This is a multi-part message in MIME format. ---------------0004050335132 Content-Type: text/plain; name="00-164.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-164.keywords" Kawasaki dynamics, random ferromagnet, spectral gap, equivalence of ensembles ---------------0004050335132 Content-Type: application/postscript; name="tot.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="tot.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: ptrf.dvi %%Pages: 36 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o tot.ps ptrf.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.04.05:1005 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 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Fq(\))g(b)s(e)f(the)h(class)f(of)h(all)e(those)j(parallelepip)s (eds)28 b(in)k Fj(R)3846 3434 y Fo(\017)3846 3492 y(L)0 3594 y Fq(suc)m(h)27 b(that)h(the)g(longest)g(side)e(is)h(smaller)f (than)h Fn(l)r Fq(,)h(and)1924 3571 y(\026)1900 3594 y Fj(R)1977 3561 y Fo(\017)1977 3619 y(L)2032 3594 y Fq(\()p Fn(l)r Fq(\))g(the)g(class)f(of)g(all)g(parallelepip)s(eds)c (in)k Fj(R)3544 3561 y Fo(\017)3544 3619 y(L)3598 3594 y Fq(\()p Fn(l)r Fq(\))h(suc)m(h)0 3721 y(that)j(the)g(shortest)f(side) g(is)f(greater)j(than)e Fn(L)1555 3688 y Fo(\017)1589 3721 y Fq(.)0 3891 y Fh(The)37 b(c)-5 b(on\014gur)g(ation)38 b(sp)-5 b(ac)g(e.)55 b Fq(Our)33 b Fh(c)-5 b(on\014gur)g(ation)39 b(sp)-5 b(ac)g(e)36 b Fq(is)d(\012)f(=)g Fn(S)2360 3858 y Ff(Z)2409 3830 y Fl(d)2446 3891 y Fq(,)j(where)f Fn(S)k Fq(=)32 b Fj(f)p Fq(0)p Fn(;)15 b Fq(1)p Fj(g)p Fq(,)38 b(or)c(\012)3434 3905 y Fo(V)3530 3891 y Fq(=)e Fn(S)3694 3858 y Fo(V)3791 3891 y Fq(for)0 4018 y(some)40 b Fn(V)62 b Fj(\032)41 b Fp(Z)529 3985 y Fo(d)567 4018 y Fq(.)69 b(The)40 b(single)e(spin)g(space)j Fn(S)k Fq(is)39 b(endo)m(w)m(ed)h (with)f(the)h(discrete)g(top)s(ology)g(and)f(\012)h(with)f(the)0 4144 y(corresp)s(onding)27 b(pro)s(duct)h(top)s(ology)-8 b(.)42 b(Giv)m(en)29 b Fn(\033)f Fj(2)d Fq(\012)k(and)g(\003)c Fj(\032)g Fp(Z)2273 4111 y Fo(d)2340 4144 y Fq(w)m(e)30 b(denote)g(b)m(y)f Fn(\033)2942 4158 y Ft(\003)3025 4144 y Fq(the)h(natural)e(pro)5 b(jection)0 4271 y(o)m(v)m(er)32 b(\012)261 4285 y Ft(\003)315 4271 y Fq(.)41 b(If)30 b Fn(U)10 b Fq(,)31 b Fn(V)51 b Fq(are)31 b(disjoin)m(t,)e Fn(\033)1257 4285 y Fo(U)1318 4271 y Fn(\034)1358 4285 y Fo(V)1452 4271 y Fq(is)g(the)i(con\014guration)f(on)h Fn(U)f Fj([)20 b Fn(V)51 b Fq(whic)m(h)29 b(is)h(equal)g(to)h Fn(\033)j Fq(on)c Fn(U)41 b Fq(and)30 b Fn(\034)0 4398 y Fq(on)j Fn(V)20 b Fq(.)48 b(Giv)m(en)33 b Fn(V)49 b Fj(2)29 b Fp(F)43 b Fq(w)m(e)34 b(de\014ne)e(the)h Fh(numb)-5 b(er)35 b(of)g(p)-5 b(articles)35 b Fn(N)2252 4412 y Fo(V)2344 4398 y Fq(:)30 b(\012)f Fj(7!)g Fp(N)45 b Fq(as)33 b Fn(N)2899 4412 y Fo(V)2963 4398 y Fq(\()p Fn(\033)s Fq(\))d(=)3218 4329 y Fg(P)3314 4425 y Fo(x)p Fm(2)p Fo(V)3482 4398 y Fn(\033)s Fq(\()p Fn(x)p Fq(\))k(while)0 4524 y(the)d Fh(density)g Fq(is)e(giv)m(en)h(b)m(y)h Fn(\032)968 4538 y Fo(V)1056 4524 y Fq(=)25 b Fn(N)1225 4538 y Fo(V)1288 4524 y Fn(=)p Fj(j)p Fn(V)c Fj(j)p Fq(.)0 4651 y(If)29 b Fn(f)38 b Fq(is)29 b(a)g(function)f(on)h(\012,)h(\001) 1015 4665 y Fo(f)1091 4651 y Fq(denotes)f(the)h(smallest)e(subset)h(of) g Fp(Z)2367 4618 y Fo(d)2434 4651 y Fq(suc)m(h)g(that)h Fn(f)10 b Fq(\()p Fn(\033)s Fq(\))30 b(dep)s(ends)d(only)h(on)i Fn(\033)3770 4665 y Ft(\001)3830 4675 y Fl(f)3875 4651 y Fq(.)0 4778 y Fn(f)45 b Fq(is)35 b(called)h Fh(lo)-5 b(c)g(al)38 b Fq(if)d(\001)824 4792 y Fo(f)907 4778 y Fq(is)g(\014nite.)57 b Fj(F)1354 4792 y Ft(\003)1444 4778 y Fq(stands)36 b(for)g(the)g Fn(\033)s Fj(\000)p Fq(algebra)g(generated)h(b)m(y)f(the)g(set)h(of)f(pro)5 b(jections)0 4904 y Fj(f)p Fn(\031)97 4918 y Fo(x)143 4904 y Fj(g)p Fq(,)27 b Fn(x)e Fj(2)g Fq(\003,)h(from)f(\012)g(to)h Fj(f)p Fq(0)p Fn(;)15 b Fq(1)p Fj(g)p Fq(,)28 b(where)d Fn(\031)1507 4918 y Fo(x)1578 4904 y Fq(:)g Fn(\033)k Fj(7!)c Fn(\033)s Fq(\()p Fn(x)p Fq(\).)39 b(When)25 b(\003)h(=)f Fp(Z)2576 4871 y Fo(d)2639 4904 y Fq(w)m(e)g(set)h Fj(F)35 b Fq(=)25 b Fj(F)3165 4923 y Ff(Z)3213 4905 y Fl(d)3275 4904 y Fq(and)g Fj(F)34 b Fq(coincides)0 5031 y(with)28 b(the)h(Borel)g Fn(\033)s Fj(\000)p Fq(algebra)g(on)g(\012)g (with)e(resp)s(ect)i(to)h(the)f(top)s(ology)h(in)m(tro)s(duced)d(ab)s (o)m(v)m(e.)42 b(By)29 b Fj(k)p Fn(f)10 b Fj(k)3449 5045 y Fm(1)3555 5031 y Fq(w)m(e)30 b(mean)0 5158 y(the)h(suprem)m(um)d (norm)i(of)g Fn(f)10 b Fq(.)40 b(The)30 b Fh(gr)-5 b(adient)32 b Fq(of)f(a)f(function)f Fn(f)40 b Fq(is)29 b(de\014ned)g(as)1445 5340 y(\()p Fj(r)1556 5354 y Fo(x)1601 5340 y Fn(f)10 b Fq(\)\()p Fn(\033)s Fq(\))26 b(=)f Fn(f)10 b Fq(\()p Fn(\033)2083 5302 y Fo(x)2129 5340 y Fq(\))20 b Fj(\000)g Fn(f)10 b Fq(\()p Fn(\033)s Fq(\))0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([5])1384 b Fq(2.2)p eop %%Page: 6 6 6 5 bop 0 91 a Fq(where)39 b Fn(\033)327 58 y Fo(x)413 91 y Fj(2)h Fq(\012)f(is)g(the)g(con\014guration)g(obtained)g(from)g Fn(\033)s Fq(,)j(b)m(y)e(\015ipping)c(the)k(spin)d(at)j(the)g(site)f Fn(x)p Fq(.)68 b(Finally)0 219 y(Osc\()p Fn(f)10 b Fq(\))25 b(=)g(sup)530 241 y Fo(\033)n(;\021)652 219 y Fj(j)p Fn(f)10 b Fq(\()p Fn(\033)s Fq(\))20 b Fj(\000)g Fn(f)10 b Fq(\()p Fn(\021)s Fq(\))p Fj(j)p Fq(.)0 705 y Fu(2.2)35 b(The)f(dilute)g(Ising)g(la)-6 b(ttice)33 b(gas.)0 962 y Fq(W)-8 b(e)30 b(consider)d(an)h(abstract)i(probabilit)m(y)c(space)j (\(\002)p Fn(;)15 b Fj(B)s Fn(;)g Fp(P)p Fq(\))29 b(and)f(a)h(set)g(of) g(i.i.d.)38 b(real)28 b(v)-5 b(alued)28 b(random)f(v)-5 b(ariables)0 1090 y(indexed)31 b(b)m(y)h(the)g(b)s(onds)f(of)h Fp(Z)1062 1057 y Fo(d)1100 1090 y Fq(,)h Fn(J)k Fq(=)28 b Fj(f)p Fn(J)1439 1104 y Fo(xy)1525 1090 y Fj(g)1570 1108 y Ft([)p Fo(x;y)r Ft(])p Fm(2)p Ff(Z)1811 1090 y Fl(d)s Fq(.)46 b Fp(E)10 b Fq(\()q Fj(\001)p Fq(\))38 b(stands)32 b(for)g(the)g(exp)s(ectation)h(with)e(resp)s(ect)h(to)h Fp(P)p Fq(.)0 1218 y(W)-8 b(e)31 b(assume)f(that)g(the)g(couplings)e Fn(J)1271 1232 y Fo(xy)1386 1218 y Fq(tak)m(e)j(only)e(t)m(w)m(o)j(v)-5 b(alues,)29 b Fn(\014)h(>)25 b Fq(0)31 b(and)e(0,)h(with)f(probabilit)m (y)e Fn(p)j Fq(and)f(1)19 b Fj(\000)g Fn(p)0 1346 y Fq(resp)s(ectiv)m (ely)-8 b(.)0 1474 y(Giv)m(en)31 b(a)g(disorder)f(con\014guration)g(w)m (e)i(declare)f(a)g(b)s(ond)f([)p Fn(x;)15 b(y)s Fq(])32 b Fh(op)-5 b(en)65 b Fq(if)30 b Fn(J)2594 1488 y Fo(xy)2706 1474 y Fq(=)c Fn(\014)36 b Fq(and)31 b Fh(close)-5 b(d)32 b Fq(otherwise.)43 b(W)-8 b(e)0 1603 y(denote)28 b(b)m(y)g Fn(C)479 1617 y Fo(x)552 1603 y Fq(the)g(cluster)f(of)h(the)g(site)g Fn(x)p Fq(,)g(namely)f(the)h(set)h(of)e(all)g(sites)h(in)e Fp(Z)2722 1570 y Fo(d)2788 1603 y Fq(whic)m(h)g(are)i(connected)h(to)f Fn(x)g Fq(b)m(y)0 1731 y(a)k(path)f(of)g(op)s(en)f(b)s(onds,)g(and)h(b) m(y)1232 1708 y(^)1211 1731 y Fn(C)1276 1745 y Fo(x)1353 1731 y Fq(the)g(connected)i(subgraph)c(of)i Fp(Z)2500 1698 y Fo(d)2570 1731 y Fq(whose)g(v)m(ertices)g(are)h(the)f(sites)g (in)f Fn(C)3854 1745 y Fo(x)0 1859 y Fq(and)g(whose)g(b)s(onds)f(are)i (the)g(op)s(en)e(b)s(onds)g(with)g(endp)s(oin)m(ts)g(in)g Fn(C)2298 1873 y Fo(x)2344 1859 y Fq(.)41 b(Notice)31 b(that)2913 1836 y(^)2893 1859 y Fn(C)2958 1873 y Fo(x)3029 1859 y Fq(=)25 b Fj(f)p Fn(x)p Fj(g)31 b Fq(if)f(all)f(the)i(b)s(onds)0 1987 y(with)e Fn(x)h Fq(as)h(one)g(endp)s(oin)m(t)d(are)j(closed.)0 2115 y(Giv)m(en)f(a)g(disorder)e(con\014guration)h Fn(J)9 b Fq(,)31 b(for)e(eac)m(h)i Fn(\033)e Fj(2)c Fq(\012)k(and)g(\003)d Fj(2)2312 2092 y Fq(^)2307 2115 y Fp(F)40 b Fq(the)30 b(Hamiltonian)e(or)i(energy)g(function)f(of)0 2243 y(the)i(particle)e (con\014guration)h Fn(\033)j Fq(in)c(the)i(graph)f(\003)g(is)g(giv)m (en)g(b)m(y)1055 2508 y Fn(H)1138 2470 y Fo(J)1131 2530 y Ft(\003)1189 2508 y Fq(\()p Fn(\033)s Fq(\))c(=)f Fj(\000)1594 2422 y Fg(X)1522 2621 y Ft([)p Fo(x;y)r Ft(])p Fm(2)p Ft(\003)1765 2631 y Fl(b)1812 2508 y Fn(J)1862 2522 y Fo(xy)1948 2508 y Fq(\(2)p Fn(\033)s Fq(\()p Fn(x)p Fq(\))d Fj(\000)d Fq(1\)\(2)p Fn(\033)s Fq(\()p Fn(y)s Fq(\))k Fj(\000)d Fq(1\))0 2887 y(Giv)m(en)25 b(a)g(collection)g(of)g(real)g(n) m(um)m(b)s(ers)f Fn(\025)p 1359 2902 54 4 v 25 w Fq(=)h Fj(f)p Fn(\025)1631 2901 y Fo(x)1677 2887 y Fj(g)1722 2905 y Fo(x)p Fm(2)p Ff(Z)1861 2887 y Fl(d)j Fq(that)e(in)d(the)j (sequel)e(will)e(b)s(e)j(referred)f(to)i(as)f(generalized)0 3015 y(c)m(hemical)30 b(p)s(oten)m(tial,)g(w)m(e)h(de\014ne)f Fn(H)1260 2965 y Fo(J)n(;\025)p 1320 2980 44 4 v 1253 3041 a Ft(\003)1367 3015 y Fq(\()p Fn(\033)s Fq(\))i(as)1310 3279 y Fn(H)1393 3230 y Fo(J)n(;\025)p 1453 3245 V 1386 3306 a Ft(\003)1501 3279 y Fq(\()p Fn(\033)s Fq(\))26 b(=)f Fn(H)1831 3242 y Fo(J)1824 3302 y Ft(\003)1882 3279 y Fq(\()p Fn(\033)s Fq(\))c Fj(\000)2143 3193 y Fg(X)2119 3388 y Fo(x)p Fm(2)p Ft(\003)2259 3397 y Fl(v)2313 3279 y Fn(\025)2366 3293 y Fo(x)2412 3279 y Fn(\033)s Fq(\()p Fn(x)p Fq(\))0 3642 y(Finally)-8 b(,)29 b(giv)m(en)h Fn(\034)36 b Fj(2)25 b Fq(\012,)30 b(w)m(e)h(let)777 3906 y Fn(H)860 3857 y Fo(\034)s(;J)n(;\025)p 979 3872 V 853 3933 a Ft(\003)1027 3906 y Fq(\()p Fn(\033)s Fq(\))26 b(=)f Fn(H)1357 3857 y Fo(J)n(;\025)p 1417 3872 V 1350 3933 a Ft(\003)1464 3906 y Fq(\()p Fn(\033)s Fq(\))c Fj(\000)1825 3820 y Fg(X)1771 4016 y Fe([)p Fl(x;y)r Fe(])p Fk(2)p Fd(Z)1971 4003 y Fl(d)1712 4073 y(x)p Fk(2)p Fe(\003)1832 4082 y Fl(v)1872 4073 y(;)d(y)8 b(=)-39 b Fk(2)p Fe(\003)2030 4082 y Fl(v)2095 3906 y Fn(J)2145 3920 y Fo(xy)2231 3906 y Fq(\(2)p Fn(\033)s Fq(\()p Fn(x)p Fq(\))22 b Fj(\000)d Fq(1\)\(2)p Fn(\034)10 b Fq(\()p Fn(y)s Fq(\))23 b Fj(\000)d Fq(1\))0 4327 y(and)30 b Fn(\034)40 b Fq(is)30 b(called)f(the)i Fh(b)-5 b(oundary)34 b(c)-5 b(ondition)p Fq(.)0 4455 y(F)d(or)25 b(eac)m(h)h(\003)f Fj(2)535 4431 y Fq(^)530 4455 y Fp(F)35 b Fq(and)24 b Fn(\034)36 b Fj(2)24 b Fq(\012)h(the)f(\(\014nite)g(v)m(olume\))h 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Fl(=apr)r(ile=)p Fe(2000)h Ft([6])1384 b Fq(2.3)p eop %%Page: 7 7 7 6 bop 0 91 a Fr(W)-9 b(arning.)53 b Fc(In)25 b(most)i(notation)f(w)m (e)h(will)c(drop)i(the)i(sup)s(erscript)c Fn(J)35 b Fc(if)25 b(that)i(do)s(es)f(not)g(generate)i(confusion)d(and)0 218 y(the)k(sup)s(erscript)e Fn(\025)p 623 233 54 4 v 29 w Fc(if)h Fn(\025)p 787 233 V 25 w Fq(=)d(0)p Fc(.)40 b(Moreo)m(v)m(er,)32 b(for)d(an)m(y)g Fq(\003)d Fj(\032)f Fp(Z)2053 185 y Fo(d)2120 218 y Fc(w)m(e)k(will)d(alw)m(a)m(ys)k(write) e Fn(\026)2998 169 y Fo(\034)s(;\025)p 3057 184 44 4 v 2998 245 a Ft(\003)3133 218 y Fc(instead)g(of)i(the)f(more)0 345 y(precise)24 b(notation)g Fn(\026)705 296 y Fo(\034)s(;\025)p 764 311 V 712 373 a Ft(^)705 389 y(\003)811 345 y Fc(.)39 b(Finally)-8 b(,)24 b(if)f(the)i(couplings)d Fn(J)1868 359 y Fo(xy)1978 345 y Fc(are)j(constan)m(t)g(and)f(equal)g(to)h Fn(\014)k Fc(for)c(all)e Fq([)p Fn(x;)15 b(y)s Fq(])26 b Fj(2)f Fq(\003)3693 359 y Fo(b)3754 345 y Fc(and)0 519 y(zero)k(if)e(either)g Fn(x)h 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y Fq(\))33 b(=)g(1)j Fc(suc)m(h)f(that,)i(for)0 3770 y(an)m(y)d Fn(J)39 b Fj(2)31 b Fq(\002)427 3784 y Ft(0)467 3770 y Fc(,)k(an)m(y)e Fn(\016)i Fj(2)30 b Fq(\(0)p Fn(;)15 b Fq(1\))35 b Fc(and)e(an)m(y)h(lo)s(cal)f(function)f Fn(f)43 b Fc(with)32 b Fq(0)f Fj(2)f Fq(\001)2575 3784 y Fo(f)2655 3770 y Fc(there)k(exists)f(a)h(p)s(ositiv)m(e)f(constan)m (t)0 3897 y Fn(C)65 3911 y Fo(f)s(;\016)195 3897 y Fc(indep)s(enden)m (t)28 b(of)j Fn(J)39 b Fc(suc)m(h)30 b(that)h(for)f(an)m(y)h(in)m (teger)f Fn(N)35 b Fj(2)25 b(f)p Fq(1)p Fn(;)15 b(:)g(:)g(:)j(;)d Fq(\(2)p Fn(L)p Fq(\))2580 3864 y Fo(d)2624 3897 y Fj(g)30 b Fc(and)g(pro)m(vided)f(that)i Fn(L)f Fc(and)g Fn(t)g Fc(are)0 4023 y(tak)m(en)h(large)g(enough)1340 4197 y Fq(V)-8 b(ar)1482 4157 y Fo(\034)1482 4220 y Ft(\003)p Fo(;N)1637 4096 y Fg(\020)1691 4197 y Fn(e)1733 4160 y Fo(tL)1810 4132 y Fl(\034)1810 4178 y Fe(\003)p Fl(;N)1936 4197 y Fn(f)1991 4096 y Fg(\021)2070 4197 y Fj(\024)25 b Fn(C)2231 4211 y Fo(f)s(;\016)2438 4136 y Fq(1)p 2372 4176 177 4 v 2372 4259 a Fn(t)2405 4233 y Fo(\013)p Fm(\000)p Fo(\016)3713 4197 y Fq(\(2)p Fn(:)p Fq(9\))0 4401 y Fc(Here)33 b Fq(\003)c(:=)f Fn(B)502 4415 y Fo(L)589 4401 y Fc(and)k Fn(\013)d Fq(=)965 4365 y Ft(1)p 965 4380 37 4 v 965 4433 a(2)1045 4401 y Fc(in)i Fn(d)e Fq(=)f(1)p Fc(,)34 b Fn(\013)29 b Fq(=)g(1)j Fc(for)h Fn(d)c(>)f Fq(1)p Fc(.)47 b(F)-8 b(urthermore)33 b(there)f(exists)g(a)h(p)s(ositiv)m(e)f (constan)m(t)0 4528 y Fn(C)65 4542 y Fo(f)s(;\016)195 4528 y Fc(suc)m(h)e(that)h(for)g(all)e Fn(L)h Fc(and)g(all)f Fn(t)1248 4740 y Fp(E)1318 4639 y Fg(\020)1378 4740 y Fq(V)-8 b(ar)1519 4701 y Fo(\034)1519 4763 y Ft(\003)p Fo(;N)1675 4639 y Fg(\020)1729 4740 y Fn(e)1771 4703 y Fo(tL)1848 4675 y Fl(\034)1848 4721 y Fe(\003)p Fl(;N)1974 4740 y Fn(f)2029 4639 y Fg(\021\021)2162 4740 y Fj(\024)40 b Fn(C)2338 4754 y Fo(f)s(;\016)2530 4679 y Fq(1)p 2465 4719 177 4 v 2465 4803 a Fn(t)2498 4776 y Fo(\013)p Fm(\000)p Fo(\016)3668 4740 y Fq(\(2)p Fn(:)p Fq(10\))0 5012 y Fh(R)-5 b(emark.)64 b Fq(An)38 b(analogous)g(result)e(w)m(as)j(pro)m(v) m(ed)f(in)e([CM2])j(for)f(translation)e(in)m(v)-5 b(arian)m(t)37 b(in)m(teraction)h(under)f(a)0 5139 y(suitable)21 b(mixing)g (condition.)36 b(W)-8 b(e)24 b(also)e(refer)h(to)g([JLQY])f(where)g(a)h (sharp)s(er)e(result)g(for)i(the)f(zero{range)i(pro)s(cess)0 5265 y(is)29 b(obtained.)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)h Ft([8])1384 b Fq(3.1)p eop %%Page: 9 9 9 8 bop 0 91 a Fr(3.)47 b(Simple)34 b(large)h(deviations)g(for)g(indep) s(enden)m(t)g(b)s(ond)g(p)s(ercolation.)0 243 y Fq(In)26 b(this)f(section)i(w)m(e)g(pro)m(v)m(e)g(some)g(v)m(ery)g(simple)d (large)j(deviations)e(results)g(for)i(indep)s(enden)m(t)d(b)s(ond)h(p)s (ercolation)0 370 y(b)s(elo)m(w)32 b(the)g(p)s(ercolation)g(threshold)f (that)i(will)d(allo)m(w)i(us)g(to)h(pro)m(v)m(e)h(some)e(sort)h(of)g (homogenization)g(prop)s(ert)m(y)0 497 y(of)h(the)h(dilute)d(Ising)h (mo)s(del)g(at)i(large)f(scale.)53 b(The)33 b(exp)s(ert)h(reader)g(ma)m (y)h(skip)e(the)h(section)g(and)g(just)f(lo)s(ok)h(at)0 624 y(the)d(\014nal)e(result)g(giv)m(en)h(in)f(Corollary)g(3.2.)0 751 y(Giv)m(en)e(an)h(in)m(teger)g Fn(n)p Fq(,)f(let)h Fn(f)37 b Fq(b)s(e)26 b(a)i(real)f(function)g(on)g(the)h(set)g(of)f (all)g(\014nite)f(connected)i(subgraphs)e(of)i Fp(Z)3605 718 y Fo(d)3670 751 y Fq(whic)m(h)0 878 y(is)d(translation)h(in)m(v)-5 b(arian)m(t,)26 b(that)h(is)e Fn(f)10 b Fq(\()p Fn(A)p Fq(\))26 b(=)f Fn(f)10 b Fq(\()p Fn(A)i Fq(+)g Fn(x)p Fq(\))26 b(for)g(all)f Fn(x)g Fj(2)g Fp(Z)2393 845 y Fo(d)2457 878 y Fq(and)h(for)g(all)f(\014nite)g(connected)j(subgraph)0 1005 y Fn(A)p Fq(,)37 b(and)d(suc)m(h)h(that)h Fj(j)p Fn(f)10 b Fq(\()p Fn(A)p Fq(\))p Fj(j)34 b(\024)f(j)p Fn(A)1197 1019 y Fo(v)1241 1005 y Fj(j)1266 972 y Fo(n)1350 1005 y Fq(where)i Fn(A)1686 1019 y Fo(v)1765 1005 y Fq(is)f(the)h(set)h (of)f(v)m(ertices)h(of)g(the)f(graph)g Fn(A)p Fq(.)55 b(Let,)37 b(for)e(a)h(\014xed)0 1132 y(\014nite)29 b(set)i(\003)g(and)e (a)i(giv)m(en)g(disorder)d(con\014guration)i Fn(J)9 b Fq(,)1447 1389 y Fn(<)25 b(f)35 b(>)1694 1403 y Ft(\003)p Fo(;J)1815 1389 y Fq(=)1956 1327 y(1)p 1922 1368 114 4 v 1922 1451 a Fj(j)p Fq(\003)p Fj(j)2066 1302 y Fg(X)2062 1497 y Fo(x)p Fm(2)p Ft(\003)2217 1389 y Fn(f)10 b Fq(\()2327 1366 y(^)2307 1389 y Fn(C)2372 1403 y Fo(x)2418 1389 y Fq(\))1260 b(\(3)p Fn(:)p Fq(1\))0 1706 y(and)30 b(let)328 1682 y(\026)308 1706 y Fn(f)35 b Fq(:=)25 b Fp(E)10 b Fq(\()q Fn(f)g Fq(\()p Fn(C)754 1720 y Ft(0)800 1706 y Fq(\)\))31 b(pro)m(vided)e(that)i Fp(E)11 b Fq(\()p Fj(j)p Fn(C)1652 1720 y Ft(0)1699 1706 y Fj(j)1724 1673 y Fo(n)1774 1706 y Fq(\))25 b Fn(<)g Fj(1)p Fq(.)0 1882 y Fr(Prop)s(osition)45 b(3.1.)53 b Fc(Assume)38 b Fn(p)g(<)g(p)1373 1896 y Fo(c)1448 1882 y Fc(and)g(let)g Fn(\017)1809 1896 y Ft(0)1889 1882 y Fq(=)2171 1847 y Ft(1)p 2009 1862 360 4 v 2009 1914 a(2)p Fo(d)p Ft(\()p Fo(n)p Ft(+1\)+1)2418 1882 y Fc(where)g Fn(n)f Fc(is)h(the)g(in)m(teger)h(go)m(v)m(erning)g (the)0 2009 y(gro)m(wth)e(of)f Fn(f)10 b Fc(.)58 b(Let)37 b Fq(\003)g Fc(b)s(e)f(a)g(parallelepip)s(ed)d(with)i(ratio)i(b)s(et)m (w)m(een)g(the)f(shortest)h(and)f(longest)g(side)g(greater)0 2136 y(than)c Fn(\017)p Fc(.)47 b(Then)31 b(for)h(an)m(y)h Fn(\017)28 b Fj(2)g Fq(\(0)p Fn(;)15 b(\017)1188 2150 y Ft(0)1230 2136 y Fq(\))33 b Fc(there)f(exist)g(constan)m(ts)i Fq(0)29 b Fn(<)f(\016)33 b Fq(=)28 b Fn(\016)s Fq(\()p Fn(n;)15 b(p;)g(\017)p Fq(\))30 b Fn(<)e Fq(1)p Fc(,)34 b Fn(m)3143 2150 y Ft(1)3212 2136 y Fq(=)28 b Fn(m)3391 2150 y Ft(1)3432 2136 y Fq(\()p Fn(p)p Fq(\))h Fn(>)f Fq(0)33 b Fc(and)0 2263 y Fn(m)80 2277 y Ft(2)146 2263 y Fq(=)25 b Fn(m)322 2277 y Ft(2)362 2263 y Fq(\()p Fn(p;)15 b(n)p Fq(\))26 b Fn(>)f Fq(0)31 b Fc(suc)m(h)f(that)983 2537 y Fn(a)p Fq(\))182 b Fp(P)1318 2409 y Fg(\022)1388 2537 y Fq(sup)1386 2613 y Fo(x)p Fm(2)p Ft(\003)1541 2537 y Fj(j)p Fn(C)1631 2551 y Fo(x)1677 2537 y Fj(j)26 b(\025)f Fn(v)1871 2409 y Fg(\023)1963 2537 y Fj(\024)g(j)p Fq(\003)p Fj(j)p Fn(e)2214 2499 y Fm(\000)p Fo(m)2336 2508 y Fe(1)2372 2499 y Fo(v)2415 2537 y Fn(;)197 b Fj(8)15 b Fn(v)28 b Fj(\025)d Fq(0)983 2787 y Fn(b)p Fq(\))182 b Fp(P)p Fq(\()p Fj(j)42 b Fn(<)25 b(f)34 b(>)1642 2801 y Ft(\003)p Fo(;J)1789 2787 y Fj(\000)1880 2763 y Fq(\026)1860 2787 y Fn(f)25 b Fj(j)g(\025)g(j)p Fq(\003)p Fj(j)2189 2750 y Fm(\000)p Fo(\017)2281 2787 y Fq(\))h Fj(\024)f Fn(e)2480 2750 y Fm(\000)p Fo(m)2602 2759 y Fe(2)2649 2750 y Fm(j)p Ft(\003)p Fm(j)2743 2722 y Fl(\016)0 3002 y Fh(Pr)-5 b(o)g(of.)0 3129 y Fq(P)m(art)31 b Fh(a\))g Fq(follo)m(ws)e(immediately)g(from)h(the)g(exp)s(onen)m(tial)g(tail)f (of)i(the)f(cluster)g(size)g(distribution)c(b)s(elo)m(w)k Fn(p)3681 3143 y Fo(c)3748 3129 y Fq(\(see)0 3256 y(e.g.)42 b([Gri]\).)0 3383 y(T)-8 b(o)29 b(pro)m(v)m(e)h(part)e Fh(b\))h Fq(w)m(e)g(\014rst)f(observ)m(e)h(that,)h(b)s(ecause)f(of)g (part)f(a\),)2327 3359 y(\026)2307 3383 y Fn(f)38 b Fq(is)28 b(w)m(ell)g(de\014ned.)39 b(Next,)30 b(giv)m(en)e Fn(\017)e Fj(2)e Fq(\(0)p Fn(;)15 b(\017)3797 3397 y Ft(0)3839 3383 y Fq(\),)0 3510 y(let)30 b Fn(l)158 3524 y Ft(1)224 3510 y Fq(=)25 b Fj(j)p Fq(\003)p Fj(j)433 3477 y Fo(\016)474 3510 y Fq(,)31 b Fn(l)557 3524 y Ft(2)623 3510 y Fq(=)25 b Fj(j)p Fq(\003)p Fj(j)832 3477 y Fo(\016)r Ft(\()p Fo(n)p Ft(+1\)+2)p Fo(\017)1219 3510 y Fq(where)30 b Fn(\016)f Fq(=)1657 3466 y Ft(1)p Fm(\000)p Fo(\017)p Ft([2)p Fo(d)p Ft(\()p Fo(n)p Ft(+1\)+1])p 1657 3489 525 4 v 1740 3541 a(1+)p Fo(d)p Ft(\()p Fo(n)p Ft(+1\))2063 3523 y Fe(2)2193 3510 y Fq(.)0 3637 y(Let)i(also)g(\002)417 3651 y Ft(1)488 3637 y Fq(b)s(e)f(the)g(ev)m(en)m(t)j(that)e Fj(j)41 b Fn(<)25 b(f)35 b(>)1519 3651 y Ft(\003)p Fo(;J)1667 3637 y Fj(\000)1757 3613 y Fq(\026)1738 3637 y Fn(f)24 b Fj(j)i(\025)f(j)p Fq(\003)p Fj(j)2067 3604 y Fm(\000)p Fo(\017)2190 3637 y Fq(and)30 b(\002)2438 3651 y Ft(2)2509 3637 y Fq(b)s(e)g(the)h(ev)m(en)m(t)h(that)f(sup)3364 3659 y Fo(x)p Fm(2)p Ft(\003)3524 3637 y Fj(j)p Fn(C)3614 3651 y Fo(x)3660 3637 y Fj(j)25 b(\024)h Fn(l)3834 3651 y Ft(1)3875 3637 y Fq(.)0 3764 y(Then)j(w)m(e)i(write)1371 3891 y Fp(P)p Fq(\(\002)1532 3905 y Ft(1)1573 3891 y Fq(\))26 b Fj(\024)f Fp(P)p Fq(\(\002)1891 3905 y Ft(1)1953 3891 y Fj(\\)19 b Fq(\002)2104 3905 y Ft(2)2145 3891 y Fq(\))h(+)g Fp(P)p Fq(\(\002)2452 3853 y Fo(c)2452 3913 y Ft(2)2494 3891 y Fq(\))1184 b(\(3)p Fn(:)p Fq(2\))0 4070 y(The)30 b(term)g Fp(P)p Fq(\(\002)565 4037 y Fo(c)565 4092 y Ft(2)607 4070 y Fq(\))g(is)g(b)s(ounded)e(from)i(ab)s(o)m(v)m(e) h(b)m(y)1356 4279 y Fp(P)p Fq(\(\002)1517 4242 y Fo(c)1517 4302 y Ft(2)1558 4279 y Fq(\))26 b Fj(\024)f(j)p Fq(\003)p Fj(j)p Fn(e)1870 4242 y Fm(\000)p Fo(m)1992 4251 y Fe(1)2028 4242 y Fo(l)2051 4251 y Fe(1)2116 4279 y Fj(\024)g Fn(e)2254 4242 y Fm(\000)p Fo(m)2376 4214 y Fk(0)2376 4260 y Fe(2)2411 4242 y Fm(j)p Ft(\003)p Fm(j)2505 4214 y Fl(\016)3713 4279 y Fq(\(3)p Fn(:)p Fq(3\))0 4489 y(b)s(ecause)30 b(of)h Fh(a\))p Fq(.)0 4616 y(In)h(order)h(to)h(estimate)g(the)g (\014rst)e(term)h(in)f(the)i(r.h.s.)48 b(of)34 b(\(3.2\),)i(consider)c (the)h(maximal)f(collection)h(of)h(cub)s(es)0 4743 y Fj(f)p Fn(Q)117 4757 y Fo(\013)169 4743 y Fj(g)22 b Fq(in)e(\003)i(suc) m(h)f(that:)37 b Fh(i\))22 b Fq(dist)o(\()p Fn(Q)1183 4757 y Fo(\013)1235 4743 y Fn(;)15 b(Q)1347 4757 y Fo(\014)1396 4743 y Fq(\))26 b(=)f(2)p Fn(l)1625 4757 y Ft(1)1687 4743 y Fq(for)d Fn(\013)k Fj(6)p Fq(=)e Fn(\014)j Fq(and)21 b Fh(ii\))h Fj(j)p Fn(Q)2455 4757 y Fo(\013)2507 4743 y Fj(j)j Fq(=)g Fn(l)2682 4710 y Fo(d)2680 4765 y Ft(2)2724 4743 y Fq(.)38 b(Let)22 b Fn(N)32 b Fq(b)s(e)20 b(the)i(n)m(um)m(b)s (er)f(of)g(suc)m(h)0 4870 y(cub)s(es.)41 b(Clearly)29 b Fj(j)p Fq(\003)21 b Fj(n)g([)839 4884 y Fo(\013)891 4870 y Fn(Q)963 4884 y Fo(\013)1014 4870 y Fj(j)26 b(\024)g Fn(C)1234 4837 y Fm(0)1270 4832 y Fo(l)1293 4841 y Fe(1)p 1270 4849 59 4 v 1270 4901 a Fo(l)1293 4910 y Fe(2)1339 4870 y Fj(j)p Fq(\003)p Fj(j)32 b Fq(for)e(a)h(suitable)f(constan)m(t)i Fn(C)2478 4837 y Fm(0)2534 4870 y Fq(so)f(that)g Fj(j)p Fq(\003)p Fj(j)c(\025)e Fn(N)10 b(l)3191 4837 y Fo(d)3189 4892 y Ft(2)3259 4870 y Fj(\025)26 b(j)p Fq(\003)p Fj(j)p Fq(\(1)c Fj(\000)e Fn(C)3734 4837 y Fm(0)3770 4832 y Fo(l)3793 4841 y Fe(1)p 3770 4849 V 3770 4901 a Fo(l)3793 4910 y Fe(2)3839 4870 y Fq(\).)0 4997 y(Therefore,)30 b(if)g(sup)658 5019 y Fo(x)p Fm(2)p Ft(\003)817 4997 y Fj(j)p Fn(C)907 5011 y Fo(x)953 4997 y Fj(j)c(\024)f Fn(l)1127 5011 y Ft(1)1125 5263 y Fj(j)40 b Fn(<)25 b(f)35 b(>)1437 5277 y Ft(\003)p Fo(;J)1584 5263 y Fj(\000)1674 5239 y Fq(\026)1655 5263 y Fn(f)24 b Fj(j)h(\024)1915 5201 y Fq(1)p 1881 5242 114 4 v 1881 5325 a Fj(j)p Fq(\003)p Fj(j)2006 5263 y(j)2061 5176 y Fg(X)2103 5367 y Fo(\013)2208 5263 y Fn(\030)2248 5277 y Fo(\013)2315 5263 y Fj(j)20 b Fq(+)g Fn(C)2523 5225 y Fm(00)2649 5201 y Fq(1)p 2580 5242 185 4 v 2580 5325 a Fj(j)p Fq(\003)p Fj(j)2693 5299 y Ft(2)p Fo(\017)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([9])1384 b Fq(3.2)p eop %%Page: 10 10 10 9 bop 0 105 a Fq(for)31 b(a)h(suitable)e(constan)m(t)i Fn(C)996 72 y Fm(00)1042 105 y Fq(,)g(where)f Fn(\030)1403 119 y Fo(\013)1481 105 y Fq(=)1579 36 y Fg(P)1714 105 y Fl(x)p Fk(2)p Fl(Q)1842 114 y(\013)1686 156 y(C)1731 165 y(x)1772 156 y Fk(\032)p Fl(Q)1870 165 y(\013)1947 4 y Fg(h)1990 105 y Fn(f)10 b Fq(\()2100 82 y(^)2080 105 y Fn(C)2145 119 y Fo(x)2190 105 y Fq(\))21 b Fj(\000)2356 81 y Fq(\026)2337 105 y Fn(f)2391 4 y Fg(i)2434 105 y Fq(.)43 b(Th)m(us,)31 b(if)g Fj(j)p Fq(\003)p Fj(j)g Fq(is)g(large)g(enough,)h(w)m(e)g(get)0 232 y(that)755 408 y Fp(P)p Fq(\(\002)916 422 y Ft(1)977 408 y Fj(\\)20 b Fq(\002)1129 422 y Ft(2)1170 408 y Fq(\))25 b Fj(\024)g Fp(P)p Fq(\()p Fj(j)1473 321 y Fg(X)1515 512 y Fo(\013)1619 408 y Fn(\030)1659 422 y Fo(\013)1726 408 y Fj(j)g(\025)1883 346 y(j)p Fq(\003)p Fj(j)1996 313 y Ft(1)p Fm(\000)p Fo(\017)p 1883 387 242 4 v 1981 470 a Fq(2)2136 408 y(\))g Fj(\024)g Fn(e)2334 370 y Fm(\000)p Fo(t)p Fm(j)p Ft(\003)p Fm(j)2512 343 y Fe(1)p Fk(\000)p Fl(\017)2644 334 y Fg(\000)2686 408 y Fp(E)2756 334 y Fg(\000)2803 408 y Fn(e)2845 370 y Ft(2)p Fo(t\030)2941 379 y Fl(\013)2993 334 y Fg(\001\001)3076 348 y Fo(N)3713 408 y Fq(\(3)p Fn(:)p Fq(4\))0 688 y(for)32 b(an)m(y)h(p)s(ositiv)m(e)e Fn(t)p Fq(,)i(b)s(ecause)g(of)g(the)f(exp)s (onen)m(tial)g(Cheb)m(yshev)f(inequalit)m(y)g(and)h(the)h(fact)g(that)g (the)g(random)0 815 y(v)-5 b(ariables)29 b Fj(f)p Fn(\030)462 829 y Fo(\013)514 815 y Fj(g)i Fq(are)g(i.i.d.)0 943 y(W)-8 b(e)32 b(no)m(w)e(c)m(ho)s(ose)h Fn(t)25 b Fq(=)1043 907 y Ft(1)p 799 922 525 4 v 799 993 a(4\()p Fo(l)886 960 y Fl(d)p Fe(\()p Fl(n)p Fe(+1\))886 1015 y(2)1092 993 y Ft(+)p Fm(j)1185 976 y Ft(\026)1170 993 y Fo(f)7 b Fm(j)p Fo(l)1257 971 y Fl(d)1257 1015 y Fe(2)1295 993 y Ft(\))1365 943 y Fq(and)30 b(observ)m(e)h(that)g(with)e(this)g(c)m (hoice)1516 1258 y Fp(E)11 b Fq(\()p Fn(e)1649 1221 y Ft(2)p Fo(t\030)1746 1230 y Fl(\013)1802 1258 y Fq(\))26 b Fj(\024)f Fn(e)2001 1221 y Ft(2)2037 1175 y Fm(p)p 2097 1175 35 4 v 46 x Fo(e)12 b(t)2171 1193 y Fe(2)2207 1221 y Ff(E)p Ft(\()p Fo(\030)2307 1193 y Fe(2)2305 1239 y Fl(a)2351 1221 y Ft(\))0 1499 y Fq(Let)31 b(us)f(estimate)h(the)f (second)h(momen)m(t)g(of)f Fn(\030)1588 1513 y Fo(\013)1640 1499 y Fq(.)0 1626 y(W)-8 b(e)69 b(denote)f(b)m(y)g Fj(A)761 1640 y Fo(x)874 1626 y Fq(the)g(set)g(of)g(all)e(connected)j(subgraphs) d(of)i Fp(Z)2687 1593 y Fo(d)2792 1626 y Fq(that)h(con)m(tain)f Fn(x)p Fq(.)152 b(Clearly)0 1753 y Fp(P)15 b Fq(\()q Fj(f)p Fn(C)216 1767 y Fo(x)288 1753 y Fq(=)24 b Fn(A)p Fj(g)d(\\)f(f)p Fn(C)708 1767 y Fo(y)777 1753 y Fq(=)25 b Fn(B)5 b Fj(g)p Fq(\))26 b(=)f Fp(P)p Fq(\()p Fn(C)1304 1767 y Fo(x)1376 1753 y Fq(=)g Fn(A)p Fq(\))15 b Fp(P)p Fq(\()p Fn(C)1745 1767 y Fo(y)1815 1753 y Fq(=)25 b Fn(B)5 b Fq(\))44 b(for)g(all)f Fn(A)48 b Fj(2)g(A)2655 1767 y Fo(x)2700 1753 y Fq(,)g Fn(B)k Fj(2)c(A)3076 1767 y Fo(y)3163 1753 y Fq(suc)m(h)c(that)h(there)f(is)0 1881 y(no)30 b(edge)h(with)e(one)i(endp)s(oin)m(t)e(in)g Fn(A)h Fq(and)g(the)h(other)f(in)f Fn(B)5 b Fq(.)0 2008 y(With)30 b(these)h(notations)f(w)m(e)h(write)142 2253 y Fp(E)11 b Fq(\()p Fn(\030)276 2215 y Ft(2)272 2275 y Fo(\013)330 2253 y Fq(\))25 b(=)548 2166 y Fg(X)486 2361 y Fo(x;y)r Fm(2)p Fo(Q)694 2370 y Fl(\013)797 2166 y Fg(X)772 2349 y Fl(A)p Fk(2A)913 2358 y Fl(x)766 2401 y(A)p Fk(\032)p Fl(Q)911 2410 y(\013)1026 2166 y Fg(X)1001 2349 y Fl(B)r Fk(2A)1144 2358 y Fl(y)995 2409 y(B)r Fk(\032)p Fl(Q)1142 2418 y(\013)1215 2253 y Fn(f)10 b Fq(\()p Fn(A)p Fq(\))15 b Fn(f)10 b Fq(\()p Fn(B)5 b Fq(\))15 b([)g Fp(P)g Fq(\()q Fj(f)p Fn(C)1893 2267 y Fo(x)1965 2253 y Fq(=)25 b Fn(A)p Fj(g)c(\\)e(f)p Fn(C)2385 2267 y Fo(y)2455 2253 y Fq(=)25 b Fn(B)5 b Fj(g)p Fq(\))20 b Fj(\000)g Fp(P)15 b Fq(\()q Fn(C)2987 2267 y Fo(x)3058 2253 y Fq(=)25 b Fn(A)p Fq(\))16 b Fp(P)f Fq(\()q Fn(C)3444 2267 y Fo(y)3513 2253 y Fq(=)25 b Fn(B)5 b Fq(\))15 b(])390 2586 y Fj(\024)548 2500 y Fg(X)486 2695 y Fo(x;y)r Fm(2)p Fo(Q)694 2704 y Fl(\013)797 2500 y Fg(X)772 2682 y Fl(A)p Fk(2A)913 2691 y Fl(x)766 2734 y(A)p Fk(\032)p Fl(Q)911 2743 y(\013)1026 2500 y Fg(X)1001 2682 y Fl(B)r Fk(2A)1144 2691 y Fl(y)995 2742 y(B)r Fk(\032)p Fl(Q)1142 2751 y(\013)1215 2586 y Fj(j)p Fn(f)10 b Fq(\()p Fn(A)p Fq(\))p Fj(j)15 b(j)p Fn(f)10 b Fq(\()p Fn(B)5 b Fq(\))p Fj(j)p Fn(\037)p Fq(\(dist\()p Fn(A;)15 b(B)5 b Fq(\))26 b Fj(\024)f Fq(1\)[)15 b Fp(P)g Fq(\()r Fj(f)p Fn(C)2673 2600 y Fo(x)2745 2586 y Fq(=)25 b Fn(A)p Fj(g)20 b(\\)g(f)p Fn(C)3165 2600 y Fo(y)3235 2586 y Fq(=)25 b Fn(B)5 b Fj(g)p Fq(\))385 2901 y(+)20 b Fp(P)15 b Fq(\()q Fn(C)647 2915 y Fo(x)718 2901 y Fq(=)25 b Fn(A)p Fq(\))16 b Fp(P)f Fq(\()q Fn(C)1104 2915 y Fo(y)1173 2901 y Fq(=)25 b Fn(B)5 b Fq(\))15 b(])390 3092 y Fj(\024)548 3006 y Fg(X)486 3201 y Fo(x;y)r Fm(2)p Fo(Q)694 3210 y Fl(\013)756 3092 y Fp(E)25 b Fq(\()7 b Fj(j)p Fn(f)j Fq(\()p Fn(C)1047 3106 y Fo(x)1092 3092 y Fq(\))p Fj(j)15 b(j)p Fn(f)10 b Fq(\()p Fn(C)1347 3106 y Fo(y)1392 3092 y Fq(\))p Fj(j)15 b Fn(\037)p Fq(\(dist\()p Fn(C)1807 3106 y Fo(x)1853 3092 y Fn(;)g(C)1958 3106 y Fo(y)2002 3092 y Fq(\))25 b Fj(\024)g Fq(1\))15 b(\))385 3382 y(+)537 3295 y Fg(X)476 3490 y Fo(x;y)r Fm(2)p Fo(Q)684 3499 y Fl(\013)786 3295 y Fg(X)762 3478 y Fl(A)p Fk(2A)903 3487 y Fl(x)756 3530 y(A)p Fk(\032)p Fl(Q)901 3539 y(\013)974 3382 y Fj(j)p Fn(f)10 b Fq(\()p Fn(A)p Fq(\))p Fj(j)15 b Fp(P)g Fq(\()r Fn(C)1404 3396 y Fo(x)1475 3382 y Fq(=)25 b Fn(A)p Fq(\))16 b Fp(E)25 b Fq(\()7 b Fj(j)p Fn(f)j Fq(\()p Fn(C)1981 3396 y Fo(y)2024 3382 y Fq(\))p Fj(j)p Fn(\037)p Fq(\(dist\()p Fn(A;)15 b(C)2532 3396 y Fo(y)2577 3382 y Fq(\))25 b Fj(\024)g Fq(1\))15 b(\))390 3693 y Fj(\024)25 b Fn(k)533 3707 y Ft(2)574 3693 y Fn(l)603 3656 y Fo(d)601 3716 y Ft(2)0 3948 y Fq(for)20 b(a)h(suitable)e(p)s (ositiv)m(e)h(constan)m(t)i Fn(k)1255 3962 y Ft(2)1296 3948 y Fq(,)h(b)s(ecause)d(as)h(w)m(e)g(are)g(b)s(elo)m(w)f(the)g(p)s (ercolation)g(threshold)f Fp(E)11 b Fq(\()p Fn(f)f Fq(\()p Fn(C)3530 3962 y Ft(0)3576 3948 y Fq(\))3611 3915 y Ft(4)3653 3948 y Fq(\))25 b Fn(<)g Fj(1)0 4076 y Fq(and)30 b Fp(P)p Fq(\()p Fn(x)d Fj($)g Fn(x)516 4043 y Fm(0)541 4076 y Fq(\))g Fj(\024)e Fn(e)741 4043 y Fm(\000)p Fo(md)p Ft(\()p Fo(x;x)1033 4015 y Fk(0)1058 4043 y Ft(\))1091 4076 y Fq(,)31 b(where)g Fn(m)26 b Fq(=)g Fn(m)p Fq(\()p Fn(p)p Fq(\))31 b(is)f(a)h(p)s(ositiv)m(e)f(constan)m(t)i(and)f Fn(x)26 b Fj($)g Fn(x)3138 4043 y Fm(0)3194 4076 y Fq(means)31 b(that)h Fn(x)f Fq(and)0 4203 y Fn(x)52 4170 y Fm(0)108 4203 y Fq(are)g(connected)g(b)m(y)f(a)h(path)f(of)g(op)s(en)g(b)s(onds) f(\(see)i([Gri]\).)41 b(Th)m(us,)29 b(with)g(the)i(ab)s(o)m(v)m(e)h(c)m (hoice)f(of)f Fn(t)g Fq(and)g Fn(\016)s Fq(,)1057 4443 y(r.h.s.)40 b(of)31 b(\(3.4\))57 b Fj(\024)25 b Fn(e)1779 4404 y Fm(\000)1836 4411 y Fq([)1862 4404 y Fo(t)p Fm(j)p Ft(\003)p Fm(j)1983 4376 y Fk(\000)p Fl(\017)2064 4404 y Fm(\000)p Fo(k)2160 4413 y Fe(3)2195 4404 y Fo(t)2222 4376 y Fe(2)2257 4411 y Fq(])2283 4404 y Fm(j)p Ft(\003)p Fm(j)2405 4443 y Fj(\024)g Fn(e)2543 4406 y Fm(\000)p Fo(m)2665 4378 y Fk(00)2665 4424 y Fe(2)2710 4406 y Fm(j)p Ft(\003)p Fm(j)2804 4378 y Fl(\016)0 4683 y Fq(for)30 b(a)h(suitable)f(constan)m(t)h Fn(m)1001 4650 y Fm(00)1001 4706 y Ft(2)1078 4683 y Fq(pro)m(vided)f(that)h Fj(j)p Fq(\003)p Fj(j)g Fq(is)f(large)g(enough.)42 b(Putting)29 b(together)k(the)d(ab)s(o)m(v)m(e)i(estimate)0 4811 y(and)e(\(3.3\))i (the)f(pro)s(of)e(is)h(concluded.)90 b Fb(\003)0 5055 y Fq(Here)31 b(is)e(a)i(simple)d(consequence)k(of)e(the)h(ab)s(o)m(v)m (e)g(large)g(deviation)e(results.)0 5213 y(Giv)m(en)i Fn(\017)26 b Fj(2)g Fq(\(0)p Fn(;)15 b(\017)571 5227 y Ft(0)613 5213 y Fq(\),)31 b(an)g(in)m(teger)g Fn(L)g Fq(and)f Fn(N)41 b Fq(real,)31 b(translation)f(in)m(v)-5 b(arian)m(t)30 b(functions)g Fj(f)p Fn(f)3045 5227 y Fo(i)3075 5213 y Fj(g)3120 5180 y Fo(N)3120 5236 y(i)p Ft(=1)3274 5213 y Fq(on)g(the)h(set)h(of)f(all)0 5340 y(\014nite)26 b(connected)i(subgraphs)e(of)h(of)g Fp(Z)1343 5307 y Fo(d)1409 5340 y Fq(suc)m(h)f(that)i(max)1973 5354 y Fo(i)p Fm(\024)p Fo(N)2140 5340 y Fj(j)p Fn(f)2210 5354 y Fo(i)2240 5340 y Fq(\()p Fn(A)p Fq(\))p Fj(j)e(\024)f(j)p Fn(A)2618 5354 y Fo(v)2662 5340 y Fj(j)2687 5307 y Ft(4)2755 5340 y Fq(for)i(an)m(y)h Fn(A)p Fq(,)g(let)f(us)g(consider)f(the)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)k Ft([10])1348 b Fq(3.3)p eop %%Page: 11 11 11 10 bop 0 91 a Fq(ev)m(en)m(t)32 b(\002\()p Fn(\017;)15 b(M)5 b(;)15 b(N)5 b(;)15 b(L)p Fq(\))27 b(=)d Fj([)954 105 y Fo(R)p Fm(2R)1120 83 y Fl(\017)1120 128 y(L)1173 91 y Fq(\002)1244 105 y Fo(R)1303 91 y Fq(\()p Fn(\017;)15 b(M)5 b(;)15 b(N)5 b(;)15 b(L)p Fq(\))32 b(where)1243 299 y(\002)1314 313 y Fo(R)1373 299 y Fq(\()p Fn(\017;)15 b(M)5 b(;)15 b(N)5 b(;)15 b(L)p Fq(\))27 b(=)277 490 y Fj(f)20 b Fq(sup)337 566 y Fo(x)p Fm(2)p Fo(R)498 490 y Fj(j)p Fn(C)588 504 y Fo(x)634 490 y Fj(j)25 b(\025)g Fn(M)h Fq(log)16 b Fn(L)f Fj(g)21 b([)f(f)c Fq(sup)1310 566 y Fo(n)p Fm(\024)p Ft(4)1470 404 y Fg(X)1463 599 y Fo(x)p Fm(2)p Fo(R)1624 490 y Fj(j)p Fn(C)1714 504 y Fo(x)1760 490 y Fj(j)1785 452 y Fo(n)1860 490 y Fj(\025)25 b Fn(M)10 b Fj(j)p Fn(R)q Fj(j)15 b(g)21 b([)f(f)g Fq(sup)2396 566 y Fo(i)p Fm(\024)p Fo(N)2558 490 y Fj(j)41 b Fn(<)25 b(f)2765 504 y Fo(i)2820 490 y Fn(>)2891 504 y Fo(R;J)3044 490 y Fj(\000)3134 466 y Fq(\026)3115 490 y Fn(f)3160 504 y Fo(i)3204 490 y Fj(j)h(\025)f(j)p Fn(R)q Fj(j)3471 452 y Fm(\000)p Fo(\017)3578 490 y Fj(g)0 769 y Fq(Then)0 940 y Fr(Corollary)93 b(3.2.)53 b Fc(Assume)81 b Fn(p)109 b(<)g(p)1507 954 y Fo(c)1544 940 y Fc(.)193 b(There)80 b(exists)h Fn(M)91 b Fc(suc)m(h)80 b(that)i(for)f(an)m(y)g Fn(a)110 b(<)f Fj(1)0 998 y Fg(P)96 1021 y Fm(1)96 1094 y Ft(1=)p Fo(L)258 1066 y Fq(sup)394 1089 y Fo(N)7 b Fm(\024)p Fo(L)566 1070 y Fl(a)625 1066 y Fp(P)15 b Fq(\()q(\002\()p Fn(\017;)g(M)5 b(;)15 b(N)5 b(;)15 b(L)p Fq(\)\))27 b Fn(<)e Fq(+)p Fj(1)p Fc(.)0 1237 y Fh(Pr)-5 b(o)g(of.)26 b Fq(It)f(follo)m(ws)f(at)i(once)f(from)f(prop)s(osition)f(3.1)j (applied)c(to)k(eac)m(h)g Fn(f)2442 1251 y Fo(i)2496 1237 y Fq(and)f(to)g(the)g(functions)e Fn(g)3353 1251 y Fo(n)3403 1237 y Fq(\()p Fn(A)p Fq(\))j(=)f Fj(j)p Fn(A)3756 1251 y Fo(v)3800 1237 y Fj(j)3825 1204 y Fo(n)3875 1237 y Fq(,)0 1364 y Fn(n)g Fq(=)g(1)p Fn(;)15 b(:)g(:)g(:)i(;)e Fq(4)31 b(and)f(the)g(fact)i(that)f(the)f(cardinalit)m(y)f(of)i Fj(R)2001 1331 y Fo(\017)2001 1389 y(L)2085 1364 y Fq(is)f(b)s(ounded)e (from)i(ab)s(o)m(v)m(e)h(b)m(y)g Fn(L)3209 1331 y Ft(2)p Fo(d)3287 1364 y Fq(.)91 b Fb(\003)0 1664 y Fr(4.)47 b(Preliminary)34 b(results)0 1813 y Fq(In)c(this)f(section)i(w)m(e)g (collect)g(sev)m(eral)g(preliminary)c(results)i(that)j(are)e(essen)m (tial)h(in)e(order)h(to)h(pro)m(v)m(e)h(that,)f(with)0 1940 y(large)j(probabilit)m(y)-8 b(,)32 b(the)i(sp)s(ectral)e(gap)i(of) g(the)f(Ka)m(w)m(asaki)h(dynamics)e(for)i(the)f(dilute)f(Ising)g(mo)s (del)g(b)s(elo)m(w)h Fn(p)3838 1954 y Fo(c)3875 1940 y Fq(,)0 2067 y(on)e(all)g(scales)g(b)s(et)m(w)m(een)h Fn(L)923 2034 y Fo(\017)989 2067 y Fq(to)g Fn(L)f Fq(can)h(b)s(e)f(b)s (ounded)e(from)i(b)s(elo)m(w)f(b)m(y)i(exactly)g(the)f(same)h(metho)s (ds)f(emplo)m(y)m(ed)0 2193 y(in)f(the)i(high)e(temp)s(erature)h (region.)44 b(More)32 b(precisely)e(w)m(e)i(will)d(form)m(ulate)i (three)h(conditions)e(on)h(the)h(disorder)0 2320 y(con\014guration)39 b(in)f(the)i(cub)s(e)e Fn(Q)1136 2334 y Fo(L)1230 2320 y Fq(whic)m(h)g(will)f(ensure)h(that,)43 b(if)38 b(satis\014ed,)j(the)f (corresp)s(onding)d(dilute)h(Ising)0 2447 y(mo)s(del)e(shares)g(all)f (the)i(relev)-5 b(an)m(t)37 b(\(for)g(our)f(purp)s(oses\))g(features)h (of)f(the)h(high)f(temp)s(erature)g(standard)g(Ising)0 2574 y(mo)s(del.)47 b(Moreo)m(v)m(er)35 b(our)d(conditions)f(will)g(b)s (e)h(meaningful)e(in)i(the)h(sense)g(that)g(the)g(probabilit)m(y)d(of)j (not)g(b)s(eing)0 2700 y(all)c(v)m(eri\014ed)h(sim)m(ultaneously)e (will)g(b)s(e)h(summable)g(in)g Fn(L)p Fq(.)0 2849 y(Let)41 b Fn(\017)210 2863 y Ft(0)293 2849 y Fq(=)501 2814 y Ft(1)p 418 2829 203 4 v 418 2881 a(10)p Fo(d)p Ft(+1)672 2849 y Fq(and)f(let)g(us)g(\014x)g(a)i(small)d(p)s(ositiv)m(e)h(n)m(um) m(b)s(er)f Fn(\017)j Fj(2)h Fq(\(0)p Fn(;)15 b(\017)2629 2863 y Ft(0)2670 2849 y Fq(\))41 b(and)g(a)g(large)g(p)s(ositiv)m(e)e (n)m(um)m(b)s(er)0 2976 y Fn(M)10 b Fq(.)71 b(F)-8 b(or)41 b(an)m(y)f(\003)i Fj(\032)g Fp(Z)831 2943 y Fo(d)869 2976 y Fq(,)h(an)m(y)d(in)m(teger)h Fn(N)52 b Fj(2)41 b(f)p Fq(1)p Fn(;)15 b Fq(2)p Fn(;)g(:)g(:)g(:)j(;)d Fj(j)p Fq(\003)p Fj(jg)p Fq(,)45 b(an)m(y)c(b)s(oundary)d(condition)h Fn(\034)52 b Fj(2)41 b Fq(\012)f(and)g(an)m(y)0 3103 y(disorder)25 b(con\014guration)h Fn(J)36 b Fq(let)27 b(also)g Fn(\025)f Fq(=)f Fn(\025)p Fq(\(\003)p Fn(;)15 b(N)5 b(;)15 b(\034)5 b(;)15 b(J)9 b Fq(\))29 b(b)s(e)d(the)h (\(unique\))f(constan)m(t)i(c)m(hemical)f(p)s(oten)m(tial)f(suc)m(h)0 3266 y(that)41 b Fn(\026)262 3223 y Fo(J)n(;\034)s(;\025)262 3293 y Ft(\003)428 3266 y Fq(\()p Fn(N)536 3280 y Ft(\003)590 3266 y Fq(\))h(=)f Fn(N)50 b Fq(and)39 b(let)h Fn(\025)1282 3280 y Ft(0)1364 3266 y Fq(=)h Fn(\025)1529 3280 y Ft(0)1570 3266 y Fq(\(\003)p Fn(;)15 b(N)10 b Fq(\))41 b(b)s(e)e(suc)m(h)h(that)g Fp(E)2491 3165 y Fg(\020)2551 3266 y Fj(j)2597 3243 y Fq(^)2576 3266 y Fn(C)2641 3280 y Ft(0)2682 3266 y Fj(j)2707 3233 y Fm(\000)p Ft(1)2805 3266 y Fn(\026)2860 3223 y Fo(\014)s(;\025)2969 3232 y Fe(0)2876 3294 y Ft(^)2860 3310 y Fo(C)2912 3319 y Fe(0)3009 3266 y Fq(\()p Fn(N)3117 3280 y Fo(C)3169 3289 y Fe(0)3210 3266 y Fq(\))3245 3165 y Fg(\021)3341 3266 y Fq(=)3478 3231 y Fo(N)p 3463 3246 93 4 v 3463 3298 a Fm(j)p Ft(\003)p Fm(j)3567 3266 y Fq(,)j(tah)m(t)e(is)0 3430 y(the)c(particle)f(densit)m(y)g(of)h(the)g (cluster)f(of)h(the)g(origin)e(a)m(v)m(eraged)k(on)e(the)g(disorder)d (is)i(equal)h(to)g Fn(N)r(=)p Fj(j)p Fq(\003)p Fj(j)p Fq(.)61 b(The)0 3557 y(existence)32 b(and)f(uniqueness)f(of)i(the)g(c)m (hemical)f(p)s(oten)m(tial)g Fn(\025)h Fq(is)f(pro)m(v)m(ed)h(in)e(the) i(app)s(endix)d(of)j([CM1],)i(for)d Fn(\025)3782 3571 y Ft(0)3855 3557 y Fq(a)0 3683 y(similar)d(reasoning)i(can)g(b)s(e)g (applied.)38 b(Then)30 b(our)g(conditions)f(read)h(as)g(follo)m(ws.)0 3854 y Fr(Assumption)35 b(1.)52 b Fc(F)-8 b(or)31 b(an)m(y)g Fn(R)26 b Fj(2)f(R)1306 3821 y Fo(\017)1306 3879 y(L)815 4038 y Fq(max)829 4096 y Fo(x)p Fm(2)p Ft(\003)999 4038 y Fj(j)p Fn(C)1089 4052 y Fo(x)1135 4038 y Fj(j)41 b(\024)25 b Fn(M)40 b Fq(log)17 b Fn(L)181 b Fq(and)227 b(max)2191 4097 y Fo(n)p Fm(\024)p Ft(4)2366 3951 y Fg(X)2359 4146 y Fo(x)p Fm(2)p Fo(R)2520 4038 y Fj(j)p Fn(C)2610 4052 y Fo(x)2656 4038 y Fj(j)2681 4000 y Fo(n)2756 4038 y Fj(\024)25 b Fn(M)g Fj(j)p Fn(R)q Fj(j)0 4331 y Fr(Assumption)35 b(2.)52 b Fc(Let)31 b Fn(h)928 4345 y Fo(x)1000 4331 y Fq(:=)25 b Fn(e)1163 4298 y Fm(\000r)1281 4307 y Fl(x)1321 4298 y Fo(H)1390 4331 y Fn(\033)s Fq(\()p Fn(x)p Fq(\))p Fc(.)42 b(Then,)29 b(for)h(an)m(y)h Fn(R)26 b Fj(2)f(R)2465 4298 y Fo(\017)2465 4356 y(L)2519 4331 y Fc(,)247 4609 y Fq(sup)295 4681 y Fo(\034)622 4609 y Fq(sup)399 4689 y Fo(N)7 b Fm(2)p Ft([)p Fo(\017)p Fm(j)p Fo(R)p Fm(j)p Fo(;)p Ft(\(1)p Fm(\000)p Fo(\017)p Ft(\))p Fm(j)p Fo(R)p Fm(j)p Ft(])997 4609 y Fj(j)15 b Fn(\026)1092 4565 y Fo(\034)s(;\025)1194 4574 y Fe(0)1092 4636 y Fo(R)1234 4609 y Fq(\()p Fn(N)1342 4623 y Fo(R)1402 4609 y Fn(;)1450 4523 y Fg(X)1442 4718 y Fo(x)p Fm(2)p Fo(R)1603 4609 y Fn(h)1655 4623 y Fo(x)1701 4609 y Fq(\))21 b Fj(\000)f(j)p Fn(R)q Fj(j)15 b Fp(E)2053 4453 y Fg( )2131 4609 y Fj(j)p Fn(C)2221 4623 y Ft(0)2262 4609 y Fj(j)2287 4571 y Fm(\000)p Ft(1)2385 4609 y Fn(\026)2440 4565 y Fo(\014)s(;\025)2549 4574 y Fe(0)2456 4636 y Ft(^)2440 4652 y Fo(C)2492 4661 y Fe(0)2589 4609 y Fq(\()p Fn(N)2697 4623 y Fo(C)2749 4632 y Fe(0)2789 4609 y Fn(;)2853 4523 y Fg(X)2829 4718 y Fo(x)p Fm(2)p Fo(C)2971 4727 y Fe(0)3023 4609 y Fn(h)3075 4623 y Fo(x)3121 4609 y Fq(\))3156 4453 y Fg(!)3258 4609 y Fj(j)26 b(\024)f(j)p Fn(R)q Fj(j)3525 4571 y Ft(1)p Fm(\000)p Fo(\017)0 4916 y Fc(Similarily)h(for)552 4892 y Fq(^)551 4916 y Fn(h)603 4930 y Fo(x)674 4916 y Fq(:=)g Fn(e)838 4883 y Fm(\000r)956 4892 y Fl(x)996 4883 y Fo(H)1065 4916 y Fq(\(1)21 b Fj(\000)f Fn(\033)s Fq(\()p Fn(x)p Fq(\)\))31 b Fc(and)1678 4892 y Fq(~)1677 4916 y Fn(h)1729 4930 y Fo(x)1801 4916 y Fq(=)25 b Fn(\033)s Fq(\()p Fn(x)p Fq(\))p Fc(.)0 5087 y Fr(Assumption)35 b(3.)52 b Fc(F)-8 b(or)31 b(an)m(y)g Fn(R)26 b Fj(2)f(R)1306 5054 y Fo(\017)1306 5112 y(L)893 5271 y Fq(sup)941 5343 y Fo(\034)1268 5271 y Fq(sup)1045 5351 y Fo(N)7 b Fm(2)p Ft([)p Fo(\017)p Fm(j)p Fo(R)p Fm(j)p Fo(;)p Ft(\(1)p Fm(\000)p Fo(\017)p Ft(\))p Fm(j)p Fo(R)p Fm(j)p Ft(])1643 5271 y Fj(j)15 b Fn(\025)p Fq(\()p Fn(R)q(;)g(N)5 b(;)15 b(\034)5 b(;)15 b(J)9 b Fq(\))22 b Fj(\000)e Fn(\025)2344 5285 y Ft(0)2385 5271 y Fq(\()p Fn(R)q(;)15 b(N)10 b Fq(\))p Fj(j)26 b(\024)f(j)p Fn(R)q Fj(j)2915 5233 y Fm(\000)p Fo(\017)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([11])1348 b Fq(4.1)p eop %%Page: 12 12 12 11 bop 0 91 a Fr(De\014nition.)53 b Fc(The)21 b(set)h(of)g(disorder) e(con\014gurations)h Fn(J)31 b Fc(that)22 b(satisfy)f(assumptions)f(1,) k(2)e(and)g(3)g(will)d(b)s(e)i(denoted)0 220 y(b)m(y)30 b Fq(\002)197 234 y Fo(g)r(ood)349 220 y Fq(\()p Fn(L;)15 b(M)5 b(;)15 b(\017)p Fq(\))p Fc(.)0 552 y Fq(Thanks)29 b(to)i(Corollary)e(3.2)j(w)m(e)f(ha)m(v)m(e)g(the)g(follo)m(wing)e (result.)0 792 y Fr(Prop)s(osition)36 b(4.1.)53 b Fc(Assume)30 b Fn(p)25 b(<)g(p)1330 806 y Fo(c)1367 792 y Fc(.)40 b(Then)86 940 y(\(i\))45 b(there)31 b(exists)f Fn(M)40 b Fc(suc)m(h)30 b(that)h(for)f(an)m(y)h Fn(\017)25 b Fj(2)g Fq(\(0)p Fn(;)15 b(\017)1857 954 y Ft(0)1899 940 y Fq(\))1965 872 y Fg(P)2061 895 y Fm(1)2061 968 y Fo(L)p Ft(=1)2222 940 y Fp(P)g Fq(\()q(\002)2399 954 y Fo(g)r(ood)2551 940 y Fq(\()p Fn(L;)g(M)5 b(;)15 b(\017)p Fq(\))2893 907 y Fo(c)2931 940 y Fq(\))26 b Fn(<)f Fj(1)p Fc(.)227 1089 y(In)30 b(particular,)f(for)h(an)m(y)h(large)f(enough)g Fn(M)41 b Fc(and)30 b(an)m(y)g Fn(\017)25 b Fj(2)g Fq(\(0)p Fn(;)15 b(\017)2414 1103 y Ft(0)2456 1089 y Fq(\))61 1237 y Fc(\(ii\))44 b(there)36 b(exists)f(a)g(set)h Fq(\002)1019 1251 y Ft(0)1093 1237 y Fj(\032)d Fq(\002)i Fc(suc)m(h)f(that)i Fp(P)p Fq(\(\002)1875 1251 y Ft(0)1917 1237 y Fq(\))d(=)g(1)j Fc(and)f(for)g(an)m(y)g Fn(J)43 b Fj(2)32 b Fq(\002)2929 1251 y Ft(0)3005 1237 y Fc(there)k(exists)e Fn(L)p Fq(\()p Fn(J)9 b Fq(\))36 b Fc(suc)m(h)227 1367 y(that)31 b Fn(J)k Fj(2)25 b Fq(\002)666 1381 y Fo(g)r(ood)817 1367 y Fq(\()p Fn(L;)15 b(M)5 b(;)15 b(\017)p Fq(\))32 b Fc(for)e(an)m(y)h Fn(L)25 b Fj(\025)g Fn(L)p Fq(\()p Fn(J)9 b Fq(\))p Fc(;)35 1515 y(\(iii\))44 b(there)31 b(exists)f Fn(\015)g Fq(=)25 b Fn(\015)5 b Fq(\()p Fn(M)10 b Fq(\))26 b Fn(>)f Fq(0)p Fc(,)31 b Fq(lim)1454 1529 y Fo(M)7 b Fm(!)p Ft(+)p Fm(1)1750 1515 y Fn(\015)e Fq(\()p Fn(M)10 b Fq(\))26 b(=)f(+)p Fj(1)p Fc(,)31 b(suc)m(h)f(that)h Fp(P)p Fq(\()p Fn(L)p Fq(\()p Fn(J)9 b Fq(\))26 b Fn(>)f(l)r Fq(\))h Fj(\024)40 b Fn(l)3345 1482 y Fm(\000)p Fo(\015)3448 1515 y Fc(.)0 1755 y Fh(Pr)-5 b(o)g(of.)31 b Fq(Once)f(p)s(oin)m(t)f(\()p Fn(i)p Fq(\))i(of)f(the)g(prop)s(osition)e(is)h(established)f(p)s(oin)m (t)h(\()p Fn(ii)p Fq(\))i(is)e(nothing)g(but)h(the)g(standard)f(Borel)0 1884 y(Can)m(telli)g(lemma.)0 2013 y(T)-8 b(o)38 b(analyze)g(the)f(con) m(v)m(ergence)j(of)d(the)h(series)1672 1945 y Fg(P)1768 2041 y Fo(L)1837 2013 y Fp(P)15 b Fq(\()q(\002)2014 2027 y Fo(g)r(ood)2166 2013 y Fq(\()p Fn(L;)g(M)5 b(;)15 b(\017)p Fq(\))2508 1980 y Fo(c)2546 2013 y Fq(\))38 b(w)m(e)g(\014rst)e(observ) m(e)i(that,)i(thanks)d(to)0 2143 y(prop)s(osition)43 b(3.1,)50 b(the)45 b(probabilit)m(y)d(that)k(assumption)d(1)i(is)f (violated)h(can)g(b)s(e)g(b)s(ounded)d(from)j(ab)s(o)m(v)m(e)h(b)m(y)0 2272 y Fn(c)15 b Fj(jR)156 2239 y Fo(\017)156 2297 y(L)211 2272 y Fj(j)g Fq(\()p Fn(L)348 2239 y Fm(\000)p Fo(mM)574 2272 y Fq(+)22 b Fn(e)709 2239 y Fm(\000)p Fo(m)831 2248 y Fe(2)867 2239 y Ft(\()p Fo(\017L)975 2211 y Fl(\017)1008 2239 y Ft(\))1036 2211 y Fl(d\016)1110 2272 y Fq(\))35 b(\(where)f Fn(m)g Fq(is)g(a)h(p)s(ositiv)m(e)e(constan)m(t)j(dep)s (ending)c(on)i Fn(p)g Fq(and)g Fn(n)g Fq(and)g(w)m(e)h(used)0 2401 y Fj(j)p Fn(R)q Fj(j)f(\025)f Fq(\()p Fn(\017L)392 2368 y Fo(\017)427 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Fl(j)2669 2162 y Fo(t)2696 2134 y Fe(2)2696 2180 y Fl(j)0 2399 y Fq(for)30 b(a)h(suitable)e(constan)m (t)i Fn(\013)26 b Fq(:=)f Fn(\013)p Fq(\()p Fn(\017;)15 b(M)10 b Fq(\),)33 b(where)c Fn(\033)1803 2366 y Ft(2)1800 2422 y Fo(j)1870 2399 y Fq(:=)c Fn(\026)2046 2349 y Fo(\034)s(;\025)p 2105 2364 44 4 v 2046 2425 a Ft(\003)2152 2399 y Fq(\()p Fn(N)2260 2413 y Ft(\003)2310 2422 y Fl(j)2349 2399 y Fn(;)15 b(N)2462 2413 y Ft(\003)2512 2422 y Fl(j)2550 2399 y Fq(\).)92 b Fb(\003)0 2757 y Fu(4.3)35 b(A)f(block)f(d)n (ynamics)g(bound)0 2963 y Fq(Here)38 b(w)m(e)g(giv)m(e)g(a)f(result)g (that)h(is)e(a)i(k)m(ey)g(step)f(in)f(our)h(recursiv)m(e)g(b)s(ound)e (of)j(the)f(sp)s(ectral)g(gap)h(of)f(Ka)m(w)m(asaki)0 3090 y(dynamics.)50 b(F)-8 b(or)34 b(simplicit)m(y)d(w)m(e)j(discuss)e (our)i(estimate)g(in)f(t)m(w)m(o)i(dimensions)c(only)i(\(see)i(ho)m(w)m (ev)m(er)g(remark)e(at)0 3217 y(the)e(end)e(of)i(section)f(3.3)i(in)d ([CM1])i(for)f(its)g(generalization)g(to)h(higher)e(dimensions\).)0 3371 y(Let)i(\003)f(b)s(e)g(an)g(elemen)m(t)h(of)969 3348 y(\026)946 3371 y Fj(R)1023 3338 y Fo(\017)1023 3396 y(L)1077 3371 y Fq(.)41 b(Without)30 b(loss)g(of)g(generalit)m(y)h (w)m(e)g(can)f(assume)g(that)544 3592 y(\003)25 b(=)728 3518 y Fg(\010)781 3592 y Fq(\()p Fn(x)868 3606 y Ft(1)909 3592 y Fn(;)15 b(x)1001 3606 y Ft(2)1043 3592 y Fq(\))25 b Fj(2)g Fp(Z)1254 3555 y Ft(2)1290 3592 y Fq(;)46 b(0)41 b Fj(\024)f Fn(x)1610 3606 y Ft(1)1691 3592 y Fj(\024)g Fn(l)1829 3606 y Ft(1)1890 3592 y Fj(\000)20 b Fq(1)p Fn(;)46 b Fq(0)41 b Fj(\024)f Fn(x)2346 3606 y Ft(2)2428 3592 y Fj(\024)g Fn(l)2566 3606 y Ft(2)2627 3592 y Fj(\000)20 b Fq(1)2793 3518 y Fg(\011)2877 3592 y Fn(;)197 b(l)3126 3606 y Ft(1)3192 3592 y Fj(\024)25 b Fn(l)3315 3606 y Ft(2)0 3813 y Fq(Let)32 b(\003)227 3827 y Ft(1)295 3813 y Fq(=)392 3740 y Fg(\010)445 3813 y Fq(\()p Fn(x)532 3827 y Ft(1)573 3813 y Fn(;)15 b(x)665 3827 y Ft(2)706 3813 y Fq(\))28 b Fj(2)e Fq(\003;)31 b(0)42 b Fj(\024)g Fn(x)1227 3827 y Ft(2)1310 3813 y Fj(\024)g Fq(\()1469 3777 y Ft(1)p 1469 3792 37 4 v 1469 3845 a(2)1537 3813 y Fq(+)20 b(2)p Fn(\017)p Fq(\))15 b Fn(l)1787 3827 y Ft(2)1860 3740 y Fg(\011)1913 3813 y Fq(,)32 b(\003)2033 3827 y Ft(2)2101 3813 y Fq(=)2199 3740 y Fg(\010)2252 3813 y Fq(\()p Fn(x)2339 3827 y Ft(1)2380 3813 y Fn(;)15 b(x)2472 3827 y Ft(2)2513 3813 y Fq(\))27 b Fj(2)g Fq(\003;)j(\()2827 3777 y Ft(1)p 2827 3792 V 2827 3845 a(2)2895 3813 y Fq(+)21 b Fn(\017)p Fq(\))15 b Fn(l)3101 3827 y Ft(2)3184 3813 y Fj(\024)42 b Fn(x)3349 3827 y Ft(2)3432 3813 y Fj(\024)g Fn(l)3572 3827 y Ft(2)3644 3740 y Fg(\011)3697 3813 y Fq(,)32 b(and)0 3940 y(let)e(\003)194 3954 y Ft(3)260 3940 y Fq(=)25 b(\003)419 3954 y Ft(1)481 3940 y Fj(\\)19 b Fq(\003)624 3954 y Ft(2)665 3940 y Fq(.)41 b(Clearly)29 b(\003)1110 3954 y Fo(i)1165 3940 y Fj(2)c(R)1328 3907 y Fo(\017)1328 3965 y(L)1383 3940 y Fq(,)30 b Fn(i)c Fq(=)f(1)p Fn(;)15 b Fq(2)p Fn(;)g Fq(3.)0 4067 y(Let)35 b(\014nally)c Fn(N)518 4081 y Fo(i)583 4067 y Fq(b)s(e)i(p)s(ossible)f (v)-5 b(alues)33 b(of)h(the)g(n)m(um)m(b)s(er)f(of)h(particles)f(in)g (\003)2580 4081 y Fo(i)2610 4067 y Fq(,)j Fn(i)31 b Fq(=)g(1)p Fn(;)15 b Fq(2)p Fn(;)g Fq(3,)37 b(and)d(let)g Fn(\027)3479 4034 y Fo(\034)3473 4093 y Ft(\003)p Fo(;)p Fa(N)3646 4067 y Fq(b)s(e)g(the)0 4194 y(m)m(ulti)29 b(canonical)h(Gibbs)f (measure)h Fn(\026)1312 4161 y Fo(\034)1312 4220 y Ft(\003)1366 4121 y Fg(\000)1408 4194 y Fj(\001)15 b(j)g Fn(N)1561 4208 y Ft(\003)1611 4217 y Fl(i)1670 4194 y Fq(=)25 b Fn(N)1839 4208 y Fo(i)1869 4194 y Fn(;)31 b(i)25 b Fq(=)g(1)p Fn(;)15 b(:)g(:)g(:)i(;)e Fq(3)2384 4121 y Fg(\001)2427 4194 y Fq(.)41 b(Then)29 b(w)m(e)i(ha)m(v)m(e)0 4374 y Fr(Prop)s(osition)39 b(4.5.)53 b Fc(Assume)32 b Fn(J)38 b Fj(2)29 b Fq(\002)1371 4388 y Fo(g)r(ood)1523 4374 y Fq(\()p Fn(M)5 b(;)15 b(\017;)g(L)p Fq(\))p Fc(.)49 b(Then,)33 b(for)f(an)m(y)h Fn(\015)h(>)29 b Fq(0)34 b Fc(there)e(exist)h Fn(L)3294 4388 y Ft(0)3364 4374 y Fq(=)c Fn(L)3526 4388 y Ft(0)3566 4374 y Fq(\()p Fn(\017;)15 b(M)5 b(;)15 b(\015)5 b Fq(\))0 4501 y Fc(suc)m(h)30 b(that,)h(if)f Fn(L)25 b Fj(\025)g Fn(L)756 4515 y Ft(0)796 4501 y Fc(,)856 4722 y Fn(\027)907 4684 y Fo(\034)901 4744 y Ft(\003)p Fo(;)p Fa(N)1055 4722 y Fq(\()q Fn(f)5 b(;)15 b(f)10 b Fq(\))24 b Fj(\024)h Fq(\(1)c(+)f Fn(\015)5 b Fq(\))p Fn(\027)1721 4684 y Fo(\034)1715 4744 y Ft(\003)p Fo(;)p Fa(N)1870 4621 y Fg(\020)1924 4722 y Fq(V)-8 b(ar)2066 4678 y Fo(\021)2066 4749 y Ft(\003)2116 4758 y Fe(1)2151 4749 y Fo(;)p Fa(N)2237 4758 y Fe(1)2277 4722 y Fq(\()p Fn(f)10 b Fq(\))20 b(+)g(V)-8 b(ar)2655 4678 y Fo(\021)2655 4749 y Ft(\003)2705 4758 y Fe(2)2740 4749 y Fo(;)p Fa(N)2826 4758 y Fe(2)2865 4722 y Fq(\()p Fn(f)10 b Fq(\))2990 4621 y Fg(\021)0 4982 y Fc(where)26 b Fq(V)-8 b(ar)401 4939 y Fo(\021)401 5009 y Ft(\003)451 5018 y Fl(i)479 5009 y Fo(;)p Fa(N)565 5018 y Fl(i)598 4982 y Fq(\()p Fn(f)10 b Fq(\))p Fc(,)27 b Fn(i)f Fq(=)f(1)p Fn(;)15 b Fq(2)p Fc(,)29 b(denotes)d(the)h(v)-5 b(ariance)27 b(of)f Fn(f)36 b Fc(w.r.t.)j(the)27 b(m)m(ulticanonical)e(measure)h(on) g Fq(\003)3666 4996 y Fo(i)3723 4982 y Fc(with)0 5110 y Fn(N)73 5124 y Fo(i)134 5110 y Fc(particles,)j Fn(N)596 5124 y Ft(3)667 5110 y Fc(of)i(whic)m(h)e(are)i(in)e Fq(\003)1352 5124 y Ft(3)1393 5110 y Fc(,)h(and)g(b)s(oundary)e (condition)h Fn(\021)34 b Fc(on)c Fn(@)2691 5077 y Ft(+)2686 5132 y Fo(r)2752 5110 y Fq(\003)2815 5124 y Fo(i)2845 5110 y Fc(.)0 5340 y Fh(Pr)-5 b(o)g(of.)32 b Fq(Thanks)d(to)i(prop)s (osition)d(4.3)k(the)e(pro)s(of)g(is)f(iden)m(tical)g(to)i(that)g(of)g (prop)s(osition)d(3.4)j(of)g([CM2].)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)f Ft([15])1348 b Fq(4.5)p eop %%Page: 16 16 16 15 bop 0 91 a Fu(4.4)35 b(On)f(the)f(distribution)h(of)f(the)h(p)-6 b(ar)g(ticle)33 b(number)0 314 y Fq(Here)f(w)m(e)f(pro)m(vide)g(some)g (simple)e(results)h(on)h(the)h(distribution)27 b(of)32 b(the)f(particle)f(n)m(um)m(b)s(ers)g(in)g(the)h(atoms)h(of)g(a)0 441 y(partition)d(of)i(a)f(giv)m(en)h(set)g(\003.)40 b(Throughout)30 b(this)f(subsection)g(the)i(setting)f(will)e(b)s(e)i (as)h(follo)m(ws.)0 602 y(Let,)j(for)e Fn(i)c Fq(=)h(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(k)s Fq(,)33 b(\003)909 616 y Fo(i)968 602 y Fj(2)28 b(R)1134 569 y Fo(\017)1134 627 y(L)1221 602 y Fq(b)s(e)j(pairwise)g(disjoin)m(t)f(and)i(let)g (\003)d(=)f Fj([)2592 616 y Fo(i)2622 602 y Fq(\003)2685 616 y Fo(i)2715 602 y Fq(.)47 b(Assume)32 b(that)g(also)h(\003)f(b)s (elongs)0 729 y(to)j Fj(R)192 696 y Fo(\017)192 754 y(L)246 729 y Fq(.)53 b(Let)35 b(also)f Fr(N)d Fq(=)h Fj(f)p Fn(N)1011 743 y Fo(i)1042 729 y Fj(g)1087 696 y Fo(k)1087 753 y(i)p Ft(=1)1244 729 y Fq(b)s(e)h(a)i(set)g(of)f(p)s(ossible)e(v)-5 b(alues)33 b(of)i Fr(N)2514 743 y Ft(\003)2600 729 y Fq(:=)c Fj(f)p Fn(N)2845 743 y Ft(\003)2895 752 y Fl(i)2929 729 y Fj(g)2974 696 y Fo(k)2974 753 y(i)p Ft(=1)3097 729 y Fq(.)52 b(Giv)m(en)34 b(a)h(b)s(oundary)0 857 y(condition)28 b Fn(\034)10 b Fq(,)30 b(let)f Fn(\025)p 633 872 54 4 v 30 w Fq(b)s(e)g(the)g(c)m(hemical)h(p)s(oten)m(tial,)f(constan)m(t)i (on)e(eac)m(h)i(atom,)g(suc)m(h)e(that)h Fn(\026)3180 807 y Fo(\034)s(;\025)p 3239 822 44 4 v 3180 884 a Ft(\003)3301 857 y Fq(\()q Fr(N)3419 871 y Fa(\003)3480 857 y Fq(\))c(=)f Fr(N)k Fq(\(see)0 984 y(app)s(endix)f(of)i([CM1])i(for)e(the)g (existence)h(of)g Fn(\025)p 1559 999 54 4 v Fq(\).)41 b(Then)29 b(w)m(e)i(ha)m(v)m(e)h(\(see)f(Corollary)e(6.3)j(in)d ([CM1]\))0 1174 y Fr(Prop)s(osition)36 b(4.6.)53 b Fc(Assume)30 b Fn(J)k Fj(2)25 b Fq(\002)1358 1188 y Fo(g)r(ood)1510 1174 y Fq(\()p Fn(M)5 b(;)15 b(\017;)g(L)p Fq(\))32 b Fc(and)d(let)i Fn(\033)2247 1141 y Ft(2)2244 1198 y Fo(i)2313 1174 y Fq(:=)25 b Fn(\026)2489 1125 y Fo(\034)s(;\025)p 2548 1140 44 4 v 2489 1201 a Ft(\003)2610 1174 y Fq(\()q Fn(N)2719 1188 y Ft(\003)2769 1197 y Fl(i)2802 1174 y Fn(;)15 b(N)2915 1188 y Ft(\003)2965 1197 y Fl(i)2998 1174 y Fq(\))q Fc(.)40 b(Then)1229 1412 y Fq(1)p 1203 1453 97 4 v 1203 1536 a Fn(C)1275 1510 y Fm(0)1406 1412 y Fq(1)p 1322 1453 214 4 v 1322 1468 a Fg(Q)1408 1563 y Fo(i)1454 1536 y Fn(\033)1506 1550 y Fo(i)1572 1474 y Fj(\024)25 b Fn(\026)1723 1424 y Fo(\034)s(;\025)p 1782 1439 44 4 v 1723 1500 a Ft(\003)1844 1474 y Fq(\()p Fr(N)1961 1488 y Ft(\003)2041 1474 y Fq(=)g Fr(N)p Fq(\))g Fj(\024)g Fn(C)2447 1436 y Fm(0)2567 1412 y Fq(1)p 2483 1453 214 4 v 2483 1468 a Fg(Q)2569 1563 y Fo(i)2614 1536 y Fn(\033)2666 1550 y Fo(i)0 1784 y Fc(for)30 b(a)h(suitable)e(constan) m(t)i Fn(C)992 1751 y Fm(0)1043 1784 y Fq(=)25 b Fn(C)1211 1751 y Fm(0)1251 1784 y Fq(\()p Fn(M)5 b(;)15 b(\017)p Fq(\))26 b Fn(>)f Fq(1)p Fc(.)0 2093 y Fq(The)30 b(next)g(result)g (concerns)g(the)h(w)m(a)m(y)g(particles)e(distribute)f(inside)g(one)j (blo)s(c)m(k)f(of)h(the)f(partition.)0 2253 y(Pic)m(k)d Fn(j)k Fj(2)25 b Fq([1)p Fn(;)15 b(:)g(:)g(:)i(;)e(k)s Fq(])28 b(and)e(divide)f(\003)1232 2267 y Fo(j)1297 2253 y Fq(in)m(to)i(t)m(w)m(o)h(disjoin)m(t)e(subsets)g Fn(V)5 b(;)31 b(W)39 b Fq(that)27 b(w)m(e)h(assume)f(to)g(b)s(e)g(also)g (elemen)m(ts)0 2381 y(of)41 b Fj(R)191 2348 y Fo(\017)191 2406 y(L)246 2381 y Fq(.)73 b(Denote)42 b(b)m(y)f Fn(N)886 2348 y Fm(\003)969 2381 y Fq(the)g(a)m(v)m(erage)j(n)m(um)m(b)s(er)c (of)h(particles)f(in)g Fn(V)61 b Fq(according)41 b(to)h Fn(\026)3132 2331 y Fo(\034)s(;\025)p 3191 2346 44 4 v 3132 2407 a Ft(\003)3279 2381 y Fq(and)f(let)g Fn(\015)5 b Fq(\()p Fn(n)p Fq(\))43 b(=)0 2508 y Fn(\027)51 2475 y Fo(\034)45 2533 y Ft(\003)p Fo(;)p Fa(N)200 2508 y Fq(\()p Fn(N)308 2522 y Fo(V)396 2508 y Fq(=)25 b Fn(n)p Fq(\).)0 2698 y Fr(Theorem)45 b(4.7.)53 b Fc(Assume)39 b Fn(J)50 b Fj(2)40 b Fq(\002)1288 2712 y Fo(g)r(ood)1440 2698 y Fq(\()p Fn(M)5 b(;)15 b(\017;)g(L)p Fq(\))p Fc(.)70 b(Then)39 b(there)h(exists)f Fn(c)2665 2712 y Ft(0)2747 2698 y Fq(=)i Fn(c)2898 2712 y Ft(0)2954 2698 y Fq(\()p Fn(M)5 b(;)15 b(\017)p Fq(\))41 b Fc(suc)m(h)e(that)i(for)e(all)0 2826 y Fn(f)c Fq(:)25 b(\012)196 2840 y Ft(\003)275 2826 y Fj(7!)g Fp(R)39 b Fc(that)31 b(dep)s(end)e(only)g(on)i Fn(N)1399 2840 y Fo(V)1462 2826 y Fq(\()p Fn(\033)s Fq(\))g Fc(the)g(follo)m(wing)e(P)m(oincar)m(\023)-43 b(e)31 b(inequalit)m(y)e(holds)497 3069 y Fn(\027)548 3032 y Fo(\034)542 3092 y Ft(\003)p Fo(;)p Fa(N)697 3069 y Fq(\()p Fn(f)5 b(;)15 b(f)10 b Fq(\))25 b Fj(\024)g Fn(c)1072 3083 y Ft(0)1128 3069 y Fq(min)o Fj(f)p Fn(N)1408 3032 y Fm(\003)1450 3069 y Fn(;)15 b Fj(j)p Fn(V)21 b Fj(j)f(\000)g Fn(N)1808 3032 y Fm(\003)1850 3069 y Fj(g)1910 2983 y Fg(X)1954 3174 y Fo(n)2057 3069 y Fn(\015)5 b Fq(\()p Fn(n)p Fq(\))21 b Fj(^)e Fn(\015)5 b Fq(\()p Fn(n)21 b Fq(+)f(1\))15 b([)q Fn(f)10 b Fq(\()p Fn(n)19 b Fq(+)h(1\))h Fj(\000)f Fn(f)10 b Fq(\()p Fn(n)p Fq(\)])3362 3024 y Ft(2)0 3466 y Fh(Pr)-5 b(o)g(of.)38 b Fq(Again,)g(thanks)f(to)g(the)g (fact)h(that)f Fn(J)45 b Fj(2)36 b Fq(\002)1809 3480 y Fo(g)r(ood)1960 3466 y Fq(\()p Fn(M)5 b(;)15 b(\017;)g(L)p Fq(\))38 b(and)f(using)e(prop)s(osition)f(4.2,)40 b(the)d(pro)s(of)f (is)0 3594 y(iden)m(tical)29 b(to)i(that)g(of)g(theorem)g(1.9)g(in)e ([CM2].)92 b Fb(\003)0 3969 y Fu(4.5)35 b(A)f(key)g(bound)f(on)h (special)f(co)n(v)-8 b(ariances.)0 4130 y Fq(The)25 b(setting)h(is)f (the)h(follo)m(wing.)38 b(W)-8 b(e)27 b(\014x)e Fn(\017)g Fj(2)g Fq(\(0)p Fn(;)15 b Fq(1\))28 b(and)d Fn(l)i Fj(2)e Fq([)p Fn(L)2184 4097 y Fo(\017)2219 4130 y Fn(;)15 b(L)p Fq(].)40 b(W)-8 b(e)27 b(then)e(consider)g(a)h(v)m(olume)g(\003)f Fj(2)g(R)3746 4097 y Fo(\017)3746 4155 y(L)3800 4130 y Fq(\()p Fn(l)r Fq(\))0 4257 y(suc)m(h)31 b(that)g(\003)c(=)f Fj([)651 4224 y Fo(k)651 4281 y(i)p Ft(=1)773 4257 y Fq(\003)836 4271 y Fo(i)867 4257 y Fq(,)31 b(where)g(\003)1250 4271 y Fo(i)1307 4257 y Fj(2)26 b(R)1471 4224 y Fo(\017)1471 4282 y(L)1525 4257 y Fq(\()p Fn(l)r Fq(\))32 b(and)e Fj(j)p Fq(\003)1921 4271 y Fo(i)1952 4257 y Fj(j)p Fn(=)p Fj(j)p Fq(\003)p Fj(j)e(\025)e Fn(\017)31 b Fq(for)g Fn(i)26 b Fq(=)h(1)p Fn(;)15 b Fj(\001)g(\001)g(\001)i Fn(;)e(k)s Fq(.)43 b(Let)32 b Fr(N)26 b Fq(:=)h Fj(f)p Fn(N)3501 4271 y Fo(i)3531 4257 y Fj(g)3576 4224 y Fo(k)3576 4281 y(i)p Ft(=1)3730 4257 y Fq(b)s(e)k(a)0 4385 y(set)i(of)g(p)s (ossible)e(v)-5 b(alues)32 b(of)h Fr(N)1056 4399 y Ft(\003)1139 4385 y Fq(:=)d Fj(f)p Fn(N)1383 4399 y Ft(\003)1433 4408 y Fl(i)1466 4385 y Fj(g)1511 4352 y Fo(k)1511 4408 y(i)p Ft(=1)1634 4385 y Fq(.)48 b(Let)34 b(also)e Fn(h)2109 4399 y Fo(x)2185 4385 y Fq(:=)d Fn(e)2352 4352 y Fm(\000r)2470 4361 y Fl(x)2511 4352 y Fo(H)2579 4385 y Fn(\033)s Fq(\()p Fn(x)p Fq(\))34 b(and)2970 4361 y(^)2969 4385 y Fn(h)3021 4399 y Fo(x)3097 4385 y Fq(:=)29 b Fn(e)3264 4352 y Fm(\000r)3382 4361 y Fl(x)3423 4352 y Fo(H)3492 4385 y Fq(\(1)22 b Fj(\000)g Fn(\033)s Fq(\()p Fn(x)p Fq(\)\))0 4512 y(and)30 b(tak)m(e)i Fn(\017)411 4479 y Fm(0)461 4512 y Fq(=)585 4476 y Ft(2)p 568 4491 70 4 v 568 4543 a Fo(db)649 4512 y Fn(\017)p Fq(,)e(where)g Fn(b)h Fq(is)e(de\014ned)g(in)g(prop)s (osition)f(5.2)k(and)e Fn(d)g Fq(is)g(the)g(dimension.)0 4702 y Fr(Lemma)36 b(4.8.)53 b Fc(Assume)33 b Fn(J)39 b Fj(2)30 b Fq(\002)1180 4716 y Fo(g)r(ood)1332 4702 y Fq(\()p Fn(M)5 b(;)15 b(\017)1537 4669 y Fm(0)1563 4702 y Fn(;)g(L)p Fq(\))p Fc(.)51 b(Then)32 b(for)h(an)m(y)h Fn(\016)g(>)c Fq(0)k Fc(there)f(exist)h(a)f(p)s(ositiv)m(e)g(constan)m (t)i Fn(C)0 4830 y Fc(and)30 b Fn(L)239 4844 y Ft(0)279 4830 y Fq(\()p Fn(M)5 b(;)15 b(\017)484 4797 y Fm(0)511 4830 y Fn(;)g(\016)s Fq(\))26 b Fn(>)f Fq(0)31 b Fc(suc)m(h)f(that)362 5191 y Fn(\027)413 5154 y Fo(\034)407 5214 y Ft(\003)p Fo(;)p Fa(N)547 5118 y Fg(\000)588 5191 y Fn(f)5 b(;)814 5130 y Fq(1)p 689 5170 296 4 v 689 5254 a Fj(j)p Fq(\003)777 5268 y Fo(i)808 5254 y Fj(jj)p Fq(\003)921 5268 y Fo(j)960 5254 y Fj(j)1032 5105 y Fg(X)1023 5287 y Fl(x)p Fk(2)p Fe(\003)1143 5302 y Fl(i)1022 5344 y(z)r Fk(2)p Fe(\003)1139 5359 y Fl(j)1199 5191 y Fn(h)1251 5205 y Fo(x)1298 5167 y Fq(^)1297 5191 y Fn(h)1349 5205 y Fo(z)1391 5118 y Fg(\001)1433 5136 y Ft(2)1499 5191 y Fj(\024)1606 5130 y Fn(C)i(l)1707 5097 y Ft(2)p 1606 5170 142 4 v 1620 5254 a Fj(j)p Fq(\003)p Fj(j)1758 5191 y Fn(\027)1809 5154 y Fo(\034)1803 5214 y Ft(\003)p Fo(;)p Fa(N)1958 5009 y Fg(0)1958 5172 y(@)2109 5105 y(X)2037 5304 y Ft([)p Fo(x;z)s Ft(])p Fm(2E)2268 5313 y Fe(\003)2328 5191 y Fn(c)2367 5205 y Fo(xz)2450 5191 y Fq(\()p Fj(r)2561 5205 y Fo(xz)2645 5191 y Fn(f)j Fq(\))2735 5154 y Ft(2)2775 5009 y Fg(1)2775 5172 y(A)2875 5191 y Fq(+)3012 5130 y Fn(\016)p 2977 5170 114 4 v 2977 5254 a Fj(j)p Fq(\003)p Fj(j)3132 5191 y Fq(V)-8 b(ar)3273 5152 y Fo(\034)3273 5214 y Ft(\003)p Fo(;)p Fa(N)3413 5191 y Fq(\()p Fn(f)10 b Fq(\))0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([16])1348 b Fq(4.6)p eop %%Page: 17 17 17 16 bop 0 91 a Fc(pro)m(vided)29 b(that)i Fn(L)25 b Fj(\025)g Fn(L)816 105 y Ft(0)857 91 y Fc(.)0 363 y Fh(Pr)-5 b(o)g(of.)31 b Fq(Fix)f Fn(\016)f(>)c Fq(0.)41 b(Due)30 b(to)g(lemma)g(A.1)h(w)m(e)f(can)g(assume,)g(without)f(loss)g(of)h (generalit)m(y)-8 b(,)31 b(that)g Fn(\027)3400 377 y Ft(\003)p Fo(;)p Fa(N)3539 363 y Fq(\()p Fn(h)3626 377 y Fo(x)3673 363 y Fq(\))25 b(=)g(0,)0 490 y Fj(8)p Fn(x)39 b Fj(2)h Fq(\003.)67 b(F)-8 b(or)40 b(notation)f(simplicit)m(y)e (de\014ne)h Fn(G)h Fq(:=)1938 454 y Ft(1)p 1895 469 122 4 v 1895 521 a Fm(j)p Ft(\003)1967 530 y Fl(i)1995 521 y Fm(j)2043 421 y Fg(P)2139 517 y Fo(x)p Fm(2)p Ft(\003)2279 526 y Fl(i)2327 490 y Fn(h)2379 504 y Fo(x)2426 490 y Fq(,)i Fj(H)g Fq(=)2776 454 y Ft(1)p 2731 469 127 4 v 2731 521 a Fm(j)p Ft(\003)2803 530 y Fl(j)2836 521 y Fm(j)2884 421 y Fg(P)2980 517 y Fo(z)s Fm(2)p Ft(\003)3117 526 y Fl(j)3171 466 y Fq(^)3170 490 y Fn(h)3222 504 y Fo(z)3264 490 y Fq(.)67 b(Let)40 b(\001)3605 503 y Ft(^)3604 519 y Fo(h)3650 490 y Fq(\()p Fn(z)t Fq(\))h(b)s(e)0 662 y(the)33 b(supp)s(ort)d(of)604 638 y(^)603 662 y Fn(h)655 676 y Fo(z)730 662 y Fq(and)i(write)g Fj(H)d Fq(=)g Fj(H)1428 629 y Ft(in)1514 662 y Fq(+)21 b Fj(H)1684 629 y Ft(ext)1820 662 y Fq(where)32 b Fj(H)2163 629 y Ft(in)2260 662 y Fq(is)g(the)h(sum)e(o)m(v)m(er)j(those)f Fn(z)t Fq('s)g(in)e(\003)3455 676 y Fo(j)3526 662 y Fq(suc)m(h)h(that)0 789 y(\001)77 802 y Ft(^)76 819 y Fo(h)123 789 y Fq(\()p Fn(z)t Fq(\))i Fj(\032)f Fq(\003)440 803 y Fo(j)513 789 y Fq(and)i Fj(H)773 756 y Ft(ext)911 789 y Fq(the)g(rest.)55 b(Then,)36 b(using)d(the)i(form)m(ula)g(relating)f(the)h(co)m(v)-5 b(ariance)37 b(of)e(t)m(w)m(o)h(functions)0 916 y Fn(f)43 b Fq(and)33 b Fn(g)k Fq(w.r.t.)51 b(the)34 b(measure)g Fn(\027)1180 883 y Fo(\034)1174 941 y Ft(\003)p Fo(;)p Fa(N)1347 916 y Fq(to)h(the)f(co)m(v)-5 b(ariance)35 b(w.r.t.)50 b(the)34 b(same)h(measure)e(conditioned)g(to)h(a)g(sub)0 1043 y Fn(\033)s Fq(-algebra,)d(w)m(e)g(get)137 1297 y Fn(\027)188 1259 y Fo(\034)182 1319 y Ft(\003)p Fo(;)p Fa(N)321 1223 y Fg(\000)363 1297 y Fn(f)5 b(;)589 1235 y Fq(1)p 463 1276 296 4 v 463 1359 a Fj(j)p Fq(\003)551 1373 y Fo(i)582 1359 y Fj(jj)p Fq(\003)695 1373 y Fo(j)734 1359 y Fj(j)806 1210 y Fg(X)797 1393 y Fl(x)p Fk(2)p Fe(\003)917 1408 y Fl(i)796 1449 y(z)r Fk(2)p Fe(\003)913 1464 y Fl(j)974 1297 y Fn(h)1026 1311 y Fo(x)1073 1273 y Fq(^)1072 1297 y Fn(h)1124 1311 y Fo(z)1166 1223 y 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y Fg(h)1177 1950 y Fn(\027)1228 1913 y Fo(\034)1222 1973 y Ft(\003)p Fo(;)p Fa(N)1377 1849 y Fg(\020)1431 1950 y Fn(f)5 b(;)30 b(\027)1587 1913 y Fo(\034)1581 1973 y Ft(\003)p Fo(;)p Fa(N)1720 1849 y Fg(\020)1775 1950 y Fn(G)14 b Fj(j)i(F)1967 1964 y Ft(\003)2017 1942 y Fl(c)2017 1987 y(i)2055 1849 y Fg(\021)2110 1950 y Fj(H)2188 1913 y Ft(in)2253 1849 y Fg(\021i)2350 1868 y Ft(2)795 2259 y Fj(\024)962 2198 y Fq(1)p 928 2239 114 4 v 928 2322 a Fj(j)p Fq(\003)p Fj(j)1067 2104 y Fg(")1120 2259 y Fn(C)1192 2222 y Fm(0)1232 2131 y Fg(\022)1310 2198 y Fj(j)p Fn(@)1388 2165 y Ft(+)1383 2220 y Fo(r)1449 2198 y Fq(\003)1512 2212 y Fo(j)1551 2198 y Fj(j)p 1310 2239 266 4 v 1367 2322 a(j)p Fq(\003)1455 2336 y Fo(j)1494 2322 y Fj(j)1587 2131 y Fg(\023)1654 2148 y Ft(2)1715 2259 y Fq(+)1817 2198 y Fn(\016)p 1816 2239 46 4 v 1816 2322 a Fq(2)1888 2104 y Fg(#)1956 2259 y Fq(V)-8 b(ar)2098 2220 y Fo(\034)2098 2282 y Ft(\003)p Fo(;)p Fa(N)2237 2259 y Fq(\()p Fn(f)10 b Fq(\))21 b(+)2500 2198 y Fn(C)7 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(used)g(the)g(h)m(yp)s(othesis)2769 2787 y Fm(j)p Ft(\003)2841 2796 y Fl(i)2869 2787 y Fm(j)p 2769 2810 122 4 v 2783 2862 a(j)p Ft(\003)p Fm(j)2927 2831 y Fj(\025)h Fn(\017)f Fq(together)h(with)e(lemma)0 3038 y(A.1)e(to)h(b)s(ound)c(the)j(term) 886 2937 y Fg(h)928 3038 y Fn(\027)979 3005 y Fo(\034)973 3063 y Ft(\003)p Fo(;)p Fa(N)1113 2937 y Fg(\020)1167 3038 y Fn(f)5 b(;)15 b(G)g Fj(j)g(F)1449 3052 y Ft(\003)1499 3029 y Fl(c)1499 3074 y(i)1538 2937 y Fg(\021i)1635 2956 y Ft(2)1696 3038 y Fq(and)20 b(prop)s(osition)e(4.4)k(and)e(the)h(fact) g(that)g Fn(J)35 b Fj(2)25 b Fq(\002)3380 3052 y Fo(g)r(ood)3531 3038 y Fq(\()p Fn(M)5 b(;)15 b(\017)3736 3005 y Fm(0)3762 3038 y Fn(;)g(L)p Fq(\))0 3165 y(to)31 b(get)621 3387 y(V)-8 b(ar)763 3348 y Fo(\034)763 3410 y Ft(\003)p Fo(;)p Fa(N)903 3387 y Fq(\()p Fn(G)p Fq(\))25 b Fj(\024)40 b Fn(C)d Fq(V)-8 b(ar)1425 3338 y Fo(\034)s(;\025)p 1484 3353 44 4 v 1425 3414 a Ft(\003)1531 3387 y Fq(\()p Fn(G)p Fq(\))25 b(=)1875 3326 y(1)p 1820 3366 155 4 v 1820 3450 a Fj(j)p Fq(\003)p Fj(j)1933 3423 y Ft(2)2020 3301 y Fg(X)2016 3496 y Fo(x)p Fm(2)p Ft(\003)2182 3312 y Fq(V)-8 b(ar)2324 3263 y Fo(\034)s(;\025)p 2383 3278 44 4 v 2324 3339 a Ft(\003)2430 3312 y Fq(\()2465 3244 y Fg(P)2561 3340 y Fo(y)r Fm(2)p Fo(C)2701 3349 y Fl(x)2742 3340 y Fm(\\)p Ft(\003)2860 3312 y Fn(h)2912 3326 y Fo(y)2956 3312 y Fq(\))p 2182 3366 810 4 v 2424 3450 a Fj(j)p Fn(C)2514 3464 y Fo(x)2580 3450 y Fj(\\)20 b Fq(\003)p Fj(j)3028 3387 y(\024)3150 3326 y Fn(C)3222 3293 y Fm(00)p 3150 3366 118 4 v 3152 3450 a Fj(j)p Fq(\003)p Fj(j)3713 3387 y Fq(\(4)p Fn(:)p Fq(4\))0 3665 y(The)30 b(third)e(term)j(in)e (the.r.h.s)h(of)h(\(4.3\))h(can)f(b)s(e)e(b)s(ounded)f(from)i(ab)s(o)m (v)m(e)i(b)m(y)1117 3871 y Fj(k)p Fn(\027)1213 3834 y Fo(\034)1207 3894 y Ft(\003)p Fo(;)p Fa(N)1347 3871 y Fq(\()p Fn(G)15 b Fj(j)g(F)1574 3885 y Ft(\003)1624 3863 y Fl(c)1624 3908 y(i)1663 3871 y Fq(\))p Fj(k)1743 3834 y Ft(2)1743 3894 y Fm(1)1851 3871 y Fq(V)-8 b(ar)1993 3832 y Fo(\034)1993 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Fq(\))0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([19])1348 b Fq(5.3)p eop %%Page: 20 20 20 19 bop 0 91 a Fn(n)55 105 y Ft(min)207 91 y Fq(:=)27 b(max)q Fj(f)p Fq(0)p Fn(;)15 b(N)32 b Fj(\000)21 b(j)p Fn(W)13 b Fj(jg)p Fq(,)32 b Fn(n)1133 105 y Ft(max)1299 91 y Fq(:=)27 b(min)o Fj(fj)p Fn(V)21 b Fj(j)p Fn(;)15 b(N)10 b Fj(g)32 b Fq(and)f(\012)c(:=)h Fj(f)p Fn(n)f Fj(2)g Fq([)p Fn(n)2633 105 y Ft(min)2758 91 y Fn(;)15 b(n)2853 105 y Ft(max)2992 91 y Fq(])27 b(:)60 b Fn(n)31 b Fq(is)15 b(an)g(in)m(teger)p Fj(g)p Fq(.)45 b(By)0 218 y(prop)s(osition)28 b(3.7)j(in)f([CM2])h(w)m(e)g(ha)m(v)m(e)754 457 y(V)-8 b(ar)908 471 y Ft(_)-33 b Fo(\027)941 383 y Fg(\000)998 457 y Fq(_)-40 b Fn(\027)5 b Fq(\()p Fn(f)25 b Fj(j)15 b Fn(N)1251 471 y Fo(V)1315 457 y Fq(\))1350 383 y Fg(\001)1417 457 y Fj(\024)40 b Fn(C)1593 471 y Fo(\015)1677 371 y Fg(X)1670 566 y Fo(n)p Fm(2)p Ft(\012)1831 457 y Fn(\015)5 b Fq(\()p Fn(n)p Fq(\))20 b Fj(^)g Fn(\015)5 b Fq(\()p Fn(n)21 b Fj(\000)e Fq(1\)[)p Fn(g)s 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y(\014)2887 326 y Fo(s)p Ft(=)p Fo(N)3042 304 y Fk(\003)3035 350 y Fl(V)3119 263 y Fq(=)25 b(0)0 512 y(so)31 b(that)g(for)f Fn(j)h Fj(\024)25 b Fn(n)f Fj(\024)h Fn(N)870 479 y Fm(\003)860 537 y Fo(V)954 512 y Fq(or)30 b Fn(j)h Fj(\025)25 b Fn(n)g Fj(\025)g Fn(N)1488 479 y Fm(\003)1478 537 y Fo(V)753 748 y Fn(\015)5 b Fq(\()p Fn(j)g Fq(\))p 747 788 178 4 v 747 872 a Fn(\015)g Fq(\()p Fn(n)p Fq(\))961 809 y Fj(\024)1057 708 y Fg(h)1100 809 y Fn(\026)1155 762 y Ft(b)p Fo(:)p Ft(c)p Fo(:)o(;\025)1335 771 y Fl(V)1390 762 y Ft(\()p Fo(n)p Ft(\))1155 836 y Fo(V)1496 809 y Fq(\()p Fn(N)1604 823 y Fo(V)1693 809 y Fq(=)25 b Fn(n)p Fq(\))15 b Fn(\026)1949 762 y Ft(b)p Fo(:)p Ft(c)p Fo(:)o(;\025)2129 771 y Fl(W)2199 762 y Ft(\()p Fo(N)7 b Fm(\000)p Fo(n)p Ft(\))1949 836 y Fo(W)2427 809 y Fq(\()p Fn(N)2535 823 y Fo(W)2643 809 y Fq(=)25 b Fn(N)30 b Fj(\000)20 b Fn(n)p Fq(\))3023 708 y Fg(i)3066 727 y Fm(\000)p Ft(1)3713 809 y Fq(\(5)p Fn(:)p Fq(6\))0 1105 y(Thanks)51 b(to)h(prop)s(osition)e 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y(Thanks)25 b(to)j(\(5.5\))g(w)m(e)f(can)g(conclude)e(that)j(there)e(exists)g(a)h (p)s(ositiv)m(e)f(n)m(umerical)f(constan)m(t)j Fn(a)e Fq(suc)m(h)g(that)h(for)g(an)m(y)0 2851 y Fn(n)e Fj(\024)g Fn(N)259 2818 y Fm(\003)249 2876 y Fo(V)1357 2956 y Fg(X)1355 3151 y Fo(j)t Fm(\024)p Fo(n)1523 2981 y Fn(\015)5 b Fq(\()p Fn(j)g Fq(\))p 1517 3021 178 4 v 1517 3105 a Fn(\015)g Fq(\()p Fn(n)p Fq(\))1730 3042 y Fj(\024)25 b(j)p Fq(\003)p Fj(j)1939 3005 y Fo(a)1999 3042 y Fq(exp)15 b(\()p Fn(c)2227 3005 y Fm(0)2268 3042 y Fq(\000)2325 3056 y Fo(J)2376 3042 y Fq(\(\003\)\))1169 b(\(5)p Fn(:)p Fq(9\))0 3363 y(and)30 b(similarly)d(for)688 3295 y Fg(P)784 3390 y Fo(j)t Fm(\025)p Fo(n)956 3319 y(\015)t Ft(\()p Fo(j)t Ft(\))p 950 3342 144 4 v 950 3395 a Fo(\015)t Ft(\()p Fo(n)p Ft(\))1105 3363 y Fq(,)j Fn(n)25 b Fj(\025)g Fn(N)1419 3330 y Fm(\003)1409 3388 y Fo(V)1472 3363 y Fq(.)0 3490 y(Supp)s(ose)i(no)m(w,)i(for)f(de\014niteness,)g(that)h (the)g(median)e(for)h Fn(\015)5 b Fq(\()p Fj(\001)p 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Fo(a)1709 5146 y Fq(exp)15 b(\()p Fn(c)1937 5109 y Fm(0)1978 5146 y Fq(\000)2035 5160 y Fo(J)2086 5146 y Fq(\(\003\)\))0 5340 y(The)30 b(lemma)g(is)f(pro)m(v)m(ed.)92 b Fb(\003)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([21])1348 b Fq(5.5)p eop %%Page: 22 22 22 21 bop 0 91 a Fh(Step)33 b(2.)41 b Fq(W)-8 b(e)31 b(can)g(no)m(w)f(turn)g(to)h(the)f(second)h(part)f(of)h(the)f(pro)s(of) g(of)g(our)g(theorem.)0 245 y(Let)1149 349 y(~)1133 372 y Fj(E)1189 335 y Fo(\034)1181 395 y Ft(\003)p Fo(;N)1321 372 y Fq(\()p Fn(f)5 b(;)15 b(f)10 b Fq(\))25 b(=)1668 311 y(1)p 1668 351 46 4 v 1668 435 a(2)1794 286 y Fg(X)1740 481 y Fo(x;y)r Fm(2)p Ft(\003)1941 490 y Fl(v)1995 372 y Fn(\027)2046 335 y Fo(\034)2040 395 y Ft(\003)p Fo(;N)2195 299 y Fg(\002)2248 372 y Fn(c)2287 386 y Fo(xy)2387 372 y Fq(\()p Fj(r)2498 386 y Fo(xy)2583 372 y Fn(f)10 b Fq(\))2673 335 y Ft(2)2729 299 y Fg(\003)0 682 y Fq(where)31 b(the)i(rates)f Fn(c)686 696 y Fo(xy)803 682 y Fq(satisfy)g(all)e(the)j 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Fo(\034)1960 5340 y Ft(\003)p Fo(;N)2116 5216 y Fg(\020)2171 5317 y Fn(\027)2222 5274 y Fo(\021)2216 5344 y Ft(\003)2266 5353 y Fe(1)2301 5344 y Fo(;N)2381 5353 y Fe(1)2435 5317 y Fq(\()q Fn(f)5 b(;)15 b(f)10 b Fq(\))19 b(+)h Fn(\027)2812 5274 y Fo(\021)2806 5344 y Ft(\003)2856 5353 y Fe(2)2891 5344 y Fo(;N)2971 5353 y Fe(2)3025 5317 y Fq(\()q Fn(f)5 b(;)15 b(f)10 b Fq(\))3240 5216 y Fg(\021)3713 5317 y Fq(\(6)p Fn(:)p Fq(5\))0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([24])1348 b Fq(6.2)p eop %%Page: 25 25 25 24 bop 0 91 a Fq(where)35 b(the)h(a)m(v)m(erage)i(is)c(o)m(v)m(er)j (the)e(random)g(v)-5 b(ariables)34 b Fn(\021)s Fq(,)j Fn(N)2122 105 y Ft(\003)2172 114 y Fe(1)2248 91 y Fq(and)d Fn(N)2502 105 y Ft(\003)2552 114 y Fe(2)2592 91 y Fq(.)56 b(As)35 b(in)f(section)i(5.1)h(of)e([CM2],)j(let)0 220 y(no)m(w)32 b(examine)f(the)h(sp)s(ectral)f(gap)i(of)e(the)h(b)s(ottom) h(rectangle)f(\003)2261 234 y Ft(1)2302 220 y Fq(,)g(the)g(reasoning)g (b)s(eing)e(similar)f(for)j(the)g(top)0 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y Fg(8)517 323 y(>)517 350 y(>)517 377 y(>)517 405 y(>)517 432 y(>)517 459 y(>)517 486 y(<)517 650 y(>)517 677 y(>)517 705 y(>)517 732 y(>)517 759 y(>)517 786 y(>)517 814 y(:)644 290 y Ft(1)p 624 305 78 4 v 624 357 a Fo(c)657 366 y Fl(n)723 282 y Fo(\015)t Ft(\()p Fo(n)p Fm(\000)p Ft(1\))p 723 305 237 4 v 769 357 a Fo(\015)t Ft(\()p Fo(n)p Ft(\))1060 240 y Fg(X)1038 435 y Fo(x)p Fm(2)p Ft(\003)1178 444 y Fe(1)997 502 y Fo(z)s Fm(2)p Ft(\003)p Fm(n)p Ft(\003)1221 511 y Fe(1)1281 326 y Fn(\027)1332 288 y Fo(\034)1326 349 y Ft(\003)p Fo(;)p Fa(N)1481 252 y Fg(\002)1519 326 y Fq(\()p Fj(r)1630 340 y Fo(z)s(x)1713 326 y Fn(f)10 b Fq(\)1)-25 b(I)1856 340 y Fo(E)1909 349 y Fl(z)r(x)1987 326 y Fn(e)2029 288 y Fm(\000r)2147 297 y Fl(xz)2221 288 y Fo(H)2281 297 y Fe(\003)2346 326 y Fj(j)15 b Fn(N)2459 340 y Ft(\003)2509 349 y Fe(1)2574 326 y Fq(=)25 b Fn(n)20 b Fj(\000)g Fq(1)2881 252 y Fg(\003)3034 326 y Fq(if)30 b Fn(n)25 b Fj(\024)g Fn(u)735 669 y Ft(1)p 624 684 258 4 v 624 736 a Fo(c)657 714 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Fq(\).)0 1189 y(Finally)36 b(w)m(e)i(a)m(v)m(erage)i(w.r.t.)62 b(to)38 b(the)g(index)e Fn(j)44 b Fq(\(see)38 b(\(6.2\)\))i(and)d(use)g (the)h(observ)-5 b(ation)37 b(that,)j(as)e Fn(j)43 b Fq(v)-5 b(aries)37 b(in)0 1316 y([1)p Fn(;)155 1280 y Ft(1)p 122 1295 103 4 v 122 1347 a(10)p Fo(\017)236 1316 y Fq(],)31 b(the)f(strips)f(\003)785 1330 y Ft(3)851 1316 y Fq(:=)c(\003)1035 1269 y Ft(\()p Fo(j)t Ft(\))1035 1340 y(3)1161 1316 y Fq(are)31 b(disjoin)m(t.)39 b(In)29 b(particular)929 1542 y(1)p 929 1583 46 4 v 929 1666 a(2)1082 1517 y Fg(X)1001 1723 y Fo(j)t Fm(2)p Ft([1)p Fo(;)1202 1698 y Fe(1)p 1173 1708 90 4 v 1173 1744 a(10)p Fl(\017)1274 1723 y Ft(])1424 1517 y Fg(X)1309 1717 y Ft([)p Fo(x;y)r Ft(])p Fm(2E)1541 1756 y Fe(\003)1582 1729 y(\()p Fl(j)s Fe(\))1582 1784 y(3)1685 1604 y Fn(\027)1736 1566 y Fo(\034)1730 1626 y Ft(\003)p Fo(;N)1885 1530 y Fg(\002)1938 1604 y Fn(c)1977 1618 y Fo(xy)2078 1604 y Fq(\()p Fj(r)2189 1618 y Fo(xy)2274 1604 y Fn(f)10 b Fq(\))2364 1566 y Ft(2)2419 1530 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Fo(\034)701 3585 y Ft(\003)p Fo(;N)841 3563 y Fq(\()p Fn(g)s(;)15 b(f)10 b Fq(\))1052 3525 y Ft(2)1120 3563 y Fj(\024)25 b Fq(2)15 b Fn(\027)1327 3525 y Fo(\034)1321 3585 y Ft(\003)p Fo(;N)1476 3462 y Fg(\020)1531 3563 y Fn(\027)1582 3525 y Fo(\034)1576 3585 y Ft(\003)p Fo(;N)1731 3563 y Fq(\()p Fn(g)s(;)g(f)26 b Fj(j)15 b(F)2028 3577 y Fo(l)2057 3563 y Fq(\))2092 3517 y Ft(2)2133 3462 y Fg(\021)2207 3563 y Fq(+)20 b(2)31 b(V)-8 b(ar)2516 3523 y Fo(\034)2516 3585 y Ft(\003)p Fo(;N)2656 3563 y Fq(\()p Fn(g)s Fq(\))31 b(V)-8 b(ar)2945 3523 y Fo(\034)2945 3585 y Ft(\003)p Fo(;N)3101 3563 y Fq(\()p Fn(f)3181 3577 y Fo(l)3208 3563 y Fq(\))1120 3806 y Fj(\024)25 b Fq(2)15 b Fn(\027)1327 3768 y Fo(\034)1321 3828 y Ft(\003)p Fo(;N)1476 3705 y Fg(\020)1531 3806 y Fn(\027)1582 3768 y Fo(\034)1576 3828 y Ft(\003)p Fo(;N)1731 3806 y Fq(\()p Fn(g)s(;)g(f)26 b Fj(j)15 b(F)2028 3820 y Fo(l)2057 3806 y Fq(\))2092 3760 y Ft(2)2133 3705 y Fg(\021)2207 3806 y Fq(+)20 b(2)p Fn(C)2408 3820 y Fo(f)2541 3744 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Fq(with)f Fn(!)k Fq(=)c(2)p Fn(\016)40 b Fq(if)c Fn(d)f Fj(\025)g Fq(2)i(and)f Fn(!)i Fq(=)d(1)25 b(+)f(2)p Fn(\016)40 b Fq(if)c Fn(d)f Fq(=)g(1,)0 4580 y(pro)m(vided)29 b(that)i Fn(l)c Fj(\025)e Fn(L)p Fq(\()p Fn(J)9 b Fq(\))929 4522 y Fe(2)p 924 4532 42 4 v 924 4569 a Fl(!)981 4580 y Fq(.)0 4754 y Fr(Lemma)31 b(6.2.)53 b Fc(Let)30 b Fn(\014)797 4768 y Fo(d)865 4754 y Fq(=)25 b(0)k Fc(if)f Fn(d)d Fj(\025)g Fq(3)p Fc(,)30 b Fn(\014)1436 4768 y Fo(d)1504 4754 y Fq(=)25 b Fn(\016)33 b Fc(if)28 b Fn(d)e Fq(=)f(2)k Fc(and)f Fn(\014)2224 4768 y Fo(d)2292 4754 y Fq(=)d(1)18 b(+)f Fn(\016)33 b Fc(if)28 b Fn(d)e Fq(=)f(1)p Fc(.)40 b(In)29 b(the)g(same)g(h)m(yp)s (otheses)0 4881 y(of)e(theorem)f(2.2)i(assume)e(that)h(for)f(some)h Fn(\014)j Fj(2)25 b Fq([)p Fn(\014)1697 4895 y Fo(d)1740 4881 y Fn(;)15 b Fq(2\))28 b Fc(and)e Fn(\013)f Fj(\025)g Fq(1)p Fc(,)j(some)f(p)s(ositiv)m(e)e(constan)m(t)j Fn(C)7 b Fq(\()p Fn(f)e(;)15 b(\016)s Fq(\))p Fc(,)28 b(all)d(pairs)0 5008 y Fn(L)p Fq(\()p Fn(J)9 b Fq(\))191 4975 y Fo(\013)269 5008 y Fj(\024)25 b Fn(l)392 5022 y Ft(1)458 5008 y Fj(\024)565 4972 y Ft(1)p 565 4987 37 4 v 565 5039 a(2)612 5008 y Fn(l)639 5022 y Ft(2)710 5008 y Fc(and)30 b(all)f Fn(N)40 b Fc(the)31 b(follo)m(wing)e(inequalit)m(y)g(holds)749 5279 y Fn(\027)800 5241 y Fo(\034)794 5302 y(B)849 5312 y Fl(l)871 5326 y Fe(2)911 5302 y Fo(;N)1001 5279 y Fq(\()p Fn(g)s(;)15 b(f)10 b Fq(\))1212 5241 y Ft(2)1280 5279 y Fj(\024)24 b Fn(C)7 b Fq(\()p Fn(f)e(;)15 b(\014)5 b Fq(\))1709 5151 y Fg(\024)1757 5279 y Fn(l)1786 5235 y Fo(\014)1784 5303 y Ft(1)1850 5279 y Fj(E)1906 5241 y Fo(\034)1898 5302 y(B)1953 5312 y Fl(l)1975 5326 y Fe(2)2015 5302 y Fo(;N)2105 5279 y Fq(\()p Fn(g)s(;)15 b(g)s Fq(\))23 b(+)2491 5218 y(1)p 2432 5258 164 4 v 2432 5352 a Fn(l)2461 5313 y Fo(d)p Fm(\000)p Fo(\016)2459 5376 y Ft(1)2636 5279 y Fq(V)-8 b(ar)2778 5239 y Fo(\034)2778 5302 y(B)2833 5312 y Fl(l)2855 5326 y Fe(2)2895 5302 y Fo(;N)2985 5279 y Fq(\()p Fn(g)s Fq(\))3101 5151 y Fg(\025)3668 5279 y Fq(\(6)p Fn(:)p Fq(25\))0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([29])1348 b Fq(6.7)p eop %%Page: 30 30 30 29 bop 0 91 a Fc(Then)28 b(there)i(exists)f(a)h(new)e(p)s(ositiv)m (e)h(constan)m(t)h Fn(C)1751 58 y Fm(0)1776 91 y Fq(\()p Fn(f)5 b(;)15 b(\014)5 b Fq(\))30 b Fc(suc)m(h)f(that)h(the)g(same)f (inequalit)m(y)f(holds)g(with)g Fn(\014)35 b Fc(and)0 219 y Fn(\013)c Fc(replaced)f(b)m(y)g Fn(\014)630 186 y Fm(0)681 219 y Fq(=)863 179 y Ft(2)p Fo(\014)p 788 198 231 4 v 788 251 a(d)p Fm(\000)p Fo(\016)r Ft(+)p Fo(\014)1059 219 y Fc(and)g Fn(\013)1294 186 y Fm(0)1345 219 y Fq(=)25 b Fn(\013)1510 175 y Ft(\()p Fo(\014)s Ft(+)p Fo(d)p Fm(\000)p Fo(\016)r Ft(\))p 1511 198 288 4 v 1636 251 a(2)1839 219 y Fc(resp)s(ectiv)m(ely)-8 b(.)0 433 y Fh(Pr)j(o)g(of.)32 b Fq(Giv)m(en)e(\(6.24\))j(the)d(pro)s (of)g(is)f(iden)m(tical)g(to)i(that)g(of)g(lemma)f(6.1)h(in)e([CM2].)93 b Fb(\003)0 647 y Fq(Notice)35 b(that)f(the)g(inequalit)m(y)e(of)i(the) g(lemma)f(holds)f(for)i Fn(\014)i Fq(=)30 b(2)k(and)f Fn(\013)f Fq(=)e(1)k(b)s(ecause)g(of)g(\(6.24\).)53 b(Therefore,)0 775 y(b)m(y)30 b(iterating)g Fn(n)f Fq(times)h(inequalit)m(y)e (\(6.25\))k(starting)e(from)g(these)g(v)-5 b(alues)30 b(for)f Fn(\013)i Fq(and)e Fn(\014)5 b Fq(,)31 b(for)f(all)e(pairs)h Fn(l)r(;)15 b(L)30 b Fq(suc)m(h)0 903 y(that)208 867 y Ft(1)p 208 882 37 4 v 208 935 a(2)255 903 y Fn(L)25 b Fj(\025)g Fn(l)i Fj(\025)e Fn(L)p Fq(\()p Fn(J)9 b Fq(\))779 870 y Fo(\013)826 879 y Fl(n)876 903 y Fq(,)31 b(w)m(e)g(\014nd)948 1230 y Fn(\027)999 1193 y Fo(\034)993 1253 y Ft(\003)p Fo(;N)1133 1230 y Fq(\()p Fn(g)s(;)15 b(f)10 b Fq(\))1344 1193 y Ft(2)1411 1230 y Fj(\024)40 b Fn(C)1587 1244 y Fo(f)s(;n)1712 1230 y Fq([)15 b Fn(l)1781 1193 y Fo(\014)1822 1202 y Fl(n)1887 1230 y Fj(E)1943 1193 y Fo(\034)1935 1253 y Ft(\003)1989 1230 y Fq(\()p Fn(g)s(;)g(g)s Fq(\))23 b(+)2374 1169 y(1)p 2315 1209 164 4 v 2315 1292 a Fn(l)2344 1266 y Fo(d)p Fm(\000)p Fo(\016)2520 1230 y Fq(V)-8 b(ar)2662 1190 y Fo(\034)2662 1253 y Ft(\003)p Fo(;N)2802 1230 y Fq(\()p Fn(f)10 b Fq(\)])0 1545 y(where)26 b Fn(\013)317 1559 y Fo(n)392 1545 y Fq(=)523 1509 y Ft(2)p 499 1524 86 4 v 499 1576 a Fo(\014)540 1585 y Fl(n)622 1545 y Fq(and)f(the)i(sequence)g Fn(\014)1371 1559 y Fo(n)1447 1545 y Fq(con)m(v)m(erges)h(to)f Fn(\014)2008 1559 y Fo(d)2077 1545 y Fq(\(de\014ned)f(in)f(lemma)h (6.2\))h(from)f(ab)s(o)m(v)m(e.)41 b(Therefore,)0 1673 y(for)30 b(an)m(y)h Fn(\016)e Fj(2)c Fq(\(0)p Fn(;)616 1637 y Ft(1)p 598 1652 73 4 v 598 1704 a(10)681 1673 y Fq(\),)31 b(there)g(exists)f(a)h(constan)m(t)g Fn(C)1763 1687 y Fo(f)s(;\016)1894 1673 y Fq(suc)m(h)f(that)964 2000 y Fn(\027)1015 1962 y Fo(\034)1009 2022 y Ft(\003)p Fo(;N)1149 2000 y Fq(\()p Fn(g)s(;)15 b(f)10 b Fq(\))1360 1962 y Ft(2)1427 2000 y Fj(\024)40 b Fn(C)1603 2014 y Fo(f)s(;\016)1718 2000 y Fq([)15 b Fn(l)1787 1962 y Fo(!)1856 2000 y Fj(E)1912 1962 y Fo(\034)1904 2022 y Ft(\003)1958 2000 y Fq(\()p Fn(g)s(;)g(g)s Fq(\))22 b(+)2343 1938 y(1)p 2284 1979 164 4 v 2284 2062 a Fn(l)2313 2036 y Fo(d)p Fm(\000)p Fo(\016)2489 2000 y Fq(V)-8 b(ar)2631 1960 y Fo(\034)2631 2022 y Ft(\003)p Fo(;N)2771 2000 y Fq(\()p Fn(f)10 b Fq(\))15 b(])732 b(\(6)p Fn(:)p Fq(26\))0 2324 y(pro)m(vided)33 b(that)589 2289 y Ft(1)p 589 2304 37 4 v 589 2356 a(2)652 2324 y Fn(L)e Fj(\025)g Fn(l)i Fj(\025)e Fn(L)p Fq(\()p Fn(J)9 b Fq(\))1217 2267 y Fe(2)p 1212 2277 42 4 v 1212 2313 a Fl(!)1269 2324 y Fq(,)35 b(where)f Fn(!)g Fq(=)d(1)23 b(+)g(2)p Fn(\016)38 b Fq(if)33 b Fn(d)f Fq(=)f(1)j(and)g Fn(!)g Fq(=)d(2)p Fn(\016)38 b Fq(if)33 b Fn(d)f Fj(\025)f Fq(2.)52 b(If)34 b(w)m(e)g(no)m(w)g(tak)m (e)0 2453 y Fn(g)29 b Fq(:=)c Fn(E)260 2467 y Fo(\025)308 2453 y Fn(f)10 b Fq(,)30 b(then)g(\(6.26\))i(giv)m(es)869 2783 y Fj(k)p Fn(E)981 2797 y Fo(\025)1030 2783 y Fn(f)10 b Fj(k)1130 2745 y Ft(2)1130 2805 y(2)1195 2783 y Fj(\024)41 b Fn(C)1379 2745 y Fm(0)1372 2805 y Fo(f)s(;\016)1487 2783 y Fq([)15 b Fn(l)1556 2745 y Fo(!)1624 2783 y Fn(\025)20 b Fq(+)1858 2721 y(1)p 1799 2762 164 4 v 1799 2845 a Fn(l)1828 2819 y Fo(d)p Fm(\000)p Fo(\016)1989 2783 y Fq(])25 b Fj(\024)40 b Fn(C)2222 2745 y Fm(0)2215 2805 y Fo(f)s(;\016)2346 2655 y Fg(\032)2429 2743 y Fn(\025)2493 2685 y Fe(1)p 2493 2695 31 4 v 2493 2732 a(2)2535 2710 y Fm(\000)p Fo(\016)2722 2743 y Fq(if)30 b Fn(d)25 b Fq(=)g(1)2429 2852 y Fn(\025)2482 2819 y Ft(1)p Fm(\000)p Fo(\016)2722 2852 y Fq(if)30 b Fn(d)25 b Fj(\025)g Fq(2)0 3138 y(pro)m(vided)i(that)i Fn(L)629 3105 y Fo(!)r Ft(+)p Fo(d)p Fm(\000)p Fo(\016)892 3138 y Fj(\025)c Fq(2)p Fn(\025)1086 3105 y Fm(\000)p Ft(1)1210 3138 y Fj(\025)g Fq(2)p Fn(L)p Fq(\()p Fn(J)9 b Fq(\))1542 3105 y Ft(2)1590 3076 y Fl(!)r Fe(+)p Fl(d)p Fk(\000)p Fl(\016)p 1590 3091 203 4 v 1670 3127 a(!)1808 3138 y Fq(.)40 b(Ab)s(o)m(v)m(e)29 b(w)m(e)g(ha)m(v)m(e)h(\014xed)d Fn(l)j Fq(equal)e(to)h Fn(\025)3165 3099 y Fm(\000)3319 3074 y Fe(1)p 3233 3084 V 3233 3121 a Fl(!)r Fe(+)p Fl(d)p Fk(\000)p Fl(\016)3479 3138 y Fq(in)e(order)h(to)0 3266 y(obtain)k(the)g(last)g(inequalit)m(y) -8 b(.)45 b(It)32 b(is)f(imp)s(ortan)m(t)h(to)g(observ)m(e)h(that)g (our)f(c)m(hoice)h(giv)m(es)f Fn(l)e Fj(\034)e Fn(L)k Fq(since)g Fn(\025)c Fj(\025)g Fn(C)7 b(L)3803 3233 y Fm(\000)p Ft(2)0 3395 y Fq(b)s(ecause)30 b(of)h(theorem)g(2.1.)42 b(This)28 b(end)i(the)g(pro)s(of)g(of)h(\(6.21\).)93 b Fb(\003)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([30])1337 b Fq(A.1)p eop %%Page: 31 31 31 30 bop 0 91 a Fr(App)s(endix.)47 b(Pro)s(of)36 b(of)f(Lu{Y)-9 b(au's)35 b(t)m(w)m(o)g(blo)s(c)m(k)h(estimate.)0 251 y Fq(Here)h(w)m(e)f(pro)m(v)m(e)h(Lu{Y)-8 b(au's)37 b(t)m(w)m(o)g(blo)s (c)m(k)f(estimate)h(whic)m(h)e(is)g(the)h(k)m(ey)h(ingredien)m(t)e(b)s (ehind)f(lemma)h(4.8.)59 b(The)0 378 y(pro)s(of)22 b(follo)m(ws)g (essen)m(tially)f(the)i(same)g(lines)e(of)h(the)h(one)g(giv)m(en)g(in)e ([CM2])i(for)g(translation)e(in)m(v)-5 b(arian)m(t)22 b(in)m(teraction)0 506 y(under)i(a)j(mixing)d(condition,)h(but)h(with)e (some)j(imp)s(ortan)m(t)e(di\013erence)g(due)g(to)i(the)f(presence)g 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Ft(=1)617 1048 y Fq(b)s(e)d(a)i(set)g(of)f(p)s(ossible)d(v)-5 b(alues)40 b(of)g Fr(N)1928 1062 y Ft(\003)2023 1048 y Fq(:=)i Fj(f)p Fn(N)2279 1062 y Ft(\003)2329 1071 y Fl(i)2362 1048 y Fj(g)2407 1015 y Fo(k)2407 1071 y(i)p Ft(=1)2530 1048 y Fq(.)70 b(Let)40 b(also)g Fn(h)3041 1062 y Fo(x)3129 1048 y Fq(:=)h Fn(e)3308 1015 y Fm(\000r)3426 1024 y Fl(x)3467 1015 y Fo(H)3536 1048 y Fn(\033)s Fq(\()p Fn(x)p Fq(\))g(and)1 1151 y(^)0 1175 y Fn(h)52 1189 y Fo(x)144 1175 y Fq(:=)k Fn(e)327 1142 y Fm(\000r)445 1151 y Fl(x)485 1142 y Fo(H)554 1175 y Fq(\(1)29 b Fj(\000)f Fn(\033)s Fq(\()p Fn(x)p Fq(\)\))44 b(and)e(tak)m(e)i Fn(\017)1453 1142 y Fm(0)1524 1175 y Fq(=)1668 1139 y Ft(2)p 1651 1154 70 4 v 1651 1207 a Fo(db)1732 1175 y Fn(\017)p Fq(,)h(where)d Fn(b)h Fq(is)e(de\014ned)h(in)f(prop)s (osition)f(5.2)k(and)d Fn(d)i Fq(is)f(the)0 1303 y(dimension.)c(W)-8 b(e)32 b(de\014ne)d Fn(G)c Fq(:=)1156 1267 y Ft(1)p 1113 1282 122 4 v 1113 1334 a Fm(j)p Ft(\003)1185 1343 y Fl(i)1213 1334 y Fm(j)1261 1235 y Fg(P)1357 1330 y Fo(x)p Fm(2)p Ft(\003)1497 1339 y Fl(i)1545 1303 y Fn(g)1588 1317 y Fo(x)1665 1303 y Fq(where)30 b Fn(g)1971 1317 y Fo(x)2042 1303 y Fq(=)25 b Fn(h)2190 1317 y Fo(x)2267 1303 y Fq(or)30 b Fn(g)2421 1317 y Fo(x)2492 1303 y Fq(=)2589 1279 y(^)2588 1303 y Fn(h)2640 1317 y Fo(x)2686 1303 y Fq(.)0 1492 y Fr(Prop)s(osition)g(A.1.)53 b Fc(Assume)24 b Fn(J)35 b Fj(2)24 b Fq(\002)1373 1506 y Fo(g)r(ood)1525 1492 y Fq(\()p Fn(M)5 b(;)15 b(\017)1730 1459 y Fm(0)1756 1492 y Fn(;)g(L)p Fq(\))p Fc(.)40 b(Then)24 b(for)h(an)m(y)g Fn(\016)k(>)c Fq(0)g Fc(there)g(exist)g(a)h(p)s(ositiv)m(e)e(constan)m (t)0 1619 y Fn(C)37 b Fc(and)30 b Fn(L)341 1633 y Ft(0)381 1619 y Fq(\()p Fn(\017)453 1586 y Fm(0)479 1619 y Fn(;)15 b(M)5 b(;)15 b(\016)s Fq(\))27 b Fn(>)e Fq(0)31 b Fc(suc)m(h)f(that) 1015 1929 y Fn(\027)1066 1891 y Fo(\034)1060 1951 y Ft(\003)p Fo(;)p Fa(N)1200 1855 y Fg(\000)1241 1929 y Fn(f)5 b(;)15 b(G)1403 1855 y Fg(\001)1444 1874 y Ft(2)1510 1929 y Fj(\024)1632 1867 y Fn(C)7 b(l)1733 1834 y Ft(2)p 1632 1908 142 4 v 1646 1991 a Fj(j)p Fq(\003)p Fj(j)1799 1929 y(E)1855 1891 y Fo(\034)1847 1951 y Ft(\003)p Fo(;)p Fa(N)1987 1929 y Fq(\()p Fn(f)e(;)15 b(f)10 b Fq(\))20 b(+)2359 1867 y Fn(\016)p 2324 1908 114 4 v 2324 1991 a Fj(j)p Fq(\003)p Fj(j)2479 1929 y Fq(V)-8 b(ar)2620 1889 y Fo(\034)2620 1951 y Ft(\003)p Fo(;)p Fa(N)2760 1929 y Fq(\()p Fn(f)10 b Fq(\))0 2229 y Fc(pro)m(vided)29 b(that)i Fn(L)25 b Fj(\025)g Fn(L)816 2243 y Ft(0)857 2229 y Fc(.)0 2418 y Fh(Pr)-5 b(o)g(of.)32 b Fq(T)-8 b(o)31 b(simplify)c(the)k(notation)g(w)m(e)g(pro)m(v)m(e)g(the)g(prop)s (osition)d(in)h(the)i(case)h Fn(k)d Fq(=)c(1.)42 b(When)30 b Fn(k)f(>)c Fq(1)31 b(the)g(pro)s(of)0 2545 y(can)g(b)s(e)e(easly)i (generalized.)0 2673 y(Fix)25 b Fn(\016)k(>)c Fq(0.)40 b(If)25 b Fn(\032)g Fj(\024)g Fn(\016)30 b Fq(or)25 b(1)11 b Fj(\000)g Fn(\032)25 b Fj(\024)g Fn(\016)30 b Fq(the)25 b(statemen)m(t)j(follo)m(ws)d(at)h(once)h(from)e(the)h(Sc)m(h)m(w)m 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v 3123 3203 a Fj(j)p Fq(\003)p Fj(j)0 3483 y Fq(for)25 b(large)g(v)-5 b(alues)24 b(of)h(the)g(\(constan)m (t\))i(c)m(hemical)d(p)s(oten)m(tial)h Fn(\025)p Fq(.)38 b(W)-8 b(e)26 b(will)d(th)m(us)h(assume,)i(without)e(further)f(notice,) 0 3611 y(that)31 b Fn(\032)25 b Fj(2)g Fq(\()p Fn(\016)n(;)15 b Fq(1)22 b Fj(\000)e Fn(\016)s Fq(\).)0 3770 y(W)-8 b(e)31 b(de\014ne)e Fj(f)p Fn(Q)535 3784 y Fo(\013)587 3770 y Fj(g)632 3784 y Fo(\013)p Fm(2)p Fo(I)800 3770 y Fq(to)h(b)s(e)g(a)g(collection)f(of)h(cub)s(es)f(of)h(side)e Fn(L)2211 3737 y Fo(\017)2241 3710 y Fk(0)2271 3770 y Fq(,)i(suc)m(h)f(that)h(for)g(an)m(y)g Fn(\013)c Fj(6)p Fq(=)f Fn(\014)34 b Fq(dist)o(\()p Fn(Q)3555 3784 y Fo(\013)3607 3770 y Fn(;)15 b(Q)3719 3784 y Fo(\014)3769 3770 y Fq(\))25 b Fj(\025)0 3898 y Fq(2)p Fn(M)h Fq(log)16 b Fn(L)p Fq(,)33 b(dist)o(\()p Fn(Q)665 3912 y Fo(\013)717 3898 y Fn(;)15 b(@)5 b Fq(\003\))29 b Fj(\025)f Fq(2)p Fn(M)d Fq(log)17 b Fn(L)32 b Fq(and)f Fj(j)p Fq(\003)22 b Fj(n)g([)1837 3912 y Fo(\013)1888 3898 y Fn(Q)1960 3912 y Fo(\013)2012 3898 y Fj(j)28 b(\024)g(j)p Fq(\003)p Fj(j)15 b Fq(log)i Fn(L=L)2594 3865 y Fo(\017)2624 3837 y Fk(0)2654 3898 y Fq(.)45 b(Clearly)31 b(suc)m(h)g(collection)h(exists.)0 4025 y(Next)21 b(w)m(e)g(observ)m(e)g(that,)i(without)c(loss)h(of)h (generalit)m(y)-8 b(,)23 b(w)m(e)e(can)f(replace)g Fn(g)2499 4039 y Fo(x)2566 4025 y Fq(b)m(y)g Fn(g)2725 4039 y Fo(x)2771 4025 y Fj(\000)p Fn(\015)5 b(\033)s Fq(\()p Fn(x)p Fq(\),)24 b Fn(\015)h Fq(b)s(eing)19 b(an)h(arbitrary)0 4153 y(constan)m(t)39 b(indep)s(enden)m(t)c(of)j Fn(x)p Fq(,)h(b)s(ecause)1466 4084 y Fg(P)1562 4180 y Fo(x)p Fm(2)p Ft(\003)1722 4153 y Fn(\033)s Fq(\()p Fn(x)p Fq(\))f(=)e Fn(N)48 b Fq(almost)37 b(surely)f(w.r.t.)62 b Fn(\027)3063 4120 y Fo(\034)3057 4178 y Ft(\003)p Fo(;N)3197 4153 y Fq(.)g(Accordingly)36 b(w)m(e)0 4280 y(de\014ne)30 b Fn(G)335 4294 y Fo(\015)406 4280 y Fq(:=)25 b Fn(G)20 b Fj(\000)g Fn(\015)788 4243 y Fo(N)846 4252 y Fe(\003)p 788 4259 104 4 v 793 4311 a Fm(j)p Ft(\003)p Fm(j)902 4280 y Fq(.)41 b(Our)29 b(c)m(hoice)i(of)g Fn(\015)k Fq(will)28 b(b)s(e)i(made)g(later.)41 b(W)-8 b(e)32 b(then)e(set)794 4599 y Fn(G)866 4562 y Ft(in)n(t)866 4622 y Fo(\015)982 4599 y Fq(:=)1149 4538 y(1)p 1115 4578 114 4 v 1115 4661 a Fj(j)p Fq(\003)p Fj(j)1350 4513 y Fg(X)1254 4713 y Fo(x)p Fm(2[)1393 4722 y Fl(\013)1440 4713 y Fo(Q)1497 4695 y Fe(in)n(t)1497 4731 y Fl(\013)1577 4525 y Fg(\000)1618 4599 y Fn(g)1661 4613 y Fo(x)1728 4599 y Fj(\000)20 b Fn(\015)5 b(\033)s Fq(\()p Fn(x)p Fq(\))2048 4525 y Fg(\001)2181 4599 y Fq(and)91 b Fn(G)2491 4562 y Ft(ext)2491 4622 y Fo(\015)2619 4599 y Fq(:=)25 b Fn(G)2812 4613 y Fo(\015)2878 4599 y Fj(\000)20 b Fn(G)3041 4562 y Ft(in)3041 4622 y Fo(\015)0 4968 y Fq(where)30 b Fn(Q)335 4935 y Ft(in)n(t)335 4990 y Fo(\013)452 4968 y Fq(=)25 b Fj(f)p Fn(x)g Fj(2)g Fn(Q)828 4982 y Fo(\013)920 4968 y Fq(:)51 b Fn(d)p Fq(\()p Fn(x;)15 b(Q)1242 4935 y Fo(c)1242 4990 y(\013)1295 4968 y Fq(\))25 b Fj(\025)g Fq(2)p Fn(M)h Fq(log)17 b Fn(L)e Fj(g)p Fq(.)41 b(Notice)31 b(that)520 5291 y(V)-8 b(ar)662 5252 y Fo(\034)662 5314 y Ft(\003)p Fo(;N)802 5291 y Fq(\()p Fn(G)909 5254 y Ft(ext)909 5314 y Fo(\015)1013 5291 y Fq(\))25 b Fj(\024)g Fn(C)1241 5254 y Fm(0)1266 5291 y Fn(\026)1321 5248 y Fo(\034)s(;\025)1321 5318 y Ft(\003)1427 5291 y Fq(\()p Fn(G)1534 5254 y Ft(ext)1534 5314 y Fo(\015)1638 5291 y Fn(;)15 b(G)1750 5254 y Ft(ext)1750 5314 y Fo(\015)1853 5291 y Fq(\))26 b Fj(\024)e Fn(C)2081 5254 y Fm(00)2138 5142 y Fg(P)2234 5238 y Fo(x)p Fm(2)p Ft(\003)p Fm(n[)2460 5247 y Fl(\013)2507 5238 y Fo(Q)2564 5219 y Fe(in)2564 5256 y Fl(\013)2640 5210 y Fj(j)p Fn(C)2730 5224 y Fo(x)2776 5210 y Fj(j)p 2138 5270 664 4 v 2393 5354 a(j)p Fq(\003)p Fj(j)2506 5327 y Ft(2)2838 5291 y Fj(\024)h Fn(C)3006 5254 y Fm(00)3063 5230 y Fq(\(log)17 b Fn(L)p Fq(\))3328 5197 y Ft(2)p 3063 5270 306 4 v 3098 5354 a Fj(j)p Fq(\003)p Fj(j)p Fn(L)3273 5327 y Fo(\017)3303 5309 y Fk(0)3690 5291 y Fq(\()p Fn(A:)p Fq(1\))0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([31])1337 b Fq(A.2)p eop %%Page: 32 32 32 31 bop 0 91 a Fq(b)s(ecause)30 b(of)h(prop)s(osition)d(4.4)j(and)f (the)h(de\014nition)d(of)i Fj(f)p Fn(Q)2022 105 y Fo(\013)2074 91 y Fj(g)2119 105 y Fo(\013)p Fm(2)p Fo(I)2258 91 y Fq(.)41 b(In)29 b(particular,)g(for)h(an)m(y)h(giv)m(en)f Fn(\016)f(>)c Fq(0,)1324 326 y Fn(\027)1375 289 y Fo(\034)1369 349 y Ft(\003)p Fo(;N)1509 253 y Fg(\000)1551 326 y Fn(f)5 b(;)15 b(G)1713 289 y Ft(ext)1713 349 y Fo(\016)1815 253 y Fg(\001)1857 272 y Ft(2)1923 326 y Fj(\024)2065 265 y Fn(\016)p 2030 306 114 4 v 2030 389 a Fj(j)p Fq(\003)p Fj(j)2169 326 y Fq(V)-8 b(ar)2311 287 y Fo(\034)2311 349 y Ft(\003)p Fo(;N)2451 326 y Fq(\()p Fn(f)10 b Fq(\))0 563 y(pro)m(vided)29 b(that)i Fn(L)f Fq(is)g(big)f(enough.)0 711 y(W)-8 b(e)30 b(no)m(w)e(turn)f(to)i(b)s(ound)e(the)h(relev)-5 b(an)m(t)29 b(part)f Fn(\027)1672 678 y Fo(\034)1666 736 y Ft(\003)p Fo(;N)1806 638 y Fg(\000)1848 711 y Fn(f)5 b(;)15 b(G)2010 678 y Ft(in)n(t)2010 734 y Fo(\015)2101 638 y Fg(\001)2143 656 y Ft(2)2183 711 y Fq(.)40 b(Let)29 b Fj(F)2474 725 y Ft(0)2544 711 y Fq(b)s(e)e(the)i Fn(\033)s Fq({algebra)g(generated)g(b)m(y)g(the)0 838 y(the)i(random)e(v)-5 b(ariables)29 b Fj(f)p Fn(\033)s Fq(\()p Fn(x)p Fq(\))p Fj(g)1134 854 y Fo(x)p Fm(2)p Ft(\003)p Fm(n[)1360 863 y Fl(\013)1408 854 y Fo(Q)1465 863 y Fl(\013)1516 838 y Fn(;)i Fj(f)p Fn(N)1690 852 y Fo(\013)1742 838 y Fj(g)1787 852 y Fo(\013)p Fm(2)p Fo(I)1925 838 y Fq(,)g(where)f Fn(N)2317 852 y Fo(\013)2369 838 y Fq(\()p Fn(\033)s Fq(\))c(:=)2641 770 y Fg(P)2737 865 y Fo(x)p Fm(2)p Fo(Q)2884 874 y Fl(\013)2950 838 y Fn(\033)s Fq(\()p Fn(x)p Fq(\).)42 b(Then)29 b(w)m(e)i(write)160 1089 y Fn(\027)211 1051 y Fo(\034)205 1111 y Ft(\003)p Fo(;N)346 1015 y Fg(\000)387 1089 y Fn(f)5 b(;)15 b(G)549 1051 y Ft(in)n(t)549 1111 y Fo(\015)640 1015 y Fg(\001)682 1034 y Ft(2)748 1089 y Fj(\024)25 b Fq(2)15 b Fn(\027)955 1051 y Fo(\034)949 1111 y Ft(\003)p Fo(;N)1090 988 y Fg(\020)1144 1089 y Fn(\027)1195 1051 y Fo(\034)1189 1111 y Ft(\003)p Fo(;N)1329 1015 y Fg(\000)1371 1089 y Fn(f)5 b(;)15 b(G)1533 1051 y Ft(in)n(t)1533 1111 y Fo(\015)1639 1089 y Fj(j)g(F)1744 1103 y Ft(0)1785 1015 y Fg(\001)1842 988 y(\021)1896 1007 y Ft(2)1957 1089 y Fq(+)20 b(2)15 b Fn(\027)2159 1051 y Fo(\034)2153 1111 y Ft(\003)p Fo(;N)2294 988 y Fg(\020)2348 1089 y Fn(f)5 b(;)15 b(\027)2489 1051 y Fo(\034)2483 1111 y Ft(\003)p Fo(;N)2623 1015 y Fg(\000)2665 1089 y Fn(G)2737 1051 y Ft(in)n(t)2737 1111 y Fo(\015)2843 1089 y Fj(j)h(F)2949 1103 y Ft(0)2990 1015 y Fg(\001)3047 988 y(\021)3101 1007 y Ft(2)748 1335 y Fj(\024)25 b Fq(2)30 b(V)-8 b(ar)1061 1295 y Fo(\034)1061 1357 y Ft(\003)p Fo(;N)1201 1335 y Fq(\()p Fn(G)1308 1297 y Ft(in)n(t)1308 1357 y Fo(\015)1400 1335 y Fq(\))p Fn(\027)1486 1297 y Fo(\034)1480 1357 y Ft(\003)p Fo(;N)1621 1234 y Fg(\020)1675 1335 y Fq(V)g(ar)1817 1295 y Fo(\034)1817 1357 y Ft(\003)p Fo(;N)1957 1261 y Fg(\000)1998 1335 y Fn(f)25 b Fj(j)15 b(F)2173 1349 y Ft(0)2214 1261 y Fg(\001)2271 1234 y(\021)2346 1335 y Fq(+)20 b(2)15 b Fn(\027)2548 1297 y Fo(\034)2542 1357 y Ft(\003)p Fo(;N)2682 1234 y Fg(\020)2737 1335 y Fn(f)5 b(;)15 b(\027)2878 1297 y Fo(\034)2872 1357 y Ft(\003)p Fo(;N)3012 1261 y Fg(\000)3053 1335 y Fn(G)3125 1297 y Ft(in)n(t)3125 1357 y Fo(\015)3232 1335 y Fj(j)g(F)3337 1349 y Ft(0)3378 1261 y Fg(\001)3435 1234 y(\021)3489 1253 y Ft(2)748 1587 y Fj(\024)855 1526 y Fn(C)7 b Fq(\()p Fn(L)p Fq(\))p 855 1566 205 4 v 900 1649 a Fj(j)p Fq(\003)p Fj(j)1070 1587 y(E)1126 1550 y Fo(\034)1118 1610 y Ft(\003)p Fo(;N)1258 1587 y Fq(\()p Fn(f)e(;)15 b(f)10 b Fq(\))20 b(+)g(2)30 b(V)-8 b(ar)1801 1547 y Fo(\034)1801 1610 y Ft(\003)p Fo(;N)1941 1587 y Fq(\()p Fn(f)10 b Fq(\))31 b(V)-8 b(ar)2238 1547 y Fo(\034)2238 1610 y Ft(\003)p Fo(;N)2394 1513 y Fg(\000)2435 1587 y Fn(\027)2486 1550 y Fo(\034)2480 1610 y Ft(\003)p Fo(;N)2621 1513 y Fg(\000)2662 1587 y Fn(G)2734 1550 y Ft(in)n(t)2734 1610 y Fo(\015)2841 1587 y Fj(j)15 b(F)2946 1601 y Ft(0)2987 1513 y Fg(\001\001)3690 1342 y Fq(\()p Fn(A:)p Fq(2\))0 1849 y(where)30 b(w)m(e)h(ha)m(v)m(e)g (used)f(\(A.1\))i(to)f(b)s(ound)d(V)-8 b(ar)1587 1809 y Fo(\034)1587 1871 y Ft(\003)p Fo(;N)1727 1849 y Fq(\()p Fn(G)1834 1816 y Ft(in)n(t)1834 1871 y Fo(\015)1926 1849 y Fq(\))30 b(b)m(y)2134 1813 y Fo(C)2190 1786 y Fk(0)p 2129 1828 93 4 v 2129 1880 a Fm(j)p Ft(\003)p Fm(j)2263 1849 y Fq(and)f(the)i(estimate)1132 2089 y Fn(\027)1183 2051 y Fo(\034)1177 2111 y Ft(\003)p Fo(;N)1317 1988 y Fg(\020)1387 2089 y Fq(V)-8 b(ar)1528 2049 y Fo(\034)1528 2111 y Ft(\003)p Fo(;N)1669 2089 y Fq(\()p Fn(f)24 b Fj(j)16 b(F)1879 2103 y Ft(0)1920 2089 y Fq(\))1970 1988 y Fg(\021)2050 2089 y Fj(\024)25 b Fn(C)7 b Fq(\()p Fn(L)p Fq(\))15 b Fj(E)2421 2051 y Fo(\034)2413 2111 y Ft(\003)p Fo(;N)2553 2089 y Fq(\()p Fn(f)5 b(;)15 b(f)10 b Fq(\))0 2304 y(for)36 b(some)g(constan)m(t)i Fn(C)7 b Fq(\()p Fn(L)p Fq(\),)37 b(v)-5 b(alid)35 b(since)g Fn(\027)1522 2271 y Fo(\034)1516 2329 y Ft(\003)p Fo(;N)1656 2304 y Fq(\()p Fj(\001)16 b(j)g(F)1838 2318 y Ft(0)1879 2304 y Fq(\))36 b(is)f(the)h(pro)s(duct)f(of)i(canonical)e(Gibbs)g(measures) h(o)m(v)m(er)0 2431 y(the)31 b(cub)s(es)e Fn(Q)479 2445 y Fo(\013)531 2431 y Fq(.)40 b(Actually)-8 b(,)31 b(using)e(prop)s (osition)f(5.2,)j(the)g(constan)m(t)h Fn(C)7 b Fq(\()p Fn(L)p Fq(\))30 b(is)f(not)i(larger)f(than)g Fn(L)3413 2398 y Fo(bd\017)3513 2370 y Fk(0)3568 2431 y Fj(\024)25 b Fn(l)3693 2398 y Ft(2)3733 2431 y Fq(.)0 2579 y(The)37 b(k)m(ey)i(p)s(oin)m(t)e(is)f(no)m(w)i(to)h(pro)m(v)m(e)f(that,)j(for)c (an)m(y)h Fn(\016)k(>)37 b Fq(0,)j(V)-8 b(ar)2276 2540 y Fo(\034)2276 2602 y Ft(\003)p Fo(;N)2431 2506 y Fg(\000)2473 2579 y Fn(\027)2524 2546 y Fo(\034)2518 2604 y Ft(\003)p Fo(;N)2658 2506 y Fg(\000)2700 2579 y Fn(G)2772 2546 y Ft(in)n(t)2772 2602 y Fo(\015)2878 2579 y Fj(j)16 b(F)2984 2593 y Ft(0)3025 2506 y Fg(\001)o(\001)3146 2579 y Fq(is)36 b(smaller)h(than)3825 2544 y Fo(\016)p 3796 2559 V 3796 2611 a Fm(j)p Ft(\003)p Fm(j)0 2706 y Fq(pro)m(vided)29 b(that)i Fn(L)f Fq(is)g(large)g(enough.)0 2833 y(Notice)h(that)g Fn(\027)533 2800 y Fo(\034)527 2858 y Ft(\003)p Fo(;N)668 2759 y Fg(\000)709 2833 y Fn(G)781 2800 y Ft(in)n(t)781 2855 y Fo(\015)888 2833 y Fj(j)15 b(F)993 2847 y Ft(0)1034 2759 y Fg(\001)1076 2833 y Fq(\()p Fn(\021)s Fq(\))31 b(is)f(the)g(sum)g(of)g(lo)s(cal)g(functions)439 3078 y Fn(\027)490 3041 y Fo(\034)484 3101 y Ft(\003)p Fo(;N)624 3005 y Fg(\000)666 3078 y Fn(G)738 3041 y Ft(in)n(t)738 3101 y Fo(\015)844 3078 y Fj(j)15 b(F)949 3092 y Ft(0)991 3005 y Fg(\001)1032 3078 y Fq(\()p Fn(\021)s Fq(\))27 b(=)1318 3017 y(1)p 1283 3057 114 4 v 1283 3141 a Fj(j)p Fq(\003)p Fj(j)1439 2992 y Fg(X)1438 3187 y Fo(\013)p Fm(2)p Fo(I)1587 3078 y Fn(\027)1638 3035 y Fo(\021)1632 3109 y(Q)1689 3118 y Fl(\013)1735 3109 y Fo(;N)1815 3118 y Fl(\013)1860 3109 y Ft(\()p Fo(\021)r Ft(\))1961 2977 y Fg(\020)2064 2992 y(X)2016 3192 y Fo(x)p Fm(2)p Fo(Q)2163 3174 y Fe(in)n(t)2163 3211 y Fl(\013)2258 3078 y Fn(g)2301 3092 y Fo(x)2368 3078 y Fj(\000)20 b Fn(\015)5 b(\033)s Fq(\()p Fn(x)p Fq(\))2703 2977 y Fg(\021)2783 3078 y Fq(:=)2949 3017 y(1)p 2915 3057 V 2915 3141 a Fj(j)p Fq(\003)p Fj(j)3071 2992 y Fg(X)3070 3187 y Fo(\013)p Fm(2)p Fo(I)3219 3078 y Fn(G)3291 3041 y Fo(\015)3291 3101 y(\013)3342 3078 y Fq(\()p Fn(\021)s Fq(\))0 3371 y(Th)m(us,)33 b(if)f(w)m(e)i(order)f(in)f(an)h(arbitrary)f(w)m(a)m(y)i (the)f(set)h Fn(I)7 b Fq(,)34 b(w)m(e)g(can)f(split)f(the)h(ab)s(o)m(v) m(e)i(sum)d(in)m(to)h(the)h(sum)e(of)h(ev)m(en)0 3498 y(and)d(o)s(dd)f Fn(\013)p Fq('s)i(and)f(apply)f(prop)s(osition)f(4.4)j (to)g(eac)m(h)h(term)e(and)g(get)808 3763 y(V)-8 b(ar)950 3723 y Fo(\034)950 3785 y Ft(\003)p Fo(;N)1105 3689 y Fg(\000)1147 3763 y Fn(\027)1198 3725 y Fo(\034)1192 3785 y Ft(\003)p Fo(;N)1332 3689 y Fg(\000)1374 3763 y Fn(G)1446 3725 y Ft(in)n(t)1446 3785 y Fo(\015)1552 3763 y Fj(j)16 b(F)1658 3777 y Ft(0)1699 3689 y Fg(\001)o(\001)1807 3763 y Fj(\024)25 b Fn(C)1975 3725 y Fm(0)2066 3701 y Fq(1)p 2011 3742 155 4 v 2011 3825 a Fj(j)p Fq(\003)p Fj(j)2124 3799 y Ft(2)2192 3763 y Fn(\026)2247 3719 y Fo(\034)s(;\025)2247 3789 y Ft(\003)2368 3607 y Fg( )2440 3676 y(X)2482 3867 y Fo(\013)2586 3763 y Fn(G)2658 3725 y Fo(\015)2658 3785 y(\013)2710 3763 y Fn(;)2750 3676 y Fg(X)2792 3867 y Fo(\013)2896 3763 y Fn(G)2968 3725 y Fo(\015)2968 3785 y(\013)3020 3607 y Fg(!)0 4033 y Fq(for)30 b(some)h(constan)m(t)h Fn(C)806 4000 y Fm(0)861 4033 y Fq(indep)s(enden)m(t)c(of)j(\003)f(and)g Fn(L)1812 4047 y Ft(0)1852 4033 y Fq(.)0 4181 y(Let)38 b(no)m(w)f Fn(\030)410 4148 y Fo(\015)406 4204 y(\013)458 4181 y Fq(\()p Fn(\021)s Fq(\))g(:=)g Fn(\026)801 4134 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r Ft(\))801 4208 y Fo(Q)858 4217 y Fl(\013)1006 4080 y Fg(\020)1061 4113 y(P)1157 4209 y Fo(x)p Fm(2)p Fo(Q)1304 4190 y Fe(in)n(t)1304 4227 y Fl(\013)1389 4181 y Fq([)p Fn(g)1457 4195 y Fo(x)1528 4181 y Fj(\000)24 b Fn(\015)5 b(\033)s Fq(\()p Fn(x)p Fq(\)])1892 4080 y Fg(\021)1948 4181 y Fq(,)39 b(where)e(the)g(c)m(hemical)g(p)s(oten)m (tial)g Fn(\025)p Fq(\()p Fn(\021)s Fq(\))h(is)e(suc)m(h)h(that)0 4372 y Fn(\026)55 4324 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r Ft(\))55 4398 y Fo(Q)112 4407 y Fl(\013)260 4372 y Fq(\()p Fn(N)368 4386 y Fo(\013)421 4372 y Fq(\))30 b(=)h Fn(N)661 4386 y Fo(\013)713 4372 y Fq(\()p Fn(\021)s Fq(\).)51 b(In)33 b(the)h(\(rare\))g(case)h(in)d(whic)m(h)g Fn(N)2086 4386 y Fo(\013)2138 4372 y Fq(\()p Fn(\021)s Fq(\))g(=)e(0)k(\()p Fn(N)2576 4386 y Fo(\013)2628 4372 y Fq(\()p Fn(\021)s Fq(\))e(=)e Fj(j)p Fn(Q)2976 4386 y Fo(\013)3028 4372 y Fj(j)p Fq(\))k(the)g(measure)g Fn(\026)3695 4324 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r Ft(\))3695 4398 y Fo(Q)3752 4407 y Fl(\013)0 4498 y Fq(will)28 b(simply)h(b)s(e)i(the)g (Dirac)g(measure)g(on)g(the)g(constan)m(t)i(con\014guration)d(iden)m (tically)f(equal)i(to)h(0)f(\(1\).)44 b(Thanks)0 4625 y(to)27 b Fh(1\))g Fq(of)f(prop)s(osition)e(4.3)j(and)f(the)g(h)m(yp)s (othesis)f Fn(J)34 b Fj(2)25 b Fq(\002)1942 4639 y Fo(g)r(ood)2094 4625 y Fq(\()p Fn(M)5 b(;)15 b(\017)2299 4592 y Fm(0)2325 4625 y Fn(;)g(L)p Fq(\))26 b(w)m(e)h(ha)m(v)m(e)g(sup)2960 4647 y Fo(\021)3020 4625 y Fj(j)p Fn(G)3117 4592 y Fo(\015)3117 4647 y(\013)3168 4625 y Fq(\()p Fn(\021)s Fq(\))12 b Fj(\000)g Fn(\030)3425 4592 y Fo(\015)3421 4647 y(\013)3473 4625 y Fq(\()p Fn(\021)s Fq(\))p Fj(j)27 b(\024)e Fn(C)3811 4592 y Fm(0)3836 4625 y Fq(.)0 4752 y(In)30 b(particular,)f(using)f (the)j(fact)g(that)g(dist)o(\()p Fn(Q)1583 4766 y Fo(\013)1635 4752 y Fn(;)15 b(Q)1747 4766 y Fo(\014)1797 4752 y Fq(\))25 b Fj(\025)g Fq(2)p Fn(M)h Fq(log)16 b Fn(L)31 b Fq(for)f(an)m(y)g Fn(\013)c Fj(6)p Fq(=)f Fn(\014)5 b Fq(,)31 b(w)m(e)g(get)601 4928 y(1)p 546 4968 V 546 5052 a Fj(j)p Fq(\003)p Fj(j)659 5025 y Ft(2)712 4989 y Fn(\026)767 4946 y Fo(\034)s(;\025)767 5016 y Ft(\003)873 4888 y Fg(\020)927 4903 y(X)969 5094 y Fo(\013)1073 4989 y Fn(G)1145 4952 y Fo(\015)1145 5012 y(\013)1217 4989 y Fj(\000)20 b Fn(\030)1352 4952 y Fo(\015)1348 5012 y(\013)1399 4989 y Fn(;)1439 4903 y Fg(X)1482 5094 y Fo(\013)1586 4989 y Fn(G)1658 4952 y Fo(\015)1658 5012 y(\013)1730 4989 y Fj(\000)g Fn(\030)1865 4952 y Fo(\015)1861 5012 y(\013)1912 4888 y Fg(\021)1992 4989 y Fj(\024)2140 4928 y Fn(C)p 2099 4968 V 2099 5052 a Fj(j)p Fq(\003)p Fj(j)2212 5025 y Ft(2)2279 4903 y Fg(X)2321 5094 y Fo(\013)2425 4989 y Fn(\026)2480 4946 y Fo(\034)s(;\025)2480 5016 y Ft(\003)2602 4989 y Fq(\()p Fn(G)2709 4952 y Fo(\015)2709 5012 y(\013)2780 4989 y Fj(\000)g Fn(\030)2915 4952 y Fo(\015)2911 5012 y(\013)2963 4989 y Fn(;)15 b(G)3075 4952 y Fo(\015)3075 5012 y(\013)3147 4989 y Fj(\000)20 b Fn(\030)3282 4952 y Fo(\015)3278 5012 y(\013)3329 4989 y Fq(\))1992 5291 y Fj(\024)25 b Fn(C)2160 5254 y Fm(0)2309 5230 y Fq(1)p 2196 5270 273 4 v 2196 5354 a Fj(j)p Fq(\003)p Fj(j)p Fn(L)2371 5327 y Fo(d\017)2439 5309 y Fk(0)2504 5291 y Fj(\024)2646 5230 y Fn(\016)p 2611 5270 114 4 v 2611 5354 a Fj(j)p Fq(\003)p Fj(j)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([32])1337 b Fq(A.3)p eop %%Page: 33 33 33 32 bop 0 105 a Fq(for)30 b Fn(L)g Fq(large)h(enough.)40 b(It)31 b(is)e(therefore)i(enough)f(to)h(b)s(ound)e Fn(\026)2134 61 y Fo(\034)s(;\025)2134 131 y Ft(\003)2240 4 y Fg(\020)2294 36 y(P)2390 132 y Fo(\013)2457 105 y Fn(\030)2501 72 y Fo(\015)2497 127 y(\013)2548 105 y Fn(;)2588 36 y Fg(P)2685 132 y Fo(\013)2752 105 y Fn(\030)2796 72 y Fo(\015)2792 127 y(\013)2843 4 y Fg(\021)2898 105 y Fq(.)0 316 y(W)-8 b(e)35 b(can)g(no)m(w)f(apply)f(the)h(P)m(oincar)m(\022)-43 b(e)36 b(inequalit)m(y)d Fn(\026)1796 272 y Fo(\034)s(;\025)1796 343 y Ft(\003)1902 316 y Fq(\()p Fn(f)5 b(;)15 b(f)10 b Fq(\))31 b Fj(\024)g Fn(C)2322 283 y Fm(0)2347 316 y Fq(\()p Fn(L)p Fq(\))p Fn(\026)2534 272 y Fo(\034)s(;\025)2534 343 y Ft(\003)2656 242 y Fg(\000)2698 248 y(P)2794 343 y Fo(x)p Fm(2)p Ft(\003)2938 316 y Fq(\()p Fj(r)3049 330 y Fo(x)3095 316 y Fn(f)10 b Fq(\))3185 283 y Ft(2)3225 242 y Fg(\001)3301 316 y Fq(where)34 b Fn(C)3640 283 y Fm(0)3665 316 y Fq(\()p Fn(L)p Fq(\))e(=)0 443 y(gap\()p Fj(L)239 410 y Fo(\034)239 468 y Ft(\003)294 443 y Fq(\))329 410 y Fm(\000)p Ft(1)461 443 y Fq(and)i Fj(L)705 410 y Fo(\034)705 468 y Ft(\003)794 443 y Fq(is)g(Glaub)s(er)f(generator.) 54 b(By)35 b(the)g(indep)s(endence)e(of)h(the)h(clusters,)g(the)g(fact) h(that)f Fn(J)41 b Fj(2)0 571 y Fq(\002)71 585 y Fo(g)r(ood)223 571 y Fq(\()p Fn(M)5 b(;)15 b(\017)428 538 y Fm(0)454 571 y Fn(;)g(L)p Fq(\))31 b(and)f(theorem)g(3.8)i(in)d([M])i(w)m(e)g (ha)m(v)m(e)1022 807 y(gap)q(\()p Fj(L)1262 769 y Fo(\034)1262 829 y Ft(\003)1316 807 y Fq(\))26 b(=)f(min)1478 866 y Fo(x)p Fm(2)p Ft(\003)1639 807 y Fq(gap)q(\()p Fj(L)1879 821 y Fo(C)1931 830 y Fl(x)1976 807 y Fq(\))h Fj(\025)f Fq(exp)o Fj(f\000)p Fn(c)p Fq(\(log)18 b Fn(L)p Fq(\))2703 741 y Fl(d)p Fk(\000)p Fe(1)p 2703 756 114 4 v 2743 791 a Fl(d)2832 807 y Fj(g)0 1088 y Fq(so)31 b(that)667 1154 y(1)p 612 1194 155 4 v 612 1278 a Fj(j)p Fq(\003)p Fj(j)725 1251 y Ft(2)793 1215 y Fn(\026)848 1172 y Fo(\034)s(;\025)848 1242 y Ft(\003)954 1114 y Fg(\020)1008 1129 y(X)1050 1320 y Fo(\013)1155 1215 y Fn(\030)1199 1178 y Fo(\015)1195 1238 y(\013)1246 1215 y Fn(;)1286 1129 y Fg(X)1329 1320 y Fo(\013)1433 1215 y Fn(\030)1477 1178 y Fo(\015)1473 1238 y(\013)1525 1114 y Fg(\021)1604 1215 y Fj(\024)1765 1154 y Fq(1)p 1711 1194 V 1711 1278 a Fj(j)p Fq(\003)p Fj(j)1824 1251 y Ft(2)1891 1215 y Fn(e)1933 1178 y Fo(c)p Ft(\(log)14 b Fo(L)p Ft(\))2190 1121 y Fl(d)p Fk(\000)p Fe(1)p 2190 1136 114 4 v 2230 1171 a Fl(d)2339 1215 y Fn(\026)2394 1172 y Fo(\034)s(;\025)2394 1242 y Ft(\003)2500 1114 y Fg(\020)2558 1129 y(X)2555 1324 y Fo(y)r Fm(2)p Ft(\003)2708 1142 y Fg(\002)2761 1215 y Fj(r)2837 1229 y Fo(y)2911 1129 y Fg(X)2910 1324 y Fo(\013)p Fm(2)p Fo(I)3059 1215 y Fn(\030)3103 1178 y Fo(\015)3099 1238 y(\013)3166 1142 y Fg(\003)3204 1161 y Ft(2)3244 1114 y Fg(\021)3690 1215 y Fq(\()p Fn(A:)p Fq(3\))0 1514 y(Observ)m(e)30 b(no)m(w)h(that,)g(b)m(y)f(construction,)g Fj(r)1509 1528 y Fo(y)1568 1514 y Fn(\030)1612 1481 y Fo(\015)1608 1536 y(\013)1685 1514 y Fq(=)25 b(0)30 b(unless)f(dist)o(\()p Fn(y)s(;)15 b(Q)2467 1528 y Fo(\013)2519 1514 y Fq(\))26 b Fj(\024)f Fn(r)s Fq(.)40 b(Th)m(us)772 1790 y Fn(\026)827 1746 y Fo(\034)s(;\025)827 1816 y Ft(\003)933 1689 y Fg(\020)991 1703 y(X)988 1898 y Fo(y)r Fm(2)p Ft(\003)1141 1716 y Fg(\002)1194 1790 y Fj(r)1270 1804 y Fo(y)1345 1703 y Fg(X)1344 1898 y Fo(\013)p Fm(2)p Fo(I)1492 1790 y Fn(\030)1536 1752 y Fo(\015)1532 1812 y(\013)1599 1716 y Fg(\003)1637 1735 y Ft(2)1677 1689 y Fg(\021)1757 1790 y Fq(=)1854 1703 y Fg(X)1853 1898 y Fo(\013)p Fm(2)p Fo(I)2162 1703 y Fg(X)2169 1886 y Fl(y)r Fk(2)p Fe(\003)2012 1953 y Ft(dist\()p Fo(y)r(;Q)2276 1962 y Fl(\013)2323 1953 y Ft(\))p Fm(\024)p Fo(r)2485 1790 y Fn(\026)2540 1746 y Fo(\034)s(;\025)2540 1816 y Ft(\003)2646 1689 y Fg(\020)2700 1716 y(\002)2753 1790 y Fj(r)2829 1804 y Fo(y)2888 1790 y Fn(\030)2932 1752 y Fo(\015)2928 1812 y(\013)2995 1716 y Fg(\003)3033 1735 y Ft(2)3073 1689 y Fg(\021)3690 1790 y Fq(\()p Fn(A:)p Fq(4\))0 2228 y(Let)23 b(us)g(estimate)g(a)h(generic)f(term)g Fn(\026)1256 2185 y Fo(\034)s(;\025)1256 2255 y Ft(\003)1377 2127 y Fg(\020)1431 2155 y(\002)1484 2228 y Fj(r)1560 2242 y Fo(y)1619 2228 y Fn(\030)1663 2195 y Fo(\015)1659 2251 y(\013)1725 2155 y Fg(\003)1763 2173 y Ft(2)1804 2127 y Fg(\021)1858 2228 y Fq(.)38 b(It)23 b(is)f(at)i(this)e(stage)i(that)g(the)f(subtraction)f (with)g(the)h(free)0 2407 y(parameter)29 b Fn(\015)34 b Fq(made)29 b(at)g(the)g(b)s(eginning)d(b)s(ecomes)j(imp)s(ortan)m(t.) 39 b(Let)2434 2384 y(~)2419 2407 y Fj(r)2495 2421 y Fo(y)2539 2407 y Fn(f)10 b Fq(\()p Fn(\033)s Fq(\))25 b(:=)g(\(1)18 b Fj(\000)e Fn(\033)s Fq(\()p Fn(y)s Fq(\)\))p Fj(r)3334 2421 y Fo(y)3394 2407 y Fn(f)26 b Fj(\000)16 b Fn(\033)s Fq(\()p Fn(y)s Fq(\))p Fj(r)3801 2421 y Fo(y)3846 2407 y Fn(f)0 2535 y Fq(and)30 b(notice)g(that)889 2588 y Fg(\002)942 2662 y Fj(r)1018 2676 y Fo(y)1077 2662 y Fn(\030)1121 2624 y Fo(\015)1117 2684 y(\013)1184 2588 y Fg(\003)1222 2607 y Ft(2)1288 2662 y Fq(=)1384 2588 y Fg(\002)1452 2639 y Fq(~)1437 2662 y Fj(r)1513 2676 y Fo(y)1571 2662 y Fn(\030)1615 2624 y Fo(\015)1611 2684 y(\013)1678 2588 y Fg(\003)1716 2607 y Ft(2)1782 2662 y Fq(=)1878 2588 y Fg(\002)1946 2639 y Fq(~)1931 2662 y Fj(r)2007 2676 y Fo(y)2066 2662 y Fn(\026)2121 2615 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r Ft(\))2121 2689 y Fo(Q)2178 2698 y Fl(\013)2326 2588 y Fg(\000)2416 2576 y(X)2368 2776 y Fo(x)p Fm(2)p Fo(Q)2515 2758 y Fe(in)n(t)2515 2794 y Fl(\013)2610 2662 y Fn(g)2653 2676 y Fo(x)2699 2561 y Fg(\021)2774 2662 y Fj(\000)20 b Fn(\015)2932 2588 y Fg(\003)2970 2607 y Ft(2)0 2991 y Fq(Let)37 b Fn(\025)222 3005 y Ft(0)297 2991 y Fq(=)e Fn(\025)456 3005 y Ft(0)497 2991 y Fq(\(\003)p Fn(;)15 b(N)10 b Fq(\))37 b(b)s(e)e(the)i(c)m(hemical)f(p)s(oten)m(tial)f(suc)m(h)h(that)h Fp(E)2316 2918 y Fg(\000)2364 2991 y Fj(j)p Fn(C)2454 3005 y Ft(0)2495 2991 y Fj(j)2520 2958 y Fm(\000)p Ft(1)2633 2991 y Fn(\026)2688 2948 y Fo(\014)s(;\025)2797 2957 y Fe(0)2688 3018 y Fo(C)2740 3027 y Fe(0)2837 2991 y Fq(\()p Fn(N)2945 3005 y Fo(C)2997 3014 y Fe(0)3038 2991 y Fq(\))3073 2918 y Fg(\001)3150 2991 y Fq(=)d Fn(N)r(=)p Fj(j)p Fq(\003)p Fj(j)k Fq(where)d Fn(C)3859 3005 y Ft(0)0 3119 y Fq(is)f(the)i(cluster)f(of)g(the)h(cen)m(ter)h(of)e(\003)h (\(see)g(section)g(3\).)56 b(The)35 b(follo)m(wing)f(lemma)h(concludes) g(the)g(pro)s(of)g(of)h(the)0 3246 y(prop)s(osition.)45 b(Equation)32 b(\(A.5\))j(b)s(elo)m(w)d(and)g(the)h(fact)g(that)g Fn(J)39 b Fj(2)29 b Fq(\002)2386 3260 y Fo(g)r(ood)2537 3246 y Fq(\()p Fn(M)5 b(;)15 b(\017)2742 3213 y Fm(0)2768 3246 y Fn(;)g(L)p Fq(\))34 b(imply)c(that)j(\(A.4\))i(can)e(b)s(e)0 3373 y(b)s(ounded)28 b(from)i(ab)s(o)m(v)m(e)i(b)m(y)f Fn(c)p Fj(j)p Fq(\003)p Fj(j)p Fq(\()1169 3333 y Ft(log)14 b Fo(L)p 1170 3352 157 4 v 1195 3413 a(L)1245 3394 y Fl(\017)1273 3381 y Fk(0)1357 3373 y Fq(+)1558 3337 y Ft(1)p 1459 3352 235 4 v 1459 3413 a Fm(j)p Fo(Q)1538 3422 y Fl(\013)1584 3413 y Fm(j)1606 3394 y Fe(2)p Fl(\017)1665 3381 y Fk(0)1705 3373 y Fq(\).)41 b(Th)m(us)30 b(the)g(righ)m(t)h(hand) e(side)h(of)g(\(A.3\))i(is)e(smaller)f(than)3825 3337 y Fo(\016)p 3796 3352 93 4 v 3796 3404 a Fm(j)p Ft(\003)p Fm(j)0 3500 y Fq(pro)m(vided)g(that)i Fn(L)f Fq(is)g(large)g(enough.)0 3683 y Fr(Lemma)j(A.2.)52 b Fc(In)30 b(the)h(same)f(setting)h(of)f (prop)s(osition)e(A.1)k(de\014ne)1208 4045 y Fn(\015)e Fq(:=)1417 3951 y Fp(E)1487 3850 y Fg(h)1536 3951 y Fj(j)p Fn(C)1626 3965 y Ft(0)1667 3951 y Fj(j)1692 3918 y Fm(\000)p Ft(1)1805 3951 y Fn(\026)1860 3910 y Fo(\025)1903 3919 y Fe(0)1860 3978 y Fo(C)1912 3987 y Fe(0)1967 3877 y Fg(\000)2009 3883 y(P)2105 3978 y Fo(x)p Fm(2)p Fo(C)2247 3987 y Fe(0)2302 3951 y Fn(g)2345 3965 y Fo(x)2391 3951 y Fn(;)15 b(N)2504 3965 y Fo(C)2556 3974 y Fe(0)2597 3877 y Fg(\001)2639 3850 y(i)p 1417 4024 1265 4 v 1532 4140 a Fp(E)1602 4039 y Fg(h)1651 4140 y Fj(j)p Fn(C)1741 4154 y Ft(0)1782 4140 y Fj(j)1807 4114 y Fm(\000)p Ft(1)1920 4140 y Fn(\026)1975 4100 y Fo(\025)2018 4109 y Fe(0)1975 4167 y Fo(C)2027 4176 y Fe(0)2082 4140 y Fq(\()p Fn(N)2190 4154 y Fo(C)2242 4163 y Fe(0)2283 4140 y Fn(;)g(N)2396 4154 y Fo(C)2448 4163 y Fe(0)2488 4140 y Fq(\))2523 4039 y Fg(i)0 4413 y Fc(Then)35 b(there)h(exist)g(t)m(w)m(o)h(p)s(ositiv)m (e)e(constan)m(ts)i Fn(k)1676 4427 y Ft(1)1717 4413 y Fn(;)15 b(k)1804 4427 y Ft(2)1881 4413 y Fc(indep)s(enden)m(t)34 b(of)i Fn(L)g Fc(and)f Fn(L)2853 4427 y Ft(0)2928 4413 y Fq(=)f Fn(L)3095 4427 y Ft(0)3136 4413 y Fq(\()p Fn(\017)3208 4380 y Fm(0)3234 4413 y Fn(;)15 b(M)5 b(;)15 b(\016)s Fq(\))38 b Fc(suc)m(h)d(that)0 4540 y Fn(\015)30 b Fj(\024)25 b Fn(k)220 4554 y Ft(1)292 4540 y Fc(and)1082 4753 y Fn(\026)1137 4710 y Fo(\034)s(;\025)1137 4780 y Ft(\003)1258 4652 y Fg(\020)1313 4680 y(\002)1381 4730 y Fq(~)1366 4753 y Fj(r)1442 4767 y Fo(y)1500 4753 y Fn(\030)1544 4716 y Fo(\015)1540 4776 y(\013)1607 4680 y Fg(\003)1645 4698 y Ft(2)1686 4652 y Fg(\021)1765 4753 y Fj(\024)1861 4598 y Fg(\()1950 4671 y Fn(k)1997 4685 y Ft(2)2038 4671 y Fj(j)p Fn(C)2128 4685 y Fo(y)2172 4671 y Fj(j)100 b Fc(if)29 b Fn(y)f Fj(2)d Fn(@)2592 4638 y Ft(+)2587 4693 y Fo(r)2653 4671 y Fn(Q)2725 4685 y Fo(\013)2041 4770 y(k)2080 4779 y Fe(2)p 1961 4786 235 4 v 1961 4847 a Fm(j)p Fo(Q)2040 4856 y Fl(\013)2085 4847 y Fm(j)2107 4828 y Fe(2)p Fl(\017)2166 4815 y Fk(0)2297 4807 y Fc(if)k Fn(y)f Fj(2)d Fn(Q)2611 4821 y Fo(\013)3690 4753 y Fq(\()p Fn(A:)p Fq(5\))0 5030 y Fc(pro)m(vided)k(that)i Fn(L)25 b Fj(\025)g Fn(L)816 5044 y Ft(0)0 5213 y Fh(Pr)-5 b(o)g(of.)26 b Fq(The)e(fact)h(that)g Fn(\015)30 b Fq(is)23 b(b)s(ounded)g(from)g (ab)s(o)m(v)m(e)j(uniformly)c(in)h Fn(L)h Fq(follo)m(ws)g(immediately)e (from)i(its)g(de\014nition)0 5340 y(and)30 b(the)g(fact)i(that)f Fp(E)10 b Fq(\()q Fj(j)p Fn(C)890 5354 y Ft(0)937 5340 y Fj(j)962 5307 y Fo(n)1011 5340 y Fq(\))26 b Fj(\024)f Fn(k)33 b Fq(for)e Fn(p)24 b(<)h(p)1600 5354 y Fo(c)1668 5340 y Fq(\(see)31 b(e.g.)42 b([Gri]\).)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([33])1337 b Fq(A.4)p eop %%Page: 34 34 34 33 bop 0 91 a Fq(W)-8 b(e)32 b(\014rst)d(consider)g(the)i(case)g Fn(y)e Fj(2)24 b Fn(@)1259 58 y Ft(+)1254 113 y Fo(r)1320 91 y Fn(Q)1392 105 y Fo(\013)1474 91 y Fq(then)1102 362 y Fj(j)1142 339 y Fq(~)1127 362 y Fj(r)1203 376 y Fo(y)1247 362 y Fn(\030)1291 325 y Fo(\015)1287 385 y(\013)1338 362 y Fj(j)i Fq(=)f Fj(j)1525 339 y Fq(~)1510 362 y Fj(r)1586 376 y Fo(y)1629 362 y Fn(\026)1684 315 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r Ft(\))1684 389 y Fo(Q)1741 398 y Fl(\013)1890 362 y Fq(\()1968 276 y Fg(X)1925 476 y Fo(x)p Fm(2)p Fo(Q)2072 458 y Fl(in)2072 494 y(\013)2157 362 y Fn(g)2200 376 y Fo(x)2246 362 y Fq(\))20 b Fj(\000)g Fn(\015)5 b Fj(j)1389 662 y(\024)1527 576 y Fg(X)1485 776 y Fo(x)p Fm(2)p Fo(Q)1632 758 y Fl(in)1632 794 y(\013)1716 662 y Fj(j)p Fn(\026)1796 615 y Fo(\025)p Ft(\()p Fo(\021)1906 588 y Fl(y)1947 615 y Ft(\))1796 689 y Fo(C)1848 698 y Fl(x)1980 662 y Fq(\()p Fn(g)2058 676 y Fo(x)2104 662 y Fq(\))21 b Fj(\000)f Fn(\026)2306 615 y Fo(\025)p Ft(\()p Fo(\021)r Ft(\))2306 689 y Fo(C)2358 698 y Fl(x)2450 662 y Fq(\()p Fn(g)2528 676 y Fo(x)2574 662 y Fq(\))p Fj(j)h Fq(+)f Fn(\015)1389 996 y Fj(\024)40 b Fn(c)1612 910 y Fg(X)1569 1110 y Fo(x)p Fm(2)p Fo(Q)1716 1092 y Fl(in)1716 1128 y(\013)1812 935 y Fj(j)p Fn(C)1902 949 y Fo(x)1948 935 y Fj(j)15 b(j)p Fn(C)2078 949 y Fo(y)2122 935 y Fj(j)p 1812 976 336 4 v 1893 1059 a(j)p Fn(Q)1990 1073 y Fo(\013)2041 1059 y Fj(j)2179 996 y Fq(+)k Fn(\015)31 b Fj(\024)40 b Fn(k)2505 1010 y Ft(2)2561 996 y Fj(j)p Fn(C)2651 1010 y Fo(y)2695 996 y Fj(j)0 1359 y Fq(where)30 b(w)m(e)h(used)e(that)i Fn(J)k Fj(2)25 b Fq(\002)1043 1373 y Fo(g)r(ood)1194 1359 y Fq(\()p Fn(M)5 b(;)15 b(\017)1399 1326 y Fm(0)1425 1359 y Fn(;)g(L)p Fq(\))31 b(in)e(the)i(last)f (inequalit)m(y)-8 b(.)0 1520 y(Let)31 b(us)f(no)m(w)g(consider)f Fn(y)f Fj(2)d Fn(Q)1054 1534 y Fo(\013)1106 1520 y Fq(.)0 1648 y(In)32 b(this)g(case,)j(under)d(the)h(\015ip)f(of)h(the)g(v)-5 b(ariable)32 b Fn(\021)s Fq(\()p Fn(y)s Fq(\),)j(the)e(v)-5 b(alue)33 b(of)g Fn(\030)2479 1615 y Fo(\015)2475 1670 y(\013)2560 1648 y Fq(c)m(hanges)h(only)e(b)s(ecause)h(the)g(n)m(um)m (b)s(er)0 1775 y(of)g(particles)f(of)h Fn(\021)j Fq(v)-5 b(aries)33 b(b)m(y)f Fj(\006)p Fq(1.)49 b(De\014ne)33 b Fn(G)1593 1789 y Fo(\013)1674 1775 y Fq(:=)1799 1707 y Fg(P)1895 1803 y Fo(x)p Fm(2)p Fo(Q)2042 1784 y Fl(in)2042 1821 y(\013)2131 1775 y Fn(g)2174 1789 y Fo(x)2220 1775 y Fq(,)h(call)e Fn(N)2521 1789 y Fo(\013)2573 1775 y Fq(\()p Fn(\021)s Fq(\))f(=)e Fn(n)j Fq(and)h(let)g Fn(\025)p Fq(\()p Fn(s)p Fq(\))c(=)h Fn(\025)p Fq(\()p Fn(\021)s(;)15 b(s)p Fq(\))34 b(b)s(e)0 1952 y(the)d(c)m(hemical)f(p)s(oten)m(tial)g (suc)m(h)g(that)h Fn(\026)1369 1905 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r(;s)p Ft(\))1369 1979 y Fo(Q)1426 1988 y Fl(\013)1630 1878 y Fg(\000)1672 1952 y Fn(N)1745 1966 y Fo(\013)1796 1878 y Fg(\001)1863 1952 y Fq(=)25 b Fn(s)30 b Fq(with)f Fn(s)c Fj(2)g Fq([0)p Fn(;)15 b Fj(j)p Fn(Q)2600 1966 y Fo(\013)2653 1952 y Fj(j)p Fq(].)41 b(Then)1464 2213 y Fn(d\025)p Fq(\()p Fn(s)p Fq(\))p 1464 2253 214 4 v 1526 2337 a Fn(ds)1714 2274 y Fq(=)2106 2213 y(1)p 1820 2253 616 4 v 1820 2359 a Fn(\026)1875 2312 y Fo(\021)r(;\025)p Ft(\()p Fo(s)p Ft(\))1875 2386 y Fo(Q)1932 2395 y Fl(\013)2075 2359 y Fq(\()p Fn(N)2183 2373 y Fo(\013)2236 2359 y Fn(;)15 b(N)2349 2373 y Fo(\013)2401 2359 y Fq(\))3690 2274 y(\()p Fn(A:)p Fq(6\))0 2615 y(so)31 b(that)173 2915 y(~)158 2938 y Fj(r)234 2952 y Fo(y)278 2938 y Fn(\026)333 2891 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r(;n)p Ft(\))333 2965 y Fo(Q)390 2974 y Fl(\013)605 2938 y Fq(\()p Fn(G)712 2952 y Fo(\013)763 2938 y Fq(\))26 b(=)f(\(1)c Fj(\000)f Fn(\021)s Fq(\()p Fn(y)s Fq(\)\))1344 2814 y Fg(Z)1435 2837 y Fo(n)p Ft(+1)1395 3021 y Fo(n)1604 2863 y Fn(\026)1659 2816 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r(;s)p Ft(\))1659 2890 y Fo(Q)1716 2899 y Fl(\013)1920 2863 y Fq(\()p Fn(G)2027 2877 y Fo(\013)2079 2863 y Fn(;)15 b(N)2192 2877 y Fo(\013)2244 2863 y Fq(\))p 1603 2917 678 4 v 1603 3023 a Fn(\026)1658 2976 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r(;s)p Ft(\))1658 3050 y Fo(Q)1715 3059 y Fl(\013)1919 3023 y Fq(\()p Fn(N)2027 3037 y Fo(\013)2079 3023 y Fn(;)g(N)2192 3037 y Fo(\013)2245 3023 y Fq(\))2291 2938 y Fn(ds)35 b Fj(\000)20 b Fn(\021)s Fq(\()p Fn(y)s Fq(\))2704 2814 y Fg(Z)2795 2837 y Fo(n)p Fm(\000)p Ft(1)2755 3021 y Fo(n)2965 2863 y Fn(\026)3020 2816 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r(;s)p Ft(\))3020 2890 y Fo(Q)3077 2899 y Fl(\013)3281 2863 y Fq(\()p Fn(G)3388 2877 y Fo(\013)3439 2863 y Fn(;)15 b(N)3552 2877 y Fo(\013)3605 2863 y Fq(\))p 2964 2917 V 2964 3023 a Fn(\026)3019 2976 y Fo(\021)r(;\025)p Ft(\()p Fo(\021)r(;s)p Ft(\))3019 3050 y Fo(Q)3076 3059 y Fl(\013)3280 3023 y Fq(\()p Fn(N)3388 3037 y Fo(\013)3440 3023 y Fn(;)g(N)3553 3037 y Fo(\013)3605 3023 y Fq(\))3652 2938 y Fn(ds)0 3303 y Fq(By)28 b(adding)e(and)h (subtracting)g(inside)e(the)j(in)m(tegrals)f(the)h(term)g Fn(\026)2284 3260 y Fo(\021)r(;\025)2284 3330 y(Q)2341 3339 y Fl(\013)2393 3230 y Fg(\000)2434 3303 y Fn(G)2506 3317 y Fo(\013)2558 3303 y Fn(;)15 b(N)2671 3317 y Fo(\013)2723 3230 y Fg(\001)2764 3303 y Fn(=\026)2864 3260 y Fo(\021)r(;\025)2864 3330 y(Q)2921 3339 y Fl(\013)2974 3230 y Fg(\000)3015 3303 y Fn(N)3088 3317 y Fo(\013)3140 3303 y Fn(;)g(N)3253 3317 y Fo(\013)3305 3230 y Fg(\001)3375 3303 y Fq(\()p Fn(\025)28 b Fq(here)f(is)g(the)0 3431 y(c)m(hemical)j(p)s(oten)m(tial) g(of)h(the)f(grand)g(canonical)g(measure)g(on)h(the)f(v)m(olume)g (\003\),)h(w)m(e)g(ha)m(v)m(e)830 3739 y Fn(\026)885 3695 y Fo(\034)s(;\025)885 3765 y Ft(\003)1006 3611 y Fg(\022)1073 3638 y(h)1131 3716 y Fq(~)1116 3739 y Fj(r)1192 3753 y Fo(y)1235 3739 y Fn(\026)1290 3692 y Fo(\021)r(;\025)p Ft(\()p Fo(s)p Ft(\))1290 3765 y Fo(Q)1347 3774 y Fl(\013)1490 3665 y Fg(\000)1532 3739 y Fn(G)1604 3753 y Fo(\013)1655 3665 y Fg(\001)1717 3739 y Fj(\000)20 b Fn(\015)1860 3638 y Fg(i)1903 3657 y Ft(2)1944 3611 y Fg(\023)2036 3739 y Fj(\024)841 4070 y Fq(2)15 b Fn(\026)956 4026 y Fo(\034)s(;\025)956 4096 y Ft(\003)1063 3969 y Fg(\020)1132 3911 y(\014)1132 3965 y(\014)1132 4020 y(\014)1132 4074 y(\014)1132 4129 y(\014)1174 3995 y Fn(\026)1229 3951 y Fo(\021)r(;\025)1229 4022 y(Q)1286 4031 y Fl(\013)1338 3922 y Fg(\000)1380 3995 y Fn(G)1452 4009 y Fo(\013)1503 3995 y Fn(;)g(N)1616 4009 y Fo(\013)1668 3922 y Fg(\001)p 1174 4049 538 4 v 1174 4148 a Fn(\026)1229 4104 y Fo(\021)r(;\025)1229 4174 y(Q)1286 4183 y Fl(\013)1337 4074 y Fg(\000)1379 4148 y Fn(N)1452 4162 y Fo(\013)1504 4148 y Fn(;)g(N)1617 4162 y Fo(\013)1669 4074 y Fg(\001)1742 4070 y Fj(\000)20 b Fn(\015)1885 3911 y Fg(\014)1885 3965 y(\014)1885 4020 y(\014)1885 4074 y(\014)1885 4129 y(\014)1915 3930 y Ft(2)1956 3969 y Fg(\021)2031 4070 y Fq(+)g(2)15 b Fn(\026)2237 4026 y Fo(\034)s(;\025)2237 4096 y Ft(\003)2343 3969 y Fg(\020)2397 3938 y(\014)2397 3992 y(\014)2397 4047 y(\014)2397 4102 y(\014)2428 3946 y(Z)2519 3968 y Fo(n)p Ft(+1)2478 4153 y Fo(n)p Fm(\000)p Ft(1)2675 4070 y Fn(R)q Fq(\()p Fn(s)p Fq(\))p Fn(ds)2948 3938 y Fg(\014)2948 3992 y(\014)2948 4047 y(\014)2948 4102 y(\014)2979 3950 y Ft(2)3019 3969 y Fg(\021)3690 3920 y Fq(\()p Fn(A:)p Fq(7\))0 4403 y(where)1138 4617 y Fn(R)q Fq(\()p Fn(s)p Fq(\))26 b(=)1454 4543 y Fn(\026)1509 4495 y Fo(\021)r(;\025)p Ft(\()p Fo(s)p Ft(\))1509 4569 y Fo(Q)1566 4578 y Fl(\013)1709 4469 y Fg(\000)1751 4543 y Fn(G)1823 4557 y Fo(\013)1874 4543 y Fn(;)15 b(N)1987 4557 y Fo(\013)2039 4469 y Fg(\001)p 1453 4596 629 4 v 1453 4702 a Fn(\026)1508 4655 y Fo(\021)r(;\025)p Ft(\()p Fo(s)p Ft(\))1508 4729 y Fo(Q)1565 4738 y Fl(\013)1708 4628 y Fg(\000)1750 4702 y Fn(N)1823 4716 y Fo(\013)1875 4702 y Fn(;)g(N)1988 4716 y Fo(\013)2040 4628 y Fg(\001)2113 4617 y Fj(\000)2215 4543 y Fn(\026)2270 4499 y Fo(\021)r(;\025)2270 4569 y(Q)2327 4578 y Fl(\013)2379 4469 y Fg(\000)2420 4543 y Fn(G)2492 4557 y Fo(\013)2544 4543 y Fn(;)g(N)2657 4557 y Fo(\013)2709 4469 y Fg(\001)p 2215 4596 537 4 v 2215 4695 a Fn(\026)2270 4651 y Fo(\034)s(;\025)2270 4722 y(Q)2327 4731 y Fl(\013)2377 4621 y Fg(\000)2419 4695 y Fn(N)2492 4709 y Fo(\013)2544 4695 y Fn(;)g(N)2657 4709 y Fo(\013)2709 4621 y Fg(\001)0 4918 y Fq(Notice)31 b(that,)h(using)c(the)j(fact)g(that)g Fn(J)k Fj(2)25 b Fq(\002)1520 4932 y Fo(g)r(ood)1671 4918 y Fq(\()p Fn(M)5 b(;)15 b(\017)1876 4885 y Fm(0)1902 4918 y Fn(;)g(L)p Fq(\))31 b(it)f(is)g(easy)h(to)g(pro)m(v)m(e)g(that)1333 5100 y Fg(\014)1333 5155 y(\014)1333 5209 y(\014)1333 5264 y(\014)1333 5318 y(\014)1375 5185 y Fn(\026)1430 5141 y Fo(\021)r(;\025)1430 5212 y(Q)1487 5221 y Fl(\013)1538 5185 y Fq(\()p Fn(G)1645 5199 y Fo(\013)1697 5185 y Fn(;)15 b(N)1810 5199 y Fo(\013)1862 5185 y Fq(\))p 1374 5238 525 4 v 1374 5337 a Fn(\026)1429 5294 y Fo(\021)r(;\025)1429 5364 y(Q)1486 5373 y Fl(\013)1538 5337 y Fq(\()p Fn(N)1646 5351 y Fo(\013)1698 5337 y Fn(;)g(N)1811 5351 y Fo(\013)1863 5337 y Fq(\))1930 5259 y Fj(\000)k Fn(\015)2072 5100 y Fg(\014)2072 5155 y(\014)2072 5209 y(\014)2072 5264 y(\014)2072 5318 y(\014)2128 5259 y Fj(\024)2376 5198 y Fn(c)p 2235 5238 322 4 v 2235 5322 a(\032)2282 5336 y Ft(0)2323 5322 y Fj(j)p Fn(Q)2420 5336 y Fo(\013)2472 5322 y Fj(j)2497 5295 y Fo(\017)2527 5277 y Fk(0)3690 5259 y Fq(\()p Fn(A:)p Fq(8\))0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)30 b Ft([34])1337 b Fq(A.5)p eop %%Page: 35 35 35 34 bop 0 91 a Fq(where)30 b Fn(\032)310 105 y Ft(0)376 91 y Fq(:=)25 b Fn(\026)552 47 y Fo(\034)s(;\025)654 56 y Fe(0)552 118 y Ft(\003)694 91 y Fq(\()p Fn(N)802 105 y Ft(\003)856 91 y Fq(\))p Fn(=)p Fj(j)p Fq(\003)p Fj(j)p Fq(.)0 218 y(Let)35 b(~)-49 b Fn(n)25 b Fq(=)g Fn(\026)394 174 y Fo(\021)r(;\025)394 245 y(Q)451 254 y Fl(\013)502 218 y Fq(\()p Fn(N)610 232 y Fo(\013)663 218 y Fq(\).)41 b(Then)29 b(using)g(\(A.6\))j(w)m(e)f(can)f(write)646 546 y Fn(R)q Fq(\()p Fn(s)p Fq(\))25 b(=)1207 485 y(1)p 961 526 538 4 v 961 624 a Fn(\026)1016 581 y Fo(\021)r(;\025)1016 651 y(Q)1073 660 y Fl(\013)1125 551 y Fg(\000)1167 624 y Fn(N)1240 638 y Fo(\013)1292 624 y Fn(;)15 b(N)1405 638 y Fo(\013)1457 551 y Fg(\001)1525 423 y(Z)1616 445 y Fo(s)1579 629 y Ft(~)-40 b Fo(n)1670 546 y Fn(ds)1760 509 y Fm(0)1796 472 y Fn(\026)1851 425 y Fo(\021)r(;\025)p Ft(\()p Fo(s)2017 397 y Fk(0)2042 425 y Ft(\))1851 499 y Fo(Q)1908 508 y Fl(\013)2075 398 y Fg(\000)2117 472 y Fn(G)2189 486 y Fo(\013)2240 472 y Fn(;)15 b(N)2353 486 y Fo(\013)2406 472 y Fn(;)g(N)2519 486 y Fo(\013)2571 398 y Fg(\001)p 1796 526 817 4 v 1878 631 a Fn(\026)1933 584 y Fo(\021)r(;\025)p Ft(\()p Fo(s)2099 566 y Fk(0)2124 584 y Ft(\))1933 658 y Fo(Q)1990 667 y Fl(\013)2157 558 y Fg(\000)2199 631 y Fn(N)2272 645 y Fo(\013)2324 631 y Fn(;)g(N)2437 645 y Fo(\013)2489 558 y Fg(\001)849 909 y Fq(+)1221 834 y Fn(\026)1276 787 y Fo(\021)r(;\025)p Ft(\()p Fo(s)p Ft(\))1276 861 y Fo(Q)1333 870 y Fl(\013)1476 760 y Fg(\000)1517 834 y Fn(G)1589 848 y Fo(\013)1640 834 y Fn(;)g(N)1753 848 y Fo(\013)1806 760 y Fg(\001)p 951 888 1166 4 v 951 994 a Fn(\026)1006 946 y Fo(\021)r(;\025)p Ft(\()p Fo(s)p Ft(\))1006 1020 y Fo(Q)1063 1029 y Fl(\013)1206 920 y Fg(\000)1248 994 y Fn(N)1321 1008 y Fo(\013)1373 994 y Fn(;)g(N)1486 1008 y Fo(\013)1538 920 y Fg(\001)1580 994 y Fn(\026)1635 950 y Fo(\021)r(;\025)1635 1020 y(Q)1692 1029 y Fl(\013)1743 920 y Fg(\000)1785 994 y Fn(N)1858 1008 y Fo(\013)1910 994 y Fn(;)g(N)2023 1008 y Fo(\013)2075 920 y Fg(\001)2143 785 y(Z)2238 807 y Ft(~)-40 b Fo(n)2193 991 y(s)2298 909 y Fn(ds)2388 871 y Fm(0)2425 834 y Fn(\026)2480 787 y Fo(\021)r(;\025)p Ft(\()p Fo(s)2646 759 y Fk(0)2671 787 y Ft(\))2480 861 y Fo(Q)2537 870 y Fl(\013)2704 760 y Fg(\000)2746 834 y Fn(N)2819 848 y Fo(\013)2871 834 y Fn(;)15 b(N)2984 848 y Fo(\013)3036 834 y Fn(;)g(N)3149 848 y Fo(\013)3201 760 y Fg(\001)p 2425 888 819 4 v 2507 994 a Fn(\026)2562 946 y Fo(\021)r(;\025)p Ft(\()p Fo(s)2728 928 y Fk(0)2754 946 y Ft(\))2562 1020 y Fo(Q)2619 1029 y Fl(\013)2787 920 y Fg(\000)2829 994 y Fn(N)2902 1008 y Fo(\013)2953 994 y Fn(;)g(N)3066 1008 y Fo(\013)3119 920 y Fg(\001)0 1216 y Fq(so)31 b(that)g(b)m(y)f(prop)s(osition)e(4.2,) k(\(A.6\))f(and)f(the)h(fact)g(that)g Fn(J)k Fj(2)24 b Fq(\002)2275 1230 y Fo(g)r(ood)2427 1216 y Fq(\()p Fn(M)5 b(;)15 b(\017)2632 1183 y Fm(0)2658 1216 y Fn(;)g(L)p Fq(\),)31 b(it)f(is)g(easy)h(to)g(see)g(that)641 1518 y Fn(\026)696 1474 y Fo(\034)s(;\025)696 1545 y Ft(\003)817 1363 y Fg( )889 1390 y(\022)956 1394 y(Z)1047 1417 y Fo(n)p Ft(+1)1006 1601 y Fo(n)p Fm(\000)p Ft(1)1203 1518 y Fn(R)q Fq(\()p Fn(s)p Fq(\))p Fn(ds)1476 1390 y Fg(\023)1543 1398 y Ft(2)1584 1363 y Fg(!)1681 1518 y Fj(\024)p Fn(c)p Fj(k)p Fn(g)s Fj(k)1927 1481 y Ft(2)1927 1541 y Fm(1)2021 1518 y Fn(\026)2076 1474 y Fo(\034)s(;\025)2076 1545 y Ft(\003)2197 1390 y Fg(\022)2275 1457 y Fq(\()p Fn(n)20 b Fj(\000)25 b Fq(~)-50 b Fn(n)p Fq(\))2566 1424 y Ft(2)p 2275 1497 332 4 v 2398 1580 a Fq(~)g Fn(n)2448 1554 y Ft(2)2618 1390 y Fg(\023)2710 1518 y Fq(=)1752 1858 y Fn(c)p Fj(k)p Fn(g)s Fj(k)1927 1820 y Ft(2)1927 1880 y Fm(1)2021 1858 y Fn(\026)2076 1814 y Fo(\034)s(;\025)2076 1884 y Ft(\003)2197 1702 y Fg( )2280 1783 y Fn(\026)2335 1740 y Fo(\021)r(;\025)2335 1810 y(Q)2392 1819 y Fl(\013)2444 1783 y Fq(\()p Fn(N)2552 1797 y Fo(\013)2604 1783 y Fn(;)15 b(N)2717 1797 y Fo(a)2762 1783 y Fq(\))p 2280 1837 517 4 v 2338 1936 a Fn(\026)2393 1892 y Fo(\021)r(;\025)2393 1962 y(Q)2450 1971 y Fl(\013)2502 1936 y Fq(\()p Fn(N)2610 1950 y Fo(\013)2662 1936 y Fq(\))2697 1909 y Ft(2)2808 1702 y Fg(!)2905 1858 y Fj(\024)3118 1796 y Fn(c)p 3027 1837 222 4 v 3027 1920 a(\032)p Fj(j)p Fn(Q)3171 1934 y Fo(\013)3223 1920 y Fj(j)3690 1689 y Fq(\()p Fn(A:)p Fq(9\))0 2162 y(Putting)28 b(together)h(\(A.8\),)i(\(A.9\),)f(the)f (fact)g(that)g Fj(j)p Fn(\032)1815 2176 y Ft(0)1872 2162 y Fj(\000)16 b Fn(\032)p Fj(j)26 b(\024)f(j)p Fq(\003)p Fj(j)2266 2129 y Fm(\000)p Fo(\017)2386 2162 y Fq(and)j Fn(\032)d(>)g(\015)34 b Fq(the)28 b(lemma)g(follo)m(ws)g(for)g Fn(L)g Fq(big)0 2289 y(enough.)91 b Fb(\003)0 2513 y Fr(References)-232 2665 y Fq([A)m(CCMM])46 b(K.S.)36 b(Alexander,)g(F.)h(Cesi,)f(L.)g(Cha)m(y)m(es,)h(C.)f(Maes,)i(F.)f (Martinelli:)d Fh(Conver)-5 b(genc)g(e)39 b(to)f(Equilibrium)g(of)227 2792 y(R)-5 b(andom)35 b(Ising)e(Mo)-5 b(dels)33 b(in)g(the)g (Gri\016ths')h(Phase.)d Fq(J.)f(Stat.)h(Ph)m(ys.)f Fr(92)h Fq(337{351)j(\(1998\).)-134 2920 y([A)m(CCN])46 b(M.)30 b(Aizenman,)e(J.T.)h(Cha)m(y)m(es,)h(L.)f(Cha)m(y)m(es)h(and)e(C.M.)i (Newman:)f Fh(Tha)j(phase)h(b)-5 b(oundary)33 b(in)e(dilute)h(and)227 3047 y(r)-5 b(andom)35 b(Ising)e(and)g(Potts)h(ferr)-5 b(omagnets.)32 b Fq(J.)e(Ph)m(ys.)g(A:)h(Math.)g(Gen.)g Fr(20)g Fq(L313{L318)i(\(1987\).)-3 3175 y([BD])47 b(L.)29 b(A.)g(Bassalygo)h(and)f(R.)g(L.)g(Dobrushin:)d Fh(Uniqueness)31 b(of)g(a)h(Gibbs)f(\014eld)h(with)g(r)-5 b(andom)34 b(p)-5 b(otential{)33 b(an)227 3302 y(elementary)h(appr)-5 b(o)g(ach)p Fq(.)34 b(Theory)c(Prob.)g(Appl.)f Fr(31)i Fq(No)g(4)g(572,)h (\(1986\).)-83 3430 y([CCM])45 b(N.)31 b(Cancrini,)d(F.)j(Cesi)e(and)h (F.)g(Martinelli:)e Fh(Kawasaki)34 b(dynamics)g(at)f(low)h(temp)-5 b(er)g(atur)g(e.)32 b Fq(J.)e(Stat.)h(Ph)m(y)-8 b(.)227 3557 y Fr(95)31 b Fq(219{275)j(\(1999\).)-63 3684 y([CM1])46 b(N.)39 b(Cancrini)c(and)j(F.)g(Martinelli:)e Fh(Comp)-5 b(arison)43 b(of)d(\014nite)f(volume)i(c)-5 b(anonic)g(al)41 b(and)g(gr)-5 b(and)41 b(c)-5 b(anonic)g(al)227 3811 y(Gibbs)33 b(me)-5 b(asur)g(es)34 b(under)f(a)g(mixing)g(c)-5 b(ondition.)32 b Fq(Mark)m(o)m(v)g(Pro)s(c.)e(Rel.)g(Fields)f Fr(6)i Fq(1{49)h(\(2000\).)-63 3939 y([CM2])46 b(N.)40 b(Cancrini)e(and)h(F.)h(Martinelli:)e Fh(On)j(the)g(sp)-5 b(e)g(ctr)g(al)44 b(gap)e(of)f(Kawasaki)i(dynamics)g(under)e(a)h (mixing)227 4066 y(c)-5 b(ondition)35 b(r)-5 b(evisite)g(d.)31 b Fq(J.)f(Math.)h(Ph)m(ys.)g Fr(41)f Fq(N.3)i(\(2000\).)-146 4194 y([CMM1])46 b(F.)74 b(Cesi,)36 b(C.)h(Maes,)h(F.)f(Martinelli:)e Fh(R)-5 b(elaxation)41 b(of)e(Disor)-5 b(der)g(e)g(d)41 b(magnets)e(in)g(the)f(Gri\016ths)i(r)-5 b(e)g(gime.)227 4321 y Fq(Comm.)30 b(Math.)i(Ph)m(ys.)e Fr(188)h Fq(135)h(\(1997\).) -146 4449 y([CMM2])46 b(F.)25 b(Cesi,)e(C.)g(Maes,)i(F.)f(Martinelli:)e Fh(R)-5 b(elaxation)29 b(to)e(e)-5 b(quilibrium)26 b(for)h(two)h (dimensional)g(disor)-5 b(der)g(e)g(d)30 b(Ising)227 4576 y(systems)k(in)e(the)h(Gri\016ths)h(phase.)e Fq(Comm.)e(Math.)h (Ph)m(ys.)g Fr(189)g Fq(323)g(\(1997\).)-71 4703 y([DKP])46 b(H.)30 b(V)-8 b(on)29 b(Dreyfus,)g(A.)g(Klein)e(and)i(J.)g(F.)g(P)m (erez:)h Fh(T)-7 b(aming)32 b(Gri\016ths)h(singularities:)f(in\014nite) f(di\013er)-5 b(entia-)227 4830 y(bility)33 b(of)g(quenche)-5 b(d)33 b(c)-5 b(orr)g(elations)35 b(functions)p Fq(.)c(Comm.)f(Math.)i (Ph)m(ys.)e Fr(170)h Fq(21,)h(\(1995\).)72 4958 y([F])46 b(J.)21 b(F)-8 b(r\177)-45 b(ohlic)m(h:)20 b Fh(Mathematic)-5 b(al)26 b(asp)-5 b(e)g(cts)26 b(of)e(the)h(physics)f(of)h(disor)-5 b(der)g(e)g(d)27 b(systems)p Fq(.)22 b(in)d(\\Critical)h(Phenomena,)227 5085 y(Random)30 b(Systems,)g(Gauge)i(Theories")e(Eds.)g(K.Osterw)m (alder)f(and)h(R.)g(Stora,)h(Elsevier)e(\(1986\).)60 5213 y([G])46 b(H.)35 b(O.)f(Georgii:)h Fh(Sp)-5 b(ontane)g(ous)39 b(magnetization)f(of)e(r)-5 b(andomly)40 b(dilute)c(ferr)-5 b(omagnets.)36 b Fq(J.Stat.)g(Ph)m(y)-8 b(.)34 b Fr(25)227 5340 y Fq(369)e(\(1981\).)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)e Ft([35])1337 b Fq(A.6)p eop %%Page: 36 36 36 35 bop -69 91 a Fq([GM1])47 b(G.)29 b(Gielis)d(and)i(C.)f(Maes:)j Fh(L)-5 b(o)g(c)g(al)32 b(analyticity)g(and)f(b)-5 b(ounds)32 b(on)f(the)g(trunc)-5 b(ate)g(d)32 b(c)-5 b(orr)g(elation)33 b(functions)e(in)227 218 y(disor)-5 b(der)g(e)g(d)36 b(systems)p Fq(.)c(Mark)m(o)m(v)g(Pro)s(c.)e(Relat.)h(Fields)e Fr(1)i Fq(459)g(\(1995\).)-69 346 y([GM2])47 b(G.)40 b(Gielis)e(and)g(C.)h(Maes:)i Fh(The)g(uniqueness)f(r)-5 b(e)g(gime)41 b(of)g(Gibbs)g(\014elds)g(with)h(unb)-5 b(ounde)g(d)42 b(disor)-5 b(der)p Fq(.)42 b(J.)227 474 y(Stat.)31 b(Ph)m(ys.)g Fr(81)g Fq(No.)g(3/4)g(829)h(\(1995\).)24 602 y([Gr])46 b(R.)33 b(Gri\016ths:)e Fh(Non{analytic)36 b(b)-5 b(ehaviour)36 b(ab)-5 b(ove)35 b(the)g(critic)-5 b(al)35 b(p)-5 b(oint)36 b(in)f(a)f(r)-5 b(andom)37 b(Ising)e(ferr)-5 b(omagnet)p Fq(.)227 729 y(Ph)m(ys.)31 b(Rev.)f(Lett.)i Fr(23)f Fq(17)g(\(1969\).)-1 857 y([Gri])45 b(G.)34 b(Grimmett:)f Fh(Per)-5 b(c)g(olation)36 b Fq(\(Grundheren)31 b(Der)j(Mathematisc)m (hen)g(Wissensc)m(haften,)f(321\))i(Springer)227 985 y(V)-8 b(erlag)31 b(\(1999\).)-38 1113 y([HSS])45 b(O.)33 b(H\177)-45 b(aggstr\177)g(om,)36 b(R.)d(H.)h(Sc)m(honmann,)e(J.)h(A.)h (Steif:)f Fh(The)j(Ising)f(mo)-5 b(del)37 b(on)f(dilute)-5 b(d)36 b(gr)-5 b(aphs)37 b(and)g(str)-5 b(ong)227 1240 y(amenability.)32 b Fq(Preprin)m(t)d(\(1999\).)-111 1368 y([JLQY])45 b(E.)29 b(Jan)m(vresse,)h(C.)f(Landim,)f(J.)h(Quastel)f (and)h(H.T.)g(Y)-8 b(au:)30 b Fh(R)-5 b(elaxation)34 b(to)e(e)-5 b(quilibrium)31 b(of)h(c)-5 b(onservative)227 1495 y(dynamics)34 b(I:)f(zer)-5 b(o{r)g(ange)35 b(pr)-5 b(o)g(c)g(ess.)32 b Fq(Ann.of)e(Prob.)g Fr(27)h Fq(No)g(1)g(325{360)i (\(1999\).)14 1624 y([L)-8 b(Y])46 b(S.)38 b(T.)f(Lu)g(and)h(H{T.)g(Y) -8 b(au:)39 b Fh(Sp)-5 b(e)g(ctr)g(al)41 b(gap)f(and)h(lo)-5 b(garithmic)41 b(Sob)-5 b(olev)41 b(ine)-5 b(quality)40 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b(quality)33 b(for)g(L)-5 b(attic)g(e)33 b(Gases)h(with)f(Mixing)e(Conditions.)h Fq(Comm.)227 2773 y(Math.)f(Ph)m(ys.)g Fr(181)g Fq(367)h(\(1996\).)0 5558 y Fe(5)p Fl(=apr)r(ile=)p Fe(2000)e Ft([36])1337 b Fq(A.7)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0004050335132--