(\016)r(=)p Fg(2)2809 3373 y Fq(:)575 b Fu(\(6.13\))118 3568 y Fh(Proof)-8 b(.)42 b Fu(By)25 b(the)h(Chebyshev)g(inequality)f(and)h(the)g (translation)f(invariance)g(of)h Fs(P)p Fu(,)g(the)g(l.h.s.)32 b(of)26 b Fp(\(6)p Fq(:)p Fp(13\))118 3682 y Fu(can)f(be)g(bounded)g (from)f(above)h(by)h Fp(exp)o(\()p Fq(s)1584 3649 y Fr(\016)r(=)p Fl(2)1693 3682 y Fp(\))p Fs(E)1783 3608 y Fk(\002)1827 3682 y Fq(\026)1882 3649 y Fr(\025)p Fl(\()p Fr(\013)p Fl(\))2027 3682 y Fp(\()p Fo(M)2171 3696 y Fl(0)2211 3682 y Fp(\()p Fq(m)p Fp(\)\))2396 3608 y Fk(\003)2435 3682 y Fq(:)118 3790 y Fu(Let)g(us)f(bound)g(the)h(term)1164 3966 y Fq(e)1206 3929 y Fr(s)1249 3879 y FA(\016)p 1248 3887 V 1248 3921 a Fg(2)1297 3966 y Fs(E)1352 3892 y Fk(\002)1411 3966 y Fq(\026)1466 3929 y Fr(\025)p Fl(\()p Fr(\013)p Fl(\))1611 3892 y Fk(\000)1653 3966 y Fq(m)1733 3929 y Fr(e)1733 3989 y(s)1790 3966 y Fo(\000)20 b Fq(m)25 b Fo(\025)2082 3896 y(p)p 2157 3896 80 4 v 2157 3966 a Fq(m)15 b(s)2295 3929 y Fn(\000)2360 3901 y FA(d)p 2360 3913 33 3 v 2361 3955 a Fg(2)2402 3929 y Fl(+)2467 3901 y FA(\016)p 2467 3913 31 3 v 2467 3955 a Fg(2)2511 3892 y Fk(\001\003)2591 3966 y Fq(:)793 b Fu(\(6.14\))118 4119 y(Thanks)36 b(again)e(to)h(Chebyshev)h(inequality)-12 b(,)37 b(for)e(any)f Fp(0)42 b Fq(<)f(t)g(<)g Fp(1)35 b Fu(\(6.14\))f(can)g(be)g(bounded)h(from)118 4227 y(above)25 b(by)1138 4385 y Fq(e)1180 4347 y Fr(s)1223 4297 y FA(\016)p 1223 4305 V 1223 4340 a Fg(2)1267 4347 y Fn(\000)p Fl(2)p Fr(t)1382 4299 y Fn(p)p 1442 4299 63 3 v 1442 4347 a Fr(ms)1547 4297 y FA(d)p 1547 4305 33 3 v 1548 4340 a Fg(2)1589 4317 y(+)1645 4297 y FA(\016)p 1645 4305 31 3 v 1645 4340 a Fg(2)1694 4385 y Fs(E)1748 4311 y Fk(\002)1819 4298 y(Y)1792 4495 y Fr(x)p Fn(2)p Fl(\003)1928 4476 y FA(e)1928 4512 y(s)1977 4385 y Fq(\026)2032 4347 y Fr(\025)p Fl(\()p Fr(\013)p Fl(\))2177 4311 y Fk(\000)2219 4385 y Fq(e)2261 4347 y Fr(t)p Fl(\()p Fr(\021)2348 4355 y FA(x)2388 4347 y Fn(\000)p Fr(m)p Fl(\))2537 4311 y Fk(\001\003)2617 4385 y Fq(:)767 b Fu(\(6.15\))118 4623 y(Using)25 b(the)h(basic)g(assumption)g(\(6.12\))e(and)h(Lagrange)f (theorem,)h(it)h(is)g(not)f(dif\002cult)h(to)g(see)e(that)875 4795 y Fq(\026)930 4758 y Fr(\025)p Fl(\()p Fr(\013)p Fl(\))1075 4722 y Fk(\000)1117 4795 y Fq(e)1159 4758 y Fr(t)p Fl(\()p Fr(\021)1246 4766 y FA(x)1286 4758 y Fn(\000)p Fr(m)p Fl(\))1435 4722 y Fk(\001)1502 4795 y Fo(\024)h Fp(\(1)c(+)f Fq(c)15 b(tms)2000 4758 y Fn(\000)2065 4731 y FA(d)p 2064 4743 33 3 v 2065 4784 a Fg(2)2106 4758 y Fl(+)2171 4731 y FA(\016)p 2171 4743 31 3 v 2171 4784 a Fg(4)2216 4795 y Fp(\))p Fq(\026)2306 4758 y Fr(\025)2347 4767 y Fg(0)2382 4758 y Fl(\()p Fr(m)p Fl(\))2503 4722 y Fk(\000)2545 4795 y Fq(e)2587 4758 y Fr(t)p Fl(\()p Fr(\021)2674 4766 y FA(x)2714 4758 y Fn(\000)p Fr(m)p Fl(\))2863 4722 y Fk(\001)118 4948 y Fu(so)26 b(that)g(\(6.15\))e(is)i (bounded)f(from)g(above)g(by)1025 5144 y Fq(e)1067 5106 y Fr(s)1110 5056 y FA(\016)p 1110 5064 V 1110 5099 a Fg(2)1155 5106 y Fn(\000)p Fl(2)p Fr(t)1270 5059 y Fn(p)p 1329 5059 63 3 v 47 x Fr(m)q(s)1435 5056 y FA(d)p 1434 5064 33 3 v 1435 5099 a Fg(2)1476 5076 y(+)1533 5056 y FA(\016)p 1533 5064 31 3 v 1533 5099 a Fg(2)1577 5106 y Fl(+)p Fr(c)12 b(tms)1805 5056 y FA(d)p 1805 5064 33 3 v 1806 5099 a Fg(2)1847 5076 y(+)1904 5056 y FA(\016)p 1904 5064 31 3 v 1904 5099 a Fg(4)1952 5144 y Fs(E)2007 5070 y Fk(\002)2051 5144 y Fq(\026)2106 5106 y Fr(\025)2147 5115 y Fg(0)2185 5070 y Fk(\000)2227 5144 y Fq(e)2269 5106 y Fr(t)p Fl(\()p Fr(\021)2356 5115 y Fg(0)2392 5106 y Fn(\000)p Fr(m)p Fl(\))2541 5070 y Fk(\001\003)2621 5093 y Fl(2)p Fr(s)2689 5070 y FA(d)2729 5144 y Fq(:)118 5305 y Fu(Since)24 b Fq(e)398 5272 y Fr(x)468 5305 y Fo(\024)h Fp(1)20 b(+)g Fq(x)g Fp(+)g(2)15 b Fq(x)995 5272 y Fl(2)1061 5305 y Fu(if)25 b Fo(j)p Fq(x)p Fo(j)h(\024)f Fp(1)p Fu(,)h(the)g(above)f(expression)f(is)j(bounded)e(from)f(above)h (by)986 5480 y Fp(exp)1125 5407 y Fk(\000)1167 5480 y Fq(s)1220 5416 y FA(\016)p 1220 5428 V 1220 5469 a Fg(2)1284 5480 y Fo(\000)20 b Fp(2)p Fq(t)1453 5410 y Fo(p)p 1529 5410 80 4 v 70 x Fq(m)15 b(s)1677 5416 y FA(d)p 1677 5428 33 3 v 1678 5469 a Fg(2)1719 5443 y Fl(+)1784 5416 y FA(\016)p 1784 5428 31 3 v 1784 5469 a Fg(2)1848 5480 y Fp(+)20 b Fq(c)15 b(t)g(ms)2174 5416 y FA(d)p 2174 5428 33 3 v 2175 5469 a Fg(2)2216 5443 y Fl(+)2281 5416 y FA(\016)p 2281 5428 31 3 v 2281 5469 a Fg(4)2346 5480 y Fp(+)20 b Fq(c)15 b(t)2524 5443 y Fl(2)2563 5480 y Fq(ms)2686 5443 y Fr(d)2726 5407 y Fk(\001)2768 5480 y Fq(:)p eop %%Page: 37 37 37 36 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(37)118 366 y Fu(The)26 b(thesis)g(follows)f(by)h(choosing)f Fq(t)g Fu(such)g(that)h Fq(t)1828 333 y Fl(2)1868 366 y Fq(m)f Fp(=)g Fq(s)2112 333 y Fn(\000)p Fr(d)p Fl(+)p Fr(\016)r(=)p Fl(2)2366 366 y Fu(.)1200 b Fe(\003)218 535 y Fu(W)-8 b(e)25 b(are)f(\002nally)g(in)g(a)h(position)h(to)f (state)g(our)f(main)g(bound)h(on)f(the)h(variance)e(appearing)h(in)h (\(6.2\).)118 666 y FE(Proposition)f(6.5.)37 b Fh(F)l(or)22 b Fq(d)k Fo(\025)f Fp(2)e Fh(there)f(exists)f Fq(s)1692 680 y Fl(0)1731 666 y Fp(\()p Fq(\016)s Fp(\))j Fh(such)e(that)h(the)f (following)e(holds)h(for)h(any)g Fq(s)j Fo(\025)g Fq(s)3483 680 y Fl(0)3522 666 y Fp(\()p Fq(\016)s Fp(\))p Fh(.)118 775 y(Let)36 b Fq(m)43 b Fo(2)g Fp(\(0)p Fq(;)626 739 y Fl(1)p 626 754 36 4 v 626 807 a(2)672 775 y Fp(\))36 b Fh(and)f(let)h Fq(n)43 b Fo(\024)g Fq(s)g Fo(\024)g Fq(k)j Fo(\024)d Fp(1000)p Fq(s)p Fh(.)65 b(Then)36 b(there)g(exists)f (a)h(measurable)f(set)h Fo(A)f Fh(with)118 909 y Fs(P)p Fp(\()p Fo(A)p Fp(\))26 b Fo(\024)f Fq(k)488 876 y Fl(2)p Fr(d)564 909 y Fq(e)606 876 y Fn(\000)12 b Fr(cs)737 852 y FA(\016)r(=)p Fg(2)862 909 y Fh(such)25 b(that)842 1069 y Fp(V)-8 b(ar)983 1097 y Fr(\026)1025 1075 y FA(\025)1062 1090 y Fg(0)1097 1075 y(\()p FA(m)p Fg(\))1223 1069 y Fp(\(Av)1376 1083 y Fr(x)p Fn(2)p Fl(\003)1512 1095 y FA(k)1554 1069 y Fq(\034)1594 1083 y Fr(x)1637 1069 y Fq(\036)1691 1083 y Fr(n;s)1791 1069 y Fp(\))25 b Fo(\024)g Fq(c)p Fp(1)-25 b(I)2040 1083 y Fn(A)2096 1064 y FA(c)2132 1069 y Fp(\()p Fq(\013)p Fp(\))p Fq(s)2303 1032 y Fn(\000)p Fl(2)p Fr(d)p Fl(+2)p Fr(\016)2579 1069 y Fp(+)20 b(1)-25 b(I)2723 1083 y Fn(A)2783 1069 y Fp(\()p Fq(\013)p Fp(\))p Fq(:)473 b Fu(\(6.16\))118 1258 y Fh(Proof)-8 b(.)42 b Fu(Let)25 b(us)h(consider)f(\002rst)g(the)h(case)f(of)g(\223low)g (density\224)h Fq(m)f Fo(\024)g Fp(4)p Fq(s)2470 1225 y Fn(\000)p Fr(d)p Fl(+)p Fr(\016)2654 1258 y Fu(.)118 1366 y(Since)i Fo(j)p Fq(\034)424 1380 y Fr(x)468 1366 y Fq(\036)522 1380 y Fr(n;s)621 1366 y Fo(j)k(\024)f Fq(c)15 b(m)912 1333 y Fr(e)912 1388 y(x;s)1008 1366 y Fu(,)30 b Fo(j)p Fp(Av)1205 1380 y Fr(x)p Fn(2)p Fl(\003)1341 1392 y FA(k)1383 1366 y Fq(\034)1423 1380 y Fr(x)1467 1366 y Fq(\036)1521 1380 y Fr(n;s)1620 1366 y Fo(j)h(\024)f Fq(c)15 b(m)1911 1380 y Fl(\003)1960 1392 y Fg(2)p FA(k)2061 1366 y Fu(and)28 b(therefore)g(the)g(l.h.s.)42 b(of)28 b(\(6.16\))f(can)h(be)118 1477 y(bounded)d(from)g(above)g(by)1043 1638 y Fq(\026)1098 1601 y Fr(\025)1139 1610 y Fg(0)1173 1601 y Fl(\()p Fr(m)p Fl(\))1295 1638 y Fp(\()p Fq(m)1410 1601 y Fl(2)1410 1661 y(\003)1459 1673 y Fg(2)p FA(k)1532 1638 y Fp(\))g Fo(\024)g Fq(c)p Fp(\()p Fq(k)1812 1601 y Fn(\000)p Fr(d)1908 1638 y Fq(m)20 b Fp(+)g Fq(m)2179 1601 y Fl(2)2219 1638 y Fp(\))25 b Fo(\024)g Fq(cs)2457 1601 y Fn(\000)p Fl(2)p Fr(d)p Fl(+2)p Fr(\016)2712 1638 y Fq(:)118 1806 y Fu(Let)h(us)f(now)g(consider)g(the)h(\223high)f (density\224)h(case)f Fq(m)g Fo(\025)g Fp(4)p Fq(s)2152 1773 y Fn(\000)p Fr(d)p Fl(+)p Fr(\016)2336 1806 y Fu(.)118 1914 y(By)30 b(the)g(equivalence)e(of)h(ensembles)g(\(see)g (proposition)i(A.4\),)g(in)e(the)h(l.h.s.)46 b(of)30 b Fp(\(6)p Fq(:)p Fp(16\))i Fq(\036)3271 1928 y Fr(n;s)3400 1914 y Fu(can)d(be)118 2036 y(substituted)e(by)712 2012 y Fp(^)700 2036 y Fq(\036)754 2050 y Fr(n;s)879 2036 y Fu(with)f(an)f(error)g(of)g(order)g Fq(s)1814 2003 y Fn(\000)p Fl(2)p Fr(d)1944 2036 y Fu(.)32 b(Therefore,)24 b(by)h(the)h(P)-5 b(oincar\264)-45 b(e)24 b(inequality)1131 2212 y Fp(V)-8 b(ar)1273 2240 y Fr(\026)1315 2218 y FA(\025)1352 2233 y Fg(0)1386 2218 y(\()p FA(m)p Fg(\))1497 2212 y Fp(\()p Fq(f)10 b Fp(\))25 b Fo(\024)g Fq(c)15 b(m)g(\026)1947 2174 y Fr(\025)1988 2183 y Fg(0)2023 2174 y Fl(\()p Fr(m)p Fl(\))2145 2212 y Fp(\()2180 2125 y Fk(X)2227 2316 y Fr(y)2327 2212 y Fo(jr)2428 2226 y Fr(y)2469 2212 y Fq(f)10 b Fo(j)2549 2174 y Fl(2)2588 2212 y Fp(\))p Fq(;)761 b Fu(\(6.17\))118 2459 y(it)26 b(is)g(enough)e(to)i(estimate)988 2654 y Fq(c)15 b(m)g(\026)1192 2616 y Fr(\025)1233 2625 y Fg(0)1268 2616 y Fl(\()p Fr(m)p Fl(\))1389 2553 y Fk(h)1483 2592 y Fp(1)p 1442 2633 126 4 v 1442 2716 a Fq(k)1492 2690 y Fl(2)p Fr(d)1629 2567 y Fk(X)1593 2764 y Fr(y)r Fn(2)p Fl(\003)1726 2776 y Fg(2)p FA(k)1795 2580 y Fk(\000)1964 2567 y(X)1852 2764 y Fr(x)p Fn(2)p Fl(\003)1988 2776 y FA(k)2026 2764 y Fn(\\)p Fl(\003)2122 2772 y FA(y)r(;s)2222 2654 y Fo(r)2298 2668 y Fr(y)2340 2654 y Fp([)p Fq(\034)2405 2668 y Fr(x)2460 2630 y Fp(^)2448 2654 y Fq(\036)2502 2668 y Fr(n;s)2602 2654 y Fp(])2642 2580 y Fk(\001)2684 2603 y Fl(2)2723 2553 y Fk(i)2766 2654 y Fq(:)618 b Fu(\(6.18\))118 2913 y(T)-9 b(o)26 b(this)h(aim)e(we)g(set)h(\(recall)d(\(6.4\)\))646 3067 y Fo(M)i Fp(:=)h Fo([)963 3081 y Fr(x)p Fn(2)p Fl(\003)1099 3093 y FA(k)1140 3067 y Fo(M)1249 3081 y Fr(x)1293 3067 y Fp(\()p Fq(m)p Fp(\))183 b Fo(A)1699 3081 y Fl(0)1763 3067 y Fp(:=)1884 2993 y Fk(\010)1937 3067 y Fq(\026)1992 3029 y Fr(\025)2033 3038 y Fg(0)2068 3029 y Fl(\()p Fr(m)p Fl(\))2189 2993 y Fk(\000)2231 3067 y Fo(M)2340 2993 y Fk(\001)2407 3067 y Fo(\025)25 b Fq(k)2553 3029 y Fr(d)2609 3067 y Fp(exp\()p Fo(\000)p Fq(s)2897 3029 y Fr(\016)r(=)p Fl(2)3005 3067 y Fp(\))3055 2993 y Fk(\011)3108 3067 y Fq(;)646 3226 y Fo(A)719 3240 y Fl(1)783 3226 y Fp(:=)h Fo([)966 3240 y Fr(x)p Fn(2)p Fl(\003)1102 3252 y FA(k)1143 3226 y Fo(A)1216 3188 y Fl(\(1\))1216 3248 y Fr(x)1310 3226 y Fp(\()p Fq(m)p Fp(\))182 b Fo(A)1715 3240 y Fl(2)1780 3226 y Fp(:=)25 b Fo([)1962 3240 y Fr(x)p Fn(2)p Fl(\003)2098 3252 y FA(k)2139 3226 y Fo(A)2212 3188 y Fl(\(2\))2212 3248 y Fr(x)2306 3226 y Fp(\()p Fq(m)p Fp(\))646 3427 y Fo(A)719 3441 y Fl(3)783 3427 y Fp(:=)h Fo([)966 3441 y Fr(y)r Fn(2)p Fl(\003)1099 3453 y Fg(2)p FA(k)1171 3326 y Fk(n)1247 3323 y(\014)1247 3377 y(\014)1247 3432 y(\014)1277 3427 y Fp(Av)1395 3441 y Fr(x)p Fn(2)p Fl(\003)1531 3453 y FA(k)1568 3441 y Fn(\\)p Fl(\003)1664 3449 y FA(y)r(;s)1754 3326 y Fk(h)1797 3427 y Fq(\034)1837 3441 y Fr(x)1906 3366 y Fq(\030)1946 3380 y Fl(0)1985 3366 y Fp(\()p Fq(m)p Fp(\))p 1906 3406 230 4 v 1917 3490 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))2145 3326 y Fk(i)2203 3323 y(\014)2203 3377 y(\014)2203 3432 y(\014)2259 3427 y Fo(\025)f(j)p Fp(\003)2443 3442 y Fr(k)2506 3427 y Fo(\\)20 b Fp(\003)2650 3441 y Fr(y)r(;s)2744 3427 y Fo(j)2769 3390 y Fn(\000)2834 3363 y Fg(1)p 2834 3375 31 3 v 2834 3416 a(2)2874 3390 y Fl(+)2955 3363 y FA(\016)p 2939 3375 63 3 v 2939 3416 a Fg(2)p FA(d)3031 3326 y Fk(o)3091 3427 y Fq(;)646 3611 y Fo(A)25 b Fp(:=)g Fo(A)938 3625 y Fl(0)997 3611 y Fo([)20 b(A)1151 3625 y Fl(1)1210 3611 y Fo([)g(A)1364 3625 y Fl(2)1423 3611 y Fo([)g(A)1577 3625 y Fl(3)1616 3611 y Fq(:)118 3769 y Fu(W)-8 b(e)26 b(\002rst)g(estimate)791 3943 y Fp(1)-25 b(I)844 3957 y Fn(A)900 3939 y FA(c)937 3943 y Fp(\()p Fq(\013)p Fp(\))p Fq(m)15 b(\026)1215 3906 y Fr(\025)1256 3915 y Fg(0)1292 3906 y Fl(\()p Fr(m)p Fl(\))1413 3843 y Fk(h)1456 3943 y Fp(1)-25 b(I)1509 3957 y Fn(M)1593 3939 y FA(c)1679 3882 y Fp(1)p 1639 3923 126 4 v 1639 4006 a Fq(k)1689 3980 y Fl(2)p Fr(d)1825 3857 y Fk(X)1790 4054 y Fr(y)r Fn(2)p Fl(\003)1923 4066 y Fg(2)p FA(k)1992 3870 y Fk(\000)2161 3857 y(X)2049 4054 y Fr(x)p Fn(2)p Fl(\003)2185 4066 y FA(k)2223 4054 y Fn(\\)p Fl(\003)2319 4062 y FA(y)r(;s)2419 3943 y Fo(r)2495 3957 y Fr(y)2536 3943 y Fp([)p Fq(\034)2601 3957 y Fr(x)2657 3919 y Fp(^)2645 3943 y Fq(\036)2699 3957 y Fr(n;s)2799 3943 y Fp(])2839 3870 y Fk(\001)2881 3892 y Fl(2)2920 3843 y Fk(i)2963 3943 y Fq(:)421 b Fu(\(6.19\))118 4202 y(By)25 b(lemma)f Fp(6)p Fq(:)p Fp(3)p Fu(,)j(for)e Fq(s)g Fu(large)f(enough)g(\(6.19\))g(can)g(be)h(bounded)g(from)g(above)g(by) 899 4356 y Fq(c)p 856 4396 V 856 4480 a(k)906 4453 y Fl(2)p Fr(d)992 4417 y Fp(1)-25 b(I)1045 4431 y Fn(A)1101 4412 y FA(c)1137 4417 y Fp(\()p Fq(\013)p Fp(\)Av)1384 4431 y Fr(y)r Fn(2)p Fl(\003)1517 4443 y FA(k)1559 4316 y Fk(h)1646 4356 y Fp(1)p 1627 4396 84 4 v 1627 4480 a Fq(s)1670 4453 y Fr(d)1848 4331 y Fk(X)1736 4528 y Fr(x)p Fn(2)p Fl(\003)1872 4540 y FA(k)1909 4528 y Fn(\\)p Fl(\003)2005 4536 y FA(y)r(;s)2106 4417 y Fq(\034)2146 4431 y Fr(x)2200 4356 y Fq(\030)2240 4370 y Fl(0)2279 4356 y Fp(\()p Fq(m)p Fp(\))p 2200 4396 230 4 v 2211 4480 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))2439 4316 y Fk(i)2482 4339 y Fl(2)2542 4417 y Fp(+)20 b Fq(c)2697 4356 y(s)2740 4323 y Fn(\000)p Fr(d)p Fl(+)p Fr(\016)p 2697 4396 227 4 v 2765 4480 a Fq(k)2815 4453 y Fr(d)3409 4417 y Fu(\(6.20\))118 4677 y(By)i(straightforward)i(computations)e(and)h(the)f(de\002nition)h (of)g Fo(A)2337 4691 y Fl(3)2398 4677 y Fu(the)f(\002rst)h(addendum)f (in)h Fp(\(6)p Fq(:)p Fp(20\))h Fu(can)118 4787 y(be)31 b(bounded)g(by)g Fq(c)15 b(k)852 4754 y Fn(\000)p Fr(d)948 4787 y Fq(s)991 4754 y Fn(\000)p Fr(d)p Fl(+)p Fr(\016)1174 4787 y Fu(.)49 b(Moreover)-10 b(,)32 b(because)e(of)h(the)g (de\002nition)h(of)f Fo(A)2882 4801 y Fl(0)2922 4787 y Fu(,)h(expression)f Fp(\(6)p Fq(:)p Fp(19\))118 4912 y Fu(with)c Fp(1)-25 b(I)374 4926 y Fn(M)458 4907 y FA(c)519 4912 y Fu(replaced)24 b(by)h Fp(1)-25 b(I)1054 4926 y Fn(M)1168 4912 y Fu(can)24 b(be)h(bounded)g(by)g Fq(c)15 b(s)2040 4879 y Fl(2)p Fr(d)2116 4912 y Fq(e)2158 4879 y Fn(\000)p Fr(s)2246 4856 y FA(\016)r(=)p Fg(2)2347 4912 y Fu(.)118 5020 y(In)25 b(conclusion)318 5195 y Fp(1)-25 b(I)371 5209 y Fn(A)427 5190 y FA(c)464 5195 y Fp(\()p Fq(\013)p Fp(\))p Fq(m)15 b(\026)742 5157 y Fr(\025)783 5166 y Fg(0)819 5157 y Fl(\()p Fr(m)p Fl(\))940 5094 y Fk(h)1033 5133 y Fp(1)p 993 5174 126 4 v 993 5257 a Fq(k)1043 5231 y Fl(2)p Fr(d)1179 5108 y Fk(X)1144 5305 y Fr(y)r Fn(2)p Fl(\003)1277 5317 y Fg(2)p FA(k)1346 5121 y Fk(\000)1515 5108 y(X)1403 5305 y Fr(x)p Fn(2)p Fl(\003)1539 5317 y FA(k)1577 5305 y Fn(\\)p Fl(\003)1673 5313 y FA(y)r(;s)1773 5195 y Fo(r)1849 5209 y Fr(y)1890 5195 y Fp([)p Fq(\034)1955 5209 y Fr(x)2011 5171 y Fp(^)1999 5195 y Fq(\036)2053 5209 y Fr(n;s)2153 5195 y Fp(])2193 5121 y Fk(\001)2235 5144 y Fl(2)2274 5094 y Fk(i)2342 5195 y Fo(\024)25 b Fq(c)2492 5121 y Fk(\002)2546 5195 y Fq(k)2596 5157 y Fn(\000)p Fr(d)2691 5195 y Fq(s)2734 5157 y Fn(\000)p Fr(d)p Fl(+)p Fr(\016)2938 5195 y Fp(+)20 b Fq(s)3072 5157 y Fl(2)p Fr(d)3147 5195 y Fq(e)3189 5157 y Fn(\000)p Fr(s)3277 5134 y FA(\016)r(=)p Fg(2)3393 5094 y Fk(i)3436 5195 y Fq(:)118 5480 y Fu(It)26 b(remains)f(to)h (prove)e(that)j Fs(P)p Fp(\()p Fo(A)p Fp(\))f Fo(\024)f Fq(k)1451 5447 y Fl(2)p Fr(d)1527 5480 y Fq(e)1569 5447 y Fn(\000)p Fr(c)11 b(s)1699 5424 y FA(\016)r(=)p Fg(2)1799 5480 y Fu(.)32 b(T)-9 b(o)26 b(this)g(aim)g(we)f(set)p eop %%Page: 38 38 38 37 bop 118 45 a Fj(38)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)901 371 y Fq(f)946 385 y Fl(1)985 371 y Fp(\()p Fq(\013)p Fp(\))27 b(:=)e(1)20 b Fo(\000)g Fq(\026)1472 333 y Fr(\025)1513 342 y Fg(0)1548 333 y Fl(\()p Fr(m)p Fl(\))1669 371 y Fp(\()p Fq(\021)1749 385 y Fl(0)1789 371 y Fp(\))15 b Fq(;)901 530 y(f)946 544 y Fl(2)985 530 y Fp(\()p Fq(\013)p Fp(\))27 b(:=)e(1)20 b Fo(\000)g Fq(\026)1472 492 y Fr(\025)1513 501 y Fg(0)1548 492 y Fl(\()p Fr(m)p Fl(\))1669 530 y Fp(\()p Fq(\021)1749 544 y Fl(0)1789 530 y Fp(;)15 b Fq(\021)1874 544 y Fl(0)1914 530 y Fp(\))p Fq(=\033)2049 492 y Fl(2)2046 552 y(0)2090 530 y Fp(\()p Fq(m)p Fp(\))g Fq(;)901 689 y(f)946 703 y Fl(3)985 689 y Fp(\()p Fq(\013)p Fp(\))27 b(:=)1261 616 y Fk(\000)1302 689 y Fq(\026)1357 652 y Fr(\025)1398 661 y Fg(0)1433 652 y Fl(\()p Fr(m)p Fl(\))1554 689 y Fp(\()p Fq(\021)1634 703 y Fr(ne)1715 689 y Fp(;)15 b Fq(\021)1800 703 y Fr(ne)1880 689 y Fp(\))20 b Fo(\000)g Fq(\026)2081 652 y Fr(\025)2122 661 y Fg(0)2157 652 y Fl(\()p Fr(m)p Fl(\))2278 689 y Fp(\()p Fq(\021)2358 703 y Fl(0)2398 689 y Fp(;)15 b Fq(\021)2483 703 y Fl(0)2523 689 y Fp(\))2558 616 y Fk(\001)2600 689 y Fq(=\037)p Fp(\()p Fq(m)p Fp(\))p Fq(:)118 874 y Fu(By)32 b(lemma)f(6.4)h Fs(P)p Fp(\()p Fo(A)874 888 y Fl(0)914 874 y Fp(\))37 b Fo(\024)g Fq(c)15 b(e)1190 841 y Fn(\000)p Fr(s)1278 817 y FA(\016)r(=)p Fg(2)1411 874 y Fu(while)33 b Fs(P)p Fp(\()p Fo(A)1824 826 y Fl(\(1\))1824 885 y Fr(x)1919 874 y Fp(\()p Fq(m)p Fp(\)\))g Fu(and)f Fs(P)p Fp(\()p Fo(A)2481 826 y Fl(\(2\))2481 885 y Fr(x)2576 874 y Fp(\()p Fq(m)p Fp(\)\))h Fu(can)f(be)g(bounded)g(from)118 996 y(above)25 b(by)g Fq(e)538 963 y Fn(\000)p Fr(c)12 b(s)669 939 y FA(\016)732 996 y Fu(by)25 b(means)g(of)g(lemma)f Fq(A:)p Fp(1)i Fu(with)h Fq(f)34 b Fp(=)25 b Fq(f)2117 1010 y Fl(1)2182 996 y Fu(and)g Fq(f)34 b Fp(=)25 b Fq(f)2576 1010 y Fl(2)2641 996 y Fu(respectively)-12 b(.)30 b(Therefore)1385 1161 y Fs(P)p Fp(\()p Fo(A)1548 1175 y Fl(1)1588 1161 y Fp(\))20 b(+)g Fs(P)p Fp(\()p Fo(A)1897 1175 y Fl(2)1937 1161 y Fp(\))26 b Fo(\024)f Fq(k)2144 1124 y Fr(d)2184 1161 y Fq(e)2226 1124 y Fn(\000)p Fr(c)12 b(s)2357 1100 y FA(\016)3409 1161 y Fu(\(6.21\))118 1311 y(In)25 b(order)g(to)h (bound)f Fs(P)p Fp(\()p Fo(A)1010 1325 y Fl(3)1050 1311 y Fp(\))g Fu(we)g(observe)g(that)868 1514 y Fp(Av)986 1528 y Fr(x)p Fn(2)p Fl(\003)1122 1540 y FA(k)1160 1528 y Fn(\\)p Fl(\003)1256 1536 y FA(y)r(;s)1345 1514 y Fq(\034)1385 1528 y Fr(x)1439 1452 y Fq(\030)1479 1466 y Fl(0)1518 1452 y Fp(\()p Fq(m)p Fp(\))p 1439 1493 230 4 v 1450 1576 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1704 1514 y(=)g(Av)1917 1547 y Fr(z)s Fn(2)p Fl(\003)2049 1514 y Fg(1)p FA(;e)2049 1557 y(n)2136 1514 y Fp(Av)2253 1528 y Fr(x)p Fn(2)p Fl(\003)2389 1540 y FA(k)2427 1528 y Fn(\\)p Fl(\003)2523 1536 y FA(y)r(;s)2613 1514 y Fq(\034)2653 1528 y Fr(x)p Fl(+)p Fr(z)2787 1514 y Fq(f)2832 1528 y Fl(3)2886 1514 y Fq(:)118 1716 y Fu(Thus)1321 1828 y Fo(A)1394 1842 y Fl(3)1459 1828 y Fo(\032)g([)1616 1842 y Fr(y)r Fn(2)p Fl(\003)1749 1854 y Fg(2)p FA(k)1841 1828 y Fo([)1902 1861 y Fr(z)s Fn(2)p Fl(\003)2034 1829 y Fg(1)p FA(;e)2034 1872 y(n)2141 1828 y Fo(A)2214 1842 y Fl(3)2253 1828 y Fp(\()p Fq(y)s(;)15 b(z)t Fp(\))118 1970 y Fu(where)814 2083 y Fo(A)887 2097 y Fl(3)926 2083 y Fp(\()p Fq(y)s(;)g(z)t Fp(\))26 b(=)f Fo(fj)p Fp(Av)1440 2097 y Fr(x)p Fn(2)p Fl(\003)1576 2109 y FA(k)1614 2097 y Fn(\\)p Fl(\003)1710 2105 y FA(y)r(;s)1800 2083 y Fq(\034)1840 2097 y Fr(x)p Fl(+)p Fr(z)1974 2083 y Fq(f)2019 2097 y Fl(3)2058 2083 y Fo(j)g(\025)g(j)p Fp(\003)2292 2098 y Fr(k)2355 2083 y Fo(\\)20 b Fp(\003)2499 2097 y Fr(y)r(;s)2593 2083 y Fo(j)2618 2045 y Fn(\000)2683 2018 y Fg(1)p 2683 2030 31 3 v 2683 2071 a(2)2724 2045 y Fl(+)2805 2018 y FA(\016)p 2788 2030 63 3 v 2788 2071 a Fg(2)p FA(d)2880 2083 y Fo(g)15 b Fq(:)118 2214 y Fu(Using)25 b(once)g(more)f(lemma)g(A.1)h(we) g(get)1342 2384 y Fs(P)p Fp(\()p Fo(A)1505 2398 y Fl(3)1545 2384 y Fp(\()p Fq(y)s(;)15 b(z)t Fp(\)\))27 b Fo(\024)e Fp(exp\()p Fo(\000)p Fq(cs)2244 2318 y FA(d)p Fi(\000)p Fg(1)p 2244 2331 111 3 v 2283 2373 a FA(d)2364 2347 y Fr(\016)2402 2384 y Fp(\))118 2534 y Fu(and)g(the)h(proof)f(is)h (complete.)2435 b Fe(\003)218 2697 y Fu(W)-8 b(e)25 b(conclude)f(this)j (part)e(with)i(a)e(slight)h(modi\002cation)g(of)f(proposition)i(6.5.) 118 2826 y FE(Proposition)k(6.6.)44 b Fh(Let)30 b Fq(n)i Fo(\024)g Fq(s)d Fh(be)g(positive)g(integers)f(and)h(let)g Fp(0)k Fq(<)f(\016)k Fo(\034)c Fp(1)p Fh(.)45 b(Let)30 b(also)e Fq(\015)38 b(>)32 b Fp(0)e Fh(and)e(set)118 2934 y Fq(J)168 2948 y Fr(s)231 2934 y Fp(=)d Fo(f)p Fp(1)p Fq(=s)505 2901 y Fr(\015)550 2934 y Fq(;)15 b Fp(2)p Fq(=s)723 2901 y Fr(\015)769 2934 y Fq(;)g(:)g(:)g(:)h Fp(1)21 b Fo(\000)f Fp(1)p Fq(=s)1220 2901 y Fr(\015)1265 2934 y Fo(g)p Fh(.)32 b(Then)25 b(there)g(exists)g(a)g(set)g Fo(A)g Fh(of)g(disorder)f(con\002gurations)f Fq(\013)j Fh(in)f Fp(\003)3589 2948 y Fl(2)p Fr(s)118 3042 y Fh(satisfying)1573 3169 y Fs(P)p Fp(\()p Fo(A)p Fp(\))i Fo(\024)d Fq(s)1936 3132 y Fr(\015)1996 3169 y Fq(e)2038 3132 y Fn(\000)p Fr(c)11 b(s)2168 3108 y FA(\016)118 3300 y Fh(and)24 b(such)h(that,)h(for)f Fq(s)g Fh(large)f(enough)g(depending)f(on)h Fq(\016)s Fh(,)779 3475 y Fp(sup)757 3553 y Fr(m)p Fn(2)p Fr(J)905 3561 y FA(s)953 3475 y Fp(V)-8 b(ar)1095 3516 y Fr(\026)1137 3481 y FA(\025)p Fg(\()p FA(m)p Fg(\))1137 3539 y(\003)1178 3554 y(2)p FA(s)1300 3374 y Fk(\020)1354 3475 y Fp(Av)1472 3427 y Fl(\()p Fr(n)p Fl(\))1472 3504 y Fr(x)p Fn(2)p Fl(\003)1608 3512 y FA(s)1645 3475 y Fq(\034)1685 3489 y Fr(x)1729 3475 y Fq(\036)1783 3489 y Fr(n;s)1882 3374 y Fk(\021)1961 3475 y Fo(\024)25 b Fq(c)p Fp(1)-25 b(I)2150 3489 y Fn(A)2206 3470 y FA(c)2242 3475 y Fp(\()p Fq(\013)p Fp(\))p Fq(s)2413 3437 y Fn(\000)p Fl(2)p Fr(d)p Fl(+2)p Fr(\016)2689 3475 y Fp(+)20 b(1)-25 b(I)2833 3489 y Fn(A)2893 3475 y Fp(\()p Fq(\013)p Fp(\))388 b Fu(\(6.22\))118 3718 y Fh(where)25 b Fp(Av)489 3671 y Fl(\()p Fr(n)p Fl(\))489 3747 y Fr(x)p Fn(2)p Fl(\003)625 3755 y FA(s)687 3718 y Fp(:=)h(Av)926 3741 y Fr(x)p Fn(2)p Fl(\003)1062 3749 y FA(s)1095 3741 y Fn(\\)p Fr(n)p Fm(Z)1235 3722 y FA(d)1269 3718 y Fh(.)118 3886 y(Proof)-8 b(.)42 b Fu(The)34 b(proposition)i(can)d(be)h(proved)h(as)f(proposition)i(6.5) e(with)h(some)f(slight)h(modi\002cations)118 3994 y(that)26 b(we)f(comment.)30 b(F)-6 b(or)24 b(any)h Fq(m)g Fo(2)g Fq(J)1427 4008 y Fr(s)1489 3994 y Fu(it)h(is)f(convenient)f(to)h (de\002ne)f Fo(M)p Fp(\()p Fq(m)p Fp(\))p Fq(;)32 b Fo(A)2878 4008 y Fl(1)2917 3994 y Fp(\()p Fq(m)p Fp(\))p Fq(;)25 b Fu(and)g Fo(A)3364 4008 y Fl(2)3404 3994 y Fp(\()p Fq(m)p Fp(\))g Fu(as)118 4102 y(done)g(respectively)f(for)h Fo(M)p Fq(;)31 b Fo(A)1208 4116 y Fl(1)1247 4102 y Fq(;)26 b Fu(and)f Fo(A)1545 4116 y Fl(2)1609 4102 y Fu(in)h(the)f(proof)h(of)f (proposition)h(6.5)f(and)g(to)h(set)1057 4270 y Fo(A)1130 4284 y Fl(0)1169 4270 y Fp(\()p Fq(m)p Fp(\))g(:=)1466 4196 y Fk(\010)1519 4270 y Fq(\026)1574 4222 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))1574 4298 y(\003)1623 4307 y Fg(2)p FA(s)1737 4196 y Fk(\000)1778 4270 y Fo(M)1887 4196 y Fk(\001)1954 4270 y Fo(\025)f Fq(s)2093 4232 y Fr(d)2148 4270 y Fp(exp\()p Fo(\000)p Fq(s)2436 4232 y Fr(\016)r(=)p Fl(2)2544 4270 y Fp(\))2594 4196 y Fk(\011)2648 4270 y Fq(;)1057 4473 y Fo(A)1130 4487 y Fl(3)1169 4473 y Fp(\()p Fq(m)p Fp(\))h(:=)1466 4372 y Fk(n)1542 4368 y(\014)1542 4423 y(\014)1542 4477 y(\014)1572 4473 y Fp(Av)1690 4425 y Fl(\()p Fr(n)p Fl(\))1690 4501 y Fr(x)p Fn(2)p Fl(\003)1826 4509 y FA(s)1863 4473 y Fq(\034)1903 4487 y Fr(x)1971 4411 y Fq(\030)2011 4425 y Fl(0)2051 4411 y Fp(\()p Fq(m)p Fp(\))p 1971 4452 230 4 v 1982 4535 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))2211 4368 y Fk(\014)2211 4423 y(\014)2211 4477 y(\014)2267 4473 y Fo(\025)f Fq(s)2406 4435 y Fn(\000)2471 4408 y FA(d)p 2470 4420 33 3 v 2471 4461 a Fg(2)2512 4435 y Fl(+)2577 4408 y FA(\016)p 2577 4420 31 3 v 2577 4461 a Fg(2)2637 4372 y Fk(o)2697 4473 y Fq(:)118 4676 y Fu(Then)f(one)f(sets)h(again)g Fo(A)p Fp(\()p Fq(m)p Fp(\))h(:=)g Fo(A)1367 4690 y Fl(0)1407 4676 y Fp(\()p Fq(m)p Fp(\))15 b Fo([)g(A)1721 4690 y Fl(1)1760 4676 y Fp(\()p Fq(m)p Fp(\))h Fo([)f(A)2075 4690 y Fl(2)2114 4676 y Fp(\()p Fq(m)p Fp(\))g Fo([)g(A)2428 4690 y Fl(3)2467 4676 y Fp(\()p Fq(m)p Fp(\))p Fu(,)25 b Fo(A)g Fp(:=)g Fo([)2947 4690 y Fr(m)p Fn(2)p Fr(J)3095 4698 y FA(s)3132 4676 y Fo(A)p Fp(\()p Fq(m)p Fp(\))p Fu(.)31 b(By)24 b(the)118 4784 y(same)h(arguments)f(as)i(in)f(the)h (proof)f(of)g(proposition)i(6.5)e(one)f(obtains)i(\(6.22\).)118 4903 y(Let)g(us)g(prove)f(the)g(estimate)h Fs(P)p Fp(\()p Fo(A)p Fp(\))h Fu(or)-10 b(,)25 b(equivalently)-12 b(,)25 b(that)i(for)e(any)h Fq(m)f Fo(2)g Fq(J)2744 4917 y Fr(s)2807 4903 y Fs(P)p Fp(\()p Fo(A)p Fp(\()p Fq(m)p Fp(\)\))i Fo(\024)f Fq(e)3321 4870 y Fn(\000)p Fr(cs)3440 4847 y FA(\016)3477 4903 y Fu(.)33 b(F)-6 b(or)118 5011 y(this)27 b(purpose,)e(given)f Fq(m)i Fo(2)e Fq(J)1134 5025 y Fr(s)1172 5011 y Fu(,)h(it)h(is)g(convenient)e(to)i(de\002ne)1084 5181 y Fo(B)s Fp(\()p Fq(m)p Fp(\))g(:=)1444 5107 y Fk(\010)1512 5104 y(\014)1512 5158 y(\014)1542 5181 y Fq(\025)1595 5195 y Fl(\003)1644 5204 y Fg(2)p FA(s)1712 5181 y Fp(\()p Fq(m)p Fp(\))21 b Fo(\000)f Fq(\025)2027 5195 y Fl(0)2066 5181 y Fp(\()p Fq(m)p Fp(\))2216 5104 y Fk(\014)2216 5158 y(\014)2272 5181 y Fo(\025)25 b Fq(s)2411 5143 y Fn(\000)2476 5116 y FA(d)p 2476 5128 33 3 v 2477 5169 a Fg(2)2518 5143 y Fl(+)2583 5116 y FA(\016)p 2582 5128 31 3 v 2582 5169 a Fg(4)2642 5107 y Fk(\011)118 5331 y Fu(and)g(write)172 5480 y Fs(P)p Fp(\()p Fo(A)p Fp(\()p Fq(m)p Fp(\)\))h Fo(\024)f Fs(P)p Fp(\()p Fo(B)s Fp(\()p Fq(m)p Fp(\)\))d(+)e Fs(P)p Fp(\()p Fo(B)1246 5443 y Fr(c)1281 5480 y Fp(\()p Fq(m)p Fp(\))g Fo(\\)g(A)1605 5494 y Fl(0)1644 5480 y Fp(\()p Fq(m)p Fp(\)\))h(+)f Fs(P)p Fp(\()p Fo(A)2104 5494 y Fl(1)2144 5480 y Fp(\()p Fq(m)p Fp(\)\))h(+)f Fs(P)p Fp(\()p Fo(A)2604 5494 y Fl(2)2644 5480 y Fp(\()p Fq(m)p Fp(\)\))h(+)f Fs(P)p Fp(\()p Fo(A)3104 5494 y Fl(3)3144 5480 y Fp(\()p Fq(m)p Fp(\)\))p Fq(:)55 b Fu(\(6.23\))p eop %%Page: 39 39 39 38 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(39)118 366 y Fu(Let)26 b(us)f(suppose)h Fp(0)f Fq(<)g(m)g Fo(\024)1100 331 y Fl(1)p 1100 346 36 4 v 1100 398 a(2)1145 366 y Fu(.)32 b(Then)25 b(lemma)f(A.3)h(implies)h(that)966 471 y Fk(\014)966 526 y(\014)997 548 y Fq(\025)1050 562 y Fl(\003)1099 571 y Fg(2)p FA(s)1167 548 y Fp(\()p Fq(m)p Fp(\))20 b Fo(\000)g Fq(\025)1481 562 y Fl(0)1521 548 y Fp(\()p Fq(m)p Fp(\))1671 471 y Fk(\014)1671 526 y(\014)1727 548 y Fo(\024)25 b Fq(c)1877 471 y Fk(\014)1877 526 y(\014)1908 548 y Fp(1)20 b Fo(\000)g Fq(m)2144 511 y Fn(\000)p Fl(1)2238 548 y Fq(\026)2293 511 y Fr(\025)2334 520 y Fg(0)2369 511 y Fl(\()p Fr(m)p Fl(\))2490 548 y Fp(\()p Fq(m)2605 562 y Fl(\003)2654 571 y Fg(2)p FA(s)2722 548 y Fp(\))2757 471 y Fk(\014)2757 526 y(\014)2788 548 y Fq(:)118 726 y Fu(Thanks)25 b(to)g(the)g(above)f(estimate)h(and)g(to)g(lemma)e(A.1)i (applied)f(with)i Fq(f)35 b Fp(:=)25 b(1)18 b Fo(\000)g Fq(m)2984 693 y Fn(\000)p Fl(1)3078 726 y Fq(\026)3133 693 y Fr(\025)3174 702 y Fg(0)3208 693 y Fl(\()p Fr(m)p Fl(\))3330 726 y Fp(\()p Fq(\021)3410 740 y Fl(0)3450 726 y Fp(\))p Fu(,)25 b(the)118 851 y(\002rst)31 b(term)e(in)h(the)h(r) -10 b(.h.s.)46 b(of)30 b(\(6.23\))f(is)i(smaller)e(than)h Fq(e)2069 818 y Fn(\000)p Fr(c)11 b(s)2199 794 y FA(\016)r(=)p Fg(2)2300 851 y Fu(.)45 b(The)31 b(second)e(term)g(is)i(smaller)e(than) 118 966 y Fq(e)160 933 y Fn(\000)p Fr(cs)279 909 y FA(\016)r(=)p Fg(2)408 966 y Fu(by)g(lemma)f(6.4.)43 b(Moreover)-10 b(,)29 b Fs(P)p Fp(\()p Fo(A)1614 980 y Fl(1)1654 966 y Fp(\()p Fq(m)p Fp(\)\))h Fu(and)f Fs(P)p Fp(\()p Fo(A)2210 980 y Fl(2)2250 966 y Fp(\()p Fq(m)p Fp(\)\))h Fu(can)e(be)h(bounded)g (by)g Fq(s)3299 933 y Fr(d)3339 966 y Fq(e)3381 933 y Fn(\000)p Fr(c)12 b(s)3512 909 y FA(\016)3579 966 y Fu(as)118 1073 y(in)25 b(the)h(proof)f(of)h(proposition)g(6.5.)118 1181 y(Finally)-12 b(,)29 b(let)f(us)h(consider)g Fs(P)p Fp(\()p Fo(A)1201 1195 y Fl(3)1240 1181 y Fp(\()p Fq(m)p Fp(\)\))p Fu(.)43 b(F)-6 b(or)29 b(simplicity)g(of)g(notation)g(we)g (restrict)g(to)g(the)g(case)f Fq(d)j Fp(=)g(1)118 1289 y Fu(and)25 b(we)g(write)274 1483 y Fq(n)429 1397 y Fk(X)344 1594 y Fr(x)p Fn(2)p Fl(\003)480 1602 y FA(s)512 1594 y Fn(\\)p Fr(n)p Fm(Z)662 1483 y Fq(\034)702 1497 y Fr(x)770 1421 y Fq(\030)810 1435 y Fl(0)849 1421 y Fp(\()p Fq(m)p Fp(\))p 770 1462 230 4 v 781 1545 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1035 1483 y(=)1131 1382 y Fk(\020)1289 1397 y(X)1185 1594 y Fr(x)p Fn(2)p Fl(\003)1321 1602 y FA(s)1354 1594 y Fn(\\)p Fl(2)p Fr(n)p Fm(Z)1585 1397 y Fk(X)1539 1593 y Fr(z)s Fn(2)p Fl(\003)1671 1571 y FA(e)1671 1615 y Fg(1)p FA(;n)1778 1483 y Fq(\034)1818 1497 y Fr(x)p Fl(+)p Fr(z)1952 1483 y Fq(f)2007 1382 y Fk(\021)2081 1483 y Fp(+)2171 1382 y Fk(\020)2406 1397 y(X)2226 1598 y Fr(x)p Fn(2)p Fl(\003)2362 1606 y FA(s)2395 1598 y Fn(\\)p Fl(\(2)p Fr(n)p Fl(+)p Fr(n)p Fl(\))p Fm(Z)2778 1397 y Fk(X)2732 1593 y Fr(z)s Fn(2)p Fl(\003)2864 1571 y FA(e)2864 1615 y Fg(1)p FA(;n)2971 1483 y Fq(\034)3011 1497 y Fr(x)p Fl(+)p Fr(z)3145 1483 y Fq(f)3200 1382 y Fk(\021)3409 1483 y Fu(\(6.24\))118 1779 y(where)33 b Fq(f)48 b Fp(:=)39 b Fq(\037)p Fp(\()p Fq(m)p Fp(\))832 1746 y Fn(\000)p Fl(1)927 1706 y Fk(\000)983 1779 y Fq(\026)1038 1746 y Fr(\025)1079 1755 y Fg(0)1114 1746 y Fl(\()p Fr(m)p Fl(\))1235 1779 y Fp(\()p Fq(\021)1315 1793 y Fl(0)1355 1779 y Fp(;)15 b Fq(\021)1440 1793 y Fl(0)1480 1779 y Fp(\))26 b Fo(\000)g Fq(\026)1693 1746 y Fr(\025)1734 1755 y Fg(0)1768 1746 y Fl(\()p Fr(m)p Fl(\))1890 1779 y Fp(\()p Fq(\021)1970 1793 y Fr(ne)2050 1779 y Fp(;)15 b Fq(\021)2135 1793 y Fr(ne)2215 1779 y Fp(\))2265 1706 y Fk(\001)2307 1779 y Fu(.)56 b(W)-8 b(e)34 b(remark)e(that)j(in)e (both)h(the)g(ad-)118 1887 y(denda)27 b(in)g(the)f(r)-10 b(.h.s.)37 b(of)27 b(\(6.24\))e(the)i(appearing)f(functions)g(have)h (disjoint)h(support)f(and)g(form)f(a)h(set)118 1995 y(of)j(cardinality) g Fq(O)s Fp(\()p Fq(k)835 1962 y Fr(d)876 1995 y Fp(\))p Fu(,)h(moreover)d Fs(E)10 b Fp(\()p Fq(f)g Fp(\))38 b(=)33 b(0)p Fu(.)44 b(Therefore,)30 b(by)g(the)f(same)h(arguments)e(used)i (in)f(the)118 2117 y(proof)c(of)h(lemma)e(A.1,)h(we)g(obtain)h(that)g Fs(P)p Fp(\()p Fo(A)1701 2131 y Fl(3)1741 2117 y Fp(\()p Fq(m)p Fp(\)\))g Fo(\024)f Fq(e)2090 2084 y Fn(\000)p Fr(cs)2209 2061 y FA(\016)2247 2117 y Fu(.)1319 b Fe(\003)118 2313 y Fu(6.2.)45 b FE(An)30 b Fq(L)521 2280 y Fn(1)625 2313 y FE(bound.)46 b Fu(W)-8 b(e)29 b(conclude)d(this)j(section)f (with)i(a)e(simple)g Fq(L)2604 2280 y Fn(1)2706 2313 y Fu(bound)g(on)g Fo(j)p Fq(\034)3181 2327 y Fr(x)3225 2313 y Fq(\036)3279 2329 y Fr(s;s)3365 2310 y Fi(0)3390 2313 y Fo(j)h Fu(when)118 2421 y Fq(s)c Fu(scales)g(as)h(an)f(inverse)f (power)h(of)h Fq(\017)p Fu(.)118 2557 y FE(L)r(emma)38 b(6.7.)47 b Fh(Let)35 b Fp(0)43 b Fq(<)f(\015)47 b(<)42 b Fp(1)36 b Fh(and)f Fp(0)42 b Fq(<)g(\016)k Fo(\034)d Fp(1)35 b Fh(and)g(set)g Fq(s)41 b Fp(=)h Fq(O)s Fp(\()p Fq(\017)2652 2524 y Fn(\000)p Fr(\015)2752 2557 y Fp(\))p Fh(.)62 b(Then,)38 b(for)d(almost)g(all)118 2664 y(con\002guration)24 b(disorder)g Fq(\013)h Fh(and)f Fq(\017)i Fh(small)e(enough,)1151 2847 y Fp(sup)1136 2933 y Fr(x)p Fn(2)p Fm(T)1273 2914 y FA(d)1273 2950 y(\017)1318 2847 y Fo(j)p Fq(\034)1383 2861 y Fr(x)1427 2847 y Fq(\036)1481 2862 y Fr(s;s)1567 2843 y Fi(0)1592 2847 y Fo(j)i(\024)f Fq(c)15 b(s)1836 2809 y Fn(\000)1901 2782 y FA(d)p 1901 2794 33 3 v 1902 2835 a Fg(2)1943 2809 y Fl(+)p Fr(\016)2126 2847 y Fo(8)p Fq(s)2220 2809 y Fn(0)2268 2847 y Fo(2)25 b Fp([)p Fq(s;)15 b(\017)2499 2809 y Fn(\000)p Fl(1)2593 2847 y Fp(])p Fq(:)766 b Fu(\(6.25\))118 3107 y Fh(Proof)-8 b(.)42 b Fu(By)25 b(the)h(equivalence)d(of)j(ensembles)e(it)i(is)g(enough)e (to)i(prove)f(\(6.25\))f(with)i Fq(\036)3067 3122 y Fr(s;s)3153 3103 y Fi(0)3204 3107 y Fu(replaced)e(by)130 3206 y Fp(^)118 3230 y Fq(\036)172 3246 y Fr(s;s)258 3227 y Fi(0)284 3230 y Fu(.)31 b(Using)26 b(lemma)d Fq(A:)p Fp(3)k Fu(we)e(get)834 3412 y Fo(j)871 3388 y Fp(^)859 3412 y Fq(\036)913 3427 y Fr(s;s)999 3408 y Fi(0)1045 3412 y Fo(\000)20 b Fq(\026)1191 3372 y Fr(\025)1232 3381 y Fg(0)1266 3372 y Fl(\()p Fr(m)1355 3348 y FA(e)1355 3402 y(s)1384 3388 y Fi(0)1412 3372 y Fl(\))1443 3338 y Fk(\000)1485 3412 y Fq(m)1565 3374 y Fl(1)p Fr(;e)1565 3434 y(s)1677 3412 y Fo(\000)g Fq(m)1848 3374 y Fl(2)p Fr(;e)1848 3434 y(s)1939 3338 y Fk(\001)1981 3412 y Fo(j)25 b(\024)g Fq(c)15 b Fo(j)p Fq(m)2286 3374 y Fr(e)2286 3435 y(s)2319 3417 y Fi(0)2346 3412 y Fp(\()p Fq(\021)s Fp(\))21 b Fo(\000)f Fq(\026)2631 3374 y Fr(\025)2672 3383 y Fg(0)2711 3412 y Fp(\()p Fq(m)2826 3374 y Fr(e)2826 3435 y(s)2859 3417 y Fi(0)2885 3412 y Fp(\))p Fo(j)464 b Fu(\(6.26\))118 3575 y(and)25 b(similarly)h(upon)e(translation)i(by)f Fq(x)p Fu(.)118 3683 y(Let)h(us)f(de\002ne)945 3846 y Fo(D)1015 3860 y Fr(x)1059 3846 y Fp(\()p Fq(m)p Fp(\))h(:=)f Fo(f)15 b(j)p Fq(m)21 b Fo(\000)f Fq(\026)1688 3808 y Fr(\025)1729 3817 y Fg(0)1763 3808 y Fl(\()p Fr(m)p Fl(\))1885 3846 y Fp(\()p Fq(m)2000 3808 y Fr(e)2000 3869 y(x;s)2093 3850 y Fi(0)2118 3846 y Fp(\))p Fo(j)26 b(\025)f Fp(\()p Fq(s)2378 3808 y Fn(0)2401 3846 y Fp(\))2436 3808 y Fn(\000)2501 3781 y FA(d)p 2501 3793 V 2502 3834 a Fg(2)2543 3808 y Fl(+)p Fr(\016)2651 3846 y Fo(g)945 4017 y(D)1018 3979 y Fn(0)1015 4039 y Fr(x)1059 4017 y Fp(\()p Fq(m)p Fp(\))h(:=)f Fo(f)15 b(j)p Fq(\026)1496 3979 y Fr(\025)1537 3988 y Fg(0)1572 3979 y Fl(\()p Fr(m)p Fl(\))1694 3943 y Fk(\000)1735 4017 y Fq(\034)1775 4031 y Fr(x)1819 4017 y Fp(\()p Fq(m)1934 3979 y Fl(1)p Fr(;e)1934 4039 y(s)2046 4017 y Fo(\000)20 b Fq(m)2217 3979 y Fl(2)p Fr(;e)2217 4039 y(s)2309 4017 y Fp(\))2344 3943 y Fk(\001)2386 4017 y Fo(j)25 b(\025)g Fq(s)2575 3979 y Fn(\000)2640 3952 y FA(d)p 2639 3964 V 2640 4005 a Fg(2)2681 3979 y Fl(+)p Fr(\016)2789 4017 y Fo(g)1137 4166 y(D)j Fp(:=)d Fo([)1417 4180 y Fr(m)1503 4166 y Fo([)1564 4188 y Fr(x)p Fn(2)p Fm(T)1701 4169 y FA(d)1701 4205 y(\017)1755 4092 y Fk(\000)1797 4166 y Fo(D)1867 4180 y Fr(x)1911 4166 y Fp(\()p Fq(m)p Fp(\))c Fo([)e(D)2235 4128 y Fn(0)2232 4188 y Fr(x)2276 4166 y Fp(\()p Fq(m)p Fp(\))2426 4092 y Fk(\001)118 4335 y Fu(where,)25 b(in)g(the)h(last)g(formula,)e Fq(m)h Fu(varies)h(among)e (all)h(possible)g(values)g(of)g Fq(m)2776 4302 y Fr(e)2776 4363 y(s)2809 4344 y Fi(0)2835 4335 y Fu(.)118 4463 y Fs(P)p Fp(\()p Fo(D)278 4477 y Fr(x)323 4463 y Fp(\()p Fq(m)p Fp(\)\))f Fu(and)g Fs(P)p Fp(\()p Fo(D)868 4430 y Fn(0)865 4485 y Fr(x)910 4463 y Fp(\()p Fq(m)p Fp(\)\))g Fu(can)f(now)g(be)g(estimated)h(from)f(above)g(by)g Fq(e)2630 4430 y Fn(\000)p Fr(cs)2749 4406 y Fg(2)p FA(\016)2841 4463 y Fu(thanks)h(to)g(lemma)e Fq(A:)p Fp(1)118 4576 y Fu(applied)e(to)g Fq(f)10 b Fp(\()p Fq(\013)p Fp(\))26 b(=)f Fq(\026)890 4543 y Fr(\025)931 4552 y Fg(0)965 4543 y Fl(\()p Fr(m)p Fl(\))1087 4576 y Fp(\()p Fq(\021)1167 4590 y Fl(0)1207 4576 y Fp(\))q Fo(\000)q Fq(m)19 b Fu(and)h Fq(f)10 b Fp(\()p Fq(\013)p Fp(\))26 b(=)f Fq(\026)1943 4543 y Fr(\025)1984 4552 y Fg(0)2018 4543 y Fl(\()p Fr(m)p Fl(\))2140 4576 y Fp(\()p Fq(\021)2220 4590 y Fl(0)2260 4576 y Fp(;)15 b Fq(\021)2345 4590 y Fl(0)2385 4576 y Fp(\))q Fo(\000)q Fs(E)2548 4503 y Fk(\002)2592 4576 y Fq(\026)2647 4543 y Fr(\025)2688 4552 y Fg(0)2722 4543 y Fl(\()p Fr(m)p Fl(\))2844 4576 y Fp(\()p Fq(\021)2924 4590 y Fl(0)2964 4576 y Fp(;)g Fq(\021)3049 4590 y Fl(0)3088 4576 y Fp(\))3123 4503 y Fk(\003)3181 4576 y Fu(respectively)-12 b(.)118 4708 y(Therefore,)27 b Fs(P)p Fp(\()p Fo(D)s Fp(\))h Fo(\024)g Fq(\017)920 4675 y Fn(\000)p Fl(2)p Fr(d)1050 4708 y Fq(e)1092 4675 y Fn(\000)p Fr(cs)1211 4651 y Fg(2)p FA(\016)1306 4708 y Fu(and)f(a)f(simple)h(use)f(of)h (Borel-Cantelli)f(lemma)f(proves)i(the)g(thesis.)3591 4815 y Fe(\003)1113 5048 y Fu(7.)47 b(C)t Ft(E)t(N)t(T)t(R)t(A)t(L)34 b Fu(L)t Ft(I)t(M)t(I)t(T)e Fu(T)t Ft(H)t(E)t(O)t(R)t(E)t(M)g Fu(V)n Ft(A)t(R)t(I)t(A)t(N)t(C)t(E)218 5210 y Fu(In)26 b(this)h(section)f(we)g(investigate)g(the)g(structure)f(of)i(the)f (space)f Fo(G)32 b Fu(that)27 b(we)f(recall)e(was)i(de\002ned)g(as)118 5318 y(\(see)f(\(4.15\)\))560 5480 y Fo(G)30 b Fp(:=)25 b Fo(f)p Fq(g)30 b Fo(2)25 b Fs(G)65 b Fp(:)40 b Fo(9)15 b Fp(\003)25 b Fo(2)g Fs(F)61 b Fu(such)26 b(that,)51 b Fo(8)p Fq(\013)26 b Fu(and)f Fo(8)p Fq(\027)31 b Fo(2)24 b(M)2560 5443 y Fr(\013)2610 5480 y Fp(\(\003\))15 b Fq(;)42 b(\027)6 b Fp(\()p Fq(g)s Fp(\))26 b(=)f(0)15 b Fo(g)p eop %%Page: 40 40 40 39 bop 118 45 a Fj(40)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(endowed)25 b(with)i(the)e(non)g(negative)f(semi-inner)h(product)1387 532 y Fq(V)1440 546 y Fr(m)1506 532 y Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))27 b(:=)i(lim)1860 593 y Fr(`)p Fn("1)2025 532 y Fq(V)2098 494 y Fl(\()p Fr(`)p Fl(\))2078 554 y Fr(m)2186 532 y Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))1067 b Fu(\(7.1\))118 740 y(where)424 903 y Fq(V)497 865 y Fl(\()p Fr(`)p Fl(\))477 925 y Fr(m)585 903 y Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))26 b(:=)g(\(2)p Fq(l)r Fp(\))1083 865 y Fn(\000)p Fr(d)1194 903 y Fs(E)1249 802 y Fk(h)1298 903 y Fq(\026)1353 865 y Fr(\025)1394 874 y Fg(0)1428 865 y Fl(\()p Fr(m)p Fl(\))1550 829 y Fk(\000)1639 816 y(X)1607 1018 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)1771 1027 y Fg(1)1819 903 y Fq(\034)1859 917 y Fr(x)1903 903 y Fq(f)5 b(;)15 b Fp(\()p Fo(\000L)2162 917 y Fl(\003)2211 929 y FA(`)2244 903 y Fp(\))2279 865 y Fn(\000)p Fl(1)2422 816 y Fk(X)2389 1018 y Fn(j)p Fr(x)p Fn(j\024)p Fr(`)2553 1027 y Fg(1)2601 903 y Fq(\034)2641 917 y Fr(x)2685 903 y Fq(g)2731 829 y Fk(\001)2773 802 y(i)2831 903 y Fq(;)107 b(m)25 b Fo(2)f Fp(\(0)p Fq(;)15 b Fp(1\))118 1199 y Fu(with)27 b Fq(`)359 1213 y Fl(1)423 1199 y Fp(:=)e Fq(`)20 b Fo(\000)693 1121 y(p)p 769 1121 38 4 v 78 x Fq(`)p Fu(.)32 b(F)-6 b(or)24 b Fq(m)h Fp(=)g(0)p Fq(;)15 b Fp(1)27 b Fu(we)e(simply)g(de\002ne)g Fq(V)2140 1166 y Fl(\()p Fr(m)p Fl(\))2262 1199 y Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))26 b(=)f Fq(V)2663 1151 y Fl(\()p Fr(m)p Fl(\))2643 1228 y Fr(`)2785 1199 y Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))26 b(=)f(0)118 1331 y Fu(In)20 b(all)g(what)g(follows)g(we) g(\002x)g(a)g(density)h Fq(m)k Fo(2)g Fp(\(0)p Fq(;)15 b Fp(1\))22 b Fu(that,)g(most)f(of)f(the)g(times,)i(will)e(not)g (appear)g(inside)118 1446 y(the)32 b(notation)g(and)f(we)g(denote)g(by) g Fs(P)1450 1413 y Fn(\003)1522 1446 y Fu(the)g(annealed)f(probability) i(measure)e(on)3042 1423 y Fp(~)3031 1446 y(\012)h Fu(characterized)118 1554 y(by)1255 1681 y Fs(P)1310 1644 y Fn(\003)1350 1681 y Fp(\()p Fq(d\013;)15 b(d\021)s Fp(\))28 b(=)d Fs(P)p Fp(\()p Fq(d\013)p Fp(\))p Fq(\026)2069 1644 y Fr(\013;\025)2175 1653 y Fg(0)2211 1644 y Fl(\()p Fr(m)p Fl(\))2333 1681 y Fp(\()p Fq(d\021)s Fp(\))p Fq(:)118 1820 y Fu(W)-8 b(e)24 b(remark)f(that)i Fs(P)808 1787 y Fn(\003)872 1820 y Fu(is)f(translation)g(invariant)h(and)f(we)f(write)h Fs(E)2384 1787 y Fn(\003)2453 1820 y Fu(for)g(the)g(corresponding)e (expecta-)118 1928 y(tion.)118 2113 y(7.1.)45 b FE(The)31 b(pre-Hilbert)e(space)h Fo(G)5 b FE(.)46 b Fu(In)30 b(what)g(follows)f (we)g(prove)f(that)j(the)e(semi\226inner)g(product)g Fq(V)118 2221 y Fu(is)34 b(well)f(de\002ned)g(and)g(that)i(the)f (subspace)f(generated)f(by)h(the)h(currents)e Fq(j)2775 2235 y Fl(0)p Fr(;e)2867 2221 y Fu(,)k Fq(e)j Fo(2)f(E)8 b Fu(,)36 b(and)e(by)f(the)118 2329 y(\003uctuations)i Fo(L)p Fq(g)s Fu(,)i Fq(g)45 b Fo(2)c Fs(G)19 b Fu(,)43 b(is)36 b(dense)e(in)h Fo(G)5 b Fu(.)60 b(T)-9 b(o)35 b(this)i(aim)e(we)f(need)g(to)h(generalize)d(the)j(standard)118 2436 y(theory)28 b(\([23)o(])g(and)f(references)f(therein\),)i(based)g (on)f(closed)g(and)h(exact)f(forms,)h(to)g(the)g(disordered)118 2544 y(case.)57 b(The)34 b(main)f(new)g(feature)h(in)f(the)h (disordered)h(case)e(is)i(a)f(richer)f(structure)g(of)h(the)g(space)f (of)118 2652 y(closed)25 b(forms)g(which)h(requires)f(a)g(proper)g (analysis.)118 2760 y(W)-8 b(e)26 b(begin)e(with)j(a)e Fh(table)g(of)f(calculus)h Fu(that)i(can)d(be)h(easily)g(checked)g(as)g (in)h(the)f(non)g(disordered)h(case.)118 2868 y(F)-6 b(or)25 b(any)g Fq(f)35 b Fo(2)24 b(G)5 b Fu(,)26 b Fq(u)g Fo(2)f Fs(G)50 b Fu(and)25 b Fq(e)h Fo(2)e(E)34 b Fu(let)789 3041 y Fq(t)822 3055 y Fr(e)859 3041 y Fp(\()p Fq(f)10 b Fp(\))25 b(:=)1148 2955 y Fk(X)1130 3160 y Fr(x)p Fn(2)p Fm(Z)1267 3141 y FA(d)1297 3041 y Fp(\()p Fq(x;)15 b(e)p Fp(\))p Fs(E)1557 3004 y Fn(\003)1603 3041 y Fp(\()p Fq(\021)1683 3055 y Fr(x)1727 3041 y Fq(;)g(f)10 b Fp(\))p Fq(;)107 b Fp(\()p Fq(f)5 b(;)15 b(u)p Fp(\))2201 3055 y Fl(0)2266 3041 y Fp(:=)2405 2955 y Fk(X)2387 3160 y Fr(x)p Fn(2)p Fm(Z)2524 3141 y FA(d)2569 3041 y Fs(E)2624 3004 y Fn(\003)2669 3041 y Fp(\()p Fq(\034)2744 3055 y Fr(x)2788 3041 y Fq(f)5 b(;)15 b(u)p Fp(\))p Fq(:)118 3301 y FE(L)r(emma)27 b(7.1.)42 b Fh(F)l(or)25 b(any)g Fq(f)34 b Fo(2)25 b(G)5 b Fh(,)26 b Fq(u)f Fo(2)g Fs(G)50 b Fh(and)24 b Fq(e;)15 b(e)1800 3268 y Fn(0)1850 3301 y Fo(2)25 b(E)615 3499 y Fq(V)20 b Fp(\()p Fq(f)5 b(;)15 b Fo(L)p Fq(u)p Fp(\))25 b(=)g Fo(\000)p Fp(\()p Fq(f)5 b(;)15 b(u)p Fp(\))1367 3513 y Fl(0)1407 3499 y Fq(;)197 b(V)20 b Fp(\()p Fo(L)p Fq(u;)15 b Fo(L)p Fq(u)p Fp(\))26 b(=)2164 3412 y Fk(X)2168 3609 y Fr(e)p Fn(2E)2320 3437 y Fp(1)p 2320 3478 46 4 v 2320 3561 a(2)2376 3499 y Fs(E)2430 3461 y Fn(\003)2476 3425 y Fk(\000)2517 3499 y Fq(c)2556 3513 y Fl(0)p Fr(;e)2649 3499 y Fp(\()p Fo(r)2760 3513 y Fl(0)p Fr(;e)2851 3499 y Fq(u)p 2851 3514 53 4 v Fp(\))2938 3461 y Fl(2)2978 3425 y Fk(\001)3020 3499 y Fq(;)615 3778 y(V)20 b Fp(\()p Fq(j)760 3792 y Fl(0)p Fr(;e)852 3778 y Fq(;)15 b(g)s Fp(\))27 b(=)e Fo(\000)p Fq(t)1200 3792 y Fr(e)1236 3778 y Fp(\()p Fq(f)10 b Fp(\))p Fq(;)248 b(V)20 b Fp(\()p Fq(j)1779 3792 y Fl(0)p Fr(;e)1872 3778 y Fq(;)15 b(j)1949 3793 y Fl(0)p Fr(;e)2037 3774 y Fi(0)2064 3778 y Fp(\))25 b(=)2230 3716 y(1)p 2230 3757 46 4 v 2230 3840 a(2)2286 3778 y Fs(E)2340 3740 y Fn(\003)2386 3704 y Fk(\000)2453 3778 y Fq(c)2492 3792 y Fl(0)p Fr(;e)2584 3778 y Fp(\()p Fo(r)2695 3792 y Fl(0)p Fr(;e)2787 3778 y Fq(\021)2832 3792 y Fl(0)2871 3778 y Fp(\))2906 3740 y Fl(2)2946 3704 y Fk(\001)2988 3778 y Fq(\016)3028 3793 y Fr(e;e)3114 3774 y Fi(0)3140 3778 y Fq(;)615 3995 y(V)20 b Fp(\()p Fq(j)760 4009 y Fl(0)p Fr(;e)852 3995 y Fq(;)15 b Fo(L)p Fq(u)p Fp(\))26 b(=)f Fo(\000)1245 3934 y Fp(1)p 1245 3975 V 1245 4058 a(2)1300 3995 y Fs(E)1355 3958 y Fn(\003)1400 3922 y Fk(\000)1442 3995 y Fq(c)1481 4009 y Fl(0)p Fr(;e)1573 3995 y Fo(r)1649 4009 y Fl(0)p Fr(;e)1740 3995 y Fq(u)p 1740 4010 53 4 v 21 w Fo(\001)20 b(r)1934 4009 y Fl(0)p Fr(;e)2026 3995 y Fq(\021)2071 4009 y Fl(0)2136 3922 y Fk(\001)2177 3995 y Fq(:)218 4183 y Fu(The)25 b(main)g(result)g(of)g(this)i(paragraph)e(is)h(the)g(following.)118 4315 y FE(Theorem)h(7.2.)185 4448 y Fh(i\))42 b(F)l(or)35 b(any)g Fq(f)5 b(;)15 b(g)45 b Fo(2)c(G)g Fh(the)35 b(limit)g Fq(V)21 b Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))42 b(:=)g(lim)1958 4463 y Fr(`)p Fn("1)2112 4448 y Fq(V)2165 4463 y Fr(`)2198 4448 y Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))36 b Fh(exists,)i(it)d(is)g (\002nite)g(and)f(it)h(de\002nes)291 4556 y(a)30 b(non)f(negative)g (semi\226inner)g(product)h(on)g Fo(G)5 b Fh(.)47 b(In)30 b(particular)g Fq(V)21 b Fp(\()p Fq(f)10 b Fp(\))33 b(:=)h(lim)2933 4571 y Fr(`)p Fn("1)3087 4556 y Fq(V)3140 4571 y Fr(`)3173 4556 y Fp(\()p Fq(f)5 b(;)15 b(f)10 b Fp(\))30 b Fh(is)g(well)291 4664 y(de\002ned.)159 4796 y(ii\))42 b(F)l(or)25 b(any)g Fq(f)35 b Fo(2)24 b(G)286 4979 y Fq(V)c Fp(\()p Fq(f)10 b Fp(\))26 b(=)40 b(sup)606 5065 y Fr(a)p 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5258 y Fq(c)2388 5272 y Fl(0)p Fr(;e)2480 5258 y Fp(\()p Fq(a)2563 5272 y Fr(e)2601 5258 y Fo(r)2677 5272 y Fr(e)2713 5258 y Fq(\021)2758 5272 y Fl(0)2818 5258 y Fo(\000)20 b(r)2985 5272 y Fr(e)3022 5258 y Fq(u)p 3022 5273 53 4 v Fp(\))3109 5220 y Fl(2)3149 5184 y Fk(\001)3190 5157 y(o)3266 5258 y Fq(:)3459 5151 y Fu(\(7.2\))p eop %%Page: 41 41 41 40 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(41)133 366 y Fh(iii\))42 b(The)25 b(subspace)1183 436 y Fk(\010)1251 424 y(X)1255 621 y Fr(e)p Fn(2E)1397 510 y Fq(a)1445 524 y Fr(e)1482 510 y Fq(j)1519 524 y Fl(0)p Fr(;e)1632 510 y Fp(+)20 b Fo(L)p Fq(u)50 b Fp(:)h Fq(a)25 b Fo(2)g Fs(R)2183 473 y Fr(d)2229 510 y Fq(;)41 b(u)25 b Fo(2)g Fs(G)2538 436 y Fk(\011)3459 510 y Fu(\(7.3\))291 748 y Fh(is)g(dense)e(in)i Fo(G)31 b Fh(endowed)22 b(of)j(the)g (semi\226inner)f(product)h Fq(V)20 b Fh(.)218 885 y Fu(Notice)i(that)h (lemma)e(7.1)g(proves)h(that)h(the)g(two)f(expressions)h(appearing)e (in)h(the)h(r)-10 b(.h.s.)31 b(of)22 b(the)h(\002rst)118 993 y(equality)j(sign)f(in)g(\(7.2\))g(are)f(equal.)118 1100 y(Before)f(proving)h(the)g(theorem)f(we)h(need)f(to)h(introduce)g (the)g(notion)g(of)g(closed)g(and)g(exact)f(forms)h(to-)118 1208 y(gether)h(with)h(their)g(generalization)d(to)j(the)f(disordered)h (case)f(and)g(prove)g(few)g(preliminary)f(results.)118 1316 y(W)-8 b(e)26 b(refer)e(the)i(reader)e(to)i([15)o(])g(for)f(a)g (complete)f(treatment.)118 1453 y FE(De\002nition)29 b(7.3.)44 b Fh(A)30 b(form)e(on)h Fp(\012)g Fh(is)g(a)g(family)g Fq(\030)35 b Fp(=)d Fo(f)p Fq(\030)1988 1468 y Fr(b)2023 1453 y Fo(g)2068 1475 y Fr(b)p Fn(\032)p Fm(Z)2204 1456 y FA(d)2267 1453 y Fh(of)d(functions)e Fq(\030)2787 1468 y Fr(b)2853 1453 y Fp(:)32 b(\012)g Fo(!)g Fs(R)r Fh(.)50 b(It)30 b(is)f(called)118 1561 y(closed)24 b(if)-8 b(,)25 b(given)g Fq(\021)j Fo(2)d Fp(\012)g Fh(and)f(bonds)g Fq(b)1400 1575 y Fl(1)1439 1561 y Fq(;)15 b(:)g(:)g(:)i(;)e(b)1680 1575 y Fr(n)1753 1561 y Fh(with)25 b Fq(\021)k Fp(=)c Fq(S)2175 1576 y Fr(b)2205 1584 y FA(n)2272 1561 y Fo(\016)20 b(\001)15 b(\001)g(\001)21 b(\016)g Fq(S)2585 1576 y Fr(b)2615 1585 y Fg(2)2674 1561 y Fo(\016)g Fq(S)2796 1576 y Fr(b)2826 1585 y Fg(1)2864 1561 y Fp(\()p Fq(\021)s Fp(\))p Fh(,)27 b(then)408 1689 y Fr(n)364 1716 y Fk(X)372 1911 y Fr(i)p Fl(=1)510 1802 y Fq(\030)550 1817 y Fr(b)580 1827 y FA(i)611 1802 y Fp(\()p Fq(\021)691 1816 y Fr(i)p Fn(\000)p Fl(1)810 1802 y Fp(\))e(=)g(0)91 b Fh(where)g Fq(\021)1467 1816 y Fl(0)1531 1802 y Fp(:=)26 b Fq(\021)s(;)41 b(\021)1812 1816 y Fr(i)1865 1802 y Fp(:=)25 b Fq(S)2042 1817 y Fr(b)2072 1827 y FA(i)2123 1802 y Fo(\016)c(\001)15 b(\001)g(\001)21 b(\016)g Fq(S)2437 1817 y Fr(b)2467 1826 y Fg(2)2525 1802 y Fo(\016)g Fq(S)2647 1817 y Fr(b)2677 1826 y Fg(1)2716 1802 y Fp(\()p Fq(\021)s Fp(\))51 b Fo(8)p Fq(i)26 b Fp(=)e(1)p Fq(;)15 b(:)g(:)g(:)j(;)d(n:)218 2057 y Fu(The)34 b(expression)854 1989 y Fk(P)950 2015 y Fr(n)950 2084 y(i)p Fl(=1)1084 2057 y Fq(\030)1124 2072 y 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Fq(w)r Fu(,)29 b Fq(x)h Fp(=)g Fq(a)22 b Fp(+)g Fq(v)s Fu(,)118 2597 y Fq(x)170 2564 y Fn(0)219 2597 y Fp(=)j Fq(a)20 b Fp(+)g Fq(w)r Fu(,)26 b Fq(b)631 2611 y Fl(1)696 2597 y Fp(=)e Fo(f)p Fq(a;)15 b(x)p Fo(g)p Fu(,)27 b Fq(b)1112 2611 y Fl(2)1177 2597 y Fp(=)e Fo(f)p Fq(x;)15 b(c)p Fo(g)p Fu(,)27 b Fq(b)1585 2564 y Fn(0)1585 2621 y Fl(1)1650 2597 y Fp(=)e Fo(f)p Fq(a;)15 b(x)1931 2564 y Fn(0)1955 2597 y Fo(g)p Fu(,)26 b Fq(b)2090 2564 y Fn(0)2090 2621 y Fl(2)2155 2597 y Fp(=)f Fo(f)p Fq(x)2348 2564 y Fn(0)2371 2597 y Fq(;)15 b(c)p Fo(g)p Fu(.)33 b(Then)637 2762 y Fq(\030)677 2777 y Fr(b)707 2786 y Fg(1)766 2762 y Fo(\016)20 b Fq(S)887 2777 y Fr(b)917 2786 y Fg(2)976 2762 y Fo(\016)h Fq(S)1098 2777 y Fr(b)1128 2786 y Fg(1)1187 2762 y Fp(+)e Fq(\030)1317 2777 y Fr(b)1347 2786 y Fg(2)1406 2762 y Fo(\016)i Fq(S)1528 2777 y Fr(b)1558 2786 y Fg(1)1617 2762 y Fp(+)f Fq(\030)1748 2777 y Fr(b)1778 2786 y Fg(1)1842 2762 y Fp(=)25 b Fq(\030)1978 2781 y Fr(b)2008 2758 y Fi(0)2008 2803 y Fg(1)2067 2762 y Fo(\016)20 b Fq(S)2188 2781 y Fr(b)2218 2758 y Fi(0)2218 2803 y Fg(2)2277 2762 y Fo(\016)h Fq(S)2399 2781 y Fr(b)2429 2758 y Fi(0)2429 2803 y Fg(1)2488 2762 y Fp(+)f Fq(\030)2619 2781 y Fr(b)2649 2758 y Fi(0)2649 2803 y Fg(2)2708 2762 y Fo(\016)g Fq(S)2829 2781 y Fr(b)2859 2758 y Fi(0)2859 2803 y Fg(1)2918 2762 y Fp(+)g Fq(\030)3049 2781 y Fr(b)3079 2758 y Fi(0)3079 2803 y Fg(1)3118 2762 y Fq(:)118 2948 y FE(P)-24 b(.)p Fd(2)p Fu(.)33 b(F)-6 b(or)25 b(any)g(couple)f(of)h (bonds)h Fq(b)1295 2962 y Fl(1)1334 2948 y Fq(;)15 b(b)1413 2962 y Fl(2)1478 2948 y Fo(\032)25 b Fs(Z)1639 2915 y Fr(d)1701 2948 y Fu(such)g(that)h Fq(b)2128 2962 y Fl(1)2188 2948 y Fo(\\)20 b Fq(b)2308 2962 y Fl(2)2372 2948 y Fp(=)25 b Fo(;)p Fu(,)1277 3113 y Fq(\030)1317 3128 y Fr(b)1347 3137 y Fg(2)1406 3113 y Fo(\016)c Fq(S)1528 3128 y Fr(b)1558 3137 y Fg(1)1617 3113 y Fp(+)f Fq(\030)1748 3128 y Fr(b)1778 3137 y Fg(1)1842 3113 y Fp(=)25 b Fq(\030)1978 3128 y Fr(b)2008 3137 y Fg(1)2067 3113 y Fo(\016)20 b Fq(S)2188 3128 y Fr(b)2218 3137 y Fg(2)2277 3113 y Fp(+)g Fq(\030)2408 3128 y Fr(b)2438 3137 y Fg(2)2477 3113 y Fq(:)118 3282 y FE(P)-24 b(.)p Fd(3)15 b Fu(.)33 b(F)-6 b(or)25 b(any)g(bond)g Fq(b)h Fo(\032)f Fs(Z)1070 3249 y Fr(d)1106 3282 y Fu(,)1576 3410 y Fq(\030)1616 3425 y Fr(b)1670 3410 y Fo(\016)c Fq(S)1792 3425 y Fr(b)1846 3410 y Fp(+)f Fq(\030)1977 3425 y Fr(b)2037 3410 y Fp(=)25 b(0)p Fq(:)118 3557 y Fu(The)33 b(above)g(characterization)f(allows)h(us)g(to)g(generalize)d (the)j(de\002nition)h(of)f(closed)f(forms)h(to)g(the)118 3665 y(disorder)26 b(case.)118 3801 y FE(De\002nition)f(7.4.)42 b Fh(A)25 b(form)g(in)g Fq(L)1216 3768 y Fl(2)1255 3801 y Fp(\()p Fs(P)1345 3768 y Fn(\003)1386 3801 y Fp(\))g Fh(is)g(a)g(family)g(of)g(functions)e Fq(\030)29 b Fp(=)c Fo(f)p Fq(\030)2594 3816 y Fr(b)2629 3801 y Fo(g)2674 3824 y Fr(b)p Fn(\032)p Fm(Z)2810 3805 y FA(d)2869 3801 y Fh(with)h Fq(\030)3106 3816 y Fr(b)3165 3801 y Fo(2)f Fq(L)3313 3768 y Fl(2)3352 3801 y Fp(\()p Fs(P)3442 3768 y Fn(\003)3483 3801 y Fp(\))p Fh(.)118 3909 y(A)36 b(form)g Fq(\030)k Fh(is)c(called)f(closed)g(if)g(it)i(satis\002es)e(properties) g FE(P)p Fh(.)p Fd(1)p Fh(,)40 b FE(P)p Fh(.)p Fd(2)d Fh(and)f FE(P)p Fh(.)p Fd(3)h Fh(where)e(equalities)g(are)h(in)118 4017 y Fq(L)180 3984 y Fl(2)219 4017 y Fp(\()p Fs(P)309 3984 y Fn(\003)350 4017 y Fp(\))p Fh(.)41 b(A)28 b(form)g Fq(\030)34 b Fp(=)c Fo(f)p Fq(\030)1005 4032 y Fr(b)1040 4017 y Fo(g)1085 4039 y Fr(b)p Fn(\032)p Fm(Z)1221 4021 y FA(d)1283 4017 y Fh(if)e(called)f(exact)h(is)g Fq(\030)1964 4032 y Fr(b)2029 4017 y Fp(=)i Fo(r)2206 4032 y Fr(b)2240 4017 y Fq(u)p 2240 4032 53 4 v 29 w Fh(for)d(some)g Fq(u)k Fo(2)f Fs(G)19 b Fh(.)46 b(A)29 b(form)e Fq(\030)33 b Fh(is)28 b(called)118 4131 y(translation)c(covariant)h(if)f Fq(\034)1071 4145 y Fr(x)1115 4131 y Fq(\030)1155 4146 y Fr(b)1214 4131 y Fp(=)h Fq(\030)1350 4146 y Fr(b)p Fl(+)p Fr(x)1504 4131 y Fh(for)g(any)g Fq(x)g Fo(2)g Fs(Z)2033 4098 y Fr(d)2069 4131 y Fh(,)g Fq(b)h Fo(\032)f Fs(Z)2345 4098 y Fr(d)2381 4131 y Fh(.)218 4268 y Fu(It)k(is)h(easy)f (to)g(check)e(that)j(exact)e(forms)h(are)f(automatically)g(closed)g (and)h(translation)g(covariant.)118 4376 y(Given)34 b(a)f(closed)h (form)f Fq(\030)k Fu(in)d Fq(L)1216 4343 y Fl(2)1256 4376 y Fp(\()p Fs(P)1346 4343 y Fn(\003)1386 4376 y Fp(\))g Fu(the)g(form)f(on)g Fp(\012)h Fo(f)p Fq(\030)2157 4391 y Fr(b)2191 4376 y Fp(\()p Fq(\013;)15 b Fo(\001)p Fp(\))p Fo(g)2429 4398 y Fr(b)p Fn(\032)p Fm(Z)2567 4379 y FA(d)2635 4376 y Fu(is)35 b(a)e(closed)h(form)f(on)g Fp(\012)h Fu(for)118 4484 y(almost)25 b(any)g(disorder)h(con\002guration)f Fq(\013)p Fu(.)118 4592 y(In)g(what)i(follows)e(by)g(a)g(form)g(we)g (will)g(always)h(mean)e(a)h(form)g(in)h Fq(L)2465 4559 y Fl(2)2504 4592 y Fp(\()p Fs(P)2594 4559 y Fn(\003)2634 4592 y Fp(\))p Fu(.)118 4728 y FE(De\002nition)33 b(7.5.)46 b Fh(A)33 b(family)g(of)f(functions)g Fq(\030)42 b Fp(=)c Fo(f)p Fq(\030)1904 4742 y Fr(e)1941 4728 y Fo(g)1986 4742 y Fr(e)p Fn(2E)2113 4728 y Fh(,)d Fq(\030)2213 4742 y Fr(e)2288 4728 y Fo(2)j Fq(L)2449 4695 y Fl(2)2488 4728 y Fp(\()p Fs(P)2578 4695 y Fn(\003)2619 4728 y Fp(\))p Fh(,)d(is)e(called)f(the)h(germ)f(of)h(the)118 4839 y(form)25 b Fq(\030)370 4806 y Fn(0)418 4839 y Fp(=)g Fo(f)p Fq(\030)603 4806 y Fn(0)599 4866 y Fr(b)634 4839 y Fo(g)679 4861 y Fr(b)p Fn(\032)p Fm(Z)815 4842 y FA(d)874 4839 y Fh(if)g Fq(\030)996 4806 y Fn(0)992 4861 y Fr(x;x)p Fl(+)p Fr(e)1208 4839 y Fp(=)g Fq(\034)1344 4853 y Fr(x)1388 4839 y Fq(\030)1428 4853 y Fr(e)1490 4839 y Fh(for)f(any)h Fq(x)g Fo(2)g Fs(Z)2018 4806 y Fr(d)2079 4839 y Fh(and)f Fq(e)i Fo(2)f(E)8 b Fh(.)218 4981 y Fu(It)32 b(follows)f(that)i Fq(\030)859 4948 y Fn(0)914 4981 y Fu(is)f(automatically)f(translation)h(covariant) g(as)g(soon)f(as)h(it)g(is)h(generated)d(by)h(a)118 5089 y(germ)24 b Fq(\030)t Fu(.)118 5197 y(W)n(ithin)i(the)e(subset)h(of)f (closed)g(and)g(translation)h(covariant)e(forms)h(we)g(consider)g(the)h (special)e(family)118 5305 y Fo(f)p Ff(U)222 5272 y Fr(e)259 5305 y Fo(g)304 5319 y Fr(e)p Fn(2E)457 5305 y Fu(de\002ned)i(by)887 5477 y Ff(U)946 5439 y Fr(e)946 5501 y(x;x)p Fl(+)p Fr(e)1134 5482 y Fi(0)1159 5477 y Fp(\()p Fq(\021)s Fp(\))i(:=)e Fq(\016)1465 5492 y Fr(e;e)1551 5473 y Fi(0)1577 5477 y Fp(\()p Fq(\021)1657 5491 y Fr(x)p Fl(+)p Fr(e)1809 5477 y Fo(\000)20 b Fq(\021)1945 5491 y Fr(x)1989 5477 y Fp(\))15 b Fq(;)107 b Fo(8)p Fq(x)25 b Fo(2)f Fs(Z)2449 5439 y Fr(d)2486 5477 y Fq(;)40 b(e;)15 b(e)2675 5439 y Fn(0)2725 5477 y Fo(2)25 b(E)8 b Fq(:)p eop %%Page: 42 42 42 41 bop 118 45 a Fj(42)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(It)24 b(is)g(simple)e(to)i(check)d(that)j(the)g(form)e Ff(U)1518 333 y Fr(e)1578 366 y Fu(is)h(not)g(exact.)30 b(Finally)-12 b(,)23 b(we)g(de\002ne)f Fp(\004)2850 380 y Fr(C)2932 366 y Fu(as)h(the)g(set)g(of)h(germs)118 474 y(of)i(closed)e(forms)i (and)441 644 y Fp(\004)502 658 y Fl(0)566 644 y Fp(:=)f Fo(f)p Fq(\030)30 b Fp(=)25 b Fo(f)p Fq(\030)983 658 y Fr(e)1020 644 y Fo(g)1065 658 y Fr(e)p Fn(2E)1233 644 y Fp(:)41 b Fo(9)p Fq(a)24 b Fo(2)h Fs(R)1568 607 y Fr(d)1614 644 y Fq(;)15 b(u)26 b Fo(2)f Fs(G)116 b Fu(with)92 b Fq(\030)2288 658 y Fr(e)2350 644 y Fp(=)24 b Fq(a)2493 658 y Fr(e)2530 644 y Ff(U)2589 607 y Fr(e)2646 644 y Fp(+)c Fo(r)2813 658 y Fr(e)2850 644 y Fq(u)p 2850 659 53 4 v 91 w Fo(8)p Fq(e)25 b Fo(2)g(E)e(g)p Fq(:)118 816 y Fu(W)-8 b(e)26 b(remark)e(that)i Fp(\004)818 830 y Fl(0)882 816 y Fo(\032)f Fp(\004)1039 830 y Fr(C)1123 816 y Fu(and)g(that)h Fp(\004)1541 830 y Fr(C)1625 816 y Fu(is)g(a)f(closed)g(subspace)g(in)g Fq(L)2600 783 y Fl(2)2639 816 y Fp(\()p Fo(\012)2745 783 y Fr(d)2786 816 y Fs(P)2841 783 y Fn(\003)2881 816 y Fp(\))p Fu(.)32 b(A)25 b(deeper)f(result)h(is)118 924 y(given)g(by)g(the)g(following)g (density)h(theorem.)118 1064 y FE(Theorem)h(7.6.)41 b Fp(\004)781 1078 y Fr(C)865 1064 y Fp(=)p 961 990 61 4 v 25 w(\004)1022 1078 y Fl(0)1087 1064 y Fh(in)25 b Fq(L)1251 1031 y Fl(2)1290 1064 y Fp(\()p Fo(\012)1396 1031 y Fr(d)1436 1064 y Fs(P)1491 1031 y Fn(\003)1531 1064 y Fp(\))p Fh(.)218 1202 y(Proof)-8 b(.)48 b Fu(The)31 b(proof)f(follows)h(closely)f(the)h(proof)f(of)h(theorem)f Fp(4)p Fq(:)p Fp(14)j Fu(in)e(appendix)g Fp(3)g Fu(of)g([23)o(])g(with) 118 1310 y(the)26 b(exception)g(of)g(the)g(last)h(step.)34 b(As)27 b(in)f([23])g(it)h(can)e(be)h(proved)g(that)h(for)f(any)g Fq(\030)31 b Fo(2)26 b Fp(\004)3126 1324 y Fr(C)3211 1310 y Fu(there)g(exists)118 1418 y(a)f(germ)f Fq(!)k Fo(2)d Fp(\004)650 1432 y Fr(C)734 1418 y Fu(with)i(the)e(following)g (properties:)188 1558 y(i\))42 b Fq(\030)25 b Fo(\000)19 b Fq(!)29 b Fo(2)p 619 1485 V 24 w Fp(\004)680 1572 y Fl(0)719 1558 y Fu(;)163 1666 y(ii\))42 b Fq(!)29 b Fu(can)d(written)g (as)h Fq(!)j Fp(=)c Fq(!)1207 1680 y Fn(\000)1286 1666 y Fp(+)21 b Fq(!)1435 1680 y Fl(+)1520 1666 y Fu(with)27 b Fq(!)1780 1680 y Fn(\006)1866 1666 y Fp(=)f Fo(f)p Fq(!)2065 1680 y Fn(\006)p Fr(;e)2176 1666 y Fo(g)2221 1680 y Fr(e)p Fn(2E)2349 1666 y Fu(,)h Fq(!)2458 1680 y Fn(\006)p Fr(;e)2568 1666 y Fp(\()p Fq(\013;)15 b(\021)s Fp(\))29 b(=)e Fq(!)2968 1680 y Fn(\006)p Fr(;e)3079 1666 y Fp(\()p Fq(\013;)15 b(\021)3257 1680 y Fl(0)3298 1666 y Fq(;)g(\021)3383 1680 y Fr(e)3420 1666 y Fp(\))27 b Fu(such)293 1774 y(that)g Fo(8)p Fq(e)e Fo(2)g(E)883 1915 y Fq(!)940 1929 y Fn(\000)p Fr(;e)1051 1915 y Fp(\()p Fq(\013;)15 b(\021)1229 1929 y Fl(0)1270 1915 y Fq(;)g(\021)1355 1929 y Fl(2)p Fr(e)1427 1915 y Fp(\))21 b Fo(\000)f Fq(!)1631 1929 y Fn(\000)p Fr(;e)1742 1915 y Fp(\()p Fq(\013;)15 b(\021)1920 1929 y Fl(0)1961 1915 y Fq(;)g(\021)2046 1929 y Fr(e)2083 1915 y Fp(\))26 b(=)f Fq(!)2297 1929 y Fn(\000)p Fr(;e)2408 1915 y Fp(\()p Fq(\013;)15 b(\021)2586 1929 y Fr(e)2624 1915 y Fq(;)g(\021)2709 1929 y Fl(2)p Fr(e)2782 1915 y Fp(\))p Fq(;)883 2055 y(!)940 2069 y Fl(+)p Fr(;e)1051 2055 y Fp(\()p Fq(\013;)g(\021)1229 2069 y Fn(\000)p Fr(e)1322 2055 y Fq(;)g(\021)1407 2069 y Fr(e)1444 2055 y Fp(\))21 b Fo(\000)f Fq(!)1648 2069 y Fl(+)p Fr(;e)1759 2055 y Fp(\()p Fq(\013;)15 b(\021)1937 2069 y Fl(0)1978 2055 y Fq(;)g(\021)2063 2069 y Fr(e)2100 2055 y Fp(\))26 b(=)f Fq(!)2314 2069 y Fl(+)p Fr(;e)2425 2055 y Fp(\()p Fq(\013;)15 b(\021)2603 2069 y Fn(\000)p Fr(e)2696 2055 y Fq(;)g(\021)2781 2069 y Fl(0)2821 2055 y Fp(\))g Fq(:)3459 1993 y Fu(\(7.4\))118 2242 y(It)30 b(remains)e(to)h(prove)f(that)i Fq(!)k Fo(2)p 1281 2169 V 31 w Fp(\004)1342 2256 y Fl(0)1381 2242 y Fu(.)42 b(Because)28 b(of)h(\(7.4\),)g Fo(8)p Fq(e)i Fo(2)g(E)37 b Fu(there)28 b(exists)i Fq(a)2991 2256 y Fn(\006)p Fr(;e)3133 2242 y Fo(2)h Fq(L)3287 2209 y Fl(2)3326 2242 y Fp(\()p Fs(P)p Fp(\))g Fu(such)118 2350 y(that)26 b Fq(!)358 2364 y Fn(\006)p Fr(;e)495 2350 y Fp(=)f Fq(a)639 2364 y Fn(\006)p Fr(;e)750 2350 y Fp(\()p Fq(\013)p Fp(\)\()p Fq(\021)958 2364 y Fr(e)1017 2350 y Fo(\000)20 b Fq(\021)1153 2364 y Fl(0)1192 2350 y Fp(\))p Fu(.)32 b(Lemma)25 b Fp(7)p Fq(:)p Fp(7)h Fu(then)f(completes)g(the)g(proof)g(of)h(the)f(theorem.) 118 2492 y FE(L)r(emma)42 b(7.7.)50 b Fh(Let)40 b Fq(!)52 b Fo(2)d Fp(\004)1112 2506 y Fr(C)1211 2492 y Fh(such)39 b(that)i(for)e(any)g Fq(e)50 b Fo(2)f(E)f Fh(there)39 b(exists)h Fq(a)2773 2506 y Fr(e)2859 2492 y Fo(2)49 b Fq(L)3031 2459 y Fl(2)3071 2492 y Fp(\()p Fs(P)p Fp(\))41 b Fh(with)f Fq(!)3505 2506 y Fr(e)3591 2492 y Fp(=)118 2603 y Fq(a)166 2617 y Fr(e)203 2603 y Fp(\()p Fq(\013)p Fp(\)\()p Fq(\021)411 2617 y Fr(e)470 2603 y Fo(\000)20 b Fq(\021)606 2617 y Fl(0)645 2603 y Fp(\))p Fh(.)32 b(Then)25 b Fq(!)j Fo(2)p 1124 2530 V 25 w Fp(\004)1185 2617 y Fl(0)1224 2603 y Fh(.)118 2792 y(Proof)-8 b(.)42 b Fu(By)29 b(subtracting)983 2724 y Fk(P)1079 2819 y Fr(e)p Fn(2E)1221 2792 y Fs(E)11 b Fp(\()p Fq(a)1359 2806 y Fr(e)1402 2792 y Fp(\))k Ff(U)1511 2759 y Fr(e)1578 2792 y Fu(from)29 b(the)h(germ)e Fq(!)s Fu(,)i(we)f(can)g(assume)g (that)i Fs(E)11 b Fp(\()p Fq(a)3260 2806 y Fr(e)3303 2792 y Fp(\))33 b(=)f(0)e Fu(for)118 2900 y(any)k Fq(e)40 b Fo(2)f(E)8 b Fu(.)58 b(In)34 b(what)h(follows)f(we)g(denote)f(the)i 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Fr(b)803 3504 y FA(i)833 3479 y Fp(\()p Fq(\013;)15 b(\021)1011 3493 y Fr(i)p Fn(\000)p Fl(1)1131 3479 y Fp(\))91 b Fu(where)f Fq(\021)1638 3493 y Fr(i)1692 3479 y Fp(:=)25 b Fq(S)1869 3494 y Fr(b)1899 3504 y FA(i)1950 3479 y Fo(\016)20 b(\001)15 b(\001)g(\001)22 b(\016)e Fq(S)2263 3494 y Fr(b)2293 3503 y Fg(2)2352 3479 y Fo(\016)h Fq(S)2474 3494 y Fr(b)2504 3503 y Fg(1)2542 3479 y Fp(\()p Fq(\021)2625 3441 y Fl(\(0\))2720 3479 y Fp(\))77 b Fo(8)p Fq(i)25 b Fp(=)g(1)p Fq(;)15 b(:)g(:)g(:)i(;)e(r)m(:)114 b Fu(\(7.5\))118 3735 y(Notice)22 b(that,)i(since)d Fo(f)p Fq(!)923 3750 y Fr(b)957 3735 y Fp(\()p Fq(\013;)15 b Fo(\001)p Fp(\))p Fo(g)1195 3757 y Fr(b)p Fn(\032)p Fm(Z)1333 3738 y FA(d)1389 3735 y Fu(is)23 b(a)f(closed)f(form)g(on)h Fp(\012)f Fu(for)h(almost)g(any)f Fq(\013)p Fu(,)j(the)e(de\002nition)g(of)g Fq(g)3617 3749 y Fr(x)118 3843 y Fu(does)29 b(not)g(depend)f(on)g(the)h (particular)g(choice)f(of)g(the)h(bonds)g Fq(b)2355 3857 y Fl(1)2395 3843 y Fq(;)15 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Fr(e)3220 5326 y Fq(h)3272 5340 y Fr(n)3335 5326 y Fq(:)99 b Fu(\(7.7\))p eop %%Page: 43 43 43 42 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(43)118 366 y Fu(and)28 b(we)f(are)g(left)g(with)h(the)g(proof)f(that)i(the)f (second)e(and)i(third)h(term)d(in)i(the)f(r)-10 b(.h.s.of)29 b(\(7.7\))e(tend)g(to)118 474 y Fp(0)32 b Fu(in)g Fq(L)366 441 y Fl(2)405 474 y Fp(\()p Fs(P)495 441 y Fn(\003)535 474 y Fp(\))p Fu(.)51 b(Let)31 b(us)h(consider)f(the)h(second)f(term)f (\(the)i(third)h(one)d(being)h(similar\).)50 b(By)32 b(Schwarz)118 582 y(inequality)26 b(and)f(the)h(identity)813 736 y Fo(r)889 750 y Fr(x;x)p Fl(+)p Fr(e)1079 736 y Fq(h)1131 750 y Fr(n)1204 736 y Fp(=)f Fo(\000)p Fq(g)1414 750 y Fr(x)1458 736 y Fp(\()p Fq(\013)p Fp(\)\()p Fq(\021)1666 750 y Fr(x)p Fl(+)p Fr(e)1819 736 y Fo(\000)20 b Fq(\021)1955 750 y Fr(x)1999 736 y Fp(\))91 b Fo(8)p Fq(x)25 b Fo(2)g Fp(\003)2402 750 y Fr(n)2474 736 y Fu(with)i Fq(x)2729 750 y 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Fs(E)1741 2611 y Fk(\020)1801 2639 y(\000)1889 2599 y Fr(k)1843 2626 y Fk(X)1847 2820 y Fr(s)p Fl(=0)1989 2712 y Fq(\034)2029 2726 y Fr(se)2114 2712 y Fq(a)2162 2726 y Fr(e)2199 2639 y Fk(\001)2240 2661 y Fl(2)2280 2611 y Fk(\021)2425 2712 y Fo(8)p Fq(e)f Fo(2)g(E)8 b Fq(:)118 2951 y Fu(T)-9 b(o)36 b(this)h(aim,)i(for)c (simplicity)i(of)e(notation,)k(we)c(\002x)h Fq(e)43 b Fo(2)f(E)h Fu(and)36 b(we)f(write)h Fq(a)2901 2965 y Fr(s)2973 2951 y Fu(in)g(place)e(of)i Fq(\034)3482 2965 y Fr(se)3551 2951 y Fq(a)3599 2965 y Fr(e)3636 2951 y Fu(.)118 3077 y(Moreover)-10 b(,)27 b(for)g(any)h Fq(r)k Fo(2)d Fs(N)40 b Fu(we)27 b(set)h Fq(a)1438 3029 y Fl(\()p Fr(r)r Fl(\))1438 3089 y Fr(s)1560 3077 y Fp(:=)i Fs(E)10 b Fp([)p Fq(a)1814 3091 y Fr(s)1871 3077 y Fo(j)15 b Fq(\013)1969 3091 y Fl(\003)2018 3099 y FA(se;r)2135 3077 y Fp(])p Fu(.)39 b(Since)27 b Fq(a)2513 3029 y Fl(\()p Fr(r)r Fl(\))2513 3089 y Fr(s)2635 3077 y Fp(=)i Fq(\034)2775 3091 y Fr(se)2844 3077 y Fq(a)2892 3029 y Fl(\()p Fr(r)r 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Fk(\020)2714 3355 y(\000)2803 3315 y Fr(k)2756 3342 y Fk(X)2760 3536 y Fr(s)p Fl(=0)2903 3428 y Fq(a)2951 3391 y Fl(\()p Fr(r)r Fl(\))2951 3451 y Fr(s)3043 3355 y Fk(\001)3085 3377 y Fl(2)3125 3327 y Fk(\021)1358 3710 y Fo(\024)25 b Fp(2)p Fs(E)1554 3609 y Fk(\020)1614 3637 y(\000)1656 3710 y Fq(a)1704 3724 y Fl(0)1764 3710 y Fo(\000)19 b Fq(a)1902 3662 y Fl(\()p Fr(r)r Fl(\))1902 3736 y(0)1995 3710 y Fp(\))2030 3673 y Fl(2)2070 3637 y Fk(\001)2132 3710 y Fp(+)2233 3649 y Fq(c)p Fp(\()p Fq(r)s Fp(\))p 2233 3689 154 4 v 2282 3773 a Fq(n)2396 3710 y(:)118 3894 y Fu(and)25 b(the)h(thesis)h(follows.)2587 b Fe(\003)218 4063 y Fu(The)23 b(connection)f(between)g(the)i(forms)f (and)g(the)h(space)f Fo(G)28 b Fu(endowed)23 b(with)i(the)e (semi\226inner)g(prod-)118 4171 y(uct)h Fq(V)c Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))26 b Fu(is)f(clari\002ed)f(by)g(next)g (proposition,)i(which)f(can)f(be)g(proved,)g(following)g([23)o(])h(and) g([37)o(],)118 4279 y(as)h(explained)e(in)i(section)f Fp(5)p Fq(:)p Fp(5)h Fu(of)g([15)o(].)118 4410 y FE(Proposition)h(7.8.) 42 b Fh(Given)25 b Fq(f)34 b Fo(2)25 b(G)31 b Fh(and)24 b Fq(e)i Fo(2)e(E)34 b Fh(there)25 b(exists)f(a)h(function)f Fq(\036)2646 4424 y Fr(e)2709 4410 y Fo(2)h Fs(G)50 b Fh(such)25 b(that)813 4564 y Fp(sup)800 4643 y Fr(\030)s Fn(2)p Fl(\004)928 4652 y Fg(0)977 4564 y Fp(\002)1048 4579 y Fr(f)1093 4564 y Fp(\()p Fq(\030)t Fp(\))h Fo(\024)f Fp(lim)15 b(inf)1388 4625 y Fr(`)p Fn("1)1597 4564 y Fq(V)1650 4579 y Fr(`)1683 4564 y Fp(\()p Fq(f)10 b Fp(\))25 b Fo(\024)g Fp(lim)15 b(sup)2001 4643 y Fr(`)p Fn("1)2222 4564 y Fq(V)2275 4579 y Fr(`)2308 4564 y Fp(\()p Fq(f)10 b Fp(\))25 b Fo(\024)47 b Fp(sup)2554 4643 y Fr(\030)s Fn(2)p Fl(\004)2682 4654 y FA(C)2749 4564 y Fp(\002)2820 4579 y Fr(f)2865 4564 y Fp(\()p Fq(\030)t Fp(\))480 b Fu(\(7.9\))118 4786 y Fh(where)1014 4924 y Fp(\002)1085 4939 y Fr(f)1130 4924 y Fp(\()p Fq(\030)t Fp(\))26 b(:=)1391 4837 y Fk(X)1396 5034 y Fr(e)p Fn(2E)1538 4924 y Fp(2)15 b Fs(E)1653 4886 y Fn(\003)1698 4850 y Fk(\000)1740 4924 y Fq(c)1779 4938 y Fl(0)p Fr(;e)1871 4924 y Fq(\036)1925 4938 y Fr(e)1962 4924 y Fq(\030)2002 4938 y Fr(e)2039 4924 y Fp(\))20 b Fo(\000)2185 4837 y Fk(X)2190 5034 y Fr(e)p Fn(2E)2342 4862 y Fp(1)p 2342 4903 46 4 v 2342 4986 a(2)2397 4924 y Fs(E)2452 4886 y Fn(\003)2497 4850 y Fk(\000)2539 4924 y Fq(c)2578 4938 y Fr(e)2615 4924 y Fq(\030)2659 4886 y Fl(2)2655 4946 y Fr(e)2699 4850 y Fk(\001)2740 4924 y Fq(:)118 5162 y Fh(Moreover)-12 b(,)25 b(given)g Fq(a)g Fo(2)g Fs(R)959 5129 y Fr(d)1031 5162 y Fh(and)f Fq(u)h Fo(2)g Fs(G)19 b Fh(,)164 5356 y Fp(\002)235 5371 y Fr(f)280 5283 y Fk(\000)321 5270 y(X)326 5467 y Fr(e)p Fn(2E)453 5356 y Fp(\()p Fo(\000)p Fq(a)607 5370 y Fr(e)644 5356 y Ff(U)703 5319 y Fr(e)752 5356 y Fp(+)13 b Fo(r)912 5370 y Fr(e)948 5356 y Fq(u)p 948 5371 53 4 v Fp(\))1035 5283 y Fk(\001)1102 5356 y Fp(=)1198 5270 y Fk(X)1202 5467 y Fr(e)p Fn(2E)1345 5356 y Fp(2)p Fq(a)1438 5370 y Fr(e)1475 5356 y Fq(t)1508 5370 y Fr(e)1545 5356 y Fp(\()p Fq(f)d Fp(\))j(+)g(2\()p Fq(f)5 b(;)15 b(u)p Fp(\))2024 5370 y Fl(0)2075 5356 y Fo(\000)2159 5270 y Fk(X)2163 5467 y Fr(e)p Fn(2E)2315 5295 y Fp(1)p 2315 5336 46 4 v 2315 5419 a(2)2371 5356 y Fs(E)2425 5319 y Fn(\003)2471 5283 y Fk(\000)2512 5356 y Fq(c)2551 5370 y Fl(0)p Fr(;e)2643 5356 y Fp(\()p Fq(a)2726 5370 y Fr(e)2764 5356 y Fo(r)2840 5370 y Fr(e)2876 5356 y Fq(\021)2921 5370 y Fl(0)2974 5356 y Fo(\000)e(r)3134 5370 y Fr(e)3169 5356 y Fq(u)p 3169 5371 53 4 v 1 w Fp(\))3257 5319 y Fl(2)3296 5283 y Fk(\001)3338 5356 y Fq(:)46 b Fu(\(7.10\))p eop %%Page: 44 44 44 43 bop 118 45 a Fj(44)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(W)-8 b(e)23 b(are)f(\002nally)g(in)h(a)g(position)h(to)f(prove)f(theorem)g (7.2.)30 b(W)-8 b(e)23 b(\002rst)g(observe)f(that)i(theorem)e(7.6)g (proves)118 474 y(that)37 b(the)g(inequalities)g(in)f(\(7.9\))g(are)g (actually)f(equalities)i(so)f(that)i Fq(V)20 b Fp(\()p Fq(f)10 b Fp(\))44 b(=)f(lim)3041 489 y Fr(`)p Fn("1)3195 474 y Fq(V)3248 489 y Fr(`)3281 474 y Fp(\()p Fq(f)10 b Fp(\))37 b Fu(exists)118 582 y(and)28 b(it)g(is)g(given)e(by)i (\(7.2\).)37 b(Moreover)-10 b(,)26 b(because)h(of)g(\(4.23\),)g Fq(V)20 b Fp(\()p Fq(f)10 b Fp(\))29 b Fq(<)f Fo(1)g Fu(so)f(that,)j(by)d(polarization,)118 690 y Fq(V)20 b Fp(\()p Fq(f)5 b(;)15 b(g)s Fp(\))32 b Fu(exists)f(\002nite)g(for)g (any)f Fq(f)5 b(;)15 b(g)38 b Fo(2)33 b(G)j Fu(and)31 b(it)g(de\002nes)g(a)g(semi\226inner)f(product.)47 b(The)31 b(density)g(of)118 798 y(the)26 b(subspace)f(\(7.3\))f(follows)h(at)h (once)e(from)h(the)h(\002rst)f(equality)h(in)f(\(7.2\).)873 b Fe(\003)118 1025 y Fu(7.2.)45 b FE(The)27 b(method)h(of)f(long)h (jumps)e(revisited.)44 b Fu(In)26 b(this)h(paragraph)f(we)g(consider) -10 b(,)26 b(for)g(any)g Fq(e)h Fo(2)f(E)8 b Fu(,)118 1133 y(a)24 b(particular)f(sequence)f Fo(f)p Fq(W)1127 1100 y Fr(e)1114 1155 y(n)1164 1133 y Fq(=n)p Fo(g)1309 1147 y Fr(n)p Fn(2)p Fm(N)1475 1133 y Fu(in)i(the)g(space)g Fo(G)29 b Fu(which)24 b(is)h(asymptotically)f(equivalent)f(to)h(the)118 1241 y(sequence)1276 1435 y Fp(2)p Fq(m)p Fp(\(1)e Fo(\000)e Fq(m)p Fp(\))p Fq(\025)1762 1398 y Fn(0)1762 1458 y Fl(0)1801 1435 y Fp(\()p Fq(m)p Fp(\))1961 1365 y Fq( )2023 1332 y Fr(e)2020 1387 y(n;n)p 1961 1415 169 4 v 2019 1498 a Fq(n)2140 1435 y(;)107 b(n)25 b Fo(2)f Fs(N)118 1625 y Fu(where)h Fq( )451 1592 y Fr(e)448 1647 y(n;n)583 1625 y Fu(has)h(been)e(de\002ned)h(in)g(\(4.32\))f(as)1079 1819 y Fq( )1141 1781 y Fr(e)1138 1841 y(n;n)1272 1819 y Fp(=)h Fq(m)1448 1781 y Fl(2)p Fr(;e)1448 1841 y(n)1560 1819 y Fo(\000)20 b Fq(m)1731 1781 y Fl(1)p Fr(;e)1731 1841 y(n)1843 1819 y Fo(\000)g Fq(\026)1989 1745 y Fk(\002)2026 1819 y Fq(m)2106 1781 y Fl(2)p Fr(;e)2106 1841 y(n)2218 1819 y Fo(\000)g Fq(m)2389 1781 y Fl(1)p Fr(;e)2389 1841 y(n)2496 1819 y Fo(j)15 b Fq(m)2616 1781 y Fr(e)2616 1841 y(n)2663 1745 y Fk(\003)118 2000 y Fu(The)26 b(functions)h Fq(W)786 1967 y Fr(e)773 2022 y(n)848 2000 y Fu(have)f(been)f (introduced)h(in)h([29)o(])f(in)h(order)e(to)i(depress)f(the)h(extra)f (\003uctuations)118 2108 y(produced)f(by)g(the)h(disorder)g(and)f(are)g (de\002ned)g(as)296 2294 y Fq(W)395 2256 y Fr(e)382 2316 y(n)456 2294 y Fp(:=)g(Av)695 2327 y Fr(x)p Fn(2)p Fl(\003)831 2295 y Fg(1)p FA(;e)831 2338 y(n)918 2294 y Fp(Av)1035 2327 y Fr(y)r Fn(2)p Fl(\003)1168 2295 y Fg(2)p FA(;e)1168 2338 y(n)1270 2294 y Fq(w)1335 2308 y Fr(x;y)1552 2294 y Fu(where)116 b Fq(w)1979 2308 y Fr(x;y)2105 2294 y Fp(:=)2226 2220 y Fk(\000)2268 2294 y Fp(1)21 b(+)e Fq(e)2466 2256 y Fn(\000)p Fl(\()p Fr(\013)2593 2264 y FA(x)2633 2256 y Fn(\000)p Fr(\013)2733 2264 y FA(y)2771 2256 y Fl(\)\()p Fr(\021)2860 2264 y FA(x)2900 2256 y Fn(\000)p Fr(\021)2990 2264 y FA(y)3028 2256 y Fl(\))3060 2220 y Fk(\001)3101 2294 y Fp(\()p Fq(\021)3181 2308 y Fr(y)3243 2294 y Fo(\000)h Fq(\021)3379 2308 y Fr(x)3423 2294 y Fp(\))p Fq(:)118 2501 y Fu(W)-8 b(e)32 b(remark)e(that,)k(for)d(any)f (bond)i Fq(b)j Fp(=)f Fo(f)p Fq(x;)15 b(y)s Fo(g)p Fu(,)34 b(the)e(quantities)g Fq(c)2438 2515 y Fr(x;y)2574 2501 y Fp(:=)j(1)25 b(+)f Fq(e)2912 2468 y Fn(\000)p Fl(\()p Fr(\013)3039 2476 y FA(x)3079 2468 y Fn(\000)p Fr(\013)3179 2476 y FA(y)3216 2468 y Fl(\)\()p Fr(\021)3305 2476 y FA(x)3345 2468 y Fn(\000)p Fr(\021)3435 2476 y FA(y)3473 2468 y Fl(\))3536 2501 y Fu(are)118 2609 y(a)h(possible)h(choice)e(of)i (transition)g(rates)f(compatible)g(with)h(our)f(general)e(assumptions)j (\(see)e(section)118 2717 y(2.2\).)31 b(Therefore,)24 b(for)h(generic)e Fq(x;)15 b(y)28 b Fo(2)d Fs(Z)1544 2684 y Fr(d)1605 2717 y Fq(c)1644 2731 y Fr(x;y)1770 2717 y Fu(can)f(be)h(thought)g(of)g(as)g(the)g(rate)g(of)g(the)g (\(long\))f(jump)118 2831 y(from)k Fq(x)h Fu(to)g Fq(y)i Fu(and)e(viceversa.)40 b(In)29 b(a)f(sense)g(the)h(rates)g Fq(c)2048 2845 y Fr(x;y)2149 2831 y Fu(,)h Fq(x;)15 b(y)34 b Fo(2)c Fs(Z)2531 2798 y Fr(d)2567 2831 y Fu(,)g(de\002ne)d(a)i(new)f (process)g(with)118 2939 y(arbitrarily)e(long)e(jumps)h(but)g(still)h (reversibile)f(w)-11 b(.r)h(.t.)32 b(the)25 b(Gibbs)i(measure)d(of)h (the)h(system.)118 3082 y FE(R)m(emark)c(7.9.)36 b Fh(The)21 b(role)f(of)h(the)f(function)g Fq(W)1654 3049 y Fr(e)1641 3105 y(n)1711 3082 y Fh(here)h(is)f(very)i(different)c(from)j(that)g (indicated)e(in)i Fu([29)o(])p Fh(.)31 b(In)118 3190 y(our)22 b(approach)f(and)g(for)g(reasons)f(that)i(will)g(appear)f (clearly)g(in)h(the)g(next)f(subsection,)h(we)f(are)g(interested)118 3319 y(in)26 b(computing)f(the)h(asymptotic)g(of)f(the)h(semi\226inner) f(product)h Fq(V)20 b Fp(\()p Fq(j)2388 3334 y Fl(0)p Fr(;e)2476 3315 y Fi(0)2503 3319 y Fq(;)2553 3264 y Fr( )2601 3241 y FA(e)2599 3281 y(n;n)p 2553 3298 146 4 v 2605 3350 a Fr(n)2709 3319 y Fp(\))26 b Fh(as)g Fq(n)g Fo(")h(1)p Fh(.)34 b(Our)27 b(strategy)118 3459 y(to)36 b(compute)f Fq(V)20 b Fp(\()p Fq(j)732 3474 y Fl(0)p Fr(;e)820 3456 y Fi(0)847 3459 y Fq(;)897 3405 y Fr( )945 3381 y FA(e)943 3421 y(n;n)p 897 3438 V 949 3491 a Fr(n)1053 3459 y Fp(\))36 b Fh(is)g(to)f(replace)g(\(in)g Fo(G)5 b Fh(\))1929 3405 y Fr( )1977 3381 y FA(e)1975 3421 y(n;n)p 1929 3438 V 1981 3491 a Fr(n)2121 3459 y Fh(with)2338 3414 y Fr(W)2415 3391 y FA(e)2405 3431 y(n)p 2338 3438 110 4 v 2371 3491 a Fr(n)2493 3459 y Fh(and)35 b(then)g(to)g(exploit)h(some)e(nice)118 3567 y(integration)24 b(by)i(parts)f(properties)f(pointed)g(out)h(in)g Fu([29)o(])h Fh(\(see)e(below\).)118 3675 y(In)30 b Fu([29)o(])g Fh(instead,)f(the)g(main)g(idea)g(is)g(\002rst)h(to)f(approximate,)i (as)e Fq(\017)j Fo(#)h Fp(0)p Fh(,)e(the)f(microscopic)e(current)h Fq(j)3569 3689 y Fl(0)p Fr(;e)118 3833 y Fh(with)i(a)g(\003uctuation)g (term)g Fo(L)p Fq(g)j Fh(plus)d(a)g(linear)f(combination)g(of)g(the) 2494 3782 y Fr(W)2571 3758 y FA(e)2600 3738 y Fi(0)2561 3805 y FA(k)p 2494 3812 133 4 v 2541 3865 a Fr(k)2636 3833 y Fh(,)j Fq(e)2736 3800 y Fn(0)2795 3833 y Fo(2)h(E)8 b Fh(,)33 b(on)d(a)h(scale)f Fq(k)k Fh(that)118 3975 y(must)26 b(diverge)e(as)g Fq(\017)i Fo(#)f Fp(0)h Fh(like)f Fq(\017)1138 3932 y Fn(\000)1242 3905 y Fg(2)p 1202 3917 110 3 v 1202 3958 a FA(d)p Fg(+2)1326 3975 y Fh(.)31 b(The)25 b(second)e(step)i(indicated)e(in)i Fu([29])g Fh(is)g(to)g(replace)3169 3924 y Fr(W)3246 3900 y FA(e)3236 3947 y(k)p 3169 3954 110 4 v 3205 4006 a Fr(k)3315 3975 y Fh(with)1280 4236 y Fp(2)p Fq(m)p Fp(\(1)c Fo(\000)f Fq(m)p Fp(\))p Fq(\025)1765 4199 y Fn(0)1765 4259 y Fl(0)1804 4236 y Fp(\()p Fq(m)p Fp(\))1964 4172 y(\()p Fq(m)2079 4127 y Fl(2)p Fr(;e)2079 4201 y(k)2192 4172 y Fo(\000)g Fq(m)2363 4127 y Fl(1)p Fr(;e)2363 4201 y(k)2455 4172 y Fp(\))p 1964 4216 526 4 v 2202 4299 a Fq(k)118 4444 y Fh(Such)j(a)g(step)f(is)g(very)h(similar)g(to)g(the)g(main)f(result)h (of)f(this)h(subsection)e(described)g(at)i(the)g(beginning)e(but,)118 4552 y(at)j(the)f(same)g(time,)h(very)g(different.)29 b(The)24 b(\002rst)g(main)f(difference)d(is)k(that)g(our)f(mesoscopic)f (scale)g Fq(n)i Fh(is)f(not)118 4660 y(linked)g(with)h Fq(\017)p Fh(.)31 b(The)25 b(second)c(difference)h(is)i(that)g(our)g (functions)f Fq( )2353 4627 y Fr(e)2350 4683 y(n;n)2484 4660 y Fh(represent)f(\(discrete\))h(gradient)g(of)118 4771 y(the)g(density)f(minus)h(their)g(canonical)e(average.)29 b(Such)23 b(a)g(counter)f(term,)i(discussed)d(at)i(length)f(in)h (section)118 4879 y(5,)i(is)g(absent)f(in)h(the)g(approach)g(of)f Fu([29)o(])p Fh(.)218 5022 y Fu(Our)h(main)g(result)g(is)h(given)e(by)i (the)f(following)g(theorem)118 5165 y FE(Theorem)i(7.10.)41 b Fh(F)l(or)25 b(any)g Fq(e)h Fo(2)f(E)1067 5397 y Fp(lim)1056 5458 y Fr(n)p Fn("1)1220 5397 y Fq(V)1293 5296 y Fk(\020)1357 5336 y Fq(W)1456 5303 y Fr(e)1443 5358 y(n)p 1357 5376 136 4 v 1398 5459 a Fq(n)1523 5397 y Fo(\000)20 b Fp(2)p Fq(m)p Fp(\(1)h Fo(\000)f Fq(m)p Fp(\))p Fq(\025)2099 5359 y Fn(0)2099 5420 y Fl(0)2138 5397 y Fp(\()p Fq(m)p Fp(\))2298 5326 y Fq( )2360 5293 y Fr(e)2357 5349 y(n;n)p 2298 5376 169 4 v 2356 5459 a Fq(n)2477 5296 y Fk(\021)2557 5397 y Fp(=)25 b(0)p Fq(:)686 b Fu(\(7.11\))p eop %%Page: 45 45 45 44 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(45)218 384 y Fu(W)-8 b(e)36 b(will)g(use)g(the)g(above)g(result)f(only)g(to)i (compute)e(the)h(limit)g(of)h Fq(V)20 b Fp(\()p Fq(j)2761 399 y Fl(0)p Fr(;e)2849 380 y Fi(0)2876 384 y Fq(;)2926 330 y Fr( )2974 306 y FA(e)2972 346 y(n;n)p 2926 363 146 4 v 2978 415 a Fr(n)3082 384 y Fp(\))36 b Fu(.)64 b(Indeed,)39 b(as)118 492 y(pointed)25 b(in)f([29],)h(the)f(function)g Fq(w)1343 506 y Fr(x;y)1468 492 y Fu(satis\002es)j(the)d(following)g (integration)g(by)g(parts)h(property:)31 b(for)118 600 y(any)25 b Fp(\003)g Fo(2)g Fs(F)36 b Fu(with)27 b Fp(\003)e Fo(3)g Fq(x;)15 b(y)28 b Fu(and)e(any)f Fq(\027)31 b Fo(2)24 b(M)p Fp(\(\003\))1299 761 y Fq(\027)6 b Fp(\()p Fq(w)1450 775 y Fr(x;y)1551 761 y Fq(g)s Fp(\))26 b(=)f Fq(\027)6 b Fp(\(\()p Fq(\021)1920 775 y Fr(x)1984 761 y Fo(\000)20 b Fq(\021)2120 775 y Fr(y)2162 761 y Fp(\))p Fo(r)2273 775 y Fr(x;y)2374 761 y Fq(g)s Fp(\))p Fq(:)118 923 y Fu(By)45 b(the)g(above)f(property)g(and)h(lemma)e(7.1)h(it)i(is)f (simple)g(to)g(check)f(that,)51 b(for)44 b(any)h Fq(e;)15 b(e)3380 890 y Fn(0)3462 923 y Fo(2)57 b(E)8 b Fu(,)118 1055 y Fq(V)191 955 y Fk(\020)246 1055 y Fq(j)283 1071 y Fl(0)p Fr(;e)371 1052 y Fi(0)397 1055 y Fq(;)448 1011 y Fr(W)525 987 y FA(e)515 1027 y(n)p 448 1035 110 4 v 481 1087 a Fr(n)567 955 y Fk(\021)647 1055 y Fp(=)25 b Fo(\000)p Fp(2)p Fq(m)p Fp(\(1)c Fo(\000)f Fq(m)p Fp(\))p Fq(\016)1286 1071 y Fr(e;e)1372 1052 y Fi(0)1398 1055 y Fu(.)32 b(Therefore,)24 b(by)h(theorem)g(7.10,)f(we)h(get)1001 1310 y Fp(lim)990 1371 y Fr(n)p Fn("1)1154 1310 y Fq(V)1227 1209 y Fk(\020)1281 1310 y Fq(j)1318 1325 y Fl(0)p Fr(;e)1406 1307 y Fi(0)1433 1310 y Fq(;)1483 1239 y( )1545 1206 y Fr(e)1542 1262 y(n;n)p 1483 1289 169 4 v 1540 1373 a Fq(n)1662 1209 y Fk(\021)1741 1310 y Fp(=)g Fo(\000)p Fq(\037)p Fp(\()p Fq(m)p Fp(\))p Fq(\016)2155 1325 y Fr(e;e)2241 1307 y Fi(0)2267 1310 y Fq(;)106 b Fo(8)p Fq(e;)15 b(e)2573 1273 y Fn(0)2623 1310 y Fo(2)24 b(E)8 b Fq(:)620 b Fu(\(7.12\))118 1539 y Fh(Proof)-8 b(.)42 b Fu(In)29 b(order)g(to)h(prove)f(theorem)f(7.10)h(it)h(is)g (convenient)e(to)i(introduce)f(some)g(notation.)44 b(First,)118 1647 y(we)28 b(\002x)h(the)f(vector)g Fq(e)j Fo(2)f(E)36 b Fu(which)29 b(will)f(be)g(often)h(omit)f(in)h(the)g(notation.)41 b(Moreover)27 b(we)h(introduce)118 1755 y(the)e(following)e (equivalence)g(relation.)118 1934 y FE(De\002nition)30 b(7.11.)45 b Fh(Given)30 b(two)g(sequences)d(of)i(functions)g Fo(f)p Fq(f)2211 1948 y Fr(n)2258 1934 y Fo(g)2303 1948 y Fr(n)p Fn(2)p Fm(N)2476 1934 y Fh(and)g Fo(f)p Fq(g)2740 1948 y Fr(n)2788 1934 y Fo(g)2833 1948 y Fr(n)p Fn(2)p Fm(N)3005 1934 y Fh(such)h(that)h Fq(f)3439 1948 y Fr(n)3516 1934 y Fh(and)118 2042 y Fq(g)161 2056 y Fr(n)234 2042 y Fh(have)25 b(support)g(in)g Fp(\003)917 2009 y Fr(e)917 2064 y(n)964 2042 y Fh(,)g(we)g(write)g Fq(f)1409 2056 y Fr(n)1481 2042 y Fo(\031)g Fq(g)1620 2056 y Fr(n)1692 2042 y Fh(if)976 2255 y Fp(lim)965 2317 y Fr(n)p 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y Fu(be)g(the)g(random)g(canonical) f(measure)g Fq(\026)p Fp([)p Fo(\001)15 b(j)g(F)2783 2655 y Fr(x;n)2890 2641 y Fp(])36 b Fu(where)e Fo(F)3296 2655 y Fr(x;n)3438 2641 y Fu(is)i(the)118 2763 y Fq(\033)s Fu(\226algebra)25 b(generated)f(by)h Fq(\034)1118 2777 y Fr(x)1162 2763 y Fq(m)1242 2719 y Fl(1)p Fr(;e)1242 2775 y(n)1333 2763 y Fu(,)h Fq(\034)1424 2777 y Fr(x)1467 2763 y Fq(m)1547 2719 y Fl(2)p Fr(;e)1547 2775 y(n)1664 2763 y Fu(and)g Fq(\021)1884 2777 y Fr(y)1950 2763 y Fu(with)h Fq(y)h Fo(62)d Fp(\003)2375 2730 y Fr(e)2375 2786 y(x;n)2481 2763 y Fu(.)32 b(Let)25 b(us)h(observe)e(that)188 2916 y(i\))42 b Fq(\026)348 2883 y Fr(\025)389 2892 y Fg(0)424 2883 y Fl(\()p Fr(m)p Fl(\))545 2916 y Fp(\()p Fq(\034)620 2930 y Fr(x)664 2916 y Fq(f)709 2930 y Fr(n)756 2916 y Fq(;)15 b(g)s Fp(\))26 b(=)f Fq(\026)1054 2883 y Fr(\025)1095 2892 y Fg(0)1130 2883 y Fl(\()p Fr(m)p Fl(\))1251 2843 y Fk(\000)1308 2916 y Fq(\027)1353 2930 y Fr(x;n)1459 2916 y Fp(\()p Fq(\034)1534 2930 y Fr(x)1578 2916 y Fq(W)1664 2930 y Fr(n)1711 2916 y Fp(;)15 b Fq(g)s Fp(\))1847 2843 y Fk(\001)1916 2916 y Fu(for)25 b(any)g(function)g Fq(g)s Fu(;)163 3051 y(ii\))42 b Fq(W)379 3065 y Fr(n)451 3051 y Fu(can)25 b(be)g(written)g(as)h(sum)f(of)h(functions)f Fq(f)34 b Fu(of)26 b(the)f(following)g(form)1292 3212 y Fq(f)35 b Fp(=)25 b(Av)1585 3246 y Fr(z)s Fn(2)p Fl(\003)1717 3213 y Fg(1)p FA(;e)1717 3256 y(n)1804 3212 y Fq(\034)1844 3226 y Fr(z)1884 3212 y Fq(h)15 b Fp(Av)2069 3246 y Fr(z)2105 3227 y Fi(0)2127 3246 y Fn(2)p Fl(\003)2223 3213 y Fg(2)p FA(;e)2223 3256 y(n)2309 3212 y Fq(\034)2349 3228 y Fr(z)2385 3209 y Fi(0)2411 3212 y Fq(h)2463 3175 y Fn(0)293 3388 y Fu(where)25 b Fq(h)h Fu(and)f Fq(h)868 3355 y Fn(0)917 3388 y Fu(depend)g(only)f(on)h Fq(\013)1616 3402 y Fl(0)1681 3388 y Fu(and)h Fq(\021)1901 3402 y Fl(0)1940 3388 y Fu(.)118 3522 y(Because)52 b(of)i Fq(i)p Fp(\))g Fu(and)f Fq(ii)p Fp(\))h Fu(and)g(thanks)g(to)g(the)f(the)h(variational)f (characterization)g(\(4.20\))f(of)118 3636 y Fq(V)171 3651 y Fr(`)204 3636 y Fp(\()p Fo(\001)p Fq(;)15 b(\026)359 3603 y Fr(\025)400 3612 y Fg(0)436 3603 y Fl(\()p Fr(m)p Fl(\))557 3636 y Fp(\))p Fu(,)26 b(it)g(is)g(enough)e(to)i(prove)f (that,)h(for)f(a)h(function)f Fq(f)34 b Fu(as)26 b(in)f Fq(ii)p Fp(\))p Fu(,)947 3841 y Fp(lim)936 3902 y Fr(n)p Fn("1)1109 3841 y Fp(lim)1100 3902 y Fr(k)r Fn("1)1339 3779 y Fp(1)p 1269 3820 185 4 v 1269 3903 a Fq(n)1324 3877 y Fl(2)1363 3903 y Fq(k)1413 3877 y Fr(d)1479 3841 y Fs(E)1534 3767 y Fk(\002)1593 3841 y Fp(sup)1594 3919 y Fr(g)r Fn(2)p Fm(G)1745 3841 y Fo(f)p Fq(\036)p Fp(\()p Fq(g)s Fp(\))p Fq(=)p Fo(D)2075 3855 y Fl(\003)2124 3867 y FA(k)2169 3841 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)2345 3803 y Fr(\025)2386 3812 y Fg(0)2421 3803 y Fl(\()p Fr(m)p Fl(\))2543 3841 y Fp(\))p Fo(g)2638 3767 y Fk(\003)2702 3841 y Fp(=)25 b(0)566 b Fu(\(7.13\))118 4070 y(where)833 4194 y Fq(\036)p Fp(\()p Fq(g)s Fp(\))27 b(:=)1151 4120 y Fk(\002)1226 4107 y(X)1189 4309 y Fn(j)p Fr(x)p Fn(j\024)p Fr(k)1361 4318 y Fg(1)1410 4194 y Fq(\026)1465 4156 y Fr(\025)1506 4165 y Fg(0)1540 4156 y Fl(\()p Fr(m)p Fl(\))1662 4120 y Fk(\000)1703 4194 y Fq(\027)1748 4208 y Fr(x;n)1854 4194 y Fp(\()p Fq(\034)1929 4208 y Fr(x)1973 4194 y Fq(f)10 b Fp(;)15 b Fq(g)s Fp(\))2149 4120 y Fk(\001)q(\003)2230 4143 y Fl(2)2269 4194 y Fq(;)106 b(k)2447 4208 y Fl(1)2512 4194 y Fp(:=)26 b Fq(k)d Fo(\000)2795 4112 y(p)p 2871 4112 51 4 v 82 x Fq(k)s(:)118 4444 y Fu(By)i(proposition)i(A.6,)e(for)g (any)g Fq(\016)k(>)c Fp(0)h Fu(there)f(exists)h Fq(`)1959 4458 y Fl(0)2024 4444 y Fo(2)f Fs(N)38 b Fu(such)25 b(that,)h(if)g Fq(n)f Fo(\025)g Fq(`)g Fo(\025)g Fq(`)3067 4458 y Fl(0)3106 4444 y Fu(,)h(then)375 4656 y Fq(\027)420 4670 y Fr(x;n)526 4656 y Fp(\()p Fq(\034)601 4670 y Fr(x)645 4656 y Fq(f)10 b Fp(;)15 b Fq(g)s Fp(\))821 4618 y Fl(2)886 4656 y Fo(\024)992 4594 y Fq(c)p Fp(\()p Fq(`)p Fp(\))p 992 4635 148 4 v 1018 4718 a Fq(n)1073 4692 y Fr(d)1150 4656 y Fo(D)s Fp(\()p Fq(g)s Fp(;)g Fq(\027)1389 4670 y Fr(x;n)1496 4656 y Fp(\))21 b(+)1678 4594 y Fq(\016)p 1653 4635 96 4 v 1653 4718 a(n)1708 4692 y Fr(d)1758 4656 y Fp(V)-8 b(ar)1899 4670 y Fr(\027)1934 4678 y FA(x;n)2034 4656 y Fp(\()p Fq(g)s Fp(\))22 b(+)2300 4594 y Fq(c)p 2273 4635 V 2273 4718 a(n)2328 4692 y Fr(d)2378 4656 y Fp(V)-8 b(ar)2519 4670 y Fr(\027)2554 4678 y FA(x;n)2654 4656 y Fp(\()p Fq(g)s Fp(\))p Fq(#)2824 4671 y Fl(\003)2873 4652 y FA(e)2873 4687 y(x;n)2970 4671 y Fr(;`)3023 4656 y Fp(\()p Fq(\013)p Fp(\))258 b Fu(\(7.14\))118 4848 y(where,)25 b(for)g(any)g(given)g Fq(\015)30 b(>)25 b Fp(0)h Fu(and)f Fq(`)g Fo(\025)g Fq(`)1575 4862 y Fl(1)1614 4848 y Fp(\()p Fq(\015)5 b Fp(\))27 b Fo(\025)d Fq(`)1896 4862 y Fl(0)1936 4848 y Fu(,)1312 5028 y Fs(P)p Fp(\()p Fq(#)1456 5043 y Fl(\003)1505 5024 y FA(e)1505 5060 y(x;n)1602 5043 y Fr(;`)1654 5028 y Fp(\()p Fq(\013)p Fp(\))j Fo(\025)e Fq(\015)5 b Fp(\))25 b Fo(\024)g Fq(e)2155 4991 y Fn(\000)p Fr(c)p Fl(\()p Fr(\015)t(;`)p Fl(\))p Fr(n)2427 4967 y FA(d)3409 5028 y Fu(\(7.15\))118 5199 y(for)j(a)g(suitable)g (constant)g Fq(c)p Fp(\()p Fq(\015)5 b(;)15 b(`)p Fp(\))p Fu(.)41 b(Using)27 b(the)i(spectral)e(gap)h(estimate)g(\(4.13\),)f(the) i(r)-10 b(.h.s.)40 b(of)28 b(\(7.14\))118 5307 y(can)d(be)g(bounded)g (by)1029 5472 y Fo(D)s Fp(\()p Fq(g)s Fp(;)15 b Fq(\027)1268 5486 y Fr(x;n)1375 5472 y Fp(\))1410 5399 y Fk(\000)1452 5472 y Fq(c)p Fp(\()p Fq(`)p Fp(\))21 b(+)f Fq(c)15 b(\016)20 b(n)1880 5435 y Fl(2)1939 5472 y Fp(+)g Fq(c)15 b(n)2139 5435 y Fl(2)2178 5472 y Fq(#)2232 5487 y Fl(\003)2281 5468 y FA(e)2281 5504 y(x;n)2377 5487 y Fr(;`)2430 5472 y Fp(\()p Fq(\013)p Fp(\))2558 5399 y Fk(\001)2601 5472 y Fq(n)2656 5435 y Fn(\000)p Fr(d)p eop %%Page: 46 46 46 45 bop 118 47 a Fj(46)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(and)25 b(therefore,)g(by)g(Schwarz)g(inequality)-12 b(,)714 535 y Fq(\036)p Fp(\()p Fq(g)s Fp(\))27 b Fo(\024)e(D)1077 549 y Fl(\003)1126 561 y FA(k)1168 535 y Fp(\()p Fq(g)s Fp(;)15 b Fq(\026)1344 497 y Fr(\025)1385 506 y Fg(0)1421 497 y Fl(\()p Fr(m)p Fl(\))1542 535 y Fp(\))1577 461 y Fk(\000)1619 535 y Fq(c)p Fp(\()p Fq(`)p Fp(\))p Fq(k)1816 497 y Fr(d)1878 535 y Fp(+)20 b Fq(c)15 b(\016)s(k)2116 497 y Fr(d)2158 535 y Fq(n)2213 497 y Fl(2)2272 535 y Fp(+)20 b Fq(c)15 b(n)2472 497 y Fl(2)2564 448 y Fk(X)2527 650 y Fn(j)p Fr(x)p Fn(j\024)p Fr(k)2699 659 y Fg(1)2747 535 y Fq(#)2801 550 y Fl(\003)2850 530 y FA(e)2850 566 y(x;n)2946 550 y Fr(;`)2999 461 y Fk(\001)3040 535 y Fq(:)118 795 y Fu(By)33 b(taking)h(the)f(limits)i Fq(\016)42 b Fo(#)e Fp(0)p Fq(;)31 b(`)38 b Fo(")h(1)p Fq(;)31 b(n)38 b Fo(")i(1)p Fq(;)30 b(k)42 b Fo(")d(1)33 b Fu(\(from)g(right)h(to)g (left\),)h(in)f(order)e(to)i(prove)118 903 y(\(7.13\))24 b(the)i(thesis)g(follows)g(since)e Fp(lim)1458 918 y Fr(n)p Fn("1)1626 903 y Fs(E)1680 829 y Fk(\000)1728 903 y Fq(#)1782 918 y Fl(\003)1831 899 y FA(e)1831 934 y(n)1873 918 y Fr(;`)1926 903 y Fp(\()p Fq(\013)p Fp(\))2054 829 y Fk(\001)2122 903 y Fp(=)h(0)h Fu(because)e(of)h(\(7.15\).)118 1119 y Fo(\017)h FE(Step)f(2)p Fu(:)31 b Fq(\026)p Fp([)p Fq(W)680 1133 y Fr(n)727 1119 y Fo(j)15 b Fq(m)847 1075 y Fl(1)p Fr(;e)847 1131 y(n)939 1119 y Fq(;)g(m)1059 1075 y Fl(2)p Fr(;e)1059 1131 y(n)1151 1119 y Fp(])25 b Fo(\031)g Fp(2)p Fq(m)p Fp(\(1)d Fo(\000)d Fq(m)p Fp(\))p Fq(\025)1782 1086 y Fn(0)1782 1143 y Fl(0)1822 1119 y Fp(\()p Fq(m)p Fp(\))p Fq( )2034 1086 y Fr(e)2031 1141 y(n;n)2141 1119 y Fq(:)118 1252 y Fu(The)39 b(proof)g(is)h(based)f(on)g (the)g(following)f(lemma,)j(which)f(follows)e(easily)h(from)g(the)g (variational)118 1363 y(characterization)25 b(of)g Fq(V)939 1378 y Fr(`)972 1363 y Fp(\()p Fo(\001)p Fq(;)15 b(\026)1127 1330 y Fr(\025)1168 1339 y Fg(0)1204 1330 y Fl(\()p Fr(m)p Fl(\))1325 1363 y Fp(\))26 b Fu(given)e(in)i(\(4.20\).)118 1529 y FE(L)r(emma)h(7.12.)41 b Fh(Let,)26 b(for)f(any)f Fq(n)h Fo(2)g Fs(N)7 b Fh(,)31 b Fq(f)1499 1543 y Fr(n)1546 1529 y Fq(;)f(h)1653 1543 y Fr(n)1726 1529 y Fo(2)25 b Fs(G)50 b Fh(be)25 b(such)g(that)185 1659 y(i\))42 b Fp(\001)367 1674 y Fr(f)401 1682 y FA(n)473 1659 y Fo(\032)25 b Fp(\003)632 1626 y Fr(e)632 1682 y(n)694 1659 y Fh(;)159 1800 y(ii\))42 b Fp(sup)428 1822 y Fr(n)490 1800 y Fo(k)p Fq(h)587 1814 y Fr(n)635 1800 y Fo(k)680 1814 y Fn(1)780 1800 y Fq(<)25 b Fo(1)91 b Fh(and)105 b Fp(lim)1436 1815 y Fr(n)p Fn("1)1604 1800 y Fq(n)1659 1767 y Fr(d)1699 1800 y Fs(E)1754 1727 y Fk(\002)1798 1800 y Fq(\026)1853 1767 y Fr(\025)1894 1776 y Fg(0)1928 1767 y Fl(\()p Fr(m)p Fl(\))2050 1727 y Fk(\000)2091 1800 y Fq(h)2143 1767 y Fl(2)2143 1823 y Fr(n)2191 1727 y Fk(\001)o(\003)2296 1800 y Fp(=)25 b(0)p Fh(;)133 1931 y(iii\))42 b Fo(j)p Fq(f)361 1945 y Fr(n)408 1931 y Fo(j)25 b(\024)g(j)p Fq(h)631 1945 y Fr(n)679 1931 y Fo(j)p Fh(.)118 2061 y(Then)g Fq(f)379 2075 y Fr(n)451 2061 y Fo(\031)g Fp(0)p Fh(.)218 2227 y Fu(Thanks)c(to)g(the)g(estimates)h(given)e(in)h (the)g(Appendix)g(it)g(can)f(be)h(proved)f(\(see)h([15)o(]\))g(that)h (condition)118 2335 y Fh(ii\))k Fu(of)f(the)h(lemma)e(is)i(satis\002ed) h(by)e(any)g(of)g(the)h(following)e(sequences:)212 2501 y Fo(f)p Fq(n)312 2464 y Fn(\000)p Fr(d)407 2501 y Fo(g)452 2515 y Fr(n)p Fn(2)p Fm(N)595 2501 y Fq(;)726 2428 y Fk(\010)794 2501 y Fp(\()p Fq(m)d Fo(\000)f Fq(m)1101 2515 y Fl(\001)1160 2523 y FA(n)1206 2501 y Fp(\))1241 2464 y Fl(2)1296 2501 y Fo(g)1341 2515 y Fr(n)p Fn(2)p Fm(N)1483 2501 y Fq(;)1615 2428 y Fk(\010)1668 2501 y Fp(1)-25 b(I)1721 2520 y Fn(fj)p Fr(m)p Fn(\000)p Fr(m)1955 2531 y Fg(\001)2005 2539 y FA(n)2052 2520 y Fn(j\025)p Fr(c)p Fn(g)2196 2428 y Fk(\011)2249 2537 y Fr(n)p Fn(2)p Fm(N)2391 2501 y Fq(;)2523 2428 y Fk(\010\000)2617 2501 y Fq(m)20 b Fo(\000)g Fq(\026)2863 2464 y Fr(\025)2904 2473 y Fg(0)2939 2464 y Fl(\()p Fr(m)p Fl(\))3060 2501 y Fp(\()p Fq(m)3175 2464 y Fr(i;e)3175 2524 y(n)3256 2501 y Fp(\))3291 2428 y Fk(\001)3333 2450 y Fl(2)3372 2428 y Fk(\011)3425 2537 y Fr(n)p Fn(2)p Fm(N)118 2681 y Fu(where)25 b Fq(i)g Fp(=)g(1)p Fq(;)15 b Fp(2)p Fu(,)27 b Fq(c)f(>)f Fp(0)g Fu(and)h Fp(\001)1205 2695 y Fr(n)1277 2681 y Fu(is)g(either)f(one)g(of)g(the)h(sets)g Fp(\003)2282 2648 y Fr(e)2282 2703 y(n)2329 2681 y Fu(,)f Fp(\003)2442 2637 y Fl(1)p Fr(;e)2442 2693 y(n)2534 2681 y Fu(,)h Fp(\003)2648 2637 y Fl(2)p Fr(;e)2648 2693 y(n)2740 2681 y Fu(.)118 2811 y(As)g(in)f([29])g(we)g(de\002ne)g(the)g(\(random)g(w) -11 b(.r)h(.t.)32 b Fq(\013)p Fu(\))26 b(function)f Fq(F)2233 2825 y Fr(n)2280 2811 y Fp(\()p Fq(m)2395 2825 y Fl(1)2435 2811 y Fq(;)15 b(m)2555 2825 y Fl(2)2594 2811 y Fp(\))26 b Fu(as)1186 2984 y Fq(F)1244 2998 y Fr(n)1291 2984 y Fp(\()p Fq(m)1406 2998 y Fl(1)1446 2984 y Fq(;)15 b(m)1566 2998 y Fl(2)1606 2984 y Fp(\))25 b(=)g Fq(\026)1817 2936 y Fr(\025)p Fl(\()p Fr(m)1947 2945 y Fg(1)1983 2936 y Fl(\))1817 3032 y(\003)1866 2999 y Fg(1)p FA(;e)1866 3042 y(n)2034 2984 y Fo(\012)20 b Fq(\026)2180 2936 y Fr(\025)p Fl(\()p Fr(m)2310 2945 y Fg(2)2346 2936 y Fl(\))2180 3032 y(\003)2229 2999 y Fg(2)p FA(;e)2229 3042 y(n)2377 2910 y Fk(\000)2419 2984 y Fq(W)2505 2998 y Fr(n)2552 2910 y Fk(\001)118 3163 y Fu(It)26 b(is)g(not)g(dif\002cult)f(to)h (show)g(that)g Fq(F)1356 3177 y Fr(n)1404 3163 y Fp(\()p Fq(m)1519 3177 y Fl(1)1558 3163 y Fq(;)15 b(m)1678 3177 y Fl(2)1718 3163 y Fp(\))26 b Fu(has)g(the)f(explicit)g(expression)166 3326 y Fq(F)224 3340 y Fr(n)272 3326 y Fp(\()p Fq(m)387 3340 y Fl(1)426 3326 y Fq(;)15 b(m)546 3340 y Fl(2)586 3326 y Fp(\))25 b(=)g Fq(m)822 3340 y Fl(1)882 3326 y Fo(\000)20 b Fq(m)1053 3340 y Fl(2)1112 3326 y Fp(+)g Fq(e)1245 3288 y Fr(\025)1286 3297 y Fg(1)p FA(;n)1378 3288 y Fl(\()p Fr(m)1467 3297 y Fg(1)1503 3288 y Fl(\))p Fn(\000)p Fr(\025)1626 3297 y Fg(2)p FA(;n)1718 3288 y Fl(\()p Fr(m)1807 3297 y Fg(2)1843 3288 y Fl(\))1874 3326 y Fp(\(1)h Fo(\000)f Fq(m)2146 3340 y Fl(1)2185 3326 y Fp(\))p Fq(m)2300 3340 y Fl(2)2360 3326 y Fo(\000)g Fq(e)2493 3288 y Fr(\025)2534 3297 y Fg(2)p FA(;n)2627 3288 y Fl(\()p Fr(m)2716 3297 y Fg(2)2751 3288 y Fl(\))p Fn(\000)p Fr(\025)2874 3297 y Fg(1)p FA(;n)2966 3288 y Fl(\()p Fr(m)3055 3297 y Fg(1)3091 3288 y Fl(\))3122 3326 y Fq(m)3202 3340 y Fl(1)3242 3326 y Fp(\(1)h Fo(\000)f Fq(m)3514 3340 y Fl(2)3553 3326 y Fp(\))p Fq(:)118 3478 y Fu(The)26 b(main)f(reason)f(to)i(introduce)f Fq(F)1374 3492 y Fr(n)1421 3478 y Fp(\()p Fq(m)1536 3492 y Fl(1)1576 3478 y Fq(;)15 b(m)1696 3492 y Fl(2)1735 3478 y Fp(\))26 b Fu(is)g(that)1206 3633 y Fq(\026)p Fp([)p Fq(W)1372 3647 y Fr(n)1418 3633 y Fo(j)15 b Fq(m)1538 3596 y Fl(1)p Fr(;e)1538 3656 y(n)1630 3633 y Fq(;)g(m)1750 3596 y Fl(2)p Fr(;e)1750 3656 y(n)1842 3633 y Fp(])26 b Fo(\031)f Fq(F)2047 3647 y Fr(n)2094 3633 y Fp(\()p Fq(m)2209 3596 y Fl(1)p Fr(;e)2209 3656 y(n)2301 3633 y Fq(;)15 b(m)2421 3596 y Fl(2)p Fr(;e)2421 3656 y(n)2513 3633 y Fp(\))p Fq(:)836 b Fu(\(7.16\))118 3786 y(This)39 b(equivalence)d(follows)i(at) g(once)f(from)g(the)h(equivalence)e(of)i(the)g(ensembles)e(together)h (with)118 3901 y(lemma)25 b(7.12)g(applied)i(to)g Fq(f)1088 3915 y Fr(n)1162 3901 y Fp(=)f Fq(\026)p Fp([)p Fq(W)1425 3915 y Fr(n)1472 3901 y Fo(j)15 b Fq(m)1592 3857 y Fl(1)p Fr(;e)1592 3913 y(n)1684 3901 y Fq(;)g(m)1804 3857 y Fl(2)p Fr(;e)1804 3913 y(n)1896 3901 y Fp(])22 b Fo(\000)e Fq(F)2092 3915 y Fr(n)2140 3901 y Fp(\()p Fq(m)2255 3857 y Fl(1)p Fr(;e)2255 3913 y(n)2347 3901 y Fq(;)15 b(m)2467 3857 y Fl(2)p Fr(;e)2467 3913 y(n)2559 3901 y Fp(\))26 b Fu(and)h Fq(h)2848 3915 y Fr(n)2922 3901 y Fp(=)g Fq(c)15 b(n)3129 3868 y Fn(\000)p Fr(d)3251 3901 y Fu(for)26 b(a)h(large)118 4009 y(enough)d(constant)h Fq(c)p Fu(.)118 4117 y(Next,)70 b(again)60 b(by)h(lemma)e(7.12)h(applied)h(with)h Fq(h)2043 4131 y Fr(n)2175 4117 y Fp(=)84 b(1)-25 b(I)2383 4150 y Fn(fj)p Fr(m)p Fn(\000)p Fr(m)2617 4118 y Fg(1)p FA(;e)2617 4161 y(n)2701 4150 y Fn(j\025)p Fr(c)2807 4158 y FA(m)2864 4150 y Fn(g)2948 4117 y Fp(+)44 b(1)-25 b(I)3116 4150 y Fn(fj)p Fr(m)p Fn(\000)p Fr(m)3350 4118 y Fg(2)p FA(;e)3350 4161 y(n)3433 4150 y Fn(j\025)p Fr(c)3539 4158 y FA(m)3597 4150 y Fn(g)3636 4117 y Fu(,)118 4243 y Fq(c)157 4257 y Fr(m)249 4243 y Fp(=)25 b(\()p Fq(m)c Fo(^)e Fp(\(1)i Fo(\000)f Fq(m)p Fp(\)\))p Fq(=)p Fp(2)r Fu(,)25 b(we)g(get)g(that)1210 4399 y Fq(F)1268 4413 y Fr(n)1315 4399 y Fp(\()p Fq(m)1430 4361 y Fl(1)p Fr(;e)1430 4421 y(n)1522 4399 y Fq(;)15 b(m)1642 4361 y Fl(2)p Fr(;e)1642 4421 y(n)1734 4399 y Fp(\))26 b Fo(\031)f Fq(F)1949 4413 y Fr(n)1996 4399 y Fp(\()p Fq(m)2111 4361 y Fl(1)p Fr(;e)2111 4421 y(n)2203 4399 y Fq(;)15 b(m)2323 4361 y Fl(2)p Fr(;e)2323 4421 y(n)2415 4399 y Fp(\)1)-25 b(I)2503 4413 y Fr(m)3409 4399 y Fu(\(7.17\))118 4551 y(where)25 b Fp(1)-25 b(I)442 4565 y Fr(m)534 4551 y Fp(:=)25 b(1)-25 b(I)708 4584 y Fn(fj)p Fr(m)p Fn(\000)p Fr(m)942 4552 y Fg(1)p FA(;e)942 4595 y(n)1025 4584 y Fn(j\024)p Fr(c)1131 4592 y FA(m)1189 4584 y Fn(g)1228 4551 y Fp(1)g(I)1281 4584 y Fn(fj)p Fr(m)p Fn(\000)p Fr(m)1515 4552 y Fg(2)p FA(;e)1515 4595 y(n)1599 4584 y Fn(j\024)p Fr(c)1705 4592 y FA(m)1763 4584 y Fn(g)1802 4551 y Fu(.)118 4718 y(Next,)25 b(by)g(T)-9 b(aylor)26 b(expansion)f(around)f(the)i(arithmetic)g(mean)e(of)h Fq(m)2490 4674 y Fl(1)p Fr(;e)2490 4730 y(n)2607 4718 y Fu(and)h Fq(m)2862 4674 y Fl(2)p Fr(;e)2862 4730 y(n)2953 4718 y Fu(,)g(we)f(write)1562 4871 y Fq(F)1620 4885 y Fr(n)1667 4871 y Fp(\()p Fq(m)1782 4834 y Fl(1)p Fr(;e)1782 4894 y(n)1874 4871 y Fq(;)15 b(m)1994 4834 y Fl(2)p Fr(;e)1994 4894 y(n)2086 4871 y Fp(\))26 b(=)166 5068 y Fq(F)224 5082 y Fr(n)271 5068 y Fp(\()p Fq(m)386 5030 y Fr(e)386 5090 y(n)433 5068 y Fq(;)15 b(m)553 5030 y Fr(e)553 5090 y(n)600 5068 y Fp(\))21 b(+)764 5006 y Fq(@)5 b(F)875 5020 y Fr(n)p 757 5047 173 4 v 757 5130 a Fq(@)g(m)890 5144 y Fl(1)939 5068 y Fp(\()p Fq(m)1054 5030 y Fr(e)1054 5090 y(n)1101 5068 y Fq(;)15 b(m)1221 5030 y Fr(e)1221 5090 y(n)1269 5068 y Fp(\)\()p Fq(m)1419 5030 y Fl(1)p Fr(;e)1419 5090 y(n)1531 5068 y Fo(\000)20 b Fq(m)1702 5030 y Fr(e)1702 5090 y(n)1749 5068 y Fp(\))g(+)1912 5006 y Fq(@)5 b(F)2023 5020 y Fr(n)p 1905 5047 V 1905 5130 a Fq(@)g(m)2038 5144 y Fl(1)2088 5068 y Fp(\()p Fq(m)2203 5030 y Fr(e)2203 5090 y(n)2250 5068 y Fq(;)15 b(m)2370 5030 y Fr(e)2370 5090 y(n)2417 5068 y Fp(\)\()p Fq(m)2567 5030 y Fl(2)p Fr(;e)2567 5090 y(n)2680 5068 y Fo(\000)k Fq(m)2850 5030 y Fr(e)2850 5090 y(n)2897 5068 y Fp(\))i(+)f Fq(R)3113 5082 y Fr(n)3160 5068 y Fp(\()p Fq(m)3275 5030 y Fl(1)p Fr(;e)3275 5090 y(n)3367 5068 y Fq(;)15 b(m)3487 5030 y Fl(2)p Fr(;e)3487 5090 y(n)3579 5068 y Fp(\))118 5264 y Fu(Then,)30 b(the)f(zero)e(order)h (contribution)h Fq(F)1548 5278 y Fr(n)1595 5264 y Fp(\()p Fq(m)1710 5231 y Fr(e)1710 5287 y(n)1757 5264 y Fq(;)15 b(m)1877 5231 y Fr(e)1877 5287 y(n)1924 5264 y Fp(\)1)-25 b(I)2013 5278 y Fr(m)2108 5264 y Fu(is)30 b(negligible,)d Fq(F)2698 5278 y Fr(n)2746 5264 y Fp(\()p Fq(m)2861 5231 y Fr(e)2861 5287 y(n)2908 5264 y Fq(;)15 b(m)3028 5231 y Fr(e)3028 5287 y(n)3075 5264 y Fp(\)1)-25 b(I)3164 5278 y Fr(m)3261 5264 y Fo(\031)31 b Fp(0)p Fu(,)f(since)118 5372 y Fq(F)176 5386 y Fr(n)224 5372 y Fp(\()p Fq(m)339 5339 y Fr(e)339 5395 y(n)386 5372 y Fq(;)15 b(m)506 5339 y Fr(e)506 5395 y(n)553 5372 y Fp(\))27 b Fo(\031)g Fp(0)f Fu(because)f(of)i(de\002nition)g(7.11)e(and)h Fq(F)2072 5386 y Fr(n)2120 5372 y Fp(\()p Fq(m)2235 5339 y Fr(e)2235 5395 y(n)2282 5372 y Fq(;)15 b(m)2402 5339 y Fr(e)2402 5395 y(n)2449 5372 y Fp(\)\(1)22 b Fo(\000)f Fp(1)-25 b(I)2731 5386 y Fr(m)2797 5372 y Fp(\))27 b Fo(\031)g Fp(0)g Fu(again)f(by)g(lemma)118 5480 y(7.12.)p eop %%Page: 47 47 47 46 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(47)118 370 y Fu(The)31 b(second)f(order)g(error)f(term,)j Fq(R)1389 384 y Fr(n)1436 370 y Fp(\()p Fq(m)1551 326 y Fl(1)p Fr(;e)1551 382 y(n)1642 370 y Fq(;)15 b(m)1762 326 y Fl(2)p Fr(;e)1762 382 y(n)1854 370 y Fp(\)1)-25 b(I)1943 384 y Fr(m)2009 370 y Fu(,)33 b(is)e(negligible)e(because)g(of)i(lemma) e(7.12)g(ap-)118 489 y(plied)22 b(with)g Fq(h)587 503 y Fr(n)660 489 y Fp(=)j Fq(c)795 415 y Fk(\002)833 489 y Fp(\()p Fq(m)948 444 y Fl(1)p Fr(;e)948 500 y(n)1046 489 y Fo(\000)7 b Fq(m)1204 456 y Fr(e)1204 511 y(n)1251 489 y Fp(\))1286 456 y Fl(2)1332 489 y Fp(+)g(\()p Fq(m)1525 444 y Fl(2)p Fr(;e)1525 500 y(n)1623 489 y Fo(\000)g Fq(m)1781 456 y Fr(e)1781 511 y(n)1827 489 y Fp(\))1862 456 y Fl(2)1902 415 y Fk(\003)1940 489 y Fu(.)30 b(Notice)21 b(that)i(it)f(is)g(here)e(that)j(the)e(characteristic)118 597 y(function)i Fp(1)-25 b(I)524 611 y Fr(m)615 597 y Fu(plays)24 b(an)f(important)h(role)f(since)g(the)h(second)g (derivatives)g(of)g Fq(F)2841 611 y Fr(n)2889 597 y Fp(\()p Fq(m)3004 611 y Fl(1)3043 597 y Fq(;)15 b(m)3163 611 y Fl(2)3203 597 y Fp(\))24 b Fu(diverge)f(as)118 705 y Fq(m)198 719 y Fr(i)251 705 y Fu(tends)j(to)g Fp(0)f Fu(or)g(to)h Fp(1)p Fu(.)118 813 y(Let)g(us)f(now)g(examine)f(the)i (relevant)e(\002rst)h(order)g(terms.)31 b(W)-8 b(e)26 b(claim)f(that)h(for)f Fq(i)h Fp(=)f(1)p Fq(;)15 b Fp(2)233 957 y Fq(@)5 b(F)344 971 y Fr(n)p 232 998 162 4 v 232 1081 a Fq(@)g(m)365 1095 y Fr(i)403 1019 y Fp(\()p Fq(m)518 981 y Fr(e)518 1041 y(n)565 1019 y Fq(;)15 b(m)685 981 y Fr(e)685 1041 y(n)732 1019 y Fp(\)\()p Fq(m)882 981 y Fr(i;e)882 1041 y(n)984 1019 y Fo(\000)k Fq(m)1154 981 y Fr(e)1154 1041 y(n)1201 1019 y Fp(\)1)-25 b(I)1290 1033 y Fr(m)1376 1019 y Fp(+)20 b(\()p Fo(\000)p Fp(1\))1653 981 y Fr(i)1682 1019 y Fp(2)p Fq(m)1807 981 y Fr(e)1807 1041 y(n)1855 1019 y Fp(\(1)h Fo(\000)f Fq(m)2127 981 y Fr(e)2127 1041 y(n)2173 1019 y Fp(\))p Fq(\025)2261 981 y Fn(0)2261 1041 y Fr(i;n)2352 1019 y Fp(\()p Fq(m)2467 981 y Fr(e)2467 1041 y(n)2514 1019 y Fp(\)\()p Fq(m)2664 981 y Fr(i;e)2664 1041 y(n)2766 1019 y Fo(\000)g Fq(m)2937 981 y Fr(e)2937 1041 y(n)2983 1019 y Fp(\)1)-25 b(I)3072 1033 y Fr(m)3164 1019 y Fo(\031)25 b Fp(0)104 b Fu(\(7.18\))118 1226 y(and)853 1351 y Fp(2)p Fq(m)978 1314 y Fr(e)978 1374 y(n)1026 1351 y Fp(\(1)21 b Fo(\000)f Fq(m)1298 1314 y Fr(e)1298 1374 y(n)1344 1351 y Fp(\)\()p Fq(\025)1467 1314 y Fn(0)1467 1374 y Fr(i;n)1559 1351 y Fp(\()p Fq(m)1674 1314 y Fr(e)1674 1374 y(n)1721 1351 y Fp(\))g Fo(\000)g Fq(\025)1920 1314 y Fn(0)1920 1374 y Fl(0)1960 1351 y Fp(\()p Fq(m)p Fp(\)\)\()p Fq(m)2260 1314 y Fr(i;e)2260 1374 y(n)2362 1351 y Fo(\000)f Fq(m)2532 1314 y Fr(e)2532 1374 y(n)2579 1351 y Fp(\)1)-25 b(I)2668 1365 y Fr(m)2759 1351 y Fo(\031)25 b Fp(0)p Fq(:)484 b Fu(\(7.19\))118 1499 y(where)25 b Fq(\025)442 1513 y Fr(i;n)558 1499 y Fp(:=)g Fq(\025)732 1533 y Fl(\003)781 1500 y FA(i;e)781 1543 y(n)860 1499 y Fu(.)118 1619 y(Before)g(proving)f(\(7.18\))g(and)i (\(7.19\))e(let)h(us)g(summarize)f(what)i(we)f(have)h(obtained)f(so)h (far)-10 b(.)32 b(Thanks)118 1727 y(to)e(\(7.16\),)f(\(7.17\),)g(the)g (above)g(discussion)i(of)e(the)g(T)-9 b(aylor)30 b(expansion)e(and)i (\(7.18\))e(together)g(with)118 1835 y(\(7.19\))791 1960 y Fq(\026)p Fp([)p Fq(W)957 1974 y Fr(n)1004 1960 y Fo(j)15 b Fq(m)1124 1923 y Fl(1)p Fr(;e)1124 1983 y(n)1216 1960 y Fq(;)g(m)1336 1923 y Fl(2)p Fr(;e)1336 1983 y(n)1428 1960 y Fp(])26 b Fo(\031)e Fp(2)p Fq(m)1699 1923 y Fr(e)1699 1983 y(n)1747 1960 y Fp(\(1)d Fo(\000)f Fq(m)2019 1923 y Fr(e)2019 1983 y(n)2065 1960 y Fp(\))p Fq(\025)2153 1923 y Fn(0)2153 1983 y Fl(0)2193 1960 y Fp(\()p Fq(m)p Fp(\)\()p Fq(m)2458 1923 y Fl(2)p Fr(;e)2458 1983 y(n)2571 1960 y Fo(\000)g Fq(m)2742 1923 y Fl(1)p Fr(;e)2742 1983 y(n)2833 1960 y Fp(\)1)-25 b(I)2922 1974 y Fr(m)118 2105 y Fu(Using)39 b(once)e(more)g(lemma)g(7.12)h(it)h(is)g(now)f(rather)h (simple)f(to)h(remove)e(the)h(factor)h Fp(1)-25 b(I)3290 2119 y Fr(m)3395 2105 y Fu(and)38 b(to)118 2213 y(replace)24 b Fq(m)510 2180 y Fr(e)510 2235 y(n)582 2213 y Fu(with)i Fq(m)p Fu(,)g(thus)g(concluding)d(the)j(proof)m(.)118 2321 y(W)-8 b(e)26 b(are)f(left)g(with)h(the)g(proof)f(of)g(\(7.18\))f (and)i(\(7.19\).)118 2459 y(Let)33 b(us)f(prove)f(\(7.18\))g(for)h Fq(i)37 b Fp(=)g(1)p Fu(.)52 b(By)32 b(computing)1978 2423 y Fr(@)t(F)2064 2431 y FA(n)p 1973 2438 139 4 v 1973 2491 a Fr(@)t(m)2076 2500 y Fg(1)2153 2459 y Fu(it)h(is)g(simple)f (to)h(check)e(that)i(the)f(l.h.s.)53 b(of)118 2575 y(\(7.18\))24 b(is)i(equal)f(to)433 2747 y Fp(\()p Fq(e)510 2709 y Fr(\025)551 2718 y Fg(1)p FA(;n)644 2709 y Fl(\()p Fr(m)733 2686 y FA(e)733 2726 y(n)776 2709 y Fl(\))p Fn(\000)p Fr(\025)899 2718 y Fg(2)p FA(;n)992 2709 y Fl(\()p Fr(m)1081 2686 y FA(e)1081 2726 y(n)1124 2709 y Fl(\))1176 2747 y Fo(\000)20 b Fp(1\))p Fq(m)1427 2709 y Fr(e)1427 2769 y(n)1474 2747 y Fp(\()p Fq(\025)1562 2709 y Fn(0)1562 2769 y Fl(1)p Fr(;n)1664 2747 y Fp(\()p Fq(m)1779 2709 y Fr(e)1779 2769 y(n)1827 2747 y Fp(\)\(1)h Fo(\000)f Fq(m)2134 2709 y Fr(e)2134 2769 y(n)2181 2747 y Fp(\))g Fo(\000)g Fp(1\)\()p Fq(m)2522 2709 y Fl(1)p Fr(;e)2522 2769 y(n)2635 2747 y Fo(\000)g Fq(m)2806 2709 y Fr(e)2806 2769 y(n)2853 2747 y Fp(\)1)-25 b(I)2941 2761 y Fr(m)3023 2747 y Fp(+)433 2906 y(\()p Fq(e)510 2868 y Fr(\025)551 2877 y Fg(2)587 2868 y Fl(\()p Fr(m)676 2845 y FA(e)676 2885 y(n)719 2868 y Fl(\))p Fn(\000)p Fr(\025)842 2877 y Fg(1)p FA(;n)934 2868 y Fl(\()p Fr(m)1023 2845 y FA(e)1023 2885 y(n)1067 2868 y Fl(\))1118 2906 y Fo(\000)20 b Fp(1\)\(1)i Fo(\000)e Fq(m)1562 2868 y Fr(e)1562 2928 y(n)1609 2906 y Fp(\)\()p Fq(\025)1732 2868 y Fn(0)1732 2928 y Fl(1)p Fr(;n)1834 2906 y Fp(\()p Fq(m)1949 2868 y Fr(e)1949 2928 y(n)1997 2906 y Fp(\))p Fq(m)2112 2868 y Fr(e)2112 2928 y(n)2179 2906 y Fo(\000)g Fp(1\)\()p Fq(m)2465 2868 y Fl(1)p Fr(;e)2465 2928 y(n)2578 2906 y Fo(\000)g Fq(m)2749 2868 y Fr(e)2749 2928 y(n)2796 2906 y Fp(\)1)-25 b(I)2884 2920 y Fr(m)2951 2906 y Fq(:)3409 2824 y Fu(\(7.20\))118 3073 y(It)25 b(is)g(enough)e(to)h(show)h(that)g(both)g(addenda)f(in)g (\(7.20\))f(are)g(equivalent)h(to)g Fp(0)h Fu(and)f(for)g(simplicity)h (we)118 3181 y(deal)g(with)i(only)d(with)j(the)f(\002rst)g(one.)31 b(Since)24 b Fp(sup)1821 3203 y Fr(n)1883 3181 y Fo(k)p Fq(\025)1981 3148 y Fn(0)1981 3206 y Fl(1)p Fr(;n)2083 3181 y Fp(\()p Fq(m)2198 3148 y Fr(e)2198 3204 y(n)2245 3181 y Fp(\)1)-25 b(I)2334 3195 y Fr(m)2400 3181 y Fo(k)2445 3195 y Fn(1)2546 3181 y Fo(\024)25 b Fq(k)2689 3195 y Fr(m)2781 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Fk(\020)1445 3659 y(X)1426 3855 y Fr(i)p Fl(=1)p Fr(;)p Fl(2)1610 3672 y Fk(\000)1652 3746 y Fq(m)1732 3708 y Fr(e)1732 3768 y(n)1799 3746 y Fo(\000)20 b Fq(\026)1945 3708 y Fr(\025)1986 3717 y Fg(0)2020 3708 y Fl(\()p Fr(m)2109 3685 y FA(e)2109 3725 y(n)2153 3708 y Fl(\))2184 3746 y Fp(\()p Fq(m)2299 3708 y Fr(i;e)2299 3768 y(n)2380 3746 y Fp(\))2415 3672 y Fk(\001)2457 3695 y Fl(2)2517 3746 y Fp(+)35 b(\()p Fq(m)2738 3708 y Fl(1)p Fr(;e)2738 3768 y(n)2850 3746 y Fo(\000)20 b Fq(m)3021 3708 y Fr(e)3021 3768 y(n)3067 3746 y Fp(\))3102 3708 y Fl(2)3157 3645 y Fk(\021)3227 3746 y Fq(:)3409 3703 y Fu(\(7.21\))118 4008 y(The)26 b(claim)e(follows)h(by)h(applying)e(lemma)g(7.12)h(with)h Fq(h)2081 4022 y Fr(n)2154 4008 y Fu(equal)e(to)i(the)f(r)-10 b(.h.s.)33 b(of)25 b(\(7.21\).)118 4143 y(Let)j(us)g(prove)f(\(7.19\).) 39 b(By)27 b(Schwarz)g(inequality)-12 b(,)29 b(it)g(is)f(enough)f(to)h (apply)g(lemma)f(7.12)f(with)j Fq(h)3488 4157 y Fr(n)3565 4143 y Fp(:=)118 4260 y(\()p Fq(\025)206 4227 y Fn(0)206 4285 y Fr(i;n)297 4260 y Fp(\()p Fq(m)412 4227 y Fr(e)412 4282 y(n)459 4260 y Fp(\))21 b Fo(\000)f Fq(\025)659 4227 y Fn(0)659 4284 y Fl(0)699 4260 y Fp(\()p Fq(m)p Fp(\)\))884 4227 y Fl(2)924 4260 y Fp(1)-25 b(I)977 4274 y Fr(m)1064 4260 y Fp(+)20 b(\()p Fq(m)1270 4216 y Fr(i;e)1270 4272 y(n)1371 4260 y Fo(\000)g Fq(m)1542 4227 y Fr(e)1542 4282 y(n)1589 4260 y Fp(\))1624 4227 y Fl(2)1664 4260 y Fu(.)33 b(In)25 b(order)h(to)g(verify)f(condition)h Fq(ii)p Fp(\))h Fu(of)f(lemma)e(7.12)h(for)118 4382 y Fq(h)170 4396 y Fr(n)217 4382 y Fu(,)h(thanks)g(to)f(the)h(boundedness) f(of)g Fp(\()p Fq(\025)1546 4349 y Fn(0)1546 4408 y Fr(i;n)1637 4382 y Fp(\()p Fq(m)1752 4349 y Fr(e)1752 4405 y(n)1799 4382 y Fp(\))20 b Fo(\000)g Fq(\025)1998 4349 y Fn(0)1998 4407 y Fl(0)2037 4382 y Fp(\()p Fq(m)p Fp(\)\))2222 4349 y Fl(2)2262 4382 y Fp(1)-25 b(I)2315 4396 y Fr(m)2407 4382 y Fu(uniformly)24 b(in)h Fq(n)p Fu(,)g(we)g(only)g(need)f(to)118 4496 y(prove)h(that)990 4630 y Fp(lim)979 4691 y Fr(n)p Fn("1)1142 4630 y Fq(n)1197 4592 y Fr(d)1238 4630 y Fs(E)1292 4556 y Fk(\002)1336 4630 y Fq(\026)1391 4592 y Fr(\025)1432 4601 y Fg(0)1467 4592 y Fl(\()p Fr(m)p Fl(\))1588 4556 y Fk(\000)1630 4630 y Fp(\()p Fq(\025)1718 4592 y Fn(0)1718 4652 y Fr(i;n)1809 4630 y Fp(\()p Fq(m)1924 4592 y Fr(e)1924 4652 y(n)1971 4630 y Fp(\))20 b Fo(\000)g Fq(\025)2170 4592 y Fn(0)2170 4652 y Fl(0)2210 4630 y Fp(\()p Fq(m)p Fp(\)\))2395 4592 y Fl(4)2435 4630 y Fp(1)-25 b(I)2488 4644 y Fr(m)2555 4556 y Fk(\001)o(\003)2659 4630 y Fp(=)25 b(0)118 4825 y Fu(or)g(equivalently)333 5013 y Fp(lim)322 5074 y Fr(n)p Fn("1)486 5013 y Fq(n)541 4976 y Fr(d)581 5013 y Fs(E)635 4912 y Fk(h)684 5013 y Fq(\026)739 4976 y Fr(\025)780 4985 y Fg(0)815 4976 y Fl(\()p Fr(m)p Fl(\))936 4912 y Fk(\020)991 4940 y(\010)1044 5013 y Fp(Av)1161 5047 y Fr(x)p Fn(2)p Fl(\003)1297 5015 y FA(i;e)1297 5058 y(n)1376 4940 y Fk(\002)1429 5013 y Fq(\026)1484 4976 y Fr(\025)1525 4986 y FA(i;n)1608 4976 y Fl(\()p Fr(m)1697 4952 y FA(e)1697 4992 y(n)1740 4976 y Fl(\))1772 5013 y Fp(\()p Fq(\021)1852 5027 y Fr(x)1896 5013 y Fp(;)15 b Fq(\021)1981 5027 y Fr(x)2026 5013 y Fp(\))20 b Fo(\000)g Fs(E)26 b Fq(\026)2297 4976 y Fr(\025)2338 4985 y Fg(0)2379 4976 y Fl(\()p Fr(m)p Fl(\))2500 5013 y Fp(\()p Fq(\021)2580 5027 y Fl(0)2620 5013 y Fp(;)15 b Fq(\021)2705 5027 y Fl(0)2745 5013 y Fp(\))2795 4940 y Fk(\003)h(\011)2902 4962 y Fl(4)2941 4912 y Fk(\021i)3063 5013 y Fp(=)25 b(0)205 b Fu(\(7.22\))118 5239 y(Let)26 b Fq(g)309 5253 y Fr(x)353 5239 y Fp(\()p Fq(\025)p Fp(\))g(:=)f Fq(\026)678 5206 y Fr(\025)723 5239 y Fp(\()p Fq(\021)803 5253 y Fr(x)848 5239 y Fp(;)15 b Fq(\021)933 5253 y Fr(x)977 5239 y Fp(\))26 b Fu(and)f(observe)f(that)j(l.h.s.)32 b(of)26 b(\(7.22\))e(is)i(bounded)f(from)g(above)f(by)1130 5430 y Fq(c)42 b Fp(lim)1199 5492 y Fr(n)p Fn("1)1363 5430 y Fq(n)1418 5393 y Fr(d)1458 5430 y Fs(E)1513 5329 y Fk(h)1562 5430 y Fq(\026)1617 5393 y Fr(\025)1658 5402 y Fg(0)1692 5393 y Fl(\()p Fr(m)p Fl(\))1814 5357 y Fk(\000)1855 5430 y Fq(A)1923 5393 y Fl(\(1\))1923 5453 y Fr(n)2038 5430 y Fp(+)20 b Fq(A)2197 5393 y Fl(\(2\))2197 5453 y Fr(n)2312 5430 y Fp(+)g Fq(A)2471 5393 y Fl(\(3\))2471 5453 y Fr(n)2565 5357 y Fk(\001)2607 5329 y(i)3409 5430 y Fu(\(7.23\))p eop %%Page: 48 48 48 47 bop 118 45 a Fj(48)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(where)896 515 y Fq(A)964 478 y Fl(\(1\))964 538 y Fr(n)1084 515 y Fp(=)1180 442 y Fk(\010)1233 515 y Fp(Av)1350 550 y Fr(x)p Fn(2)p Fl(\003)1486 517 y FA(i;e)1486 560 y(n)1565 515 y Fp([)c Fq(g)1648 529 y Fr(x)1692 515 y Fp(\()p Fq(\025)1780 529 y Fr(i;n)1871 515 y Fp(\()p Fq(m)1986 478 y Fr(e)1986 538 y(n)2034 515 y Fp(\)\))20 b Fo(\000)g Fq(g)2258 529 y Fr(x)2303 515 y Fp(\()p Fq(\025)2391 529 y Fr(i;n)2482 515 y Fp(\()p Fq(m)p Fp(\)\))15 b(])2707 442 y Fk(\011)2761 464 y Fl(4)896 689 y Fq(A)964 652 y Fl(\(2\))964 712 y Fr(n)1084 689 y Fp(=)1180 616 y Fk(\010)1233 689 y Fp(Av)1350 723 y Fr(x)p Fn(2)p Fl(\003)1486 691 y FA(i;e)1486 734 y(n)1565 689 y Fp([)g Fq(g)1648 703 y Fr(x)1692 689 y Fp(\()p Fq(\025)1780 703 y Fr(i;n)1871 689 y Fp(\()p Fq(m)p Fp(\)\))21 b Fo(\000)f Fq(g)2211 703 y Fr(x)2256 689 y Fp(\()p Fq(\025)2344 703 y Fl(0)2383 689 y Fp(\()p Fq(m)p Fp(\)\))15 b(])2623 616 y Fk(\011)2678 638 y Fl(4)896 863 y Fq(A)964 825 y Fl(\(3\))964 885 y Fr(n)1084 863 y Fp(=)1180 789 y Fk(\010)1233 863 y Fp(Av)1350 897 y Fr(x)p Fn(2)p Fl(\003)1486 865 y FA(i;e)1486 908 y(n)1565 863 y Fp([)g Fq(g)1648 877 y Fr(x)1692 863 y Fp(\()p Fq(\025)1780 877 y Fl(0)1820 863 y Fp(\()p Fq(m)p Fp(\)\))21 b Fo(\000)f Fs(E)26 b Fq(\026)2242 825 y Fr(\025)2283 834 y Fg(0)2324 825 y Fl(\()p Fr(m)p Fl(\))2445 863 y Fp(\()p Fq(\021)2525 877 y Fl(0)2565 863 y Fp(;)15 b Fq(\021)2650 877 y Fl(0)2690 863 y Fp(\))g(])2765 789 y Fk(\011)2819 812 y Fl(4)2858 863 y Fq(:)118 1055 y Fu(By)27 b(lemma)f Fq(A:)p Fp(3)p Fu(,)j Fq(A)802 1007 y Fl(\(1\))802 1067 y Fr(n)925 1055 y Fo(\024)f Fq(c)15 b Fp(\()p Fq(m)1193 1022 y Fr(e)1193 1077 y(n)1262 1055 y Fo(\000)21 b Fq(m)p Fp(\))1469 1022 y Fl(4)1536 1055 y Fu(and)27 b Fq(A)1780 1007 y Fl(\(2\))1780 1067 y Fr(n)1903 1055 y Fo(\024)h Fq(c)p Fp(\()p Fq(m)22 b Fo(\000)f Fq(\026)2325 1022 y Fr(\025)2366 1031 y Fg(0)2401 1022 y Fl(\()p Fr(m)p Fl(\))2522 1055 y Fp(\()p Fq(m)2637 1010 y Fr(i;e)2637 1067 y(n)2718 1055 y Fp(\)\))2788 1022 y Fl(4)2828 1055 y Fu(.)37 b(A)n(t)28 b(this)h(point)f(\(7.23\))118 1163 y(follows)d(by)h(simple)f(considerations)h(for)f(sum)g(of)g(centered)f (independent)h(random)f(variables.)3591 1271 y Fe(\003)118 1439 y Fu(7.3.)45 b FE(The)f(subspace)f(orthogonal)i(to)f(the)f (\003uctuations.)i Fu(Here)40 b(we)i(introduce)f(a)h(convenient)118 1547 y(Hilbert)27 b(space)g Fo(H)h Fu(containing)f Fo(G)33 b Fu(and)27 b(we)g(describe)g(the)h(orthogonal)e(subspace)h(in)h Fo(H)g Fu(of)f(the)h(space)118 1655 y(of)e(\003uctuations)f Fo(fL)p Fq(g)44 b Fp(:)d Fq(g)28 b Fo(2)d Fs(G)19 b Fo(g)p Fu(.)118 1783 y FE(De\002nition)33 b(7.13.)46 b Fh(Let)33 b Fo(N)51 b Fp(:=)38 b Fo(f)p Fq(g)k Fo(2)c(G)58 b Fp(:)c Fq(V)20 b Fp(\()p Fq(g)s Fp(\))39 b(=)f(0)p Fo(g)c Fh(and)e(let)g Fo(H)i Fh(be)f(the)f(completion)g(of)g(the)h(pre-)118 1891 y(Hilbert)39 b(space)e Fo(G)5 b Fq(=)p Fo(N)13 b Fh(.)73 b(W)n(ith)40 b(an)e(abuse)g(of)g(notation,)j(we)d(write)h Fq(V)59 b Fh(for)38 b(the)h(scalar)f(product)g(in)h Fo(H)118 1999 y Fh(induced)24 b(by)h(the)g(semi-inner)f(product)h Fq(V)45 b Fh(in)25 b Fo(G)5 b Fh(.)218 2127 y Fu(The)25 b(sets)975 2238 y Fo(L)p Fs(G)50 b Fp(:=)25 b Fo(fL)p Fq(g)44 b Fp(:)d Fq(g)29 b Fo(2)c Fs(G)18 b Fo(g)q Fq(;)203 b Fo(LG)31 b Fp(:=)25 b Fo(fL)p Fq(g)44 b Fp(:)d Fq(g)28 b Fo(2)d(G)5 b(g)118 2368 y Fu(can)29 b(be)h(considered)g(as)g(subsets) h(of)f Fo(H)h Fu(in)f(a)g(natural)f(way)-12 b(.)46 b(Our)30 b(main)g(result)f(proves)h(that)h(for)f(any)118 2476 y Fq(e)35 b Fo(2)f(E)k Fu(the)31 b(sequence)e Fo(f)p Fq( )1033 2443 y Fr(e)1030 2499 y(n;n)1140 2476 y Fq(=n)p Fo(g)1285 2490 y Fr(n)p Fn(2)p Fm(N)1459 2476 y Fu(converges)f(in)j Fo(H)g Fu(to)g(some)g(limit)g(point)g Fq( )2957 2490 y Fr(e)3025 2476 y Fu(and)f(that)i(the)f(set)118 2599 y Fo(f)p Fq( )222 2613 y Fr(e)260 2599 y Fo(g)305 2613 y Fr(e)p Fn(2E)457 2599 y Fu(forms)25 b(a)h(basis)g(of)g Fo(L)p Fs(G)1229 2566 y Fn(?)1294 2599 y Fu(.)32 b(The)25 b(Cauchy)g(property)g(of)h(the)f(sequence)f Fo(f)p Fq( )2957 2566 y Fr(e)2954 2621 y(n;n)3064 2599 y Fq(=n)p Fo(g)3209 2613 y Fr(n)p Fn(2)p Fm(N)3377 2599 y Fu(follows)118 2710 y(by)g(a)h(telescopic)f(estimate)h(based)f(on)g(the)h(variance)f (bounds)g(discussed)i(in)e(subsection)h(6.1.)30 b(T)-9 b(o)25 b(this)118 2818 y(aim)h(the)f(following)g(lemma)e(is)k(crucial.) 118 2946 y FE(L)r(emma)g(7.14.)41 b Fh(Given)26 b Fq(k)i Fo(2)d Fs(N)38 b Fh(let)25 b Fq(f)34 b Fo(2)25 b(G)31 b Fh(be)25 b(such)g(that)h Fp(\001)2136 2961 y Fr(f)2206 2946 y Fo(\032)f Fp(\003)2365 2961 y Fr(k)2408 2946 y Fh(.)31 b(Then)1093 3107 y Fq(V)20 b Fp(\()p Fq(f)10 b Fp(\))25 b Fo(\024)g Fq(c)15 b(k)1516 3069 y Fr(d)p Fl(+2)1647 3107 y Fs(E)1702 3033 y Fk(\000)1750 3107 y Fp(V)-8 b(ar)1891 3135 y Fr(\026)1933 3113 y FA(\025)1970 3128 y Fg(0)2005 3113 y(\()p FA(m)p Fg(\))2116 3107 y Fp(\(Av)2269 3121 y Fr(x)p Fn(2)p Fl(\003)2405 3133 y FA(k)2447 3107 y Fq(\034)2487 3121 y Fr(x)2530 3107 y Fq(f)10 b Fp(\))2620 3033 y Fk(\001)2662 3107 y Fq(:)118 3278 y Fh(Proof)-8 b(.)42 b Fu(W)-8 b(e)28 b(\002rst)g(estimate)g Fq(V)1126 3293 y Fr(`)1159 3278 y Fp(\()p Fq(f)10 b Fp(\))27 b Fu(for)g Fq(`)i Fo(\035)g Fp(1)f Fu(by)g(means)e(of)i(lemma)e(6.1.)38 b(T)-9 b(o)27 b(this)i(aim)f(we)f(partition)118 3386 y(the)f(cube)e Fp(\003)544 3401 y Fr(`)573 3410 y Fg(1)637 3386 y Fu(into)i(non)e(overlapping)g(cubes)h Fo(f)p Fp(\003)1847 3401 y Fr(x)1887 3411 y FA(i)1913 3401 y Fr(;k)1976 3386 y Fo(g)2021 3400 y Fr(i)p Fn(2)p Fr(I)2157 3386 y Fu(of)h(side)g Fp(2)p Fq(k)e Fp(+)c(1)25 b Fu(and)h(write)1187 3540 y Fp(Av)1304 3554 y Fr(x)p Fn(2)p Fl(\003)1440 3566 y FA(`)1466 3581 y Fg(1)1509 3540 y Fq(\034)1549 3554 y Fr(x)1592 3540 y Fq(f)35 b Fp(=)25 b(Av)1885 3554 y Fr(i)p Fn(2)p Fr(I)1996 3467 y Fk(\000)2038 3540 y Fp(Av)2155 3554 y Fr(x)p Fn(2)p Fl(\003)2291 3566 y FA(x)2326 3582 y(i)2352 3566 y(;k)2413 3540 y Fq(\034)2453 3554 y Fr(x)2497 3540 y Fq(f)2552 3467 y Fk(\001)118 3698 y Fu(Therefore,)g(by)g (applying)g(lemma)f(6.1)g(with)j Fp(\003)e(=)g(\003)1938 3713 y Fr(`)1997 3698 y Fu(and)g Fp(\003)2234 3712 y Fr(i)2287 3698 y Fp(=)g(\003)2446 3713 y Fr(x)2486 3723 y FA(i)2513 3713 y Fr(;)p Fl(2)p Fr(k)2610 3698 y Fu(,)h(we)f(obtain) 998 3859 y Fq(V)1051 3874 y Fr(`)1084 3859 y Fp(\()p Fq(f)10 b Fp(\))25 b Fo(\024)g Fq(c)15 b(k)1434 3822 y Fr(d)p Fl(+2)1565 3859 y Fp(Av)1683 3873 y Fr(i)p Fn(2)p Fr(I)1794 3859 y Fp(V)-8 b(ar)1935 3887 y Fr(\026)1977 3865 y FA(\025)2014 3880 y Fg(0)2049 3865 y(\()p FA(m)p Fg(\))2160 3785 y Fk(\000)2202 3859 y Fp(Av)2319 3873 y Fr(x)p Fn(2)p Fl(\003)2455 3885 y FA(x)2490 3901 y(i)2516 3885 y(;k)2577 3859 y Fq(\034)2617 3873 y Fr(x)2661 3859 y Fq(f)2716 3785 y Fk(\001)2757 3859 y Fq(:)118 4017 y Fu(It)26 b(is)g(enough)e(now)h(to)h(take)f(the)h(expectation)f(w)-11 b(.r)h(.t.)32 b Fq(\013)26 b Fu(and)f(then)g(the)h(limit)g Fq(`)f Fo(")g(1)p Fu(.)558 b Fe(\003)218 4179 y Fu(Lemma)32 b(7.14)h(and)g(proposition)h(6.5)f(allow)g(us)g(to)h(prove)f(the)g(key) g(technical)g(estimate)g(of)h(this)118 4287 y(subsection:)118 4415 y FE(L)r(emma)27 b(7.15.)41 b Fh(Let)26 b Fq(d)f Fo(\025)g Fp(2)p Fh(,)h Fq(n)f Fo(\024)g Fq(s)g Fo(\024)g Fq(k)j Fo(\024)d Fp(100)p Fq(s)i Fh(be)d(positive)h(integers)f(and)g Fp(0)i Fq(<)f(\016)k Fo(\034)c Fp(1)p Fh(.)32 b(Then)1212 4568 y Fq(V)20 b Fp(\()p Fq(\036)1374 4531 y Fr(e)1374 4591 y(n;s)1494 4568 y Fo(\000)g Fq(\036)1639 4531 y Fr(e)1639 4591 y(n;k)1744 4568 y Fp(\))26 b Fo(\024)f Fq(c)15 b(s)1998 4531 y Fl(2)p Fn(\000)p Fr(d)p Fl(+)p Fr(\016)2308 4568 y Fo(8)p Fq(e)25 b Fo(2)g(E)849 b Fu(\(7.24\))118 4722 y Fh(for)25 b(any)f Fq(s)h Fh(large)g(enough)f(\()p Fq(s)h Fo(\025)g Fq(s)1252 4736 y Fl(0)1291 4722 y Fp(\()p Fq(\016)s Fp(\))p Fh(\).)118 4883 y(Proof)-8 b(.)42 b Fu(Since)24 b Fq(\036)673 4850 y Fr(e)673 4906 y(n;s)792 4883 y Fo(\000)c Fq(\036)937 4850 y Fr(e)937 4911 y(n;k)1068 4883 y Fo(2)25 b(G)30 b Fu(has)c(support)g(in)f Fp(\003)1890 4898 y Fr(k)1933 4883 y Fu(,)h(by)f(lemma)f(7.14)g(we)h(obtain)753 5069 y Fq(V)20 b Fp(\()p Fq(\036)915 5031 y Fr(e)915 5091 y(n;s)1035 5069 y Fo(\000)g Fq(\036)1180 5031 y Fr(e)1180 5092 y(n;k)1285 5069 y Fp(\))26 b Fo(\024)f Fq(c)15 b(k)1546 5031 y Fr(d)p Fl(+2)1716 4982 y Fk(X)1692 5180 y Fr(r)r Fl(=)p Fr(s;k)1887 5069 y Fs(E)1942 4995 y Fk(\000)1989 5069 y Fp(V)-8 b(ar)2131 5097 y Fr(\026)2173 5075 y FA(\025)2210 5090 y Fg(0)2245 5075 y(\()p FA(m)p Fg(\))2355 5069 y Fp(\(Av)2508 5083 y Fr(x)p Fn(2)p Fl(\003)2644 5095 y FA(k)2686 5069 y Fq(\034)2726 5083 y Fr(x)2770 5069 y Fq(\036)2824 5031 y Fr(e)2824 5091 y(n;r)2924 5069 y Fp(\))2959 4995 y Fk(\001)3001 5069 y Fq(:)118 5318 y Fu(The)26 b(thesis)g(now)f(follows)g(from)g(proposition)i(6.5.) 1698 b Fe(\003)218 5480 y Fu(W)-8 b(e)25 b(also)h(need)e(a)i(density)g (result.)p eop %%Page: 49 49 49 48 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(49)118 366 y FE(L)r(emma)27 b(7.16.)41 b Fo(LG)31 b Fh(and)24 b Fo(L)p Fs(G)50 b Fh(have)25 b(the)g(same)f(closure)g(in)h Fo(H)q Fh(.)118 538 y(Proof)-8 b(.)42 b Fu(W)-8 b(e)31 b(\002x)h Fq(g)39 b Fo(2)c Fs(G)56 b Fu(and)32 b(we)f(prove)f(that)j Fo(L)p Fq(g)39 b Fp(=)c(lim)2091 553 y Fr(s)p Fn("1)2249 538 y Fo(L)p Fp(\()p Fq(g)28 b Fo(\000)c Fq(g)2556 552 y Fr(s)2593 538 y Fp(\))32 b Fu(where)f Fq(g)2980 552 y Fr(s)3052 538 y Fp(=)36 b Fq(\026)p Fp([)p Fq(g)18 b Fo(j)d Fq(m)3420 552 y Fr(s)3457 538 y Fp(])p Fu(,)34 b(i.e.)118 646 y(that)f Fp(lim)435 661 y Fr(s)p Fn("1)592 646 y Fq(V)21 b Fp(\()p Fo(L)p Fq(g)807 660 y Fr(s)844 646 y Fp(\))37 b(=)f(0)p Fu(.)53 b(T)-9 b(o)33 b(this)g(aim)f(we)g (de\002ne)g Fq(X)2127 660 y Fr(s)2201 646 y Fp(:=)k Fo(f)p Fq(x)i Fp(:)e Fq(s)25 b Fo(\000)f Fp(1)38 b Fo(\024)e(j)p Fq(x)p Fo(j)h(\024)g Fq(s)24 b Fp(+)h(1)p Fo(g)p Fu(.)53 b(Then)118 754 y(lemma)24 b Fp(7)p Fq(:)p Fp(1)i Fu(implies)g(that)1093 951 y Fq(V)20 b Fp(\()p Fo(L)p Fq(g)1307 965 y Fr(s)1344 951 y Fp(\))26 b(=)1501 865 y Fk(X)1505 1061 y Fr(e)p Fn(2E)1657 889 y Fp(1)p 1657 930 46 4 v 1657 1013 a(2)1712 951 y Fs(E)1767 913 y Fn(\003)1813 877 y Fk(\000)1854 951 y Fq(c)1893 965 y Fl(0)p Fr(;e)1985 951 y Fp(\()2044 865 y Fk(X)2020 1061 y Fr(x)p Fn(2)p Fr(X)2165 1069 y FA(s)2214 951 y Fo(r)2290 965 y Fl(0)p Fr(;e)2381 951 y Fq(\034)2421 965 y Fr(x)2465 951 y Fq(g)2508 965 y Fr(s)2545 951 y Fp(\))2580 913 y Fl(2)2620 877 y Fk(\001)2662 951 y Fq(:)722 b Fu(\(7.25\))118 1233 y(Let)28 b Fp(^)-49 b Fq(g)307 1247 y Fr(s)345 1233 y Fp(\()p Fq(\013;)15 b(\021)s Fp(\))27 b(:=)e Fq(\026)764 1185 y Fr(\025)p Fl(\()p Fr(m)894 1193 y FA(s)928 1185 y Fl(\()p Fr(\021)r Fl(\)\))764 1262 y(\003)813 1270 y FA(s)1052 1233 y Fp(\()p Fq(g)s Fp(\))p Fq(:)h Fu(By)e(the)g(equivalence)e(of)j(ensembles)d 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y Fu(.)118 1774 y FE(Proposition)i(7.17.)41 b Fh(Let)26 b Fq(d)f Fo(\025)g Fp(3)h Fh(and)e Fq(e)i Fo(2)f(E)8 b Fh(.)31 b(Then)25 b(the)g(sequence)1282 1931 y Fq( )1344 1893 y Fr(e)1341 1953 y Fl(1)p Fr(;s)1458 1931 y Fp(=)g Fq(\021)1599 1945 y Fr(e)1656 1931 y Fo(\000)20 b Fq(\021)1792 1945 y Fl(0)1852 1931 y Fo(\000)g Fq(\026)p Fp([)p Fq(\021)2068 1945 y Fr(e)2125 1931 y Fo(\000)g Fq(\021)2261 1945 y Fl(0)2315 1931 y Fo(j)15 b Fq(m)2435 1893 y Fr(e)2435 1953 y(s)2473 1931 y Fp(])118 2102 y Fh(converges)23 b(to)i(some)g(element)e Fq( )1205 2116 y Fr(e)1268 2102 y Fo(2)i(L)p Fs(G)1481 2069 y Fn(?)1571 2102 y Fh(as)g Fq(s)g Fo(")h(1)p Fh(.)118 2210 y(Moreover)-12 b(,)1422 2378 y Fp(lim)1416 2439 y Fr(s)p Fn("1)1580 2307 y Fq( )1642 2274 y Fr(e)1639 2330 y(n;s)p 1580 2357 159 4 v 1632 2440 a Fq(n)1773 2378 y Fp(=)25 b Fq( )1928 2392 y Fr(e)2056 2378 y Fo(8)p Fq(n)g Fo(2)g Fs(N)6 b Fq(:)1052 b Fu(\(7.26\))118 2609 y Fh(Proof)-8 b(.)42 b Fu(W)-8 b(e)25 b(\002x)h Fp(0)f Fq(<)g(\016)k Fo(\034)c Fp(1)p Fu(.)32 b(By)26 b(lemma)e(7.15,)g(if)i Fq(i)f Fo(2)g Fs(N)38 b Fu(is)26 b(large)e(enough)g(and)i Fq(i)2903 2576 y Fl(3)2968 2609 y Fo(\024)f Fq(s)g Fo(\024)f Fp(\()p Fq(i)d Fp(+)f(1\))3485 2576 y Fl(3)3526 2609 y Fu(,)1322 2776 y Fq(V)g Fp(\()p Fq( )1492 2738 y Fr(e)1489 2804 y Fl(1)p Fr(;i)1568 2785 y Fg(3)1627 2776 y Fo(\000)g Fq( )1780 2738 y Fr(e)1777 2798 y Fl(1)p Fr(;s)1869 2776 y Fp(\))26 b Fo(\024)f Fq(c)15 b(i)2123 2738 y Fl(3\(2)p Fn(\000)p Fr(d)p Fl(+)p Fr(\016)r Fl(\))2433 2776 y Fq(:)118 2942 y Fu(Since)26 b Fq(d)j Fo(\025)f Fp(3)p Fu(,)g(it)g(is)g(enough)e (to)i(prove)e(that)j(the)e(sequence)f Fo(f)p Fq( )2319 2909 y Fr(e)2316 2974 y Fl(1)p Fr(;i)2395 2955 y Fg(3)2434 2942 y Fo(g)2479 2956 y Fr(i)p Fn(2)p Fm(N)2630 2942 y Fu(is)i(Cauchy)-12 b(.)37 b(This)28 b(follows)f(by)118 3063 y(applying)e(again)g(lemma)f(7.15)g(to)i(get)979 3177 y Fn(1)949 3204 y Fk(X)957 3400 y Fr(i)p Fl(=1)1095 3291 y Fq(V)1178 3226 y Fg(1)p 1178 3238 31 3 v 1178 3279 a(2)1223 3291 y Fp(\()p Fq( )1320 3253 y Fr(e)1317 3319 y Fl(1)p Fr(;i)1396 3300 y Fg(3)1455 3291 y Fo(\000)20 b Fq( )1608 3253 y Fr(e)1605 3319 y Fl(1)p Fr(;)p Fl(\()p Fr(i)p Fl(+1\))1828 3300 y Fg(3)1868 3291 y Fp(\))26 b Fo(\024)2055 3177 y Fn(1)2025 3204 y Fk(X)2033 3400 y Fr(i)p Fl(=1)2171 3291 y Fq(c)15 b(i)2279 3226 y Fg(3)p 2279 3238 V 2279 3279 a(2)2319 3253 y Fl(\(2)p Fn(\000)p Fr(d)p Fl(+)p Fr(\016)r Fl(\))2619 3291 y Fq(<)25 b Fo(1)p Fq(:)118 3545 y Fu(Next)37 b(we)g(prove)g(that)i Fq( )1009 3559 y Fr(e)1046 3545 y Fu(,)i(the)d(limit)g(point)g(of)g Fo(f)p Fq( )1964 3512 y Fr(e)1961 3569 y Fl(1)p Fr(;s)2053 3545 y Fo(g)2098 3559 y Fr(s)p Fn(2)p Fm(N)2230 3545 y Fu(,)j(belongs)c(to)h Fo(L)p Fs(G)2882 3512 y Fn(?)2947 3545 y Fu(.)68 b(T)-9 b(o)38 b(this)h(aim,)i(by)118 3658 y(lemmas)24 b(7.1)h(and)g(7.16,)g(we)g(need)f(to)i(show)g(that)1060 3825 y Fp(lim)1054 3887 y Fr(s)p Fn("1)1225 3739 y Fk(X)1207 3944 y Fr(x)p Fn(2)p Fm(Z)1344 3925 y FA(d)1390 3825 y Fs(E)10 b Fp([)16 b Fq(\026)1540 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4477 y Fr(e)1531 4533 y(s)1568 4510 y Fp(])p Fu(.)32 b(T)-9 b(o)26 b(this)h(aim)e(we)g(set)640 4688 y Fp(\001)716 4702 y Fr(s)777 4688 y Fp(:=)h Fo(f)p Fq(x)f Fo(2)g Fs(Z)1172 4650 y Fr(d)1249 4688 y Fp(:)40 b(\()p Fq(x)21 b Fp(+)f(\001)1589 4702 y Fr(g)1629 4688 y Fp(\))g Fo(\\)g Fp(\003)1828 4650 y Fr(e)1828 4710 y(s)1890 4688 y Fo(6)p Fp(=)25 b Fo(;)h Fu(and)f Fp(\()p Fq(x)c Fp(+)f(\001)2506 4702 y Fr(g)2546 4688 y Fp(\))g Fo(\\)g Fp(\(\003)2780 4650 y Fr(e)2780 4710 y(s)2817 4688 y Fp(\))2852 4650 y Fr(c)2913 4688 y Fo(6)p Fp(=)25 b Fo(;)15 b(g)p Fq(:)118 4844 y Fu(Since)24 b Fq(g)29 b Fo(2)c(G)5 b Fu(,)866 4890 y Fk(X)848 5095 y Fr(x)p Fn(2)p Fm(Z)985 5076 y FA(d)1030 4976 y Fs(E)11 b Fp([)k Fq(\026)1180 4939 y Fr(\025)1221 4948 y Fg(0)1262 4939 y Fl(\()p Fr(m)p Fl(\))1383 4976 y Fp(\()p Fq(\036)1472 4939 y Fr(e)1472 4999 y Fl(1)p Fr(;s)1564 4976 y Fq(;)g(\034)1644 4990 y Fr(x)1688 4976 y Fq(g)s Fp(\))g(])27 b(=)e Fs(E)11 b Fp([)k Fq(\026)2082 4939 y Fr(\025)2123 4948 y Fg(0)2163 4939 y Fl(\()p Fr(m)p Fl(\))2285 4976 y Fp(\()p Fq(\036)2374 4939 y Fr(e)2374 4999 y Fl(1)p Fr(;s)2466 4976 y Fq(;)2530 4890 y Fk(X)2506 5087 y Fr(x)p Fn(2)p Fl(\001)2652 5095 y FA(s)2701 4976 y Fq(\034)2741 4990 y Fr(x)2784 4976 y Fq(g)s Fp(\))g(])p Fq(:)479 b Fu(\(7.27\))118 5213 y(W)-8 b(e)26 b(estimate)g(the)f(r)-10 b(.h.s.)33 b(of)25 b(\(7.27\))f(by)h(Schwarz)f(inequality)-12 b(.)32 b(Let)26 b(us)f(observe)f(that)1198 5386 y Fs(E)h Fp([)q(V)-8 b(a)q(r)1440 5414 y Fr(\026)1482 5392 y FA(\025)1519 5407 y Fg(0)1554 5392 y(\()p FA(m)p Fg(\))1665 5312 y Fk(\000)1730 5299 y(X)1706 5496 y Fr(x)p Fn(2)p Fl(\001)1852 5504 y FA(s)1900 5386 y Fq(\034)1940 5400 y Fr(x)1984 5386 y Fq(g)2030 5312 y Fk(\001)2087 5386 y Fp(])26 b Fo(\024)f Fq(c)2273 5400 y Fr(g)2329 5386 y Fq(s)2372 5348 y Fn(\000)p Fr(d)p Fl(+1)2557 5386 y Fq(:)827 b Fu(\(7.28\))p eop %%Page: 50 50 50 49 bop 118 45 a Fj(50)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(for)k(some)g (\002nite)h(constant)f Fq(c)1110 380 y Fr(g)1151 366 y Fu(.)30 b(Therefore,)23 b(in)h(order)f(to)g(conclude)f(the)h(proof)m (,)h(it)g(is)g(enough)e(to)i(show)1358 529 y Fs(E)11 b Fp([)k(V)-7 b(ar)1601 558 y Fr(\026)1643 535 y FA(\025)1680 550 y Fg(0)1715 535 y(\()p FA(m)p Fg(\))1825 529 y Fp(\()p Fq(\036)1914 492 y Fr(e)1914 552 y Fl(1)p Fr(;s)2007 529 y Fp(\))15 b(])26 b Fo(\024)f Fq(c)15 b(s)2301 492 y Fn(\000)p Fr(d)2396 529 y Fq(:)988 b Fu(\(7.29\))118 691 y(By)40 b(the)h(equivalence)d(of)j(ensemble)d(\(see)i(lemma)f (A.5\))h(and)g(P)-5 b(oincar\264)-45 b(e)40 b(inequality)g(for)h (Glauber)118 799 y(dynamics,)26 b(we)f(obtain)271 959 y Fp(V)-8 b(ar)413 987 y Fr(\026)455 965 y FA(\025)492 980 y Fg(0)527 965 y(\()p FA(m)p Fg(\))638 959 y Fp(\()p Fq(\036)727 921 y Fr(e)727 981 y Fl(1)p Fr(;s)819 959 y Fp(\))25 b Fo(\024)g Fq(c)15 b(s)1072 921 y Fn(\000)p Fl(2)p Fr(d)1223 959 y Fp(+)20 b Fq(c)15 b Fp(V)-8 b(ar)1510 987 y Fr(\026)1552 965 y FA(\025)1589 980 y Fg(0)1624 965 y(\()p FA(m)p Fg(\))1735 959 y Fp(\()1782 935 y(^)1770 959 y Fq(\036)1824 921 y Fr(e)1824 981 y Fl(1)p Fr(;s)1916 959 y Fp(\))26 b Fo(\024)f Fq(c)15 b(s)2170 921 y Fn(\000)p Fl(2)p Fr(d)2320 959 y Fp(+)20 b Fq(c)15 b(s)2508 921 y Fr(d)2549 959 y Fq(\026)2604 921 y Fr(\025)2645 930 y Fg(0)2679 921 y Fl(\()p Fr(m)p Fl(\))2801 885 y Fk(\000)2843 959 y Fp(\()p Fo(r)2954 973 y Fl(0)3005 935 y Fp(^)2993 959 y Fq(\036)3047 921 y Fr(e)3047 981 y Fl(1)p Fr(;s)3139 959 y Fp(\))3174 921 y Fl(2)3214 885 y Fk(\001)3409 959 y Fu(\(7.30\))118 1140 y(where)401 1116 y Fp(^)389 1140 y Fq(\036)443 1107 y Fr(e)443 1164 y Fl(1)p Fr(;s)560 1140 y Fu(has)26 b(been)e(de\002ned)h(in)g(\(6.3\).)118 1265 y(By)g(lemma)f(A.3)h(the)h(last)g(term)e(in)i(\(7.30\))e(is)i (bounded)f(by)g Fq(c)15 b(s)2282 1232 y Fn(\000)p Fr(d)2403 1265 y Fu(thus)26 b(proving)e(\(7.29\).)118 1373 y(Finally)h(we)g (prove)f(\(7.26\).)31 b(T)-9 b(o)25 b(this)i(aim,)f(by)f(writing)522 1611 y Fq( )584 1573 y Fr(e)581 1633 y(n;s)706 1611 y Fp(=)816 1549 y(1)p 811 1590 55 4 v 811 1673 a Fq(n)891 1497 y Fr(n)p Fn(\000)p Fl(1)892 1524 y Fk(X)894 1719 y Fr(v)r Fl(=0)1039 1611 y Fp(Av)1157 1644 y Fr(x)p Fn(2)p Fl(\003)1293 1612 y Fg(1)p FA(;e)1293 1655 y(n)1379 1537 y Fk(\000)1421 1611 y Fq(\021)1466 1629 y Fr(x)p Fl(+\()p Fr(v)r Fl(+1\))p Fr(e)1800 1611 y Fo(\000)20 b Fq(\021)1936 1625 y Fr(x)p Fl(+)p Fr(v)r(e)2124 1611 y Fo(\000)g Fq(\026)p Fp([)p Fq(\021)2340 1629 y Fr(x)p Fl(+\()p Fr(v)r Fl(+1\))p Fr(e)2674 1611 y Fo(\000)g Fq(\021)2810 1625 y Fr(x)p Fl(+)p Fr(v)r(e)2993 1611 y Fo(j)15 b Fq(m)3113 1573 y Fr(e)3113 1633 y(s)3150 1611 y Fp(])3190 1537 y Fk(\001)3232 1611 y Fq(;)118 1860 y Fu(and)26 b(by)h(the)f(observation)h(that)g Fq(\034)1278 1874 y Fr(x)1322 1860 y Fq(f)36 b Fp(=)26 b Fq(f)36 b Fu(for)26 b(any)g Fq(f)36 b Fo(2)27 b(H)f Fu(and)h Fq(x)g Fo(2)f Fs(Z)2568 1827 y Fr(d)2604 1860 y Fu(,)h(it)g(is)g(enough)e(to)i(prove)e(that)118 1968 y(for)g(any)g(given)g Fq(x)g Fo(2)g Fs(Z)892 1935 y Fr(d)1177 2082 y Fq(V)c Fp(\()p Fq(\026)p Fp([)p Fq(\021)1411 2096 y Fr(e)1468 2082 y Fo(\000)f Fq(\021)1604 2096 y Fl(0)1659 2082 y Fo(j)15 b Fq(m)1779 2045 y Fr(e)1779 2105 y(x;s)1875 2082 y Fp(])21 b Fo(\000)f Fq(\026)p Fp([)p Fq(\021)2137 2096 y Fr(e)2194 2082 y Fo(\000)g Fq(\021)2330 2096 y Fl(0)2384 2082 y Fo(j)15 b Fq(m)2504 2045 y Fr(e)2504 2105 y(s)2542 2082 y Fp(]\))807 b Fu(\(7.31\))118 2219 y(goes)36 b(to)h Fp(0)g Fu(as)g Fq(s)44 b Fo(")h(1)p Fu(.)65 b(As)37 b(in)g(the)g(proof)f(of)h(lemma)e(7.15)h(\(7.31\))f(is) j(bounded)e(from)g(above)g(by)118 2327 y Fq(c)p Fp(\()p Fq(\016)s Fp(\))p Fq(s)313 2294 y Fl(2)p Fn(\000)p Fr(d)p Fl(+)p Fr(\016)559 2327 y Fu(for)25 b(any)g Fp(0)h Fq(<)f(\016)k Fo(\034)c Fp(1)p Fu(.)2304 b Fe(\003)218 2517 y Fu(W)-8 b(e)25 b(are)g(now)g(able)f(to)i(exhibit)g(a)f(basis)h(of)g Fo(L)p Fs(G)1823 2484 y Fn(?)1913 2517 y Fu(related)f(to)g(the)h (functions)2873 2462 y Fr( )2921 2439 y FA(e)2919 2479 y(n;n)p 2873 2496 146 4 v 2924 2548 a Fr(n)3054 2517 y Fu(with)g Fq(n)f Fo(2)g Fs(N)37 b Fu(and)118 2625 y Fq(e)26 b Fo(2)f(E)8 b Fu(.)118 2755 y FE(Theorem)27 b(7.18.)41 b Fh(Let)26 b Fq(d)f Fo(\025)g Fp(3)p Fh(.)32 b(Then)1441 2954 y Fp(lim)1429 3015 y Fr(n)p Fn("1)1603 2883 y Fq( )1665 2850 y Fr(e)1662 2906 y(n;n)p 1603 2933 169 4 v 1660 3016 a Fq(n)1807 2954 y Fp(=)25 b Fq( )1962 2968 y Fr(e)2090 2954 y Fo(8)p Fq(e)g Fo(2)g(E)1067 b Fu(\(7.32\))118 3157 y Fh(where)25 b Fq( )431 3171 y Fr(e)493 3157 y Fh(is)g(as)g(in)f(proposition)g(7.17.)31 b(Moreover)-12 b(,)1170 3309 y Fq(V)20 b Fp(\()p Fq(j)1315 3324 y Fl(0)p Fr(;e)1403 3306 y Fi(0)1430 3309 y Fq(;)15 b( )1529 3323 y Fr(e)1566 3309 y Fp(\))26 b(=)f Fo(\000)p Fq(\037)p Fp(\()p Fq(m)p Fp(\))p Fq(\016)2041 3324 y Fr(e)2074 3306 y Fi(0)2096 3324 y Fr(;e)2244 3309 y Fo(8)p Fq(e;)15 b(e)2419 3272 y Fn(0)2468 3309 y Fo(2)25 b(E)807 b Fu(\(7.33\))118 3469 y Fh(and)24 b Fo(f)p Fq( )393 3483 y Fr(e)431 3469 y Fo(g)476 3483 y Fr(e)p Fn(2E)628 3469 y Fh(forms)h(a)g(basis)f(of)h Fo(L)p Fs(G)1380 3436 y Fn(?)1445 3469 y Fh(.)118 3636 y(Proof)-8 b(.)42 b Fu(F)-6 b(or)28 b(any)g Fq(n)j Fo(2)f Fs(N)41 b Fu(let)29 b Fq(k)1155 3650 y Fr(n)1233 3636 y Fo(2)h Fs(N)42 b Fu(be)28 b(such)g(that)i Fp(\()p Fq(k)2018 3650 y Fr(n)2088 3636 y Fo(\000)22 b Fp(1\))2261 3603 y Fl(3)2332 3636 y Fq(<)31 b(n)f Fo(\024)h Fq(k)2671 3603 y Fl(3)2668 3658 y Fr(n)2715 3636 y Fu(.)42 b(Then,)29 b(by)g(lemma)e(7.15,)118 3744 y Fq(V)20 b Fp(\()p Fq( )288 3711 y Fr(e)285 3766 y(n;n)395 3744 y Fq(=n)h Fo(\000)e Fq( )668 3711 y Fr(e)665 3776 y(n;k)767 3757 y Fg(3)765 3793 y FA(n)811 3744 y Fq(=n)p Fp(\))26 b Fo(#)g Fp(0)g Fu(as)f Fq(n)g Fo(")h(1)p Fu(.)31 b(Therefore,)25 b(thanks)h(to)f(\(7.26\),)265 3997 y Fp(lim)254 4058 y Fr(n)p Fn("1)418 3997 y Fq(V)501 3932 y Fg(1)p 501 3944 31 3 v 501 3985 a(2)546 3997 y Fp(\()p Fq( )640 4011 y Fr(e)697 3997 y Fo(\000)798 3926 y Fq( )860 3893 y Fr(e)857 3949 y(n;n)p 798 3976 169 4 v 855 4059 a Fq(n)977 3997 y Fp(\))h(=)36 b(lim)1134 4058 y Fr(n)p Fn("1)1297 3997 y Fq(V)1381 3932 y Fg(1)p 1381 3944 31 3 v 1381 3985 a(2)1425 3997 y Fp(\()p Fq( )1519 4011 y Fr(e)1577 3997 y Fo(\000)1678 3921 y Fq( )1740 3888 y Fr(e)1737 3949 y(n;k)1837 3957 y FA(n)p 1678 3976 206 4 v 1753 4059 a Fq(n)1893 3997 y Fp(\))25 b Fo(\024)36 b Fp(lim)2049 4058 y Fr(n)p Fn("1)2228 3935 y Fp(1)p 2223 3976 55 4 v 2223 4059 a Fq(n)2333 3883 y Fn(1)2303 3911 y Fk(X)2311 4106 y Fr(i)p Fl(=1)2449 3997 y Fq(V)2532 3932 y Fg(1)p 2532 3944 31 3 v 2532 3985 a(2)2577 3997 y Fp(\()p Fq( )2674 3959 y Fr(e)2671 4025 y(n;i)2758 4006 y Fg(3)2817 3997 y Fo(\000)20 b Fq( )2970 3959 y Fr(e)2967 4025 y(n;)p Fl(\()p Fr(i)p Fl(+1\))3198 4006 y Fg(3)3237 3997 y Fp(\))137 b Fu(\(7.34\))118 4235 y(and)25 b(the)h(last)g(series)f(is)h(converging)d(by)i(lemma)f(7.15.)31 b(Thus)26 b(\(7.32\))e(follows.)118 4343 y(A)n(t)30 b(this)g(point,)h (\(7.33\))d(follows)h(from)f(theorem)h(7.10.)41 b(Let)30 b(us)f(prove)f(that)i Fo(f)p Fq( )2899 4357 y Fr(e)2937 4343 y Fo(g)2982 4357 y Fr(e)p Fn(2E)3138 4343 y Fu(forms)f(a)g(basis) 118 4453 y(of)c Fo(L)p Fs(G)348 4420 y Fn(?)413 4453 y Fu(.)32 b(Let)25 b Fq(P)38 b Fu(be)24 b(the)h(orthogonal)f (projection)g(of)h Fo(H)h Fu(on)e Fo(L)p Fs(G)2323 4420 y Fn(?)2388 4453 y Fu(.)32 b(Then,)25 b Fo(L)p Fs(G)2824 4420 y Fn(?)2914 4453 y Fu(has)g(dimension)g(non)118 4561 y(larger)j(than)i Fq(d)g Fu(since,)g(by)g(theorem)f(7.2,)h(it)h (is)f(generated)e(by)i Fo(f)p Fq(P)13 b(j)2467 4575 y Fl(0)p Fr(;e)2559 4561 y Fo(g)2604 4575 y Fr(e)p Fn(2E)2732 4561 y Fu(.)44 b(By)30 b(\(7.33\))e Fo(f)p Fq( )3317 4575 y Fr(e)3355 4561 y Fo(g)3400 4575 y Fn(2E)3524 4561 y Fu(is)i(a)118 4675 y(set)c(of)f Fq(d)h Fu(independent)f(vectors)f (belonging)g(to)h Fo(L)p Fs(G)1909 4642 y Fn(?)1999 4675 y Fu(and)g(therefore)g(a)g(basis)i(of)e Fo(L)p Fs(G)3083 4642 y Fn(?)3148 4675 y Fu(.)418 b Fe(\003)118 4841 y FE(R)m(emark)28 b(7.19.)42 b Fh(Let)27 b(us)f(make)f(an)h(observation)f (which)h(will)g(reveal)g(useful)g(in)g(the)g(proof)f(of)h(the)g(conti-) 118 4949 y(nuity)g(of)e(the)h(diffusion)f(matrix)i Fq(D)s Fp(\()p Fq(m)p Fp(\))f Fh(\(see)f(next)h(subsection\).)118 5057 y(Since)h(the)i(constant)e Fq(c)i Fh(appering)e(in)h(\(7.24\))g (does)f(not)h(depend)e(on)i(the)g(density)g Fq(m)g Fh(and)f(thanks)h (to)h(the)118 5165 y(estimate)d(\(7.34\),)f(the)h(statement)g(\(7.32\)) f(in)h(the)g(above)g(theorem)f(can)h(be)f(strengthed)g(as)1113 5372 y Fp(lim)1102 5433 y Fr(n)p Fn("1)1325 5372 y Fp(sup)1266 5455 y Fr(m)p Fn(2)p Fl(\(0)p Fr(;)p Fl(1\))1536 5372 y Fq(V)1589 5386 y Fr(m)1655 5298 y Fk(\000)1707 5301 y Fq( )1769 5268 y Fr(e)1766 5324 y(n;n)p 1707 5351 169 4 v 1764 5434 a Fq(n)1906 5372 y Fo(\000)c Fq( )2056 5386 y Fr(e)2093 5298 y Fk(\001)2160 5372 y Fp(=)25 b(0)91 b Fo(8)p Fq(e)25 b Fo(2)g(E)8 b Fq(:)p eop %%Page: 51 51 51 50 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(51)118 366 y Fu(7.4.)45 b FE(Decomposition)36 b(of)h(currents.)45 b Fu(In)35 b(this)h(subsection)f(we)g(prove)f(the)h(characterization)f (and)118 474 y(the)25 b(regularity)e(of)i(the)g(diffusion)h(matrix)e Fq(D)s Fp(\()p Fq(m)p Fp(\))h Fu(stated)h(in)e(theorem)g(2.1)g(and)g (we)h(prove)f(also)g(theo-)118 582 y(rem)f(7.22,)g(which)i(is)g (crucial)e(for)g(the)i(estimate)f(of)g Fp(\012)1975 596 y Fl(0)2038 582 y Fu(\(see)g(subsection)g(4.3\).)30 b(In)24 b(what)h(follows,)g(we)118 690 y(assume)g Fq(d)h Fo(\025)f Fp(3)p Fu(.)118 825 y(Denoting)35 b(by)h Fq(P)50 b Fu(the)36 b(orthogonal)f(projection)h(of)g Fo(H)h Fu(on)e Fo(L)p Fs(G)2315 792 y Fn(?)2380 825 y Fu(,)k(thanks)e(to)f(theorem)f(7.18,)j (for)e(a)118 933 y(suitable)26 b Fq(d)20 b Fo(\002)g Fq(d)26 b Fu(matrix)f Fq(D)s Fp(\()p Fq(m)p Fp(\))h Fu(we)f(can)f (write)875 1114 y Fq(j)912 1128 y Fl(0)p Fr(;e)1030 1114 y Fp(=)h Fo(\000)1219 1028 y Fk(X)1212 1226 y Fr(e)1245 1207 y Fi(0)1267 1226 y Fn(2E)1372 1114 y Fq(D)1447 1130 y Fr(e;e)1533 1111 y Fi(0)1559 1114 y Fp(\()p Fq(m)p Fp(\))p Fq( )1768 1130 y Fr(e)1801 1111 y Fi(0)1848 1114 y Fp(+)20 b(\(1)h Fo(\000)f Fq(P)13 b Fp(\)\()p Fq(j)2309 1128 y Fl(0)p Fr(;e)2402 1114 y Fp(\))182 b Fo(8)p Fq(e)25 b Fo(2)g(E)8 b Fq(:)505 b Fu(\(7.35\))118 1369 y(By)32 b(taking)f(the)h(scalar)f(product)h(of)g(both)g(sides)h(of)e(\(7.35\))g (with)i Fq(j)2475 1385 y Fl(0)p Fr(;e)2563 1366 y Fi(0)2589 1369 y Fu(,)h(thanks)e(to)g(lemma)e(7.1)h(and)118 1477 y(\(7.33\),)24 b(we)h(obtain)1220 1638 y Fq(D)1295 1653 y Fr(e;e)1381 1634 y Fi(0)1407 1638 y Fp(\()p Fq(m)p Fp(\))h(=)1770 1576 y(1)p 1689 1617 208 4 v 1689 1700 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1906 1638 y Fq(V)1959 1652 y Fr(m)2026 1638 y Fp(\()p Fq(P)13 b(j)2169 1652 y Fl(0)p Fr(;e)2261 1638 y Fq(;)i(P)e(j)2409 1653 y Fl(0)p Fr(;e)2497 1634 y Fi(0)2524 1638 y Fp(\))118 1835 y Fu(thus)32 b(proving)e(that)i Fq(D)s Fp(\()p Fq(m)p Fp(\))g Fu(is)g(a)f (non-negative)e(symmetric)i(matrix.)49 b(In)31 b(particular)-10 b(,)32 b Fq(D)s Fp(\()p Fq(m)p Fp(\))f Fu(can)g(be)118 1943 y(characterized)24 b(as)i(the)g(unique)e(symmetric)h Fq(d)c Fo(\002)e Fq(d)26 b Fu(matrix)f(such)g(that)929 2146 y Fp(\()p Fq(a;)15 b(D)s Fp(\()p Fq(m)p Fp(\))p Fq(a)p Fp(\))27 b(=)1577 2084 y(1)p 1496 2125 V 1496 2208 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1713 2146 y Fq(V)1766 2160 y Fr(m)1833 2072 y Fk(\000)1874 2146 y Fq(P)13 b Fp(\()1980 2060 y Fk(X)1984 2256 y Fr(e)p Fn(2E)2127 2146 y Fq(a)2175 2160 y Fr(e)2212 2146 y Fq(j)2249 2160 y Fl(0)p Fr(;e)2341 2146 y Fp(\))2376 2072 y Fk(\001)2509 2146 y Fo(8)p Fq(a)25 b Fo(2)g Fs(R)2779 2108 y Fr(d)2825 2146 y Fq(:)559 b Fu(\(7.36\))118 2400 y(Since)24 b(the)i(r)-10 b(.h.s.)32 b(of)26 b(\(7.36\))e(can)g(be)h(written)h(as)1309 2595 y Fp(inf)1297 2656 y Fr(g)r Fn(2)p Fm(G)1537 2534 y Fp(1)p 1456 2574 V 1456 2657 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1674 2595 y Fq(V)1727 2609 y Fr(m)1793 2521 y Fk(\000)1835 2509 y(X)1839 2706 y Fr(e)p Fn(2E)1981 2595 y Fq(a)2029 2609 y Fr(e)2066 2595 y Fq(j)2103 2609 y Fl(0)p Fr(;e)2216 2595 y Fo(\000)20 b(L)p Fq(g)2416 2521 y Fk(\001)2458 2595 y Fq(;)118 2850 y Fu(by)25 b(lemma)f(7.1)h(the)g(matrix)h Fq(D)s Fp(\()p Fq(m)p Fp(\))f Fu(corresponds)g(to)h(the)f(one)g (described)g(in)g(proposition)i(2.1.)118 2985 y(In)e(the)h(following)e (lemmas)h(we)g(describe)g(some)g(properties)g(of)g(the)h(diffusion)h (matrix)e Fq(D)s Fp(\()p Fq(m)p Fp(\))p Fu(.)118 3120 y FE(L)r(emma)i(7.20.)41 b Fh(There)25 b(exists)g Fq(c)h(>)f Fp(0)h Fh(such)f(that)g Fq(c)p Fp(1)-25 b(I)26 b Fo(\024)f Fq(D)s Fp(\()p Fq(m)p Fp(\))g Fo(\024)g Fq(c)2390 3087 y Fn(\000)p Fl(1)2485 3120 y Fp(1)-25 b(I)25 b Fh(for)g(any)g Fq(m)g Fo(2)g Fp(\(0)p Fq(;)15 b Fp(1\))p Fh(.)118 3302 y(Proof)-8 b(.)42 b Fu(Given)34 b Fq(a)41 b Fo(2)g Fs(R)895 3269 y Fr(d)977 3302 y Fu(we)34 b(set)h Fq(w)44 b Fp(:=)1518 3234 y Fk(P)1613 3329 y Fr(e)p Fn(2E)1756 3302 y Fq(a)1804 3316 y Fr(e)1841 3302 y Fq( )1900 3316 y Fr(e)1971 3302 y Fu(and)35 b Fq(v)44 b Fp(:=)2380 3234 y Fk(P)2476 3329 y Fr(e)p Fn(2E)2618 3302 y Fq(a)2666 3316 y Fr(e)2703 3302 y Fq(P)13 b(j)2811 3316 y Fl(0)p Fr(;e)2904 3302 y Fu(.)59 b(Then)35 b(\(7.36\))e(and)118 3410 y(lemma)24 b(7.1)h(imply)g(the)g(upper)g(bound)696 3613 y Fp(\()p Fq(a;)15 b(D)s Fp(\()p Fq(m)p Fp(\))p Fq(a)p Fp(\))26 b(=)1343 3551 y(1)p 1262 3592 V 1262 3675 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1480 3613 y Fq(V)1533 3627 y Fr(m)1599 3613 y Fp(\()p Fq(v)s(;)15 b(v)s Fp(\))27 b Fo(\024)2017 3551 y Fp(1)p 1936 3592 V 1936 3675 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))2154 3613 y Fq(V)2207 3627 y Fr(m)2273 3613 y Fp(\()2308 3527 y Fk(X)2312 3723 y Fr(e)p Fn(2E)2455 3613 y Fq(a)2503 3627 y Fr(e)2540 3613 y Fq(j)2577 3627 y Fl(0)p Fr(;e)2669 3613 y Fp(\))f Fo(\024)f Fq(c)15 b Fo(k)p Fq(a)p Fo(k)3018 3575 y Fl(2)3059 3613 y Fq(:)118 3867 y Fu(In)92 b(order)e(to)i(prove)f(the)h(lower)e(bound)h(we)g (observe)g(that,)109 b(by)92 b(theorem)e(7.18,)118 3975 y Fq(V)171 3989 y Fr(m)238 3975 y Fp(\()p Fq(v)s(;)15 b(w)r Fp(\))27 b(=)e Fo(\000)p Fq(\037)p Fp(\()p Fq(m)p Fp(\))p Fo(k)p Fq(a)p Fo(k)1001 3942 y Fl(2)1079 3975 y Fu(while,)41 b(thanks)e(to)f(\(4.33\),)h Fq(V)2151 3989 y Fr(m)2218 3975 y Fp(\()p Fq(w)r Fp(\))47 b Fo(\024)f Fq(c)15 b(m)p Fp(\(1)30 b Fo(\000)e Fq(m)p Fp(\))p Fo(k)p Fq(a)p Fo(k)3115 3942 y Fl(2)3155 3975 y Fu(.)69 b(Therefore,)118 4083 y(by)25 b(Schwarz)f(inequality)-12 b(,)1118 4303 y Fp(\()p Fq(a;)15 b(D)s Fp(\()p Fq(m)p Fp(\))p Fq(a)p Fp(\))27 b Fo(\025)1766 4241 y Fp(1)p 1685 4282 V 1685 4365 a Fq(\037)p Fp(\()p Fq(m)p Fp(\))1912 4241 y Fq(V)1965 4255 y Fr(m)2032 4241 y Fp(\()p Fq(v)s(;)15 b(w)r Fp(\))2256 4208 y Fl(2)p 1912 4282 386 4 v 1976 4365 a Fq(V)2029 4379 y Fr(m)2095 4365 y Fp(\()p Fq(w)r Fp(\))2332 4303 y Fo(\025)25 b Fq(c)15 b Fo(k)p Fq(a)p Fo(k)2620 4265 y Fl(2)118 4518 y Fu(thus)26 b(proving)f(the)g(lemma.)2502 b Fe(\003)118 4697 y FE(L)r(emma)27 b(7.21.)41 b Fq(D)s Fp(\()p Fq(m)p Fp(\))26 b Fh(is)f(a)g(continuous)f(function)g(on)g Fp(\(0)p Fq(;)15 b Fp(1\))p Fh(.)118 4901 y(Proof)-8 b(.)42 b Fu(Let)28 b Fp(0)k Fq(<)e(\014)k Fu(and)29 b Fp(0)i Fq(<)g(\016)j(<)1336 4865 y Fl(1)p 1336 4880 36 4 v 1336 4932 a(2)1381 4901 y Fu(.)41 b(W)-8 b(e)30 b(observe)d(that)j (the)f(limit)g(point)g Fq( )2777 4915 y Fr(e)2843 4901 y Fu(of)g(the)g(sequence)3506 4846 y Fr( )3554 4823 y FA(e)3552 4863 y(n;n)p 3506 4880 146 4 v 3557 4932 a Fr(n)118 5009 y Fu(depends)d(on)g(the)g(closure)f(of)h Fo(G)5 b Fq(=)p Fo(N)40 b Fu(and)27 b(therefore)e(on)h Fq(m)p Fu(.)33 b(Therefore,)26 b(it)h(is)g(convenient)d(to)j(denote)118 5138 y(it)33 b(by)f Fq( )394 5090 y Fl(\()p Fr(m)p Fl(\))391 5149 y Fr(e)516 5138 y Fu(.)51 b(Moreover)-10 b(,)32 b(thanks)g(to)g(remark)f(7.19)g(and)h(lemma)e(7.20,)j(there)e(exists)i Fq(n)3212 5152 y Fl(0)3287 5138 y Fo(2)j Fs(N)45 b Fu(such)118 5245 y(that)785 5405 y Fo(k)p Fq(D)s Fo(k)953 5419 y Fn(1)1103 5405 y Fp(sup)1044 5488 y Fr(m)p Fn(2)p Fl(\(0)p Fr(;)p Fl(1\))1314 5405 y Fq(V)1367 5419 y Fr(m)1433 5332 y Fk(\000)1475 5405 y Fq( )1537 5368 y Fl(\()p Fr(m)p Fl(\))1534 5428 y Fr(e)1679 5405 y Fo(\000)1780 5334 y Fq( )1842 5301 y Fr(e)1839 5357 y(n;n)p 1780 5384 169 4 v 1837 5468 a Fq(n)1958 5332 y Fk(\001)2010 5327 y Fg(1)p 2010 5339 31 3 v 2010 5380 a(2)2080 5405 y Fo(\024)25 b Fq(\014)96 b Fo(8)p Fq(e)25 b Fo(2)g(E)99 b(8)p Fq(n)24 b Fo(\025)h Fq(n)2955 5419 y Fl(0)3409 5405 y Fu(\(7.37\))p eop %%Page: 52 52 52 51 bop 118 45 a Fj(52)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(where,)25 b Fo(k)p Fq(D)s Fo(k)582 380 y Fn(1)683 366 y Fp(:=)g(sup)941 388 y Fr(e;e)1027 369 y Fi(0)1048 388 y Fn(2E)1158 366 y Fo(k)p Fq(D)1278 382 y Fr(e;e)1364 363 y Fi(0)1390 366 y Fo(k)1435 380 y Fn(1)1510 366 y Fu(.)118 478 y(T)-9 b(ogether)26 b(with)h(\(7.35\),)f(this)h(implies)g(that,)g(for)f(any)g (given)g Fq(m)g Fo(2)h Fp(\(0)p Fq(;)15 b Fp(1\))p Fu(,)28 b(we)e(can)g(\002nd)g Fq(g)3215 492 y Fr(m)3309 478 y Fo(2)g(G)32 b Fu(such)118 586 y(that)997 751 y Fq(V)1050 765 y Fr(m)1117 677 y Fk(\000)1159 751 y Fq(j)1196 765 y Fl(0)p Fr(;e)1308 751 y Fp(+)1406 664 y Fk(X)1399 862 y Fr(e)1432 843 y Fi(0)1454 862 y Fn(2E)1559 751 y Fq(D)1634 766 y Fr(e;e)1720 747 y Fi(0)1746 751 y Fp(\()p Fq(m)p Fp(\))1906 680 y Fq( )1968 647 y Fr(e)2001 623 y Fi(0)1965 702 y Fr(n)2008 711 y Fg(0)2043 702 y Fr(;n)2106 711 y Fg(0)p 1906 730 238 4 v 1978 813 a Fq(n)2033 827 y Fl(0)2174 751 y Fp(+)20 b Fo(L)p Fq(g)2371 765 y Fr(m)2438 677 y Fk(\001)2490 672 y Fg(1)p 2490 684 31 3 v 2490 726 a(2)2559 751 y Fo(\024)25 b Fp(2)p Fq(\014)5 b(:)628 b Fu(\(7.38\))118 972 y(Since)26 b Fq(\025)411 986 y Fl(0)450 972 y Fp(\()p Fq(m)p Fp(\))i Fu(is)g(a)f(smooth)g(function)g (of)g Fq(m)h Fo(2)g Fp(\(0)p Fq(;)15 b Fp(1\))29 b Fu(and)e(thanks)h (to)f(Lemma)f(7.1,)h(\(7.38\))f(remains)118 1080 y(valid)g(if)f Fq(V)469 1094 y Fr(m)561 1080 y Fu(is)h(replaced)e(by)h Fq(V)1183 1095 y Fr(m)1245 1076 y Fi(0)1272 1080 y Fu(,)g(where)f Fq(m)1672 1047 y Fn(0)1721 1080 y Fu(is)i(arbitrary)f(inside)h(an)f (open)f(interval)g Fq(I)3150 1094 y Fr(m)3242 1080 y Fu(containing)118 1188 y Fq(m)p Fu(.)31 b(In)22 b(what)i(follows)f(we)f (restrict)h(to)g(the)g(density)g(interval)g Fp([)p Fq(\016)n(;)15 b Fp(1)d Fo(\000)g Fq(\016)s Fp(])p Fu(.)32 b(Thanks)23 b(to)g(compactness)f(and)118 1302 y(interpolation)j(and)g(thanks)g(to)h (\(4.33\),)e(there)g(exists)i(a)f(continuos)f(matrix)h Fq(D)2843 1269 y Fl(\()p Fr(\014)s Fl(\))2945 1302 y Fp(\()p Fo(\001)p Fp(\))h Fu(and)f(a)f(family)h(of)118 1430 y(functions)g Fq(g)555 1382 y Fl(\()p Fr(\014)s Fl(\))552 1442 y Fr(m)658 1430 y Fu(,)h Fq(m)f Fo(2)f Fp([)p Fq(\016)n(;)15 b Fp(1)23 b Fo(\000)c Fq(\016)s Fp(])p Fu(,)27 b(such)e(that)i Fo(k)p Fq(D)1792 1382 y Fl(\()p Fr(\014)s Fl(\))1789 1460 y Fr(e;e)1875 1441 y Fi(0)1901 1430 y Fo(k)1946 1444 y Fn(1)2046 1430 y Fo(\024)e(k)p Fq(D)2262 1446 y Fr(e;e)2348 1427 y Fi(0)2375 1430 y Fo(k)2420 1444 y Fn(1)2520 1430 y Fu(and)659 1675 y Fq(V)712 1689 y Fr(m)779 1601 y Fk(\000)821 1675 y Fq(j)858 1689 y Fl(0)p Fr(;e)970 1675 y Fp(+)1068 1589 y Fk(X)1061 1787 y Fr(e)1094 1768 y Fi(0)1116 1787 y Fn(2E)1221 1675 y Fq(D)1299 1627 y Fl(\()p Fr(\014)s Fl(\))1296 1705 y Fr(e;e)1382 1686 y Fi(0)1408 1675 y Fp(\()p Fq(m)p Fp(\))1568 1604 y Fq( )1630 1571 y Fr(e)1663 1548 y Fi(0)1627 1627 y Fr(n)1670 1636 y Fg(0)1705 1627 y Fr(;n)1768 1636 y Fg(0)p 1569 1654 238 4 v 1640 1737 a Fq(n)1695 1751 y Fl(0)1836 1675 y Fp(+)20 b Fo(L)p Fq(g)2036 1638 y Fl(\()p Fr(\014)s Fl(\))2033 1698 y Fr(m)2139 1601 y Fk(\001)2190 1597 y Fg(1)p 2190 1609 31 3 v 2190 1650 a(2)2260 1675 y Fo(\024)25 b Fp(3)p Fq(\014)97 b Fo(8)p Fq(m)24 b Fo(2)h Fp([)p Fq(\016)n(;)15 b Fp(1)22 b Fo(\000)e Fq(\016)s Fp(])118 1914 y Fu(and)25 b(therefore)696 2072 y Fq(V)749 2086 y Fr(m)816 1999 y Fk(\000)857 2072 y Fq(j)894 2086 y Fl(0)p Fr(;e)1007 2072 y Fp(+)1105 1986 y Fk(X)1098 2184 y Fr(e)1131 2165 y Fi(0)1153 2184 y Fn(2E)1258 2072 y Fq(D)1336 2024 y Fl(\()p Fr(\014)s Fl(\))1333 2102 y Fr(e;e)1419 2083 y Fi(0)1445 2072 y Fp(\()p Fq(m)p Fp(\))p Fq( )1657 2024 y Fl(\()p Fr(m)p Fl(\))1654 2102 y Fr(e)1687 2083 y Fi(0)1800 2072 y Fp(+)19 b Fo(L)p Fq(g)1999 2035 y Fl(\()p Fr(\014)s Fl(\))1996 2095 y Fr(m)2102 1999 y Fk(\001)2153 1994 y Fg(1)p 2153 2006 V 2153 2047 a(2)2223 2072 y Fo(\024)25 b Fp(4)p Fq(\014)97 b Fo(8)p Fq(m)24 b Fo(2)h Fp([)p Fq(\016)n(;)15 b Fp(1)22 b Fo(\000)e Fq(\016)s Fp(])327 b Fu(\(7.39\))118 2312 y(From)25 b(the)g(above)g(formula)f(and)i (\(7.35\),)e(we)h(have)431 2472 y Fq(P)13 b(j)539 2486 y Fl(0)p Fr(;e)657 2472 y Fp(=)25 b Fo(\000)846 2386 y Fk(X)839 2584 y Fr(e)872 2565 y Fi(0)894 2584 y Fn(2E)999 2472 y Fq(D)1074 2487 y Fr(e;e)1160 2468 y Fi(0)1186 2472 y Fp(\()p Fq(m)p Fp(\))p Fq( )1398 2424 y Fl(\()p Fr(m)p Fl(\))1395 2502 y Fr(e)1428 2483 y Fi(0)1546 2472 y Fp(=)g Fo(\000)1734 2386 y Fk(X)1728 2584 y Fr(e)1761 2565 y Fi(0)1783 2584 y Fn(2E)1888 2472 y Fq(D)1966 2424 y Fl(\()p Fr(\014)s Fl(\))1963 2502 y Fr(e;e)2049 2483 y Fi(0)2075 2472 y Fp(\()p Fq(m)p Fp(\))p Fq( )2287 2424 y Fl(\()p Fr(m)p Fl(\))2284 2502 y Fr(e)2317 2483 y Fi(0)2429 2472 y Fp(+)20 b Fq(\030)2564 2435 y Fl(\()p Fr(m)p Fl(\))2560 2495 y Fr(e)2776 2472 y Fo(8)p Fq(m)25 b Fo(2)g Fp([)p Fq(\016)n(;)15 b Fp(1)22 b Fo(\000)e Fq(\016)s Fp(])118 2742 y Fu(where)39 b Fq(V)456 2756 y Fr(m)522 2742 y Fp(\()p Fq(\030)601 2694 y Fl(\()p Fr(m)p Fl(\))597 2754 y Fr(e)723 2742 y Fp(\))768 2679 y Fg(1)p 768 2691 V 768 2732 a(2)862 2742 y Fo(\024)48 b Fp(4)p Fq(\014)5 b Fu(.)75 b(By)39 b(taking)g(the)h(scalar)f(product)g(with)h Fq(j)2643 2757 y Fl(0)p Fr(;e)2731 2738 y Fi(0)2797 2742 y Fu(we)f(obtain)h(\(thanks)g(to)118 2850 y(theorem)25 b(7.18\))684 3007 y Fo(j)p Fq(\037)p Fp(\()p Fq(m)p Fp(\))916 2933 y Fk(\000)958 3007 y Fq(D)1033 3022 y Fr(e;e)1119 3003 y Fi(0)1145 3007 y Fp(\()p Fq(m)p Fp(\))c Fo(\000)f Fq(D)1485 2959 y Fl(\()p Fr(\014)s Fl(\))1482 3037 y Fr(e;e)1568 3018 y Fi(0)1594 3007 y Fp(\()p Fq(m)p Fp(\))1744 2933 y Fk(\001)1786 3007 y Fo(j)25 b(\024)g Fp(4)p Fq(V)2030 3021 y Fr(m)2097 3007 y Fp(\()p Fq(j)2169 3021 y Fl(0)p Fr(;e)2262 3007 y Fp(\))2307 2942 y Fg(1)p 2307 2954 V 2307 2996 a(2)2352 3007 y Fq(\014)96 b Fo(8)p Fq(m)24 b Fo(2)h Fp([)p Fq(\016)n(;)15 b Fp(1)22 b Fo(\000)e Fq(\016)s Fp(])p Fq(;)118 3194 y Fu(that)34 b(is)g Fo(j)p Fq(D)504 3210 y Fr(e;e)590 3191 y Fi(0)616 3194 y Fp(\()p Fq(m)p Fp(\))26 b Fo(\000)g Fq(D)967 3147 y Fl(\()p Fr(\014)s Fl(\))964 3224 y Fr(e;e)1050 3205 y Fi(0)1075 3194 y Fp(\()p Fq(m)p Fp(\))p Fo(j)40 b(\024)e Fq(c)p Fp(\()p Fq(\016)s Fp(\))p Fq(\014)5 b Fu(,)37 b(thus)d(proving)f(that)h Fq(D)2475 3210 y Fr(e;e)2561 3191 y Fi(0)2587 3194 y Fp(\()p Fo(\001)p Fp(\))g Fu(is)g(continuous)f(on)f Fp([)p Fq(\016)n(;)15 b Fp(1)28 b Fo(\000)118 3314 y Fq(\016)s Fp(])p Fu(.)3380 b Fe(\003)218 3473 y Fu(W)-8 b(e)25 b(are)g(now)g(able)g(to)g(prove)g(our)g(main)g(result.)118 3600 y FE(Theorem)i(7.22.)41 b Fh(Let)26 b Fq(d)f Fo(\025)g Fp(3)p Fh(.)32 b(Then)25 b(given)f Fq(\016)29 b(>)c Fp(0)728 3822 y(inf)716 3883 y Fr(g)r Fn(2)p Fm(G)866 3822 y Fp(lim)15 b(sup)931 3901 y Fr(n)p Fn("1)1252 3822 y Fp(sup)1159 3905 y Fr(m)p Fn(2)p Fl([)p Fr(\016)o(;)p Fl(1)p Fn(\000)p Fr(\016)r Fl(])1496 3822 y Fq(V)1549 3836 y Fr(m)1616 3721 y Fk(\020)1670 3822 y Fq(j)1707 3836 y Fl(0)p Fr(;e)1820 3822 y Fp(+)20 b Fo(L)p Fq(g)j Fp(+)2138 3736 y Fk(X)2131 3934 y Fr(e)2164 3915 y Fi(0)2186 3934 y Fn(2E)2291 3822 y Fq(D)2366 3837 y Fr(e;e)2452 3818 y Fi(0)2478 3822 y Fp(\()p Fq(m)p Fp(\))2638 3751 y Fq( )2700 3718 y Fr(e)2733 3695 y Fi(0)2697 3774 y Fr(n;n)p 2638 3801 169 4 v 2696 3884 a Fq(n)2817 3721 y Fk(\021)2897 3822 y Fp(=)i(0)p Fq(:)346 b Fu(\(7.40\))118 4062 y Fh(Moreover)-12 b(,)25 b(if)g Fq(D)j Fh(has)d(continuous)f(extension)f(to)i Fo(f)p Fp(0)p Fq(;)15 b Fp(1)p Fo(g)p Fh(,)28 b(\(7.40\))d(is)g(valid)f (with)i Fq(\016)j Fp(=)c(0)p Fh(.)118 4221 y(Proof)-8 b(.)42 b Fu(\(7.40\))33 b(is)j(a)e(simple)h(consequence)d(of)j(the)g (estimates)h(exhibited)f(in)g(the)g(proof)g(of)g(lemma)118 4347 y(7.21.)c(Let)25 b(us)g(observe)g(that,)i(given)d Fq(\014)31 b(>)25 b Fp(0)p Fu(,)h(by)f(de\002ning)g Fq(g)2178 4299 y Fl(\()p Fr(\014)s Fl(\))2175 4359 y Fr(m)2305 4347 y Fu(as)h(in)g(the)f(above)g(proof)m(,)g(then)707 4569 y Fp(lim)15 b(sup)772 4648 y 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5024 y(to)e(extend)e(\(7.41\))g(to)i (all)f Fp([0)p Fq(;)15 b Fp(1])p Fu(.)2364 b Fe(\003)1657 5210 y Fu(A)t Ft(P)t(P)t(E)t(N)t(D)t(I)t(X)30 b Fu(A)218 5372 y(In)h(this)i(\002nal)e(appendix)g(we)g(have)g(collected)f (several)g(technical)h(results)g(used)h(in)f(the)h(previous)118 5480 y(sections.)p eop %%Page: 53 53 53 52 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(53)118 366 y Fu(A.1.)46 b FE(L)r(arge)27 b(deviations)e(estimates.)44 b Fu(.)118 507 y FE(L)r(emma)50 b(A.1.)55 b Fh(Let)48 b Fq(f)72 b Fp(=)62 b Fq(f)10 b Fp(\()p Fq(\013)p Fp(\))49 b Fh(be)e(a)g(mean-zero)f(local)h(function)g(and)f Fp(\003)63 b Fo(2)g Fs(F)58 b Fh(be)48 b(such)f(that)118 615 y Fp(\(\001)229 630 y Fr(f)295 615 y Fp(+)20 b Fq(x)p Fp(\))g Fo(\\)g Fp(\(\001)685 630 y Fr(f)751 615 y Fp(+)g Fq(y)s Fp(\))25 b(=)g Fo(;)h Fh(for)e(any)h Fq(x;)15 b(y)29 b Fo(2)24 b Fp(\003)p Fh(.)32 b(Then)1033 864 y Fs(P)p Fp([)15 b Fo(j)p Fp(Av)1272 878 y Fr(x)p Fn(2)p Fl(\003)1412 864 y Fq(\034)1452 878 y Fr(x)1495 864 y Fq(f)25 b Fo(j)g(\025)g Fq(\016)19 b Fp(])26 b Fo(\024)f Fp(2)p Fq(e)2004 804 y Fn(\000)2095 770 y FA(\016)2125 749 y Fg(2)2159 770 y Fi(j)p Fg(\003)p Fi(j)p 2069 788 195 3 v 2069 841 a Fg(4)p Fi(k)p FA(f)6 b Fi(k)2199 824 y Fg(2)2199 855 y Fi(1)2460 864 y Fo(8)p Fq(\016)29 b(>)c Fp(0)p Fq(:)118 1071 y Fh(Proof)-8 b(.)42 b Fu(Given)25 b Fq(t)g(>)g Fp(0)p Fu(,)h(since)f Fs(E)10 b Fp(\()q Fq(f)g Fp(\))30 b(=)25 b(0)h Fu(and)f Fq(e)1703 1038 y Fr(x)1768 1071 y Fo(\000)20 b Fq(x)25 b Fo(\024)g Fq(e)2074 1038 y Fr(x)2114 1014 y Fg(2)2178 1071 y Fu(for)g(any)g Fq(x)g Fo(\025)g Fp(0)p Fu(,)1034 1319 y Fq(e)1076 1281 y Fr(tf)1173 1319 y Fp(=)1300 1205 y Fn(1)1270 1232 y Fk(X)1269 1427 y Fr(n)p Fl(=0)1427 1257 y Fp(\()p Fq(tf)10 b Fp(\))1585 1224 y Fr(n)p 1427 1298 205 4 v 1489 1381 a Fq(n)p Fp(!)1667 1319 y Fo(\024)25 b Fq(e)1805 1281 y Fr(t)p Fn(k)p Fr(f)7 b Fn(k)1941 1289 y Fi(1)2032 1319 y Fo(\000)20 b(k)p Fq(f)10 b Fo(k)2268 1333 y Fn(1)2343 1319 y Fq(t)25 b Fo(\024)g Fq(e)2539 1281 y Fn(k)p Fr(f)7 b Fn(k)2650 1258 y Fg(2)2650 1298 y Fi(1)2720 1319 y Fq(:)118 1577 y Fu(Therefore,)25 b(thanks)h(to)f(the)h(conditions)g(on)f Fq(f)34 b Fu(and)26 b Fp(\003)p Fu(,)326 1762 y Fs(P)p Fp([)15 b(Av)540 1776 y Fr(x)p Fn(2)p Fl(\003)679 1762 y Fq(\034)719 1776 y Fr(x)763 1762 y Fq(f)35 b Fo(\025)25 b Fq(\016)19 b Fp(])25 b Fo(\024)g Fq(e)1186 1725 y Fn(\000)p Fr(t\016)1305 1762 y Fs(E)10 b Fp(\()q Fq(e)1437 1725 y Fr(t)18 b Fl(Av)1571 1736 y FA(x)p Fi(2)p Fg(\003)1693 1725 y Fr(\034)1724 1733 y FA(x)1763 1725 y Fr(f)1809 1762 y Fp(\))25 b(=)g Fq(e)2007 1725 y Fn(\000)p Fr(t\016)2126 1762 y Fp([)p Fs(E)11 b Fp(\()p Fq(e)2283 1725 y Fr(t)q(f)c Fn(j)p Fl(\003)p Fn(j)2439 1701 y Fi(\000)p Fg(1)2531 1762 y Fp(\)])2591 1725 y Fn(j)p Fl(\003)p Fn(j)2709 1762 y Fo(\024)25 b Fq(e)2847 1725 y Fn(\000)p Fr(t\016)r Fl(+)p Fr(t)3040 1701 y Fg(2)3076 1725 y Fn(k)p Fr(f)7 b Fn(k)3187 1701 y Fg(2)3187 1741 y Fi(1)3253 1725 y Fn(j)p Fl(\003)p Fn(j)3342 1701 y Fi(\000)p Fg(1)3428 1762 y Fq(:)118 1936 y Fu(The)33 b(thesis)i(follows)e(by)g(taking)h Fq(t)k Fp(:=)g Fq(\016)s Fo(j)p Fp(\003)p Fo(j)p Fq(=)p Fp(\(2)p Fo(k)p Fq(f)10 b Fo(k)1917 1903 y Fl(2)1917 1958 y Fn(1)1995 1936 y Fp(\))33 b Fu(and)h(by)f(considering)f(the)i (above)f(estimates)118 2044 y(with)27 b Fq(f)34 b Fu(replaced)24 b(by)h Fo(\000)p Fq(f)10 b Fu(.)2558 b Fe(\003)118 2262 y Fu(A.2.)46 b FE(Equilibrium)25 b(bounds.)118 2402 y(L)r(emma)31 b(A.2.)45 b Fh(Given)29 b Fp(\003)k Fo(2)f Fs(F)40 b Fh(and)28 b Fq(\025)33 b Fo(2)f Fs(R)38 b Fh(we)29 b(de\002ne)f Fq(m)k Fp(:=)g Fq(\026)2322 2369 y Fr(\025)2368 2402 y Fp(\()p Fq(m)2483 2416 y Fl(\003)2536 2402 y Fp(\))e Fh(and)e Fq(a)2824 2416 y Fr(m)2923 2402 y Fp(:=)33 b(min)n(\()p Fq(m;)15 b Fp(1)24 b Fo(\000)f Fq(m)p Fp(\))p Fh(.)118 2510 y(Then,)i(for)g(any)g Fp(\001)g Fo(\032)g Fp(\003)g Fh(and)f(any)h(function)f Fq(f)34 b Fh(such)25 b(that)h Fp(\001)2158 2525 y Fr(f)2228 2510 y Fo(\032)f Fp(\003)p Fh(,)929 2692 y Fq(a)p Fp(\))h Fq(c)p Fo(j)p Fp(\001)p Fo(j)p Fq(m)g Fo(\024)f Fq(\026)p Fp(\()p Fq(N)1568 2706 y Fl(\001)1631 2692 y Fp(\))g Fo(\024)g Fq(c)1826 2655 y Fn(\000)p Fl(1)1921 2692 y Fo(j)p Fp(\001)p Fo(j)p Fq(m;)929 2845 y(b)p Fp(\))h Fq(c)15 b Fo(j)p Fp(\001)p Fo(j)p Fp(\(1)22 b Fo(\000)e Fq(m)p Fp(\))25 b Fo(\024)g Fq(\026)p Fp(\()p Fo(j)p Fp(\001)p Fo(j)c(\000)e Fq(N)2038 2859 y Fl(\001)2101 2845 y Fp(\))26 b Fo(\024)f Fq(c)2297 2807 y Fn(\000)p Fl(1)2392 2845 y Fo(j)p Fp(\001)p Fo(j)p Fp(\(1)c Fo(\000)f Fq(m)p Fp(\))p Fq(;)929 2997 y(c)p Fp(\))26 b Fq(c)p Fo(j)p Fp(\001)p Fo(j)p Fq(a)1242 3011 y Fr(m)1335 2997 y Fo(\024)f Fq(\026)p Fp(\()p Fq(N)1594 3011 y Fl(\001)1657 2997 y Fp(;)15 b Fq(N)1770 3011 y Fl(\001)1833 2997 y Fp(\))26 b Fo(\024)f Fq(c)2029 2959 y Fn(\000)p Fl(1)2123 2997 y Fo(j)p Fp(\001)p Fo(j)p Fq(a)2297 3011 y Fr(m)2364 2997 y Fq(;)929 3183 y(d)p Fp(\))h Fo(j)p Fq(\026)p Fp(\()p Fq(f)10 b Fp(;)15 b Fq(N)1320 3197 y Fl(\003)1373 3183 y Fp(\))p Fo(j)26 b(\024)f Fq(c)p Fo(k)p Fq(f)10 b Fo(k)1739 3197 y Fn(1)1830 3183 y Fp(min)1981 3082 y Fk(\020)2035 3183 y Fo(j)p Fp(\001)2136 3198 y Fr(f)2182 3183 y Fo(j)p Fq(a)2255 3197 y Fr(m)2322 3183 y Fq(;)2362 3076 y Fk(q)p 2453 3076 287 4 v 107 x Fo(j)p Fp(\001)2554 3198 y Fr(f)2599 3183 y Fo(j)p Fq(a)2672 3197 y Fr(m)2754 3082 y Fk(\021)2809 3183 y Fq(:)118 3415 y Fh(Proof)-8 b(.)42 b Fu(In)25 b(what)h(follows)f(we)g(assume)g Fq(m)g Fo(\024)1688 3380 y Fl(1)p 1688 3395 36 4 v 1688 3447 a(2)1734 3415 y Fu(.)118 3523 y Fq(a)p Fp(\))h Fu(and)g Fq(b)p Fp(\))g Fu(can)f(be)h(easily)g(derived)g(from)f(the)h(boundedness)g(of)g(the)g (random)f(\002eld)h Fq(\013)p Fu(.)33 b(Let)26 b(us)g(prove)118 3631 y Fq(c)p Fp(\))p Fu(.)34 b(The)26 b(upper)g(bound)f(follows)h(by)g (observing)f(that)i Fq(\026)p Fp(\()p Fq(N)2150 3645 y Fl(\001)2214 3631 y Fp(;)15 b Fq(N)2327 3645 y Fl(\001)2390 3631 y Fp(\))27 b Fo(\024)f Fq(\026)p Fp(\(\001\))g Fu(and)g(by)g (applying)f Fq(a)p Fp(\))p Fu(.)34 b(In)118 3741 y(order)26 b(to)h(prove)e(the)i(lower)f(bound,)g(let)g(us)h(introduce)f(the)g(set) h Fq(W)38 b Fp(:=)25 b Fo(f)p Fq(x)h Fo(2)e Fp(\003)i(:)f Fq(\026)p Fp(\()p Fq(\021)3087 3755 y Fr(x)3131 3741 y Fp(\))h Fo(\024)3298 3705 y Fl(1)p 3298 3720 V 3298 3772 a(2)3343 3741 y Fo(g)q Fu(.)35 b(Since)118 3849 y Fo(j)p Fq(W)13 b Fo(j)25 b(\025)g(j)p Fp(\003)p Fo(j)p Fq(=)p Fp(2)i Fu(and)e(thanks)h(to)g Fq(a)p Fp(\))p Fu(,)966 4066 y Fq(\026)p Fp(\()p Fq(N)1129 4080 y Fl(\003)1182 4066 y Fp(;)15 b Fq(N)1295 4080 y Fl(\003)1349 4066 y Fp(\))26 b Fo(\025)f Fq(\026)p Fp(\()p Fq(N)1669 4080 y Fr(W)1749 4066 y Fp(;)15 b Fq(N)1862 4080 y Fr(W)1943 4066 y Fp(\))26 b Fo(\025)2110 4004 y Fp(1)p 2110 4045 46 4 v 2110 4128 a(2)2165 4066 y Fq(\026)p Fp(\()p Fq(N)2328 4080 y Fr(W)2409 4066 y Fp(\))f Fo(\025)g Fq(c)15 b(m)p Fo(j)p Fp(\003)p Fo(j)118 4270 y Fu(thus)25 b(proving)e(the)h(lower)f (bound)h(in)g Fq(c)p Fp(\))g Fu(with)h Fp(\001)f Fu(replaced)e(by)i Fp(\003)p Fu(.)31 b(In)24 b(order)g(to)g(consider)f(the)i(general)118 4378 y(case,)39 b(we)c(de\002ne)g Fq(m)860 4345 y Fn(0)927 4378 y Fp(=)42 b Fq(\026)p Fp(\()p Fq(m)1210 4392 y Fl(\001)1273 4378 y Fp(\))p Fu(.)64 b(Then)36 b(by)f(the)i(previous)e(arguments,)j Fq(\026)p Fp(\()p Fq(N)2946 4392 y Fl(\001)3009 4378 y Fp(;)15 b Fq(N)3122 4392 y Fl(\001)3185 4378 y Fp(\))44 b Fo(\025)f Fq(c)15 b(m)3512 4345 y Fn(0)3535 4378 y Fo(j)p Fp(\001)p Fo(j)118 4486 y Fu(which,)26 b(by)g Fq(a)p Fp(\))p Fu(,)g(is)g(bounded)f(from)f(below)h(by)g Fq(c)15 b(m)p Fo(j)p Fp(\001)p Fo(j)p Fu(.)118 4594 y(Let)26 b(us)f(prove)g Fq(d)p Fp(\))p Fu(.)32 b(By)25 b(Schwarz)f(inequality)i (and)f Fq(c)p Fp(\))749 4805 y Fo(j)p Fq(\026)p Fp(\()p Fq(f)10 b Fp(;)15 b Fq(N)1032 4819 y Fl(\003)1085 4805 y Fp(\))p Fo(j)26 b(\024)f Fq(\026)p Fp(\()p Fq(f)10 b Fp(;)15 b Fq(f)10 b Fp(\))1552 4740 y Fg(1)p 1552 4752 31 3 v 1552 4794 a(2)1596 4805 y Fq(\026)p Fp(\()p Fq(N)1759 4819 y Fl(\001)1818 4831 y FA(f)1862 4805 y Fp(;)15 b Fq(N)1975 4819 y Fl(\001)2034 4831 y FA(f)2078 4805 y Fp(\))2123 4740 y Fg(1)p 2124 4752 V 2124 4794 a(2)2193 4805 y Fo(\024)25 b Fq(c)15 b(\026)p Fp(\()p Fq(f)10 b Fp(;)15 b Fq(f)10 b Fp(\))2628 4740 y Fg(1)p 2628 4752 V 2628 4794 a(2)2673 4698 y Fk(q)p 2764 4698 267 4 v 107 x Fq(m)15 b Fo(j)p Fp(\001)2960 4820 y Fr(f)3005 4805 y Fo(j)118 5008 y Fu(Since)25 b Fq(\026)p Fp(\()p Fq(f)10 b Fp(;)15 b Fq(f)10 b Fp(\))27 b Fo(\024)g(k)p Fq(f)10 b Fo(k)902 4975 y Fl(2)902 5031 y Fn(1)977 5008 y Fu(,)28 b(it)f(remains)f(to)h(prove)f(that)h Fq(\026)p Fp(\()p Fq(f)10 b Fp(;)15 b Fq(f)10 b Fp(\))27 b Fo(\024)g Fq(c)15 b(m)p Fo(k)p Fq(f)10 b Fo(k)2673 4975 y Fl(2)2673 5031 y Fn(1)2749 5008 y Fo(j)p Fp(\001)2850 5023 y Fr(f)2895 5008 y Fo(j)p Fu(.)36 b(T)-9 b(o)27 b(this)h(aim)e(let)h Fq(\021)3622 4975 y Fn(\003)118 5116 y Fu(be)e(the)h(con\002guration)e (with)i(no)f(particle.)31 b(Then,)26 b(thanks)g(to)f Fq(a)p Fp(\))p Fu(,)755 5307 y Fq(\026)p Fp(\()p Fq(f)10 b Fp(;)15 b Fq(f)10 b Fp(\))25 b Fo(\024)g Fq(\026)1206 5233 y Fk(\000)o(\000)1289 5307 y Fq(f)k Fo(\000)20 b Fq(f)10 b Fp(\()p Fq(\021)1592 5269 y Fn(\003)1632 5307 y Fp(\))1667 5233 y Fk(\001)1709 5256 y Fl(2)1748 5233 y Fk(\001)1815 5307 y Fo(\024)25 b(k)p Fq(f)10 b Fo(k)2056 5269 y Fl(2)2056 5329 y Fn(1)2131 5307 y Fq(\026)p Fp(\()p Fq(N)2294 5321 y Fl(\001)2353 5333 y FA(f)2397 5307 y Fp(\))25 b Fo(\024)g Fq(c)15 b Fo(k)p Fq(f)10 b Fo(k)2752 5269 y Fl(2)2752 5329 y Fn(1)2828 5307 y Fo(j)p Fp(\001)2929 5322 y Fr(f)2974 5307 y Fo(j)p Fq(:)3591 5480 y Fe(\003)p eop %%Page: 54 54 54 53 bop 118 45 a Fj(54)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y FE(L)r(emma)27 b(A.3.)42 b Fh(F)l(or)25 b(any)g Fq(\025;)15 b(\025)1123 333 y Fn(0)1172 366 y Fo(2)25 b Fs(R)r Fh(,)32 b Fp(\003)25 b Fo(2)g Fs(F)36 b Fh(and)24 b(any)h(function)f Fq(f)34 b Fh(with)26 b Fp(\001)2660 381 y Fr(f)2730 366 y Fo(\032)f Fp(\003)p Fh(,)729 558 y Fo(j)p Fq(\026)809 521 y Fr(\025)850 497 y Fi(0)876 558 y Fp(\()p Fq(f)10 b Fp(\))20 b Fo(\000)g Fq(\026)1167 521 y Fr(\025)1212 558 y Fp(\()p Fq(f)10 b Fp(\))p Fo(j)26 b(\024)f Fq(c)p Fo(k)p Fq(f)10 b Fo(k)1668 572 y Fn(1)1743 558 y Fo(j)p Fp(\001)1844 573 y Fr(f)1890 558 y Fo(j)15 b(j)p Fq(\026)2010 521 y Fr(\025)2051 497 y Fi(0)2078 558 y Fp(\()p Fq(m)2193 572 y Fl(\003)2246 558 y Fp(\))21 b Fo(\000)e Fq(\026)2447 521 y Fr(\025)2493 558 y Fp(\()p Fq(m)2608 572 y Fl(\003)2661 558 y Fp(\))p Fo(j)p Fq(;)706 b Fu(\(A.1\))729 719 y Fo(j)p Fq(\026)809 681 y Fr(\025)850 658 y Fi(0)876 719 y Fp(\()p Fq(\021)956 733 y Fr(x)1001 719 y Fp(;)15 b Fq(\021)1086 733 y Fr(x)1130 719 y Fp(\))21 b Fo(\000)f Fq(\026)1332 681 y Fr(\025)1377 719 y Fp(\()p Fq(\021)1457 733 y Fr(x)1501 719 y Fp(;)15 b Fq(\021)1586 733 y Fr(x)1630 719 y Fp(\))p Fo(j)26 b(\024)f Fq(c)15 b Fo(j)p Fq(\026)1946 681 y Fr(\025)1987 658 y Fi(0)2014 719 y Fp(\()p Fq(m)2129 733 y Fl(\003)2183 719 y Fp(\))20 b Fo(\000)g Fq(\026)2384 681 y Fr(\025)2429 719 y Fp(\()p Fq(m)2544 733 y Fl(\003)2598 719 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b(for)g(any)f Fp(\003)i Fo(2)f Fs(F)36 b Fh(and)24 b(any)g Fq(m)h Fo(2)g Fp(\(0)p Fq(;)15 b Fp(1\))p Fh(,)934 1895 y Fo(j)p Fq(\025)1012 1909 y Fl(\003)1065 1895 y Fp(\()p Fq(m)p Fp(\))21 b Fo(\000)f Fq(\025)1380 1909 y Fl(0)1419 1895 y Fp(\()p Fq(m)p Fp(\))p Fo(j)26 b(\024)1900 1833 y Fq(c)p 1726 1874 387 4 v 1726 1957 a(m)p Fp(\(1)21 b Fo(\000)f Fq(m)p Fp(\))2123 1895 y Fo(j)p Fq(m)h Fo(\000)e Fq(\026)2394 1857 y Fr(\025)2435 1866 y Fg(0)2470 1857 y Fl(\()p Fr(m)p Fl(\))2591 1895 y Fp(\()p Fq(m)2706 1909 y Fl(\003)2760 1895 y Fp(\))p Fo(j)p Fq(:)607 b Fu(\(A.6\))118 2144 y Fh(Proof)-8 b(.)42 b Fu(It)26 b(is)g(simple)f(to) h(derive)f(\(A.2\),)g(\(A.4\))g(and)g(\(A.5\))g(from)g(\(A.1\))g(and)g (\(A.3\).)118 2257 y(Let)h(us)f(prove)g(\(A.1\).)31 b(By)25 b(setting)h Fq(\025)p Fp(\()p Fq(s)p Fp(\))f(:=)h Fq(\025)1677 2271 y Fl(\003)1730 2257 y Fp(\()p Fq(s)p Fp(\))p Fu(,)g Fq(m)f Fp(:=)g Fq(\026)2175 2224 y Fr(\025)2220 2257 y Fp(\()p Fq(m)2335 2271 y Fl(\003)2388 2257 y Fp(\))h Fu(and)f 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Fq(\026)1577 5065 y Fr(\025)p Fl(\()p Fr(m)1707 5075 y FA(i)1735 5065 y Fl(\))1577 5141 y(\001)1636 5151 y FA(i)1857 5113 y Fu(where)90 b Fq(m)2273 5127 y Fr(i)2327 5113 y Fp(:=)2484 5051 y Fq(N)2557 5065 y Fr(i)p 2458 5092 155 4 v 2458 5175 a Fo(j)p Fp(\001)2559 5189 y Fr(i)2587 5175 y Fo(j)2622 5113 y Fq(;)1137 5301 y Fp(\026)-50 b Fq(\027)31 b Fp(:=)25 b Fq(\026)p Fp(\()p Fo(\001)15 b(j)g Fq(N)1572 5315 y Fl(\001)1631 5325 y FA(i)1688 5301 y Fp(=)25 b Fq(N)1857 5315 y Fr(i)1935 5301 y Fo(8)p Fq(i)h Fp(=)f(1)p Fq(;)15 b(:)g(:)g(:)i(;)e(k)s Fp(\))p Fq(:)118 5480 y Fu(Then)25 b(we)g(have)g(the)h(following)e(main)h(results)h(\(for)f(the)g(latter)g (see)g(also)h(proposition)g Fp(3)p Fq(:)p Fp(3)g Fu(in)g([7]\).)p eop %%Page: 55 55 55 54 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(55)118 366 y FE(Proposition)27 b(A.4.)42 b Fh(\(Equivalence)23 b(of)i(ensembles\))118 474 y(Let)d Fq(\015)5 b(;)15 b(\016)30 b Fo(2)25 b Fp(\(0)p Fq(;)15 b Fp(1\))24 b Fh(and)d Fq(f)31 b Fh(be)22 b(a)f(local)h(function)e(such)i(that)h Fo(j)p Fp(\001)2159 488 y Fr(i)2187 474 y Fo(j)i(\025)g Fq(\016)s Fo(j)p Fp(\001)p Fo(j)p Fh(,)f(for)e(any)f Fq(i)26 b Fp(=)f(1)p Fq(;)15 b(:)g(:)g(:)i(;)e(k)s Fh(,)23 b Fp(\001)3419 489 y Fr(f)3490 474 y Fo(\032)i Fp(\001)118 586 y Fh(and)f Fo(j)p Fp(\001)390 601 y Fr(f)436 586 y Fo(j)h(\024)g(j)p Fp(\001)p Fo(j)708 553 y Fl(1)p Fn(\000)p Fr(\015)843 586 y Fh(.)118 694 y(Then)g(there)g(exist)g(constants)e Fq(c)1172 708 y Fl(1)1212 694 y Fq(;)15 b(c)1291 708 y Fl(2)1331 694 y Fh(,)26 b(depending)c(respectively)i(on)h Fq(\015)5 b(;)15 b(\016)n(;)g(k)30 b Fh(and)24 b Fq(\016)n(;)15 b(k)s Fh(,)27 b(such)e(that)981 910 y Fo(j)p Fp(\001)p Fo(j)g(\025)g Fq(c)1267 924 y Fl(1)1423 910 y Fo(\))116 b(j)5 b Fp(\026)-50 b Fq(\027)6 b Fp(\()p Fq(f)k Fp(\))20 b Fo(\000)27 b Fp(\026)-52 b Fq(\026)p Fp(\()p Fq(f)10 b Fp(\))p Fo(j)25 b(\024)g Fq(c)2307 924 y Fl(2)2362 910 y Fo(k)p Fq(f)10 b Fo(k)2507 924 y Fn(1)2592 846 y Fo(j)p Fp(\001)2693 861 y Fr(f)2739 846 y Fo(j)p 2592 889 172 4 v 2615 972 a(j)p Fp(\001)p Fo(j)2774 910 y Fq(:)118 1127 y FE(L)r(emma)22 b(A.5.)34 b Fh(Let)21 b Fq(\016)29 b Fo(2)c Fp(\(0)p Fq(;)15 b Fp(1\))22 b Fh(and)d Fq(f)29 b Fh(be)20 b(a)g(local)f(function)g(such)h(that)g Fp(\001)2552 1142 y Fr(f)2623 1127 y Fo(\032)25 b Fp(\001)20 b Fh(and)e Fo(j)p Fp(\001)3081 1141 y Fr(i)3130 1127 y Fo(n)j Fp(\001)3272 1142 y Fr(f)3317 1127 y Fo(j)k(\025)g Fq(\016)s Fo(j)p Fp(\001)3607 1141 y Fr(i)3636 1127 y Fo(j)118 1235 y Fh(for)g(any)f Fq(i)i Fp(=)f(1)p Fq(;)15 b Fo(\001)g(\001)g(\001)32 b Fq(;)15 b(k)s Fh(.)118 1343 y(Then)25 b(there)g(exist)g(constants)e Fq(c)1172 1357 y Fl(1)1212 1343 y Fq(;)15 b(c)1291 1357 y Fl(2)1331 1343 y Fh(,)26 b(depending)c(respectively)i(on)h Fq(k)j Fh(and)c Fq(k)s(;)15 b(\016)s Fh(,)27 b(such)e(that)331 1506 y Fo(j)p Fp(\001)432 1520 y Fr(i)461 1506 y Fo(j)g(\025)g Fq(c)646 1520 y Fl(1)777 1506 y Fo(8)p Fq(i)g Fp(=)g(1)p Fq(;)15 b(:)g(:)g(:)i(;)e(k)120 b Fo(\))h Fp(\026)-50 b Fq(\027)5 b Fp(\()p Fo(j)p Fq(f)10 b Fo(j)p Fp(\))26 b Fo(\024)f Fq(c)1987 1520 y Fl(2)2049 1506 y Fp(\026)-52 b Fq(\026)p Fp(\()p Fo(j)p Fq(f)10 b Fo(j)p Fp(\))91 b Fh(and)f Fp(V)-8 b(ar)2745 1520 y Fl(\026)-38 b Fr(\027)2785 1506 y Fp(\()p Fq(f)10 b Fp(\))25 b Fo(\024)g Fq(c)3070 1520 y Fl(2)3110 1506 y Fp(V)-8 b(ar)3257 1520 y Fl(\026)-41 b Fr(\026)3298 1506 y Fp(\()p Fq(f)10 b Fp(\))p Fq(;)118 1704 y Fu(A.4.)46 b FE(Some)32 b(special)g(equilibrium)f(covariances.) 46 b Fu(In)32 b(this)g(paragraph)f(we)g(estimate)g(the)h(canon-)118 1812 y(ical)j(covariance)f(between)g(a)h(generic)e(function)i(and)g(a)g (function)g(which)g(can)g(be)f(written)i(as)f(the)118 1920 y(spatial)27 b(average)e(of)i(local)e(functions.)34 b(W)-8 b(e)27 b(observe)e(that)j(the)e(bound)g(we)g(provide)g(differs)h (from)f(the)118 2028 y(standard)f(Lu-)-7 b(Y)e(au's)25 b(T)-7 b(wo)24 b(Blocks)g(Estimate)h(\(see)e([27]\))h(by)f(an)h (additional)h(term)e(depending)h(on)f(the)118 2136 y(random)i(\002eld)g Fq(\013)h Fu(and)f(satisfying)i(a)e(large)f(deviations)j(estimate.)118 2352 y(In)42 b(what)g(follows)f(we)g(\002x)g(functions)h Fq(h;)15 b(h)1653 2319 y Fn(0)1729 2352 y Fo(2)52 b Fs(G)19 b Fu(,)51 b(depending)41 b(only)g(on)g Fq(\013)2856 2366 y Fl(0)2936 2352 y Fu(and)h Fq(\021)3172 2366 y Fl(0)3211 2352 y Fu(,)k(such)41 b(that)118 2460 y Fo(k)p Fq(h)p Fo(k)260 2474 y Fn(1)336 2460 y Fq(;)15 b Fo(k)p Fq(h)473 2427 y Fn(0)498 2460 y Fo(k)543 2474 y Fn(1)645 2460 y Fo(\024)27 b Fp(1)p Fu(.)36 b(Moreover)-10 b(,)26 b(for)g(any)h (positive)h(integer)d Fq(L)p Fu(,)i(we)g(denote)f(by)g Fq(R)2962 2474 y Fr(L)3041 2460 y Fu(the)h(set)g(of)g(boxes)118 2568 y(with)g(sides)f(of)f(length)g(in)g Fp([)p Fq(L;)15 b Fp(100)p Fq(L)p Fp(])p Fu(.)118 2705 y FE(Proposition)27 b(A.6.)42 b Fh(Given)25 b Fp(0)h Fq(<)f(\016)k(<)1421 2669 y Fl(1)p 1421 2684 36 4 v 1421 2736 a(2)1491 2705 y Fh(there)c(exists)g Fq(`)1979 2719 y Fl(0)2044 2705 y Fo(2)f Fs(N)38 b Fh(having)25 b(the)g(following)f(property)-5 b(.)118 2813 y(Let)31 b Fq(`;)15 b(L)34 b Fo(2)g Fs(N)43 b Fh(be)30 b(such)g(that)h Fq(`)1170 2827 y Fl(0)1243 2813 y Fo(\024)j Fq(`)g Fo(\024)g Fq(L)c Fh(and)g(let)g Fq(V)5 b(;)15 b(W)47 b Fo(2)34 b Fq(R)2313 2827 y Fr(L)2395 2813 y Fh(with)d Fq(V)44 b Fo(\\)23 b Fq(W)47 b Fp(=)34 b Fo(;)p Fh(.)47 b(Then,)32 b(for)e(any)118 2921 y Fq(\027)h Fo(2)25 b(M)p Fp(\()p Fq(V)c Fp(\))k Fh(and)f(any)h(function)f Fq(g)29 b Fo(2)c Fs(G)19 b Fh(,)330 3137 y Fq(\027)6 b Fp(\(Av)534 3151 y Fr(v)r Fn(2)p Fr(V)678 3137 y Fq(\034)718 3151 y Fr(v)759 3137 y Fq(h)p Fp(;)15 b Fq(g)s Fp(\))932 3100 y Fl(2)998 3137 y Fo(\024)1104 3076 y Fq(c)p Fp(\()p Fq(`)p Fp(\))p 1104 3116 148 4 v 1116 3200 a Fo(j)p Fq(V)21 b Fo(j)1262 3137 y(D)s Fp(\()p Fq(g)s Fp(;)15 b Fq(\027)6 b Fp(\))22 b(+)1685 3076 y Fq(c\016)p 1664 3116 124 4 v 1664 3200 a Fo(j)p Fq(V)f Fo(j)1798 3137 y Fp(V)-8 b(ar)1940 3151 y Fr(\027)1983 3137 y Fp(\()p Fq(g)s Fp(\))21 b(+)2264 3076 y Fq(c)p 2221 3116 V 2221 3200 a Fo(j)p Fq(V)g Fo(j)2355 3137 y Fp(V)-8 b(ar)2497 3151 y Fr(\027)2540 3137 y Fp(\()p Fq(g)s Fp(\)1)-25 b(I)2710 3156 y Fn(f)p Fr(m)p Fn(2)p Fr(I)2885 3168 y FA(\016)2920 3156 y Fn(g)2959 3137 y Fq(#)3013 3152 y Fr(V)t(;`)3110 3137 y Fp(\()p Fq(\013)p Fp(\))214 b Fu(\(A.7\))118 3355 y Fh(where)25 b Fq(m)g Fp(:=)h Fq(\027)6 b Fp(\()p Fq(m)765 3369 y Fr(V)825 3355 y Fp(\))26 b Fh(and)f Fq(I)1098 3370 y Fr(\016)1161 3355 y Fp(:=)h([)p Fq(\016)n(;)15 b Fp(1)22 b Fo(\000)e Fq(\016)s Fp(])p Fh(.)33 b(Moreover)-12 b(,)25 b(for)g(any)g Fq(\015)31 b(>)25 b Fp(0)h Fh(there)f(exists)g Fq(`)3101 3369 y Fl(1)3166 3355 y Fp(=)g Fq(`)3300 3369 y Fl(1)3339 3355 y Fp(\()p Fq(\015)5 b Fp(\))27 b Fo(\025)e Fq(`)3622 3369 y Fl(0)118 3463 y Fh(such)g(that)960 3588 y Fq(`)998 3602 y Fl(1)1063 3588 y Fo(\024)g Fq(`)g Fo(\024)g Fq(L)116 b Fo(\))g Fs(P)p Fp(\()15 b Fq(#)1862 3603 y Fr(V)t(;`)1960 3588 y Fp(\()p Fq(\013)p Fp(\))27 b Fo(\025)e Fq(\015)20 b Fp(\))26 b Fo(\024)e Fq(e)2476 3551 y Fn(\000)p Fr(c)p Fl(\()p Fr(\015)t(;`)p Fl(\))p Fr(L)2753 3527 y FA(d)2794 3588 y Fq(:)633 b Fu(\(A.8\))118 3733 y Fh(Finally)-5 b(,)25 b(for)g(any)f Fq(\027)31 b Fo(2)25 b(M)p Fp(\()p Fq(V)41 b Fo([)20 b Fq(W)13 b Fp(\))25 b Fh(and)f(any)h(function)f Fq(g)29 b Fo(2)c Fs(G)18 b Fh(,)373 3900 y Fq(\027)6 b Fp(\(Av)576 3914 y Fr(v)r Fn(2)p Fr(V)721 3900 y Fq(\034)761 3914 y Fr(v)802 3900 y Fq(h)p Fo(\001)p Fp(Av)997 3914 y Fr(w)r Fn(2)p Fr(W)1177 3900 y Fq(\034)1217 3914 y Fr(w)1273 3900 y Fq(h)1325 3863 y Fn(0)1349 3900 y Fp(;)15 b Fq(g)s Fp(\))1470 3863 y Fl(2)1536 3900 y Fo(\024)889 4040 y Fq(c)p Fp(\()p Fq(`)p Fp(\))p 889 4081 148 4 v 906 4164 a Fo(j)p Fp(\003)p Fo(j)1047 4102 y(D)s Fp(\()p Fq(g)s Fp(;)g Fq(\027)6 b Fp(\))21 b(+)1464 4040 y Fq(c\016)p 1449 4081 114 4 v 1449 4164 a Fo(j)p Fp(\003)p Fo(j)1573 4102 y Fp(V)-8 b(ar)1714 4116 y Fr(\027)1758 4102 y Fp(\()p Fq(g)s Fp(\))21 b(+)2033 4040 y Fq(c)p 1996 4081 V 1996 4164 a Fo(j)p Fp(\003)p Fo(j)2120 4102 y Fp(V)-8 b(ar)2261 4116 y Fr(\027)2305 4102 y Fp(\()p Fq(g)s Fp(\))2421 4028 y Fk(\000)2464 4102 y Fq(#)2518 4117 y Fr(V)t(;`)2615 4102 y Fp(\()p Fq(\013)p Fp(\))21 b(+)f Fq(#)2909 4117 y Fr(W)n(;`)3026 4102 y Fp(\()p Fq(\013)p Fp(\))3154 4028 y Fk(\001)3452 4027 y Fu(\(A.9\))118 4336 y Fh(Proof)-8 b(.)42 b Fu(W)-8 b(e)26 b(\002rst)h(prove)f(\(A.7\))f(by)i(referring,)d (for)j(many)e(steps,)j(to)e(the)g(proof)h(of)f(proposition)h(A.1)f(in) 118 4444 y([7].)32 b(Let)25 b(us)h(\002st)g(introduce)f(some)f(useful)h (notation.)118 4552 y(W)-8 b(e)35 b(\002x)f(a)g(partition)h Fq(V)61 b Fp(=)40 b Fo([)1149 4566 y Fr(i)p Fn(2)p Fr(I)1259 4552 y Fq(Q)1331 4566 y Fr(i)1359 4552 y Fu(,)d(with)e Fq(Q)1704 4566 y Fr(i)1772 4552 y Fo(2)40 b Fq(R)1942 4567 y Fr(`)1975 4552 y Fu(,)d(and)d(de\002ne)g Fq(N)2572 4566 y Fr(i)2640 4552 y Fp(:=)40 b Fq(N)2849 4566 y Fr(Q)2905 4576 y FA(i)2935 4552 y Fu(,)d Fq(m)3077 4566 y Fr(i)3145 4552 y Fp(:=)j Fq(N)3354 4566 y Fr(Q)3410 4576 y FA(i)3440 4552 y Fq(=)p Fo(j)p Fq(Q)3582 4566 y Fr(i)3611 4552 y Fo(j)p Fu(,)118 4660 y Fq(h)170 4674 y Fr(i)224 4660 y Fp(:=)345 4592 y Fk(P)441 4687 y Fr(x)p Fn(2)p Fr(Q)584 4697 y FA(i)629 4660 y Fq(\034)669 4674 y Fr(x)713 4660 y Fq(h)p Fu(,)26 b Fo(F)34 b Fp(:=)25 b Fq(\033)s Fp(\()p Fq(m)1206 4674 y Fr(i)1250 4660 y Fo(j)15 b Fq(i)26 b Fo(2)f Fq(I)7 b Fp(\))26 b Fu(and)f(for)g Fq(s)g Fo(2)g Fp([0)p Fq(;)15 b Fp(1])171 4938 y Fq(A)239 4952 y Fr(i)267 4938 y Fp(\()p Fq(m)p Fp(\))26 b(:=)585 4874 y Fq(\026)640 4826 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))640 4903 y Fr(V)802 4874 y Fp(\()p Fq(h)889 4888 y Fr(i)918 4874 y Fp(;)15 b Fq(N)1031 4888 y Fr(i)1060 4874 y Fp(\))p 574 4917 531 4 v 574 5026 a Fq(\026)629 4978 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))629 5055 y Fr(V)792 5026 y Fp(\()p Fq(N)900 5040 y Fr(i)928 5026 y Fp(;)g Fq(N)1041 5040 y Fr(i)1070 5026 y Fp(\))1135 4938 y Fo(\000)1236 4877 y Fs(E)26 b Fq(\026)1361 4844 y Fr(\025)1402 4853 y Fg(0)1443 4844 y Fl(\()p Fr(m)p Fl(\))1564 4877 y Fp(\()p Fq(h)1651 4891 y Fl(0)1691 4877 y Fp(;)15 b Fq(\021)1776 4891 y Fl(0)1816 4877 y Fp(\))p 1236 4917 616 4 v 1240 5004 a Fs(E)26 b Fq(\026)1365 4978 y Fr(\025)1406 4987 y Fg(0)1446 4978 y Fl(\()p Fr(m)p Fl(\))1568 5004 y Fp(\()p Fq(\021)1648 5018 y Fl(0)1688 5004 y Fp(;)15 b Fq(\021)1773 5018 y Fl(0)1813 5004 y Fp(\))2043 4938 y Fq(B)2112 4952 y Fr(i)2140 4938 y Fp(\()p Fq(s)p Fp(\))26 b(:=)2421 4861 y Fq(\026)2476 4813 y Fr(\025)p Fl(\()p Fr(s)p Fl(\))2476 4890 y Fr(Q)2532 4900 y FA(i)2608 4861 y Fp(\()p Fq(h)2695 4875 y Fr(i)2724 4861 y Fp(;)15 b Fq(N)2837 4875 y Fr(i)2866 4861 y Fp(\))p 2410 4917 502 4 v 2410 5026 a Fq(\026)2465 4978 y Fr(\025)p Fl(\()p Fr(s)p Fl(\))2465 5055 y Fr(Q)2521 5065 y FA(i)2598 5026 y Fp(\()p Fq(N)2706 5040 y Fr(i)2734 5026 y Fp(;)g Fq(N)2847 5040 y Fr(i)2876 5026 y Fp(\))2942 4938 y Fo(\000)3053 4874 y Fq(\026)3108 4826 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))3108 4903 y Fr(V)3270 4874 y Fp(\()p Fq(h)3357 4888 y Fr(i)3386 4874 y Fp(;)g Fq(N)3499 4888 y Fr(i)3528 4874 y Fp(\))p 3042 4917 531 4 v 3042 5026 a Fq(\026)3097 4978 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))3097 5055 y Fr(V)3260 5026 y Fp(\()p Fq(N)3368 5040 y Fr(i)3396 5026 y Fp(;)g Fq(N)3509 5040 y Fr(i)3538 5026 y Fp(\))3583 4938 y Fq(:)118 5199 y Fu(As)24 b(in)f([7],)h(if)g Fq(m)h Fo(62)g Fq(I)821 5214 y Fr(\016)882 5199 y Fu(then)e(it)h(is)g(enough)e(to)i(apply)f(Schwarz) f(inequality)h(and)g(lemma)f(A.5)h(to)h(obtain)118 5307 y(the)i(thesis,)h(otherwise)e(it)i(is)f(convenient)d(to)j(bound)f(the)h (l.h.s.)32 b(of)25 b(\(A.7\))g(as)339 5480 y Fq(\027)6 b Fp(\(Av)543 5494 y Fr(v)r Fn(2)p Fr(V)687 5480 y Fq(\034)727 5494 y Fr(v)768 5480 y Fq(h)p Fp(;)15 b Fq(g)s Fp(\))941 5443 y Fl(2)1007 5480 y Fo(\024)25 b Fp(2)p Fq(\027)1199 5407 y Fk(\000)1256 5480 y Fq(\027)6 b Fp(\(Av)1460 5494 y Fr(v)r Fn(2)p Fr(V)1605 5480 y Fq(\034)1645 5494 y Fr(v)1685 5480 y Fq(h)p Fp(;)15 b Fq(g)20 b Fo(j)15 b(F)9 b Fp(\))2004 5407 y Fk(\001)2047 5429 y Fl(2)2106 5480 y Fp(+)20 b(2)p Fq(\027)2293 5407 y Fk(\000)2350 5480 y Fq(\027)6 b Fp(\(Av)2554 5494 y Fr(v)r Fn(2)p Fr(V)2698 5480 y Fq(\034)2738 5494 y Fr(v)2779 5480 y Fq(h)15 b Fo(j)g(F)9 b Fp(\);)15 b Fq(g)3099 5407 y Fk(\001)3141 5429 y Fl(2)3401 5480 y Fu(\(A.10\))p eop %%Page: 56 56 56 55 bop 118 47 a Fj(56)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 366 y Fu(As)30 b(in)f([7)o(])h(we)e(can)h(bound)f(the)i(\002rst)f(addendum)g(in)g(the) g(r)-10 b(.h.s.)44 b(of)29 b(\(A.10\))f(by)h Fq(c)p Fp(\()p Fq(`)p Fp(\))p Fo(D)s Fp(\()p Fq(g)s Fp(;)15 b Fq(\027)6 b Fp(\))31 b Fu(and)e(the)118 474 y(second)c(one)f(by)1123 636 y Fq(c)p Fp(V)-8 b(ar)1304 650 y Fr(\027)1347 636 y Fp(\()p Fq(g)s Fp(\))1463 535 y Fk(\020)1595 574 y Fp(1)p 1529 615 179 4 v 1529 698 a Fq(`)1567 672 y Fr(d)1607 698 y Fq(L)1669 672 y Fr(\016)1737 636 y Fp(+)1866 574 y(1)p 1838 615 103 4 v 1838 698 a Fq(L)1900 672 y Fr(d)1965 550 y Fk(X)1977 746 y Fr(i)p Fn(2)p Fr(I)2112 636 y Fp(V)g(ar)2253 677 y Fr(\026)2295 642 y FA(\025)p Fg(\()p FA(m)p Fg(\))2295 700 y FA(V)2443 636 y Fp(\()p Fq(\030)2522 592 y Fr(\015)2518 663 y(i)2567 636 y Fp(\))2602 535 y Fk(\021)3401 636 y Fu(\(A.11\))118 898 y(where,)41 b(for)c(an)h(arbitrarily)g(\002xed)g Fq(\015)5 b Fu(,)41 b Fq(\030)1545 854 y Fr(\015)1541 925 y(i)1590 898 y Fp(\()p Fq(\021)s Fp(\))47 b(:=)f Fq(\026)1952 850 y Fr(\025)p Fl(\()p Fr(m)2082 860 y FA(i)2110 850 y Fl(\()p Fr(\021)r Fl(\)\))1952 926 y Fr(Q)2008 936 y FA(i)2234 898 y Fp(\()p Fq(h)2321 912 y Fr(i)2378 898 y Fo(\000)29 b Fq(\015)5 b(N)2603 912 y Fr(i)2631 898 y Fp(\))p Fu(.)70 b(Let)38 b(us)g(explain)f(how)h(to) 118 1015 y(proceed.)31 b(Thanks)26 b(to)f(P)-5 b(oincar\264)-45 b(e)25 b(inequality)g(for)g(Glauber)g(dynamics)g(we)g(obtain)1141 1192 y Fp(V)-8 b(ar)1283 1234 y Fr(\026)1325 1198 y FA(\025)p Fg(\()p FA(m)p Fg(\))1325 1257 y FA(V)1473 1192 y Fp(\()p Fq(\030)1552 1148 y Fr(\015)1548 1219 y(i)1597 1192 y Fp(\))25 b Fo(\024)g Fq(c)1842 1106 y Fk(X)1823 1303 y Fr(x)p Fn(2)p Fr(Q)1966 1313 y FA(i)2007 1192 y Fq(\026)2062 1144 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))2062 1221 y Fr(V)2224 1192 y Fp(\(\()p Fo(r)2370 1206 y Fr(x)2414 1192 y Fq(\030)2458 1148 y Fr(\015)2454 1219 y(i)2503 1192 y Fp(\))2538 1155 y Fl(2)2578 1192 y Fp(\))p Fq(:)763 b Fu(\(A.12\))118 1489 y(By)25 b(choosing)g Fq(\015)31 b Fp(=)803 1445 y Fm(E)21 b Fr(\026)901 1421 y FA(\025)938 1436 y Fg(0)972 1421 y(\()p FA(m)p Fg(\))1079 1445 y Fl(\()p Fr(h)1147 1454 y Fg(0)1182 1445 y Fl(;)p Fr(\021)1237 1454 y Fg(0)1271 1445 y Fl(\))p 803 1469 497 4 v 806 1532 a Fm(E)f Fr(\026)903 1510 y FA(\025)940 1525 y Fg(0)975 1510 y(\()p FA(m)p Fg(\))1082 1532 y Fl(\()p Fr(\021)1144 1541 y Fg(0)1179 1532 y Fl(;)p Fr(\021)1234 1541 y Fg(0)1269 1532 y Fl(\))1334 1489 y Fu(it)26 b(is)g(simple)f(to)h(check)e(that)1153 1755 y Fo(r)1229 1769 y Fr(x)1273 1755 y Fq(\030)1317 1710 y Fr(\015)1313 1782 y(i)1386 1755 y Fp(=)h Fq(A)1550 1769 y Fr(i)1599 1755 y Fp(+)20 b(\()p Fo(\000)p Fp(1\))1876 1717 y Fr(\021)1911 1725 y FA(x)1970 1631 y Fk(Z)2061 1657 y Fr(m)2123 1667 y FA(i)2150 1657 y Fl(\()p Fr(\021)2214 1634 y FA(x)2254 1657 y Fl(\))2021 1837 y Fr(m)2083 1847 y FA(i)2110 1837 y Fl(\()p Fr(\021)r Fl(\))2301 1755 y Fq(B)2370 1769 y Fr(i)2398 1755 y Fp(\()p Fq(s)p Fp(\))p Fq(ds:)118 1978 y Fu(By)25 b(writing)201 2260 y Fq(B)270 2274 y Fr(i)298 2260 y Fp(\()p Fq(s)p Fp(\))h(=)733 2198 y Fo(j)p Fq(Q)830 2212 y Fr(i)859 2198 y Fo(j)p 543 2239 532 4 v 543 2358 a Fq(\026)598 2300 y Fr(\025)639 2314 y Fg(\()667 2300 y Fr(m)p Fl(\))598 2387 y Fr(V)761 2358 y Fp(\()p Fq(N)869 2372 y Fr(i)897 2358 y Fp(;)15 b Fq(N)1010 2372 y Fr(i)1039 2358 y Fp(\))1084 2159 y Fk(\020)1139 2136 y(Z)1230 2162 y Fr(s)1189 2342 y(\026)1231 2307 y FA(\025)p Fg(\()p FA(m)p Fg(\))1231 2365 y FA(V)1375 2342 y Fl(\()p Fr(m)1464 2352 y FA(i)1491 2342 y Fl(\))1548 2183 y Fq(\026)1603 2135 y Fr(\025)p Fl(\()p Fr(s)1704 2112 y Fi(0)1726 2135 y Fl(\))1603 2212 y Fr(Q)1659 2222 y FA(i)1758 2183 y Fp(\()p Fq(h)1845 2197 y Fr(i)1874 2183 y Fp(;)g Fq(N)1987 2197 y Fr(i)2015 2183 y Fp(;)g Fq(N)2128 2197 y Fr(i)2157 2183 y Fp(\))p 1548 2239 645 4 v 1608 2348 a Fq(\026)1663 2300 y Fr(\025)p Fl(\()p Fr(s)1764 2281 y Fi(0)1787 2300 y Fl(\))1663 2376 y Fr(Q)1719 2386 y FA(i)1818 2348 y Fp(\()p Fq(N)1926 2362 y Fr(i)1955 2348 y Fp(;)g Fq(N)2068 2362 y Fr(i)2096 2348 y Fp(\))2202 2260 y Fq(ds)2292 2222 y Fn(0)2315 2260 y Fp(+)1785 2523 y Fq(\026)1840 2476 y Fr(\025)p Fl(\()p Fr(s)p Fl(\))1840 2552 y Fr(Q)1896 2562 y FA(i)1973 2523 y Fp(\()p Fq(h)2060 2537 y Fr(i)2089 2523 y Fp(;)g Fq(N)2202 2537 y Fr(i)2231 2523 y Fp(\))p 1775 2580 502 4 v 1775 2688 a Fq(\026)1830 2640 y Fr(\025)p Fl(\()p Fr(s)p Fl(\))1830 2717 y Fr(Q)1886 2727 y FA(i)1963 2688 y Fp(\()p Fq(N)2071 2702 y Fr(i)2099 2688 y Fp(;)g Fq(N)2212 2702 y Fr(i)2241 2688 y Fp(\))2301 2477 y Fk(Z)2392 2503 y Fr(s)2352 2683 y(\026)2394 2648 y FA(\025)p Fg(\()p FA(m)p Fg(\))2394 2706 y FA(V)2538 2683 y Fl(\()p Fr(m)2627 2693 y FA(i)2654 2683 y Fl(\))2711 2523 y Fq(\026)2766 2476 y Fr(\025)p Fl(\()p Fr(s)2867 2452 y Fi(0)2889 2476 y Fl(\))2766 2552 y Fr(Q)2822 2562 y 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3073 y Fr(\015)1177 3144 y(i)1226 3117 y Fo(j)25 b(\024)g Fq(A)1440 3131 y Fr(i)1489 3117 y Fp(+)1592 3056 y Fq(c)p 1590 3096 44 4 v 1590 3180 a(\016)1644 3040 y Fk(\014)1644 3095 y(\014)1689 3117 y Fq(m)1769 3131 y Fr(i)1797 3117 y Fp(\()p Fq(\021)s Fp(\))c Fo(\000)f Fq(\026)2082 3069 y Fr(\025)p Fl(\()p Fr(m)p Fl(\))2082 3146 y Fr(V)2245 3117 y Fp(\()p Fq(m)2360 3131 y Fr(i)2388 3117 y Fp(\))2423 3040 y Fk(\014)2423 3095 y(\014)2474 3117 y Fp(+)2577 3056 y Fq(c)p 2575 3096 V 2575 3180 a(\016)2628 3117 y(`)2666 3080 y Fn(\000)p Fr(d)2762 3117 y Fq(:)614 b Fu(\(A.13\))118 3287 y(By)25 b(\(A.11\),)g(\(A.12\))g(and)g(\(A.13\))g(it)h(is)g(simple)f(to)g (conclude)f(the)h(proof)h(if)g Fq(`)f Fu(is)h(large)e(enough)g(and)467 3486 y Fq(#)521 3501 y Fr(V)t(;`)618 3486 y Fp(\()p Fq(\013)p Fp(\))i(:=)72 b(sup)894 3565 y Fr(m)p Fn(2)p Fr(M)1071 3576 y FA(V)1139 3486 y Fp(Av)1257 3500 y Fr(i)p Fn(2)p Fr(I)1368 3486 y Fq(A)1436 3500 y Fr(i)1464 3486 y Fp(\()p Fq(m)p Fp(\))1614 3449 y Fl(2)1745 3486 y Fu(where)90 b Fq(M)2169 3500 y Fr(V)2256 3486 y Fp(=)2352 3413 y Fk(\010)2454 3425 y Fp(1)p 2415 3466 124 4 v 2415 3549 a Fo(j)p Fq(V)20 b Fo(j)2548 3486 y Fq(;)2638 3425 y Fp(2)p 2599 3466 V 2599 3549 a Fo(j)p Fq(V)g Fo(j)2732 3486 y Fq(;)15 b(:)g(:)g(:)i(;)e Fp(1)21 b Fo(\000)3140 3425 y Fp(1)p 3101 3466 V 3101 3549 a Fo(j)p Fq(V)f Fo(j)3235 3413 y Fk(\011)3288 3486 y Fq(:)118 3737 y Fu(By)25 b(standard)i (arguments)d(\(as)h(for)g(lemma)f Fp(3)p Fq(:)p Fp(9)j Fu(in)e([7]\))g(\(A.9\))g(can)g(be)g(derived)g(from)g(\(A.7\).)118 3845 y(Let)h(us)f(prove)g(\(A.8\).)31 b(By)25 b(lemmas)f(A.2)i(and)f (A.3)374 4028 y Fo(j)p Fq(A)467 4042 y Fr(i)495 4028 y Fo(j)h(\024)825 3967 y Fq(c)p 652 4007 387 4 v 652 4091 a(m)p Fp(\(1)20 b Fo(\000)g Fq(m)p Fp(\))1048 3927 y Fk(\020)1118 4028 y Fo(j)p Fq(\026)1198 3991 y Fr(\025)1239 4000 y Fg(0)1274 3991 y Fl(\()p Fr(m)p Fl(\))1395 4028 y Fp(\()p Fq(h)1482 4042 y Fr(i)1511 4028 y Fp(;)15 b Fq(m)1631 4042 y Fr(i)1659 4028 y Fp(\))21 b Fo(\000)f Fs(E)26 b Fq(\026)1931 3991 y Fr(\025)1972 4000 y Fg(0)2012 3991 y Fl(\()p Fr(m)p Fl(\))2134 4028 y Fp(\()p Fq(h)2221 4042 y Fl(0)2261 4028 y Fp(;)15 b Fq(\021)2346 4042 y Fl(0)2386 4028 y Fp(\))p Fo(j)p Fp(+)1118 4253 y Fo(j)p Fp(\()p Fq(\026)1233 4216 y Fr(\025)1274 4225 y Fg(0)1309 4216 y Fl(\()p Fr(m)p Fl(\))1431 4253 y Fp(\()p Fq(N)1539 4267 y Fr(i)1567 4253 y Fp(;)g Fq(m)1687 4267 y Fr(i)1716 4253 y Fp(\))20 b Fo(\000)g Fs(E)26 b Fq(\026)1987 4216 y Fr(\025)2028 4225 y Fg(0)2068 4216 y Fl(\()p Fr(m)p Fl(\))2190 4253 y Fp(\()p Fq(\021)2270 4267 y Fl(0)2310 4253 y Fp(;)15 b Fq(\021)2395 4267 y Fl(0)2435 4253 y Fp(\))p Fo(j)20 b Fp(+)g Fo(j)p Fq(m)h Fo(\000)f Fq(\026)2878 4216 y Fr(\025)2919 4225 y Fg(0)2953 4216 y Fl(\()p Fr(m)p Fl(\))3075 4253 y Fp(\()p Fq(m)3190 4267 y Fr(V)3251 4253 y Fp(\))p Fo(j)3326 4152 y Fk(\021)3381 4253 y Fq(:)118 4440 y Fu(Therefore)29 b(it)h(is)g(enough)e(to)i(prove)f(that)h(given)f (a)g(function)g Fq(f)42 b Fp(=)32 b Fq(f)10 b Fp(\()p Fq(\013)2601 4454 y Fl(0)2640 4440 y Fp(\))30 b Fu(with)g Fo(k)p Fq(f)10 b Fo(k)3056 4454 y Fn(1)3163 4440 y Fo(\024)32 b Fp(1)e Fu(then)g(for)118 4548 y(any)25 b Fq(\015)30 b(>)25 b Fp(0)h Fu(there)f(exists)h Fq(`)1043 4562 y Fl(1)1108 4548 y Fp(=)f Fq(`)1242 4562 y Fl(1)1281 4548 y Fp(\()p Fq(\015)5 b Fp(\))26 b Fu(such)f(that)715 4762 y Fs(P)770 4689 y Fk(\000)812 4762 y Fp(Av)930 4776 y Fr(i)p Fn(2)p Fr(I)1041 4762 y Fp(\(Av)1194 4776 y Fr(x)p Fn(2)p Fr(Q)1337 4786 y FA(i)1366 4762 y Fq(\034)1406 4776 y Fr(x)1450 4762 y Fq(f)k Fo(\000)20 b Fs(E)11 b Fp(\()p Fq(f)f Fp(\)\))1830 4725 y Fl(2)1901 4762 y Fo(\025)25 b Fq(\015)2064 4689 y Fk(\001)2131 4762 y Fo(\024)g Fp(2)p Fq(e)2314 4716 y Fn(\000)2379 4685 y FA(c\015)2442 4664 y Fg(2)2478 4685 y FA(L)2520 4664 y(d)p 2380 4701 177 3 v 2437 4753 a(`)2463 4739 y(d)2752 4762 y Fo(8)p Fq(`)g Fo(\025)g Fq(`)3000 4776 y Fl(1)3039 4762 y Fq(:)118 4929 y Fu(T)-9 b(o)24 b(this)g(aim)g(we)e(de\002ne)h Fq(f)1024 4943 y Fr(i)1077 4929 y Fp(:=)i(\(Av)1351 4943 y Fr(x)p Fn(2)p Fr(Q)1494 4953 y FA(i)1524 4929 y Fq(\034)1564 4943 y Fr(x)1607 4929 y Fq(f)e Fo(\000)13 b Fs(E)d Fp(\()p Fq(f)g Fp(\)\))1973 4896 y Fl(2)2042 4929 y Fu(and)2234 4905 y Fp(\026)2214 4929 y Fq(f)2259 4943 y Fr(i)2312 4929 y Fp(:=)25 b Fq(f)2478 4943 y Fr(i)2519 4929 y Fo(\000)13 b Fs(E)d Fp(\()q Fq(f)2738 4943 y Fr(i)2771 4929 y Fp(\))p Fu(.)31 b(Then)23 b(by)g(lemma)f Fq(A:)p Fp(1)p Fu(,)118 5037 y(for)j(any)g Fp(0)h Fq(<)f(\016)k(<)c Fp(1)p Fu(,)744 5212 y Fs(E)10 b Fp(\()p Fq(f)878 5226 y Fr(i)912 5212 y Fp(\))26 b Fo(\024)f Fs(P)1124 5138 y Fk(\000)1181 5212 y Fo(j)p Fp(Av)1324 5226 y Fr(x)p Fn(2)p Fr(Q)1467 5236 y FA(i)1497 5212 y Fq(\034)1537 5226 y Fr(x)1580 5212 y Fq(f)30 b Fo(\000)20 b Fs(E)10 b Fp(\()q Fq(f)g Fp(\))p Fo(j)31 b(\025)25 b Fq(\016)2137 5138 y Fk(\001)2199 5212 y Fp(+)19 b Fq(\016)2332 5174 y Fl(2)2398 5212 y Fo(\024)25 b Fp(2)p Fq(e)2581 5174 y Fn(\000)p Fr(c)12 b(\016)2712 5151 y Fg(2)2747 5174 y Fr(`)2776 5151 y FA(d)2837 5212 y Fp(+)19 b Fq(\016)2970 5174 y Fl(2)3011 5212 y Fq(:)118 5372 y Fu(Therefore,)30 b(by)g(choosing)g Fq(\016)j Fu(small)d(enough)e(and)i Fq(`)g Fu(large)e(enough,)i Fs(E)11 b Fp(\()p Fq(f)2667 5386 y Fr(i)2701 5372 y Fp(\))33 b Fo(\024)2883 5331 y Fr(\015)p 2883 5351 41 4 v 2885 5404 a Fl(2)2963 5372 y Fu(for)c(any)h Fq(i)j Fo(2)f Fq(I)37 b Fu(and)118 5480 y(\(by)25 b(applying)g(again)g(lemma)f(A.1\)) p eop %%Page: 57 57 57 56 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(57)958 484 y Fs(P)p Fp(\()15 b(Av)1182 498 y Fr(i)p Fn(2)p Fr(I)1293 484 y Fq(f)1338 498 y Fr(i)1391 484 y Fo(\025)24 b Fq(\015)d Fp(\))k Fo(\024)g Fs(P)p Fp(\()15 b(Av)1934 498 y Fr(i)p Fn(2)p Fr(I)2064 460 y Fp(\026)2045 484 y Fq(f)2090 498 y Fr(i)2143 484 y Fo(\025)2249 422 y Fq(\015)p 2249 463 53 4 v 2252 546 a Fp(2)2326 484 y(\))25 b Fo(\024)g Fp(2)p Fq(e)2569 446 y Fn(\000)p Fr(c)12 b(\015)2707 423 y Fg(2)2742 446 y Fn(j)p Fr(I)5 b Fn(j)3401 484 y Fu(\(A.14\))118 649 y(thus)26 b(concluding)d(the)j(proof)m(.)2432 b Fe(\003)118 827 y Fu(A.5.)46 b FE(Moving)26 b(P)m(article)g(L)r(emma.)46 b Fu(Given)25 b Fq(x;)15 b(y)29 b Fo(2)c Fs(Z)1958 794 y Fr(d)2019 827 y Fu(we)g(de\002ne)912 981 y Fq(z)954 995 y Fr(i)1008 981 y Fp(:=)g(\()p Fq(y)1209 995 y Fl(1)1248 981 y Fq(;)15 b(y)1333 995 y Fl(2)1373 981 y Fq(;)g(:)g(:)g(:)h(;)f(y) 1619 995 y Fr(i)1647 981 y Fq(;)g(x)1739 995 y Fr(i)p Fl(+1)1858 981 y Fq(;)g(:)g(:)g(:)i(;)e(x)2112 996 y Fr(d)2153 981 y Fp(\))182 b Fo(8)p Fq(i)25 b Fp(=)g(0)p Fq(;)15 b(:)g(:)g(:)i(;)e(d)118 1135 y Fu(and)32 b(write)h Fq(\015)584 1149 y Fr(x;y)717 1135 y Fu(for)f(the)h(path)g(connecting)d Fq(z)1738 1149 y Fl(0)1815 1135 y Fp(=)37 b Fq(x)32 b Fu(to)h Fq(z)2161 1149 y Fl(1)2233 1135 y Fu(by)f(moving)f(along)g(the) i(\002rst)g(direction,)118 1243 y(then)26 b(connecting)e Fq(z)823 1257 y Fl(1)889 1243 y Fu(to)j Fq(z)1037 1257 y Fl(2)1102 1243 y Fu(by)f(moving)g(along)f(the)h(second)g(direction)g (and)g(so)h(on)e(until)h(arriving)g(to)118 1351 y Fq(z)160 1366 y Fr(d)226 1351 y Fp(=)f Fq(y)s Fu(.)31 b(W)-8 b(e)26 b(denote)f(by)g Fo(j)p Fq(\015)1061 1365 y Fr(x;y)1162 1351 y Fo(j)h Fu(the)f(length)g(of)g(the)h(path)g Fq(\015)2146 1365 y Fr(x;y)2247 1351 y Fu(.)118 1482 y FE(L)r(emma)h(A.7.)42 b Fh(\(Moving)25 b(P)m(articles)f(Lemma\))118 1590 y(Given)h(a)g(box)h Fp(\003)f Fh(and)f Fq(\027)31 b Fo(2)25 b(M)p Fp(\(\003\))h Fh(then)676 1763 y Fq(\027)727 1689 y Fk(\000)784 1763 y Fp(\()p Fo(r)895 1777 y Fr(x;y)996 1763 y Fq(f)10 b Fp(\))1086 1725 y Fl(2)1140 1689 y Fk(\001)1207 1763 y Fo(\024)25 b Fq(c)15 b Fo(j)p Fq(\015)1429 1777 y Fr(x;y)1530 1763 y Fo(j)1607 1676 y Fk(X)1570 1874 y Fr(b)p Fn(2)p Fr(\015)1683 1882 y FA(x;y)1791 1763 y Fq(\027)1842 1689 y Fk(\000)1898 1763 y Fp(\()p Fo(r)2009 1778 y Fr(b)2044 1763 y Fq(f)10 b Fp(\))2134 1725 y Fl(2)2188 1689 y Fk(\001)2411 1763 y Fo(8)p Fq(x;)15 b(y)28 b Fo(2)d Fp(\003)p Fq(;)41 b(f)34 b Fo(2)25 b Fs(G)19 b Fq(:)218 2023 y Fu(The)k(above)f(lemma)f (is)j(well)e(known)g(for)h(non)f(disordered)h(systems)h(\(see)e(for)h (example)e([35)o(]\).)31 b(W)-8 b(e)118 2131 y(learned)24 b(from)h(J.Quastel)h(the)g(generalization)d(to)j(the)g(disordered)f (case.)118 2309 y(A.6.)46 b FE(An)22 b(application)g(of)h(F)-6 b(eynman-K)r(ac)23 b(formula.)46 b Fu(The)22 b(following)e(proposition) j(can)d(be)h(derived)118 2417 y(from)k(the)g(F)-6 b(eynman-Kac)23 b(formula)i(as)g(explained)g(in)g([23].)32 b(W)-8 b(e)25 b(report)g(only)g(the)g(statement.)118 2548 y(Let)d Fq(X)29 b Fu(be)21 b(a)h(\002nite)g(set)g(on)g(which)g(it)g(is)h(de\002ned)e(a) h(probability)h(measure)d Fq(\027)27 b Fu(and)22 b(a)g(Markov)f (generator)118 2656 y Ff(L)30 b Fu(reversible)g(w)-11 b(.r)h(.t.)49 b Fq(\027)6 b Fu(.)47 b(W)-8 b(e)31 b(denote)g(by)f Fs(E)1620 2670 y Fr(\027)1700 2656 y Fu(the)h(expectation)f(w)-11 b(.r)h(.t.)49 b(the)31 b(Markov)f(process)h(having)118 2764 y(in\002nitesimal)26 b(generator)e Ff(L)g Fu(and)i(initial)g (distribution)h Fq(\027)k Fu(and)26 b(by)f Fq(x)2467 2778 y Fr(t)2522 2764 y Fu(the)g(con\002guration)f(at)i(time)g Fq(t)p Fu(.)118 2895 y FE(Proposition)k(A.8.)44 b Fh(Let)28 b Fq(V)50 b Fp(:)30 b Fs(R)1202 2909 y Fl(+)1289 2895 y Fo(\002)22 b Fq(X)37 b Fo(!)30 b Fs(R)37 b Fh(be)28 b(a)g(bounded)e(measurable)h(function)g(and)g(let,)h(for)g(any)118 3003 y Fq(t)d(>)g Fp(0)p Fh(,)1257 3119 y Fp(\000)1314 3133 y Fr(t)1369 3119 y Fp(:=)g(sup)14 b Fq(spec)1812 3139 y Fr(L)1860 3120 y Fg(2)1894 3139 y Fl(\()p Fr(\027)t Fl(\))1992 3119 y Fo(f)p Ff(L)21 b Fp(+)f Fq(V)g Fp(\()p Fq(t;)15 b Fo(\001)p Fp(\))p Fo(g)p Fq(:)118 3258 y Fh(Then)846 3413 y Fs(E)900 3427 y Fr(\027)950 3312 y Fk(h)1008 3413 y Fp(exp)1146 3312 y Fk(n)1207 3289 y(Z)1298 3316 y Fr(t)1258 3496 y Fl(0)1343 3413 y Fq(V)20 b Fp(\()p Fq(s;)15 b(x)1586 3427 y Fr(s)1623 3413 y Fp(\))p Fq(ds)1748 3312 y Fk(oi)1877 3413 y Fo(\024)25 b Fp(exp)2112 3312 y Fk(n)2173 3289 y(Z)2264 3316 y Fr(t)2223 3496 y Fl(0)2308 3413 y Fp(\000)2365 3427 y Fr(s)2417 3413 y Fq(ds)2507 3312 y Fk(o)2659 3413 y Fo(8)p Fq(t)f(>)h Fp(0)p Fq(:)118 3642 y Fu(A.7.)46 b FE(T)-7 b(wo)24 b(Blocks)h(Estimate.)45 b Fu(F)-6 b(or)23 b(a)g(treatment)g(of)g(the)h(T)-7 b(wo)24 b(Blocks)f(estimate)h(in)f (non)g(disordered)118 3750 y(systems)j(see)f([23)o(])h(and)f(reference) e(therein.)31 b(Let)26 b(us)f(state)h(and)g(prove)e(a)i(generalized)d (version.)118 3881 y FE(Proposition)k(A.9.)42 b Fh(Given)25 b Fq(\015)30 b(>)25 b Fp(0)p Fh(,)h(for)f(almost)g(any)f(disorder)g (con\002guration)g Fq(\013)386 4039 y Fp(lim)15 b(sup)330 4118 y Fr(a)p Fn(#)p Fl(0)p Fr(;k)r Fn("1)p Fr(;\017)p Fn(#)p Fl(0)803 4039 y Fp(sup)736 4122 y Fr(w)r Fl(:)p Fn(j)p Fr(w)r Fn(j\024)967 4094 y FA(a)p 964 4106 34 3 v 968 4148 a(\017)1023 4039 y Fp(sup)f Fq(spec)1345 4059 y Fr(L)1393 4040 y Fg(2)1427 4059 y Fl(\()p Fr(\026)1496 4067 y FA(\017)1528 4059 y Fl(\))1559 4039 y Fo(f)p Fp(Av)1722 4061 y Fr(x)p Fn(2)p Fm(T)1859 4042 y FA(d)1859 4078 y(\017)1893 4039 y Fo(j)p Fq(m)1998 4054 y Fr(x;k)2121 4039 y Fo(\000)20 b Fq(m)2292 4054 y Fr(x)p Fl(+)p Fr(w)r(;k)2501 4039 y Fo(j)g Fp(+)g Fq(\015)5 b(\017)2726 4001 y Fr(d)p Fn(\000)p Fl(2)2857 4039 y Fo(L)2920 4053 y Fr(\017)2952 4039 y Fo(g)26 b(\024)f Fp(0)p Fq(:)212 b Fu(\(A.15\))118 4293 y Fh(Proof)-8 b(.)42 b Fu(W)-8 b(e)39 b(extend)f(to)h(the)g (disordered)g(case)f(the)h(proof)g(of)g(the)f(T)-7 b(wo)40 b(Blocks)e(estimate)h(of)g([37)o(])118 4401 y(thanks)g(to)f(the)g (ergodicity)g(of)g(the)g(random)f(\002eld)h Fq(\013)p Fu(.)69 b(T)-9 b(o)38 b(this)i(aim)e(let)f(us)h(introduce)f(the)h (scale)118 4509 y(parameter)23 b Fq(`)g Fu(with)i Fq(`)g Fo(")h(1)e Fu(after)f Fq(k)29 b Fo(")d(1)p Fu(.)k(Then,)25 b(with)f(a)g(negligible)e(error)h(of)h(order)f Fq(O)s Fp(\()p Fq(`=k)s Fp(\))p Fu(,)i(for)f(any)118 4619 y Fq(x)33 b Fo(2)g Fs(T)356 4586 y Fr(d)356 4641 y(\017)428 4619 y Fu(we)c(can)g(substitute)j Fq(m)1237 4634 y Fr(x;k)1368 4619 y Fu(with)f Fp(Av)1693 4633 y Fr(y)r Fn(2)p Fl(\003)1826 4645 y FA(k)1868 4619 y Fq(m)1948 4634 y Fr(x)p Fl(+)p Fr(y)r(;`)2133 4619 y Fu(.)45 b(Therefore,)30 b(thanks)h(to)f(the)g (subadditivy)118 4727 y(of)c Fp(sup)14 b Fq(spec)p Fu(,)25 b(the)h(l.h.s.)32 b(of)26 b(\(A.15\))e(can)h(be)f(bounded)h(from)g (above)g(\(with)i(an)e(error)f Fq(O)s Fp(\()p Fq(\016)s Fp(\))p Fu(\))i(by)293 4888 y Fp(sup)225 4970 y Fr(w)r Fl(:)p Fn(j)p Fr(w)r Fn(j\024)456 4943 y FA(a)p 454 4955 V 458 4996 a(\017)513 4888 y Fp(Av)630 4902 y Fr(y)r Fn(2)p Fl(\003)763 4914 y FA(k)805 4888 y Fp(Av)923 4906 y Fr(y)960 4887 y Fi(0)983 4906 y Fn(2)p Fl(\003)1079 4918 y FA(k)1128 4906 y Fl(:)12 b Fn(j)p Fr(w)r Fl(+)p Fr(y)1325 4887 y Fi(0)1346 4906 y Fn(\000)p Fr(y)r Fn(j)p Fr(>)p Fl(2)p Fr(`)1606 4888 y Fp(sup)j Fq(spec)1662 4972 y Fr(L)1710 4953 y Fg(2)1745 4972 y Fl(\()p Fr(\026)1814 4980 y FA(\017)1845 4972 y Fl(\))1929 4888 y Fo(f)p Fp(Av)2091 4910 y Fr(x)p Fn(2)p Fm(T)2228 4891 y FA(d)2228 4926 y(\017)2263 4888 y Fo(j)p Fq(m)2368 4903 y Fr(x)p Fl(+)p Fr(y)r(;`)2572 4888 y Fo(\000)20 b Fq(m)2743 4903 y Fr(x)p Fl(+)p Fr(w)r Fl(+)p Fr(y)2983 4884 y Fi(0)3005 4903 y Fr(;`)3057 4888 y Fo(j)h Fp(+)f Fq(\015)5 b(\017)3283 4850 y Fr(d)p Fn(\000)p Fl(2)3413 4888 y Fo(L)3476 4902 y Fr(\017)3509 4888 y Fo(g)118 5130 y Fu(where)37 b(the)g(additional)i (restriction)f Fo(j)p Fq(w)31 b Fp(+)d Fq(y)1713 5097 y Fn(0)1764 5130 y Fo(\000)g Fq(y)s Fo(j)45 b Fq(>)h Fp(2)p Fq(`)37 b Fu(is)h(painless.)68 b(By)37 b(renaming)f(the)i(index) 118 5238 y(variables,)26 b(it)g(is)g(enough)e(to)h(show)h(that)h(given) d Fq(\015)31 b(>)25 b Fp(0)p Fu(,)h(for)f(almost)g(any)g(disorder)h (con\002guration)e Fq(\013)p Fu(,)436 5396 y Fp(lim)15 b(sup)384 5475 y Fr(`)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)908 5396 y Fp(sup)781 5478 y Fr(w)r Fl(:2)p Fr(`<)p Fn(j)p Fr(w)r Fn(j\024)1131 5451 y FA(a)p 1128 5463 V 1132 5504 a(\017)1187 5396 y Fp(sup)f Fq(spec)1243 5480 y Fr(L)1291 5461 y Fg(2)1325 5480 y Fl(\()p Fr(\026)1394 5488 y FA(\017)1425 5480 y Fl(\))1524 5396 y Fo(f)p Fp(Av)1687 5418 y Fr(x)p Fn(2)p Fm(T)1824 5399 y FA(d)1824 5435 y(\017)1858 5396 y Fo(j)p Fq(m)1963 5411 y Fr(x;`)2076 5396 y Fo(\000)20 b Fq(m)2247 5411 y Fr(x)p Fl(+)p Fr(w)r(;`)2446 5396 y Fo(j)h Fp(+)f Fq(\015)5 b(\017)2672 5358 y Fr(d)p Fn(\000)p Fl(2)2802 5396 y Fo(L)2865 5410 y Fr(\017)2898 5396 y Fo(g)25 b(\024)g Fp(0)p Fq(:)267 b Fu(\(A.16\))p eop %%Page: 58 58 58 57 bop 118 45 a Fj(58)1151 b(A.)19 b(F)-6 b(AGGIONA)f(TO)17 b(AND)i(F)-13 b(.)19 b(MAR)n(TINELLI)118 370 y Fu(F)-6 b(or)24 b(any)g Fq(u;)15 b(v)29 b Fo(2)c Fs(Z)752 337 y Fr(d)813 370 y Fu(let)f(us)g(de\002ne)1340 347 y Fp(^)1319 370 y Fo(L)1382 384 y Fr(u;v)1508 370 y Fp(=)h(\(1)18 b(+)f Fq(e)1832 337 y Fn(\000)p Fl(\()p Fr(\013)1959 345 y FA(u)2001 337 y Fn(\000)p Fr(\013)2101 345 y FA(v)2139 337 y Fl(\)\()p Fr(\021)2228 345 y FA(u)2270 337 y Fn(\000)p Fr(\021)2360 345 y FA(v)2397 337 y Fl(\))2429 370 y Fp(\))p Fo(r)2540 384 y Fr(u;v)2641 370 y Fu(.)31 b(It)26 b(is)f(simple)f(to)h (check)e(that)139 468 y Fp(^)118 491 y Fo(L)181 505 y Fr(u;v)312 491 y Fu(is)31 b(self)n(-adjoint)g(w)-11 b(.r)h(.t.)46 b(Gibbs)31 b(measures.)44 b(Then,)31 b(given)e Fq(w)k Fu(as)d(above,)h(thanks)g(to)f(the)g(Moving)118 598 y(P)-5 b(article)29 b(lemma)f(\(see)g(lemma)g(A.7\))i(and)f(the)h(properties)f (of)h(the)f(transition)i(rates,)g(it)f(is)g(simple)f(to)118 706 y(prove)c(that)860 824 y Fp(Av)977 846 y Fr(x)p Fn(2)p Fm(T)1114 827 y FA(d)1114 863 y(\017)1149 824 y Fp(Av)1266 838 y Fr(u)p Fn(2)p Fl(\003)1403 850 y FA(x;`)1491 824 y Fp(Av)1608 838 y Fr(v)r Fn(2)p Fl(\003)1740 850 y FA(x)p Fg(+)p FA(w)q(;`)1921 824 y Fp(\()p Fo(\000)2049 801 y Fp(^)2027 824 y Fo(L)2090 838 y Fr(u;v)2191 824 y Fp(\))h Fo(\024)f Fq(c)15 b(a)2450 786 y Fl(2)2490 824 y Fq(\017)2527 786 y Fr(d)p Fn(\000)p Fl(2)2657 824 y Fp(\()p Fo(\000L)2826 838 y Fr(\017)2859 824 y Fp(\))p Fq(:)482 b Fu(\(A.17\))118 960 y(Therefore,)25 b(by)g(localizing)f(as)i(in)f(\(4.12\),)f(the)i Fq(supspec)e Fu(in)i(\(A.16\))e(is)i(bounded)f(by)324 1116 y Fp(Av)442 1138 y Fr(x)p Fn(2)p Fm(T)579 1119 y FA(d)579 1155 y(\017)628 1116 y Fp(sup)677 1188 y Fr(\027)795 1116 y Fp(sup)14 b Fq(spec)1117 1135 y Fr(L)1165 1117 y Fg(2)1200 1135 y Fl(\()p Fr(\027)t Fl(\))1298 1116 y Fo(fj)p Fq(m)1448 1131 y Fr(x;`)1561 1116 y Fo(\000)20 b Fq(m)1732 1131 y Fr(x)p Fl(+)p Fr(w)r(;`)1931 1116 y Fo(j)h Fp(+)f Fq(c)15 b(\015)5 b(a)2222 1078 y Fn(\000)p Fl(2)2332 1116 y Fp(Av)2449 1130 y Fr(u)p Fn(2)p Fl(\003)2586 1142 y FA(x;`)2674 1116 y Fp(Av)2791 1130 y Fr(v)r Fn(2)p Fl(\003)2923 1142 y FA(x)p Fg(+)p FA(w)q(;`)3104 1116 y Fp(\()p Fo(\000)3232 1093 y Fp(^)3210 1116 y Fo(L)3273 1130 y Fr(u;v)3374 1116 y Fp(\))p Fo(g)118 1314 y Fu(where)27 b Fq(\027)33 b Fu(varies)28 b(in)f Fo(M)p Fp(\(\003)1040 1329 y Fr(x;`)1155 1314 y Fo([)21 b Fp(\003)1300 1329 y Fr(x)p Fl(+)p Fr(w)r(;`)1500 1314 y Fp(\))p Fu(.)39 b(Thanks)28 b(to)g(perturbation)f(theory)g(\(see)g(proposition)i(4.2\)) 118 1422 y(we)c(only)g(need)f(to)i(prove)e(that,)j(for)e(almost)h(any)f (disorder)h(con\002guration)e Fq(\013)p Fu(,)924 1571 y Fp(lim)15 b(sup)873 1650 y Fr(`)p Fn("1)p Fr(;a)p Fn(#)p Fl(0)p Fr(;\017)p Fn(#)p Fl(0)1396 1571 y Fp(sup)1269 1654 y Fr(w)r Fl(:2)p Fr(`<)p Fn(j)p Fr(w)r Fn(j\024)1619 1627 y FA(a)p 1617 1639 34 3 v 1621 1680 a(\017)1676 1571 y Fp(Av)1793 1593 y Fr(x)p Fn(2)p Fm(T)1930 1574 y FA(d)1930 1610 y(\017)1979 1571 y Fp(sup)2028 1643 y Fr(\027)2147 1571 y Fq(\027)6 b Fp(\()p Fo(j)p Fq(m)2338 1586 y Fr(x;`)2450 1571 y Fo(\000)20 b Fq(m)2621 1586 y Fr(x)p Fl(+)p Fr(w)r(;`)2821 1571 y Fo(j)p Fp(\))p Fq(:)495 b Fu(\(A.18\))118 1805 y(W)-8 b(e)26 b(observe)e(that)j(by)e (lemma)f(A.5)h(in)g(the)h(above)f(expression)g(we)g(can)118 1913 y(substitute)33 b Fq(\027)j Fu(with)d(the)e(grand)g(canonical)f (measure)g Fq(\026)h Fu(such)g(that)h Fq(\026)p Fp(\()p Fq(m)2676 1927 y Fl(\003)2729 1913 y Fp(\))j(=)g Fq(\027)6 b Fp(\()p Fq(m)3071 1927 y Fl(\003)3124 1913 y Fp(\))32 b Fu(where)e Fp(\003)35 b(:=)118 2021 y(\003)181 2036 y Fr(x;`)294 2021 y Fo([)20 b Fp(\003)438 2036 y Fr(x)p Fl(+)p Fr(w)r(;`)637 2021 y Fu(.)118 2129 y(Let)31 b(us)f(introduce)f (the)i(scale)e(parameter)g Fq(s)h Fu(with)h Fq(s)j Fo(")f(1)d Fu(after)h Fq(`)i Fo(")h(1)p Fu(.)46 b(Then,)31 b(by)g(approximating) 118 2237 y Fq(m)198 2252 y Fr(x;`)316 2237 y Fu(with)26 b Fp(Av)635 2251 y Fr(y)r Fn(2)p Fl(\003)768 2263 y FA(x;`)856 2237 y Fq(m)936 2251 y Fr(y)r(;s)1055 2237 y Fu(and)g(thanks)g(to)f (lemma)f(A.3)134 2409 y Fq(\026)p Fp(\()p Fo(j)p Fq(m)329 2424 y Fr(x;`)442 2409 y Fo(\000)c Fq(m)613 2424 y Fr(x)p Fl(+)p Fr(w)r(;`)812 2409 y Fo(j)p Fp(\))26 b Fo(\024)f Fq(c)15 b Fp(Av)1166 2423 y Fr(y)r Fn(2)p Fl(\003)1299 2435 y FA(x;`)1387 2409 y Fq(\026)1442 2372 y Fr(\025)1483 2381 y Fg(0)1517 2372 y Fl(\()p Fr(m)p Fl(\))1639 2409 y Fp(\()p Fo(j)p Fq(m)1779 2423 y Fr(y)r(;s)1893 2409 y Fo(\000)20 b Fq(m)2064 2423 y Fr(y)r Fl(+)p Fr(w)r(;s)2265 2409 y Fo(j)p Fp(\))h(+)e Fq(c)c(s)2533 2372 y Fr(d)2574 2332 y Fk(\014)2574 2386 y(\014)2604 2409 y Fq(m)20 b Fo(\000)g Fq(\026)2850 2372 y Fr(\025)2891 2381 y Fg(0)2926 2372 y Fl(\()p Fr(m)p Fl(\))3047 2409 y Fp(\()p Fq(m)3162 2423 y Fl(\003)3215 2409 y Fp(\))3250 2332 y Fk(\014)3250 2386 y(\014)3281 2409 y Fp(+)p Fq(O)s Fp(\()p Fq(s=`)p Fp(\))p Fq(:)118 2562 y Fu(where)29 b Fq(m)j Fp(=)f Fq(\026)p Fp(\()p Fq(m)777 2576 y Fl(\003)830 2562 y Fp(\))i(=)e Fq(\027)6 b Fp(\()p Fq(m)1166 2576 y Fl(\003)1219 2562 y Fp(\))30 b Fu(and)g Fp(\003)f Fu(is)h(de\002ned)f(as)h(above.)43 b(Therefore,)30 b(it)g(is)g(enough)e(to)i(prove)118 2670 y(that)c(for)g(almost)f(all)g(disorder)h(con\002guration)e Fq(\013)868 2826 y Fp(lim)15 b(sup)802 2905 y Fr(s)p Fn("1)p Fr(;`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)1228 2826 y Fp(Av)1346 2849 y Fr(x)p Fn(2)p Fm(T)1483 2830 y FA(d)1483 2865 y(\017)1532 2826 y Fp(sup)1569 2899 y Fr(m)1684 2826 y Fp(Av)1802 2840 y Fr(y)r Fn(2)p Fl(\003)1935 2852 y FA(`)1969 2826 y Fq(\026)2024 2789 y Fr(\025)2065 2798 y Fg(0)2099 2789 y Fl(\()p Fr(m)p Fl(\))2221 2826 y Fp(\()p Fo(j)p Fq(m)2361 2840 y Fr(x)p Fl(+)p Fr(y)r(;s)2570 2826 y Fo(\000)20 b Fq(m)p Fo(j)p Fp(\))25 b(=)g(0)p Fq(;)802 3050 y Fp(lim)15 b(sup)814 3129 y Fr(`)p Fn("1)p Fr(;\017)p Fn(#)p Fl(0)1095 3050 y Fp(Av)1212 3072 y Fr(x)p Fn(2)p Fm(T)1349 3053 y FA(d)1349 3089 y(\017)1399 3050 y Fp(sup)1436 3122 y Fr(m)1551 3050 y Fo(j)p Fq(m)20 b Fo(\000)g Fq(\026)1822 3013 y Fr(\025)1863 3022 y Fg(0)1898 3013 y Fl(\()p Fr(m)p Fl(\))2019 3050 y Fp(\()p Fq(m)2134 3065 y Fr(x;`)2227 3050 y Fp(\))p Fo(j)25 b Fp(=)g(0)p Fq(:)118 3283 y Fu(Since)35 b Fs(E)11 b Fq(\026)477 3250 y Fr(\025)518 3259 y Fg(0)558 3250 y Fl(\()p Fr(m)p Fl(\))680 3283 y Fp(\()p Fq(m)795 3297 y Fr(x;n)901 3283 y Fp(\))44 b(=)e Fq(m)36 b Fu(for)g(any)g(integer)f Fq(n)h Fu(and)g(any)f(site)i Fq(x)p Fu(,)i(the)d(above)g(limits)h(follow)e(by)118 3391 y(straightforward)h(arguments)e(from)g(the)i(ergodicity)e(of)i (the)f(random)f(\002eld)h Fq(\013)g Fu(and)g(the)g(technical)118 3499 y(estimate)26 b(\(A.3\).)2878 b Fe(\003)1644 3691 y Fu(R)t Ft(E)t(F)t(E)t(R)t(E)t(N)t(C)t(E)t(S)160 3836 y FD([1])37 b(C.)30 b(An\264)-37 b(e,)32 b(S.)d(Blach\036)-37 b(ere,)31 b(D.)f(Chafa)-8 b(\250)-29 b(\021,)32 b(P)-19 b(.)29 b(F)-5 b(oug\036)-37 b(eres,)30 b(I.)g(Gentil,)i(F)-14 b(.)29 b(Malrieu,)i(C.)e(R)m(oberto)i(and)f(G.)f(Scheffer)-8 b(.)29 b Fw(Sur)h(les)301 3928 y(in)m(\264)-34 b(egalit)m(\264)g(es)20 b(de)h(Sobolev)e(logarithmiques)p FD(.)h(Soci\264)-37 b(et\264)g(e)20 b(Math\264)-37 b(ematique)19 b(de)i(France,)f(P)l (aris,)f Fx(2000)p FD(.)160 4019 y([2])37 b(C.)27 b(Bernardin.)g Fw(R)o(egularity)g(of)f(the)h(diffusion)g(coef\002cient)f(for)g(a)h (lattice)g(gas)g(reversible)e(under)i(Bernoulli)e(measures.)301 4110 y FD(Stoch.)c(Proc.)g(and)g(their)f(Appl.)h(V)-7 b(ol.)21 b(101.)e(p.43/68)g(\(2002\))160 4202 y([3])37 b(I.)23 b(Benjamini,)g(P)-19 b(.)21 b(F)-5 b(errari,)21 b(C.)i(Landim.)f Fw(Asymmetric)h(conservative)e(processes)f(with)k (random)d(rates)p FD(.)g(Stochastic)i(Pro-)301 4293 y(cesses)e(Appl.)f Fb(6)p Fx(1)i FD(\(1996\))d(181-204)f(.)160 4384 y([4])37 b(P)-19 b(.)20 b(Billingsley)-10 b(.)21 b Fw(Convergence)e(of)h (probability)g(Measures)p FD(.)f(John)j(W)o(iley)f(&)f(Sons,)h(New)f(Y) -7 b(ork.)160 4476 y([5])37 b(H.)21 b(B)q(\250)-38 b(ottger)-8 b(,)20 b(V)-17 b(.V)g(.)22 b(Bryksin.)e Fw(Hopping)f(conduction)i(in)f (solids)p FD(.)g(Akademie)f(V)-8 b(erlag.)20 b(Berlin.)h(1985.)160 4567 y([6])37 b(N.)30 b(Cancrini,)i(F)-14 b(.)30 b(Martinelli.)e Fw(Comparision)g(of)h(\002nite)g(volume)g(canonical)g(and)h(grand)f (canonical)h(Gibbs)f(measures)301 4658 y(under)21 b(a)g(mixing)f (condition)g FD(.)h(Markov)f(Proc.)g(R)m(el.)h(Fields,)f Fb(6)p FD(,)h(\()p Fx(2000)p FD(\))i(no.)p Fx(1)p FD(,)e Fx(23)d Fy(\000)f Fx(72)p FD(.)160 4750 y([7])37 b(N.)21 b(Cancrini,)g(F)-14 b(.)20 b(Martinelli.)g Fw(On)g(the)g(spectral)g (gap)g(of)g(K)o(awasaki)h(dynamics)g(under)f(a)h(mixing)g(condition)f (revisited)p FD(.)f(J.)301 4841 y(Math.)i(Phys.)f Fb(41)p FD(,)h(\()p Fx(2000)p FD(\))i(no.)p Fx(3)p FD(,)e Fx(1391)e Fy(\000)d Fx(1423)p FD(.)160 4932 y([8])37 b(N.)19 b(Cancrini,)g(F)-14 b(.)18 b(Martinelli.)g Fw(Diffusive)g(scaling)h(of)e(the)i(spectral)e (gap)h(for)f(the)h(dilute)h(Ising)f(lattice-gas)g(dynamics)i(below)301 5024 y(the)h(percolation)e(threshold)p FD(.)g(Probab.)h(Theory)g(R)m (elated)h(Fields)g Fb(120)p FD(,)g(\()p Fx(2001)p FD(\))i(no.)e Fx(4)p FD(,)g Fx(497)d Fy(\000)f Fx(534)p FD(.)160 5115 y([9])37 b(N.)17 b(Cancrini,)h(F)-14 b(.)17 b(Cesi,)g(F)-14 b(.)16 b(Martinelli.)g Fw(The)h(spectral)e(gap)i(for)f(the)g(K)o (awasaki)i(dynamics)f(at)g(low)f(temperature)p FD(.)f(J.)i(Statist.)301 5206 y(Phys.)k Fb(95)p FD(,)g(\()p Fx(1999)p FD(\),)h(no.)f Fx(1)d Fy(\000)f Fx(2)p FD(,)k Fx(215)d Fy(\000)f Fx(271)p FD(.)118 5298 y([10])37 b(P)-19 b(.)20 b(Caputo.)p Fw(private)g (communication)118 5389 y FD([11])37 b(P)-19 b(.)20 b(Caputo.)p Fw(Uniform)f(P)l(oincar)m(\264)-34 b(e)21 b(inequalities)f(for)g (unbounded)f(conservative)g(spin)h(systems:)k(the)c(non)f(interacting)i (case.)301 5480 y FD(Preprint)f(2002)p eop %%Page: 59 59 59 58 bop 1015 45 a Fj(HYDROD)o(YNAMIC)23 b(LIMIT)e(OF)j(A)g (DISORDERED)f(LA)-7 b(T)q(TICE)23 b(GAS)824 b(59)118 366 y FD([12])37 b(P)-19 b(.)20 b(Caputo)g(and)h(F)-14 b(.)20 b(Martinelli.)f Fw(R)o(elaxation)g(time)h(of)f(anisotropic)h (simple)f(exclusion)g(processes)f(and)i(quantum)g(Heisen-)301 458 y(berg)g(models)p FD(.)f(T)-7 b(o)21 b(appear)f(in)h(Ann.)g(Appl.)g (Prob.)118 549 y([13])37 b(E.)24 b(Carlen,)h(M.C.)e(Caravalho)h(and)h (M.)e(Loss.)h Fw(Determination)f(of)g(the)h(spectral)f(gap)h(for)f(K)o (ac's)g(master)h(equation)g(and)301 640 y(related)c(stochastic)h (evolutions)p FD(.)e(Preprint)g Fx(2002)p FD(.)118 732 y([14])37 b(A.De)23 b(Masi,)h(P)-19 b(.A.)23 b(F)-5 b(errari,)23 b(S.)g(Goldstein,)i(W)-13 b(.D.)23 b(W)o(ick.)h Fw(An)g(invariance)g (principle)f(for)g(reversible)f(Markov)h(processes.)301 823 y(Applications)30 b(to)g(random)f(motions)g(in)h(random)g (enviroments)e FD(Journal)i(of)h(Statistical)e(Physics.)i(V)-7 b(ol.)30 b(55.)g(N.3/4.)301 914 y(1989.)118 1006 y([15])37 b(A.)28 b(F)-6 b(aggionato.)29 b Fw(Hydrodynamic)h(limit)f(of)e(a)i (disordered)e(system)p FD(.)g(Ph.D.)h(Thesis.)f(Scuola)h(Normale)g (Superiore)e(di)301 1097 y(Pisa.)20 b Fx(2002)j FD (http://mpej.unige.ch/mp)p 1481 1097 23 4 v 26 w(arc/c/03/03-37.ps.gz.) 118 1188 y([16])37 b(J.)20 b(Fritz.)e Fw(Hydrodynamics)i(in)g(a)f (symmetric)h(random)e(medium.)h FD(Comm.)h(Math.)f(Phys.)f Fb(125)q FD(,)h(\()p Fx(1989)p FD(\),)j(no.)e Fx(1)p FD(,)f Fx(13)12 b Fy(\000)g Fx(25)p FD(.)118 1280 y([17])37 b(P)-19 b(.)20 b(Gartner)-8 b(,)20 b(R.)g(Pitis.)g Fw(Occupacy)q (-correlation)h(corrections)f(in)g(hoppying)g FD(Phys.)h(R)m(ev)-9 b(.)20 b(B.)h Fb(45)g FD(\(1992\))118 1371 y([18])37 b(M.Z.)21 b(Guo,)g(G.C.)g(P)l(apanicolaou,)h(S.R.S.)e(V)-8 b(aradhan.)22 b Fw(Nonlinear)d(diffusion)j(limit)f(for)g(a)h(system)f (with)h(nearest)e(neigh-)301 1462 y(bor)g(interactions)p FD(.)g(Comm.)h(Math.)f(Phys.)g Fb(118)q FD(,)h(\()p Fx(1988)p FD(\),)h Fx(31)c Fy(\000)f Fx(59)p FD(.)118 1554 y([19])37 b(K.)22 b(W)-13 b(.)21 b(K)o(ehr)-8 b(,)20 b(O)l(.)h(P)l(aetzold.)g Fw(Collective)f(and)i(tracer)f(diffusion)h(of)f(lattice)g(gases)g(in)h (lattices)f(with)i(site\226energy)e(disorder)p FD(.)301 1645 y(Physica)g(A)g(190)e(\(1992\))g(1\22612.)118 1736 y([20])37 b(K.)21 b(W)-13 b(.)22 b(K)o(ehr)-8 b(,)19 b(T)-11 b(.)21 b(W)o(ichmann,)h(in)g Fw(Diffusion)f(processes:)k (experiment,)19 b(theory)l(,)h(simulations)p FD(.)g(Ed.)i(A.)e(P)l (ekalski.)g(Lecture)301 1828 y(Notes)h(in)g(Physics)g(V)-7 b(ol.)21 b(438)f(\(Springer)-8 b(,)19 b(1994\))g(p.)i(179/)118 1919 y([21])37 b(K.)23 b(W)-13 b(.)22 b(K)o(ehr)-8 b(,)21 b(T)-11 b(.)22 b(W)o(ichmann.)p Fw(Diffusion)h(coef\002cients)f(of)g (single)f(and)i(many)f(particles)g(in)h(lattices)f(with)i(different)d (forms)301 2010 y(of)g(disorder)p FD(.)e(cond-mat/9602121.)118 2102 y([22])37 b(K.)21 b(W)-13 b(.)21 b(K)o(ehr)-8 b(,)19 b(O)l(.)h(P)l(aetzold,)g(T)-11 b(.)21 b(W)o(ichmann.)g Fw(Collective)f(diffusion)h(of)f(lattice)h(gases)f(on)g(linear)h (chains)g(with)h(site-energy)301 2193 y(disorder)p FD(.)e(Physics)h (Letters)e(A.)h Fb(182)q FD(,)g Fx(\(1993\))i(135)d Fy(\000)d Fx(139)p FD(.)118 2284 y([23])37 b(C.)29 b(Kipnis,)h(C.)e(Landim.)h Fw(Scaling)f(limits)h(of)e(interacting)i(particle)e(systems)p FD(.)g(Grundleheren)h(der)g(mathematischen)301 2376 y(W)o (issenschaften)21 b(V)-7 b(ol.)p Fx(320)p FD(,)22 b(Springer)e(V)-8 b(erlag.)20 b Fx(\(1999\))p FD(.)118 2467 y([24])37 b(S.)23 b(Kirckpatrick.)e Fw(Classical)i(transport)d(in)j(disordered)f(media:) 30 b(scaling)23 b(and)f(effective-medium)h(theories)p FD(.)e(Phys.)i(R)m(ev)-9 b(.)301 2558 y(Lett.)20 b Fb(27)q FD(,)g Fx(\(1971\))j(1722)p FD(.)118 2650 y([25])37 b(A.)30 b(K)o(oukkous.)e Fw(Hydrodynamic)j(behavior)f(of)f(symmetric)h (zero-range)e(processes)f(with)k(random)e(rates.)g FD(Stochastic)301 2741 y(Processes)20 b(Appl.)h Fb(84)p FD(,)g Fx(\(1999\))h(297)c Fy(\000)f Fx(312)p FD(.)118 2832 y([26])37 b(T)-11 b(.M.)20 b(Liggett.)g Fw(Interacting)h(particle)f(systems)p FD(.)f (Springer\226V)-8 b(erlag.)20 b(New)g(Y)-7 b(ork.)118 2924 y([27])37 b(S.T)-11 b(.)27 b(Lu,)h(H.T)-11 b(.)27 b(Y)-7 b(au.)27 b Fw(Spectral)g(gap)g(and)h(logarithmic)g(Sobolev)e (inequality)i(for)f(K)o(awasaki)i(and)e(Glauber)g(dynamics)p FD(.)301 3015 y(Comm.)21 b(Math.)g(Phys.)f Fb(156)q FD(,)g Fx(\(1993\))i(399)c Fy(\000)f Fx(433)p FD(.)118 3106 y([28])37 b(K.)21 b(Nagy)-10 b(.)20 b Fw(Symmetric)i(random)e(walk)h (in)g(random)e(enviroment)p FD(.)g(Preprint.)118 3198 y([29])37 b(J.)17 b(Quastel.)f Fw(Diffusion)g(in)h(disordered)f(media)p FD(.)h(In)g(T)-11 b(.)16 b(Funaki)h(and)g(W)-13 b(.)17 b(W)-5 b(oyczinky)-10 b(,)18 b(editors,)f Fw(Proceedings)f(on)g (stochastic)301 3289 y(method)i(for)g(nonlinear)e(P)-19 b(.D.E.)p FD(,)19 b(IMA)f(volumes)g(in)g(Mathematics)g Fb(77)q FD(,)g Fx(\(1995\))i(65)9 b Fy(\000)g Fx(79)p FD(.)20 b(Springer)d(V)-8 b(erlag,)19 b(New)f(Y)-7 b(ork.)118 3380 y([30])37 b(J.)26 b(Quastel.)e Fw(Diffusion)i(of)f(color)f(in)i (the)g(simple)e(exclusion)h(process)p FD(.)f(Comm.)i(Pure)e(Appl.)i (Math.)f Fb(45)p FD(,)i(\()p Fx(1992)p FD(\))h(no.)p Fx(6)p FD(,)301 3472 y Fx(623)19 b Fy(\000)e Fx(679)p FD(.)118 3563 y([31])37 b(J.)21 b(Quastel.)e(Private)h(communication.) 118 3654 y([32])37 b(P)-19 b(.M.)30 b(Richards.)i Fw(Theory)f(of)g (one-dimensional)d(hopping)j(conductivity)h(and)g(diffusion)p FD(.)e(Phys.)h(R)m(ev)-9 b(.)32 b(B)f Fb(16)q FD(,)j(\()p Fx(1997)p FD(\))301 3745 y Fx(1393)19 b Fy(\000)e Fx(1409)p FD(.)118 3837 y([33])37 b(T)-11 b(.)18 b(Seppalainen.)g Fw(R)o(ecent)f(results)f(and)j(open)d(problems)g(on)h(the)h (hydrodynamics)h(of)e(disordered)g(asymmetric)i(exclusion)301 3928 y(and)i(zero-range)e(processes.)f FD(R)m(esenhas)j(IME-USP)e(4)i (\(1999\),)e(1-15.)118 4019 y([34])37 b(H.)21 b(Spohn.)g Fw(Large)g(Scale)g(Dynamics)g(of)f(Interacting)h(P)m(articles)p FD(,)g(Springer)e(V)-8 b(erlag.)20 b(Berlin.)118 4111 y([35])37 b(H.)21 b(Spohn,)g(H.T)-11 b(.)20 b(Y)-7 b(au.)19 b Fw(Bulk)i(diffusivity)h(of)e(lattice)g(gases)g(close)f(to)h (criticality)p FD(.)j(J.)d(Stat.)g(Phys.)g Fb(79)p FD(,)h Fx(\(1995\))h(231)17 b Fy(\000)f Fx(241)p FD(.)118 4202 y([36])37 b(S.R.S.)21 b(V)-8 b(aradhan.)22 b Fw(Nonlinear)f(diffusion)h (limit)g(for)g(a)g(system)g(with)h(nearest)e(neighbor)g(interactions)g Fz(I)6 b(I)g FD(.)21 b(In)i(K.)f(D.)f(El-)301 4293 y(worthy)f(and)g(N.) g(Ikeda,)g(editors,)f Fw(Asymptotic)h(Problems)f(in)h(Probability)g (Theory:)25 b(Stochastic)c(Models)d(and)i(Diffusion)301 4385 y(on)e(Fractals)p FD(.)h(V)-7 b(ol.)19 b Fx(283)h FD(of)f(Pitman)g(R)m(esearch)f(Notes)h(in)g(Mathematics,)f(pages)h Fx(75)10 b Fy(\000)g Fx(128)p FD(.)21 b(John)f(W)o(iley)e Fx(&)h FD(Sons,)g(New)301 4476 y(Y)-7 b(ork.)118 4567 y([37])37 b(S.R.S.)20 b(V)-8 b(aradhan,)20 b(H.T)-11 b(.)20 b(Y)-7 b(au.)20 b Fw(Diffusive)h(limit)g(of)f(lattice)h(gases)f (with)h(mixing)g(conditions)p FD(.)e(Asian)i(J.)g(Math.)f Fb(1)p FD(,)h(\()p Fx(1997)p FD(\))301 4659 y(no.)h Fx(4)p FD(,)f Fx(623)d Fy(\000)f Fx(678)p FD(.)118 4750 y([38])37 b(W)-13 b(.)31 b(D.)f(W)o(ick.)h Fw(Hydrodynamic)g(limit)g(of)f (non-gradient)f(interacting)h(particle)h(process)p FD(.)d(J.)j(Stat.)f (Phys.)g Fb(54)p FD(,)j Fx(\(1989\))301 4841 y(832)19 b Fy(\000)e Fx(892)p FD(.)220 5013 y(F)m FC(A)t(K)r(U)t(L)o(T)453 5009 y FD(\250)453 5013 y FC(A)t(T)j FD(I)t(I)i(-)g(M)t FC(A)m(T)t(H)t(E)t(M)t(A)m(T)t(I)t(K)e(U)t(N)t(D)i FD(N)t FC(A)m(T)t(U)t(R)r(W)t(I)t(S)t(S)t(E)t(N)t(S)t(C)t(H)t(A)t(F)t(T)t(E)t (N)t FD(,)c(T)t FC(E)t(C)t(H)t(N)t(I)t(S)t(C)t(H)t(E)h FD(U)t FC(N)t(I)t(V)t(E)t(R)t(S)t(I)t(T)2870 5009 y FD(\250)2870 5013 y FC(A)t(T)g FD(B)t FC(E)t(R)t(L)t(I)t(N)t FD(,)i(S)t FC(T)t(R)t(A)t(S)t(S)t(E)f(D)t(E)t(S)120 5104 y FD(1)t(7)t(.)28 b(J)t FC(U)t(N)t(I)c FD(1)t(3)t(6)t(,)e(1)t(0)t(6)t(2)t(3)f(B)t FC(E)t(R)t(L)t(I)t(N)t FD(,)i(G)t FC(E)t(R)t(M)t(A)t(N)t(Y)218 5195 y Fw(E-mail)d(address)p FD(:)25 b Fa(faggiona@math.tu-berlin.de) 220 5351 y FD(D)t FC(I)t(P)n(A)t(R)r(T)t(I)t(M)t(E)t(N)t(T)t(O)c(D)t(I) k FD(M)t FC(A)m(T)t(E)t(M)t(A)m(T)t(I)t(C)s(A)t FD(,)d(U)t FC(N)t(I)t(V)t(E)t(R)t(S)t(I)t(T)m(A)-5 b FD(')22 b FC(D)t(I)j FD(R)t FC(O)t(M)t(A)f FD(T)t FC(R)t(E)t FD(,)f(L)t(.)t FC(G)t(O)g FD(S)t(.)h(M)t FC(U)t(R)t(I)t(A)t(L)t(D)t(O)f FD(1)t(,)g(0)t(0)t(1)t(4)t(6)e(R)t FC(O)t(M)t(A)t FD(,)i(I)t FC(T)m(A)t(L)n(Y)218 5442 y Fw(E-mail)d(address)p FD(:)25 b Fa(martin@mat.uniroma3.it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0302071939552--