This is a multi-part message in MIME format. ---------------0107110806230 Content-Type: text/plain; name="01-260.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-260.keywords" Gibbs vs non-Gibbs, almost quasilocality, variational principle. ---------------0107110806230 Content-Type: application/postscript; name="varjuin.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="varjuin.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: varjuin.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o varjuin.ps varjuin %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.07.11:1318 %%BeginProcSet: texc.pro /TeXDict 250 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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14065A48130E147EB512FE181F7E9E1D>I E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 319 469 a Fu(V)-6 b(ariational)22 b(principle)f(and)g(almost)g (quasilo)r(calit)n(y)587 560 y(for)h(renormalized)f(measures.)761 686 y Ft(Rob)q(erto)c(F)l(ern\023)-24 b(andez,)481 744 y(Lab)q(oratoire)18 b(de)e(Math)o(\023)-23 b(ematiques)14 b(Raph)o(\177)-23 b(el)16 b(Salem,)446 803 y(Univ)o(ersit)o(\023)-23 b(e)14 b(de)i(Rouen,)g(UFR)f(Sciences,)g(site)g(Colb)q(ert,)605 861 y(76821)j(Mon)o(t)e(Sain)o(t-Aignan,)g(F)l(rance.)524 919 y(E-mail:)k(Rob)q(erto.F)l(ernandez@univ-rouen.fr)813 977 y(Arnaud)c(Le)g(Ny)l(,)481 1035 y(Lab)q(oratoire)i(de)e(Math)o (\023)-23 b(ematiques)14 b(Raph)o(\177)-23 b(el)16 b(Salem,)446 1093 y(Univ)o(ersit)o(\023)-23 b(e)14 b(de)i(Rouen,)g(UFR)f(Sciences,)g (site)g(Colb)q(ert,)605 1151 y(76821)j(Mon)o(t)e(Sain)o(t-Aignan,)g(F)l (rance.)587 1209 y(E-mail:)j(Arnaud.Len)o(y@univ-rouen.fr)833 1267 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1048 a Fm(j)p Fr(\003)871 1055 y Fh(n)894 1048 y Fm(j)907 1035 y Fh(d)947 1017 y Fr(log)1013 955 y Fc(Z)1066 1017 y Fr(exp)1136 967 y Fc(\020)1175 974 y(X)1163 1072 y Fh(x)p Fe(2)p Fj(\003)1231 1076 y Fb(n)1260 1017 y Fg(\034)1280 1024 y Fh(x)1301 1017 y Fg(f)1328 967 y Fc(\021)1363 1017 y Fg(d\027)224 1162 y Fr(whenev)o(er)20 b(it)f(exists.)32 b(This)19 b(limit)i(exists,)f(for)e(ev)o(ery)h (quasilo)q(cal)i(function)f Fg(f)5 b Fr(,)20 b(if)f Fg(\027)224 1219 y Fr(is)k(Gibbsian)g(or)f(asymptotically)g(decoupled)j([24)o(],)e (yielding)h(a)e(con)o(v)o(ex)g(function)224 1275 y Fg(p)p Fr(\()8 b Fm(\001)g(j)p Fg(\027)s Fr(\).)295 1332 y(F)l(or)14 b(our)g(purp)q(oses,)h(it)g(is)g(imp)q(ortan)o(t)f(to)g(separate)g(the) g(di\013eren)o(t)h(ingredien)o(ts)h(of)224 1388 y(the)f(usual)h(v)m (ariational)h(principle)h(in)e(statistical)f(mec)o(hanics.)224 1482 y Fs(De\014nition)k(2.17)f(\(Sp)q(eci\014cation-i)q(ndep)q(enden)o (t)j(v)m(ariational)f(principle\))k Ff(A)224 1538 y(me)n(asur)n(e)12 b Fg(\027)k Fm(2)d(M)532 1519 y Fj(+)532 1552 y(1)p Fh(;inv)613 1538 y Fr(\(\012\))e Ff(satis\014es)g(a)h(variational)g(principle)g(if) g(the)g(r)n(elative)g(entr)n(opy)224 1602 y Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))k Ff(and)g(the)g(pr)n(essur)n(e)f Fg(p)p Fr(\()p Fg(f)5 b Fm(j)p Fg(\027)s Fr(\))15 b Ff(exist)h(for)g (al)r(l)g Fg(\026)d Fm(2)g(M)1226 1583 y Fj(+)1226 1615 y(1)p Fh(;inv)1306 1602 y Fr(\(\012\))j Ff(and)f(al)r(l)h Fg(f)i Fm(2)13 b(F)1659 1609 y Fj(lo)q(c)1705 1602 y Ff(,)224 1658 y(and)j(they)h(ar)n(e)f(c)n(onjugate)g(c)n(onvex)g (functions)f(in)h(the)h(sense)d(that)605 1769 y Fg(p)p Fr(\()p Fg(f)5 b Fm(j)p Fg(\027)s Fr(\))25 b(=)95 b(sup)814 1816 y Fh(\026)p Fe(2M)901 1802 y Fd(+)901 1828 y(1)p Fb(;)p Fd(in)o(v)969 1816 y Fj(\(\012\))1022 1719 y Fc(h)1044 1769 y Fg(\026)p Fr(\()p Fg(f)5 b Fr(\))10 b Fm(\000)g Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))1316 1719 y Fc(i)1602 1769 y Fr(\(2.18\))224 1919 y Ff(for)17 b(al)r(l)f Fg(f)i Fm(2)13 b(F)478 1926 y Fj(lo)q(c)523 1919 y Ff(,)k(and)654 2029 y Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))25 b(=)47 b(sup)867 2069 y Fh(f)t Fe(2F)937 2075 y Fd(lo)q(c)977 1979 y Fc(h)998 2029 y Fg(\026)p Fr(\()p Fg(f)5 b Fr(\))10 b Fm(\000)h Fg(p)p Fr(\()p Fg(f)5 b Fm(j)p Fg(\027)s Fr(\))1268 1979 y Fc(i)1602 2029 y Fr(\(2.19\))224 2169 y Ff(for)17 b(al)r(l)f Fg(\026)d Fm(2)g(M)500 2150 y Fj(+)500 2183 y(1)p Fh(;inv)581 2169 y Fr(\(\012\))p Ff(.)295 2263 y Fr(Gibbs)18 b(measures)e(satisfy)h (this)h(sp)q(eci\014cation-indep)r(ende)q(n)o(t)h(principle.)29 b(P\014ster)224 2319 y([24)o(,)15 b(Section)h(3.1])e(has)h(recen)o(tly) h(extended)h(its)e(v)m(alidit)o(y)i(to)e(asymptotically)h(decou-)224 2376 y(pled)23 b(measures.)37 b(In)22 b(these)f(cases)g Fg(h)p Fr(\()8 b Fm(\001)g(j)p Fg(\027)s Fr(\))20 b(is)h(the)g(rate)g (function)h(for)e(a)h(\(lev)o(el)h(3\))224 2432 y(large-deviation)17 b(principle)h(for)c Fg(\027)s Fr(.)960 2613 y(7)p eop %%Page: 8 8 8 7 bop 224 286 a Fs(De\014nition)19 b(2.20)f(\(V)l(ariational)h (principle)g(relativ)o(e)f(to)g(a)g(sp)q(eci\014cation\))25 b Ff(L)n(et)224 343 y Fg(\015)18 b Ff(b)n(e)c(a)i(sp)n(e)n(ci\014c)n (ation)d(and)j Fg(\027)f Fm(2)e(G)807 350 y Fj(in)o(v)857 343 y Fr(\()p Fg(\015)s Fr(\))p Ff(.)19 b(We)d(say)f(that)h(a)f (variational)h(principle)e(o)n(c-)224 399 y(curs)i(for)h Fg(\027)j Ff(and)c Fg(\015)i Ff(if)e(for)h(al)r(l)f Fg(\026)d Fm(2)g(M)888 380 y Fj(+)888 413 y(1)p Fh(;inv)969 399 y Fr(\(\012\))673 491 y Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))f(=)h(0)29 b Fm(\()-8 b(\))30 b Fg(\026)13 b Fm(2)g(G)1134 498 y Fj(in)o(v)1183 491 y Fr(\()p Fg(\015)s Fr(\))f Fg(:)331 b Fr(\(2.21\))295 587 y(The)19 b(equiv)m(alence)i (\(2.21\))16 b(holds)k(for)e(Gibbs)h(measures)f Fg(\027)s Fr(,)i(while)g(the)e(implica-)224 644 y(tion)g(to)e(the)h(righ)o(t)g (is)h(v)m(alid,)h(more)e(generally)l(,)h(for)f(measures)g Fg(\027)k Fr(consisten)o(t)c(with)g Fg(\015)224 700 y Fr(quasilo)q(cal)22 b(\(see)f([12)o(],)g(Chapter)g(10\).)35 b(In)22 b([19)o(])e(the)h(implication)i(to)d(the)g(left)h(w)o(as)224 757 y(extended)16 b(to)e(blo)q(c)o(k-transformed)h(measures)g (satisfying)g(appropriate)g(supp)q(ort)f(h)o(y-)224 813 y(p)q(othesis.)19 b(Belo)o(w)12 b(w)o(e)f(extend)g(the)g(implication)i (to)e(the)g(righ)o(t)g(to)f(some)h(non-Gibbsian)224 870 y(\(non-quasilo)q(cal\))17 b(measures.)224 990 y Fl(2.3)56 b(T)-5 b(ransformations)18 b(of)h(measures)224 1076 y Fs(De\014nition)g(2.22)k Ff(A)18 b Fs(renormalization)23 b(transformation)c Fg(T)25 b Ff(fr)n(om)19 b Fr(\(\012)p Fg(;)8 b Fm(F)d Fr(\))16 b Ff(to)224 1132 y Fr(\(\012)275 1116 y Fe(0)286 1132 y Fg(;)8 b Fm(F)345 1116 y Fe(0)355 1132 y Fr(\))18 b Ff(is)g(a)g(pr)n(ob)n(ability)f(kernel)g Fg(T)6 b Fr(\()i Fm(\001)g(j)g(\001)g Fr(\))15 b Ff(on)j Fr(\(\012)p Fg(;)8 b Fm(F)1172 1116 y Fe(0)1181 1132 y Fr(\))p Ff(.)26 b(That)18 b(is,)g(for)g(e)n(ach)g Fg(!)g Fm(2)e Fr(\012)224 1188 y Fg(T)6 b Fr(\()i Fm(\001)g(j)p Fg(!)r Fr(\))13 b Ff(is)j(a)g(pr)n(ob)n(ability)f(me)n(asur)n(e)h(on)f Fr(\(\012)975 1172 y Fe(0)986 1188 y Fg(;)8 b Fm(F)1045 1172 y Fe(0)1055 1188 y Fr(\))16 b Ff(and)f(for)h(e)n(ach)g Fg(A)1384 1172 y Fe(0)1408 1188 y Fm(2)d(F)1489 1172 y Fe(0)1516 1188 y Fg(T)6 b Fr(\()p Fg(A)1601 1172 y Fe(0)1612 1188 y Fm(j)i(\001)g Fr(\))14 b Ff(is)224 1245 y Fm(F)5 b Ff(-me)n(asur)n(able.)295 1301 y(The)11 b(tr)n(ansformation) h(is)g(a)g Fs(blo)q(c)o(k-spin)h(transformation)g Ff(if)f Fr(\012)1437 1285 y Fe(0)1460 1301 y Ff(is)g(of)g(the)g(form)224 1358 y Fr(\(\012)275 1341 y Fe(0)275 1370 y Fj(0)294 1358 y Fr(\))312 1341 y Fk(L)337 1330 y Fa(0)366 1358 y Ff(and)18 b(ther)n(e)g(exists)f Fg(\013)g(>)f Fr(0)i Ff(\(c)n(ompr)n(ession)f(factor\))h(such)h(that)g(the)f(fol)r(lowing) 224 1414 y(two)f(pr)n(op)n(erties)f(hold)264 1500 y(\(i\))22 b Fr(Strict)15 b(lo)q(calit)o(y)p Ff(:)22 b(F)m(or)16 b(every)g Fg(n)p Ff(,)h Fg(A)941 1483 y Fe(0)965 1500 y Fm(2)c(F)1046 1483 y Fe(0)1057 1507 y Fj(\003)1081 1498 y Fa(0)1081 1516 y Fb(n)1121 1500 y Ff(implies)j Fg(T)1310 1483 y Fe(\000)p Fj(1)1357 1500 y Fr(\()p Fg(A)1409 1483 y Fe(0)1420 1500 y Fr(\))c Fm(2)h(F)1526 1509 y Fj(\003)1550 1498 y Fa(0)1550 1522 y Fd([)p Fb(\013n)p Fd(])1611 1500 y Ff(.)250 1602 y(\(ii\))22 b Fr(F)l(actorization)p Ff(:)32 b(Ther)n(e)22 b(exists)f(a)i(distanc)n(e)e Fr(dist)i Ff(in)f Fi(L)1329 1585 y Fe(0)1361 1602 y Ff(such)g(that)h(if)f Fg(A)1653 1585 y Fe(0)1688 1602 y Fm(2)338 1658 y(F)376 1642 y Fe(0)387 1666 y Fh(D)417 1656 y Fa(0)454 1658 y Ff(and)h Fg(B)585 1642 y Fe(0)624 1658 y Fm(2)j(F)718 1642 y Fe(0)729 1666 y Fh(E)757 1656 y Fa(0)794 1658 y Ff(with)e Fr(dist)q(\()p Fg(D)1031 1642 y Fe(0)1042 1658 y Fg(;)8 b(E)1100 1642 y Fe(0)1110 1658 y Fr(\))26 b Fg(>)g(\013)p Ff(,)g(then)e Fg(T)6 b Fr(\()p Fg(A)1478 1642 y Fe(0)1505 1658 y Fm(\\)16 b Fg(B)1587 1642 y Fe(0)1599 1658 y Fm(j)8 b(\001)g Fr(\))24 b(=)338 1715 y Fg(T)6 b Fr(\()p Fg(A)423 1698 y Fe(0)434 1715 y Fm(j)i(\001)g Fr(\))g Fg(T)e Fr(\()p Fg(B)589 1698 y Fe(0)599 1715 y Fm(j)i(\001)g Fr(\))p Ff(.)224 1811 y Fr(A)i(renormalization)h (transformation)e(is)i Ff(deterministic)h Fr(if)f(it)f(is)h(of)e(the)i (form)e Fg(T)d Fr(\()i Fm(\001)g(j)p Fg(!)r Fr(\))i(=)224 1867 y Fg(\016)244 1876 y Fh(t)p Fj(\()p Fh(!)q Fj(\))310 1867 y Fr(\()e Fm(\001)g Fr(\))13 b(for)i(some)f Fg(t)g Fr(:)e(\012)g Fm(!)h Fr(\012)762 1851 y Fe(0)774 1867 y Fr(.)295 1924 y(A)k(renormalization)h(transformation)e Fg(T)24 b Fr(induces)19 b(a)e(transformation)f Fg(\026)h Fm(7!)f Fg(\026T)224 1980 y Fr(on)f(measures,)g(with)h(\()p Fg(\026T)6 b Fr(\)\()p Fg(f)740 1964 y Fe(0)751 1980 y Fr(\))12 b(=)h Fg(\026)p Fr([)p Fg(T)6 b Fr(\()p Fg(f)947 1964 y Fe(0)958 1980 y Fr(\)])15 b(for)f(eac)o(h)i Fg(f)1203 1964 y Fe(0)1227 1980 y Fm(2)d(F)1308 1964 y Fe(0)1319 1980 y Fr(.)295 2037 y(In)19 b(most)f(applications,)i(blo)q(c)o(k-spin) h(transformations)c(ha)o(v)o(e)h(a)h(pro)q(duct)g(form:)224 2093 y Fg(T)6 b Fr(\()p Fg(d!)329 2076 y Fe(0)340 2093 y Fm(j)p Fg(!)r Fr(\))24 b(=)486 2059 y Fc(Q)529 2106 y Fh(x)549 2097 y Fa(0)570 2093 y Fg(T)597 2101 y Fh(x)617 2091 y Fa(0)629 2093 y Fr(\()p Fg(d!)699 2101 y Fh(x)719 2091 y Fa(0)732 2093 y Fm(j)p Fg(!)r Fr(\),)13 b(where)g Fg(T)975 2100 y Fh(x)997 2093 y Fr(\()p Fm(f)p Fg(!)1066 2101 y Fh(x)1086 2091 y Fa(0)1099 2093 y Fm(gj)8 b(\001)g Fr(\))i Fm(2)j(F)1268 2100 y Fh(B)1295 2110 y Fb(x)1312 2103 y Fa(0)1327 2093 y Fr(,)h(for)e(a)i(family)g(of)f(sets)224 2149 y Fm(f)p Fg(B)281 2157 y Fh(x)301 2148 y Fa(0)335 2149 y Fm(\032)20 b Fi(L)f Fr(:)g Fg(x)499 2133 y Fe(0)531 2149 y Fm(2)h Fi(L)613 2133 y Fe(0)623 2149 y Fm(g)g Fr(|the)g Ff(blo)n(cks)t Fr(|)f(with)h(b)q(ounded)h(diameter)f(whose)f (union)224 2206 y(co)o(v)o(ers)c Fi(L)p Fr(.)k(T)l(ransformations)c(of) g(this)h(sort)e(are)i(called)h Ff(r)n(e)n(al-sp)n(ac)n(e)e(r)n (enormalization)224 2262 y(tr)n(ansformations)j Fr(in)d(ph)o(ysics.)20 b(The)14 b(transformations)f(de\014ning)j(cellular)g(automata)224 2319 y(\(with)k(lo)q(cal)i(rules\))e(\014t)g(also)g(in)o(to)g(this)h (framew)o(ork.)33 b(The)21 b(corresp)q(onding)g(blo)q(c)o(ks)224 2375 y(o)o(v)o(erlap)15 b(and)g(the)h(compression)f(factor)g(ma)o(y)f (b)q(e)i(c)o(hosen)f(arbitrarily)h(close)g(to)f(one.)295 2432 y(W)l(e)g(brie\015y)i(remind)f(the)g(reader)f(of)g(some)g(of)g (the)h(transformations)e(considered)224 2488 y(in)i(the)f(sequel:)960 2613 y(8)p eop %%Page: 9 9 9 8 bop 292 286 a Fm(\017)23 b Ff(Pr)n(oje)n(ctions)10 b(and)i(de)n(cimations)t Fr(:)17 b(Giv)o(en)11 b Fg(D)i Fm(\032)g Fi(L)p Fr(,)d(this)h(is)g(the)f(\(pro)q(duct\))g(deter-)338 343 y(ministic)k(transformation)d(de\014ned)i(b)o(y)f Fg(t)p Fr(\()p Fg(!)r Fr(\))h(=)g(\()p Fg(!)1216 350 y Fh(x)1238 343 y Fr(\))1256 350 y Fh(x)p Fe(2)p Fh(D)1331 343 y Fr(.)19 b(The)12 b Ff(de)n(cimation)h(of)338 399 y(sp)n(acing)18 b Fg(b)13 b Fm(2)g Fi(N)p Fr(,)h(for)h(whic)o(h)h Fi(L)c Fr(=)h Fi(Z)963 383 y Fh(d)980 399 y Fr(,)j Fg(D)d Fr(=)h Fg(b)p Fi(Z)1162 383 y Fh(d)1179 399 y Fr(,)h(is)h(a)f(blo)q(c)o (k)h(transformation,)338 456 y(while)f Ff(Schonmann)-5 b('s)13 b(example)k Fr([25)o(],)c(corresp)q(onding)h(to)f Fg(D)g Fr(=)h(h)o(yp)q(erplane,)h(is)338 512 y(not)g(b)q(ecause)h(it)f (fails)h(to)f(b)q(e)h(strictly)f(lo)q(cal.)292 606 y Fm(\017)23 b Ff(Kadano\013)10 b Fr(:)17 b(This)11 b(is)f(a)g(pro)q (duct)h(blo)q(c)o(k)g(transformation)d(de\014ned)k(b)o(y)e Fg(T)1553 613 y Fh(x)1575 606 y Fr(\()p Fg(d!)1645 613 y Fh(x)1666 606 y Fm(j)p Fg(!)r Fr(\))i(=)338 663 y(exp\()p Fg(p)c(!)486 646 y Fe(0)484 674 y Fh(x)513 628 y Fc(P)561 676 y Fh(y)q Fe(2)p Fh(B)630 680 y Fb(x)659 663 y Fg(!)687 670 y Fh(y)708 663 y Fr(\))p Fg(=)p Fr(norm)o(,)j(for)f(a)g(giv)o(en)h (c)o(hoice)h(of)e(parameter)g Fg(p)g Fr(and)h(blo)q(c)o(ks)338 719 y Fg(B)372 726 y Fh(x)394 719 y Fr(.)24 b(In)18 b(the)e(limit)i Fg(p)d Fm(!)h(1)h Fr(one)f(obtains)h(the)g Ff(majority)h(tr)n (ansformation)i Fr(for)338 775 y(the)i(giv)o(en)g(blo)q(c)o(ks.)41 b(If)22 b Fg(B)810 782 y Fh(x)856 775 y Fr(=)i 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Fm(\000)p Fr(")h(in)g(the)g(sequel,)g(resp) q(ectiv)o(ely)i(equal)e(to)f(\\plus")h(and)g(to)f(\\min)o(us")h(at)f (eac)o(h)g(site.)224 1811 y(F)l(or)21 b(brevit)o(y)l(,)j(quasilo)q (calit)o(y)f(in)g(the)f(\\+",)h(resp.)e(\\)p Fm(\000)p Fr(",)i(direction)g(will)g(b)q(e)g(called)224 1867 y Ff(right)e(c)n(ontinuity)p Fr(,)f(resp.)f Ff(left)h(c)n(ontinuity)p Fr(.)32 b(The)20 b(partial)g(order)g(determines)g(a)f(no-)224 1924 y(tion)e(of)g(monotonicit)o(y)g(for)g(functions)g(on)g(\012.)25 b(A)18 b(sp)q(eci\014cation)g Fg(\031)h Fr(is)f Ff(monotonicity)224 1980 y(pr)n(eserving)h Fr(if)e(for)f(eac)o(h)g(\014nite)i(\003)c Fm(\032)g Fi(L)p Fr(,)h Fg(\031)961 1987 y Fj(\003)987 1980 y Fg(f)22 b Fr(is)17 b(increasing)g(whenev)o(er)g Fg(f)k Fr(is.)j(These)224 2037 y(sp)q(eci\014cations)19 b(ha)o(v)o(e)e(a)g(n)o(um)o(b)q(er)h(of)e(useful)j(prop)q(erties.)26 b(In)18 b(the)g(follo)o(wing)g(lemma,)224 2093 y(w)o(e)11 b(summarize)h(the)g(prop)q(erties)g(of)f(monotonicit)o(y)g(preserving)h 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(if)h(so)f(is)h Fg(\015)s Ff(.)960 2613 y Fr(9)p eop %%Page: 10 10 10 9 bop 338 286 a Ff(F)m(urthermor)n(e,)14 b Fg(\015)639 270 y Fj(\()p Fe(\000)p Fj(\))695 286 y Fr(\()p Fg(f)5 b Fr(\))12 b Fm(\024)h Fg(\015)s Fr(\()p Fg(f)5 b Fr(\))11 b Fm(\024)i Fg(\015)994 270 y Fj(\(+\))1050 286 y Fr(\()p Fg(f)5 b Fr(\))13 b Ff(for)h(any)f(lo)n(c)n(al)g(incr)n(e)n(asing)e Fg(f)5 b Ff(,)15 b(and)338 343 y(the)h(sp)n(e)n(ci\014c)n(ations)e Fg(\015)713 326 y Fj(\(+\))769 343 y Ff(,)j Fg(\015)827 326 y Fj(\()p Fe(\000)p Fj(\))899 343 y Ff(and)f Fg(\015)i Ff(ar)n(e)f(c)n(ontinuous)e(on)i(the)f(set)388 454 y Fr(\012)421 461 y Fe(\006)475 454 y Fr(=)536 403 y Fc(n)566 454 y Fg(!)f Fm(2)e Fr(\012)f(:)g Fg(\015)749 435 y Fj(\(+\))805 454 y Fr(\()p Fg(f)5 b Fm(j)p Fg(!)r Fr(\))12 b(=)h Fg(\015)998 435 y Fj(\()p Fe(\000)p Fj(\))1054 454 y Fr(\()p Fg(f)5 b Fm(j)p Fg(!)r Fr(\))12 b Fm(8)p Fg(f)18 b Fm(2)13 b(F)1313 461 y Fj(lo)q(c)1359 454 y Fg(;)8 b Fr(\003)j Fm(2)i(S)1497 403 y Fc(o)1540 454 y Fg(:)49 b 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Fr(\()p Fg(f)5 b Fm(jF)871 1141 y Fh(D)902 1134 y Fr(\))16 b Ff(for)h(e)n(ach)f Fg(f)22 b Ff(incr)n(e)n(asing.)255 1228 y(\(d\))g(F)m(or)11 b(e)n(ach)h(\(in\014nite\))e Fg(D)j Fm(\032)g Fi(L)e Ff(ther)n(e)h(exist)f(monotonicity)h(pr)n (eserving)f(sp)n(e)n(ci\014c)n(a-)338 1292 y(tions)i Fr(\000)476 1276 y Fj(\()p Fh(D)q(;)p Fe(\006)p Fj(\))587 1292 y Ff(such)h(that)g(the)g(pr)n(oje)n(ctions)f Fg(\026)1105 1273 y Fe(\006)1105 1306 y Fh(D)1150 1292 y Fm(2)g(G)s Fr(\(\000)1269 1276 y Fj(\()p Fh(D)q(;)p Fe(\006)p Fj(\))1366 1292 y Fr(\))g Ff(and)h Fr(\000)1511 1268 y Fj(\()p Fh(D)q(;)p Fe(\000)p Fj(\))1511 1306 y(\003)1608 1292 y Fr(\()p Fg(f)5 b Fr(\))12 b Fm(\024)338 1359 y Fr(\000)366 1335 y Fj(\()p Fh(D)q(;)p Fj(+\))366 1374 y(\003)463 1359 y Fr(\()p Fg(f)5 b Fr(\))17 b Ff(for)h(e)n(ach)f Fg(f)22 b Ff(incr)n(e)n(asing.)h([By)17 b(\(a\))g(and)g(\(c\))g Fr(\000)1357 1343 y Fj(\()p Fh(D)q(;)p Fj(+\))1471 1359 y Ff(\()p Fr(\000)1518 1343 y Fj(\()p Fh(D)q(;)p Fe(\000)p Fj(\))1615 1359 y Ff(\))g(c)n(an)338 1416 y(b)n(e)g(chosen)f(to)i(b)n (e)e(right)i(\(left\))e(c)n(ontinuous)h(and)g(extende)n(d)g(to)g(a)h (glob)n(al)e(sp)n(e)n(ci-)338 1472 y(\014c)n(ation)f(on)h Fr(\012)597 1479 y Fh(D)645 1472 y Ff(with)h(the)g(same)f(pr)n(op)n (erties.])295 1579 y Fr(Mo)q(dels)24 b(satisfying)f(the)g(FK)o(G)g (prop)q(ert)o(y)g([10)o(])g(are)g(the)g(standard)g(source)g(of)224 1635 y(monotonicit)o(y-preserving)f(sp)q(eci\014cations.)39 b(This)22 b(class)g(of)e(mo)q(dels)i(includes)h(the)224 1691 y(ferromagnets)17 b(with)h(t)o(w)o(o-)e(and)i(one-b)q(o)q(dy)h(in) o(teractions)f(\(eg.)f(Ising\).)29 b(Item)18 b(\(d\))f(of)224 1748 y(the)f(lemma)g(is)g(p)q(oten)o(tially)h(relev)m(an)o(t)g(for)e (renormalized)i(measures)e(b)q(ecause)i(of)e(the)224 1804 y(fact)j(that)g(a)g(transformed)f(measure)h Fg(\026T)25 b Fr(can)19 b(b)q(e)g(seen)f(as)g(the)h(pro)s(jection)f(on)g(the)224 1861 y(primed)e(v)m(ariables)h(of)e(the)g(measure)g Fg(\026)c Fm(\002)f Fg(T)21 b Fr(on)15 b(\012)10 b Fm(\002)h Fr(\012)1190 1844 y Fe(0)1216 1861 y Fr(de\014ned)17 b(b)o(y)603 1963 y(\()p Fg(\026)10 b Fm(\002)h Fg(T)6 b Fr(\)\()p Fg(d!)r(;)i(d!)902 1944 y Fe(0)911 1963 y Fr(\))25 b(=)g Fg(T)6 b Fr(\()p Fg(d!)1119 1944 y Fe(0)1130 1963 y Fm(j)p Fg(!)r Fr(\))i Fg(\026)p Fr(\()p Fg(d!)r Fr(\))j Fg(:)262 b Fr(\(2.25\))224 2065 y(T)l(o)14 b(apply)h(\(d\))f(of)g(the)g(lemma,)h(ho)o(w)o(ev)o (er,)e(one)h(has)g(to)g(\014nd)h(a)f(suitable)i(sp)q(eci\014cation)224 2122 y(for)h(this)h(measure)f Fg(\026)12 b Fm(\002)g Fg(T)6 b Fr(.)26 b(If)17 b Fg(\026)g Fm(2)f(G)s Fr(\()p Fg(\015)s Fr(\))g(and)h Fg(T)23 b Fr(is)18 b(a)f(pro)q(duct)h (transformation,)e(a)224 2178 y(natural)f(candidate)h(is)g(the)f (family)h Fg(\015)c Fm(\012)f Fg(T)21 b Fr(of)15 b(sto)q(c)o(hastic)g (k)o(ernels)224 2284 y(\()p Fg(\015)s Fm(\012)p Fg(T)6 b Fr(\))355 2292 y Fj(\003)p Fe(\002)p Fj(\003)430 2283 y Fa(0)443 2284 y Fr(\()p Fg(d!)513 2291 y Fj(\003)540 2284 y Fg(;)i(d!)615 2266 y Fe(0)613 2296 y Fj(\003)637 2287 y Fa(0)649 2284 y Fm(j)p Fg(!)690 2291 y Fj(\003)714 2282 y Fd(c)733 2284 y Fg(;)g(!)784 2266 y Fe(0)782 2298 y Fj(\(\003)820 2288 y Fa(0)831 2298 y Fj(\))845 2288 y Fd(c)862 2284 y Fr(\))25 b(=)1040 2241 y Fc(Y)1000 2342 y Fh(x)1020 2331 y Fa(0)1032 2342 y Fj(:)5 b Fh(x)1067 2331 y Fa(0)1078 2342 y Fe(2)p Fj(\003)1126 2331 y Fa(0)971 2374 y Fj(or)14 b Fh(B)1044 2384 y Fb(x)1061 2377 y Fa(0)1074 2374 y Fe(\\)p Fj(\003)p Fe(6)p Fj(=)p Fe(;)1180 2284 y Fg(T)1207 2292 y Fh(x)1227 2283 y Fa(0)1240 2284 y Fr(\()p Fg(d!)1310 2292 y Fh(x)1330 2283 y Fa(0)1342 2284 y Fm(j)p Fg(!)1383 2291 y Fh(B)1410 2301 y Fb(x)1427 2294 y Fa(0)1443 2284 y Fr(\))8 b Fg(\015)1493 2291 y Fj(\003)1518 2284 y Fr(\()p Fg(d!)1588 2291 y Fj(\003)1614 2284 y Fm(j)p Fg(!)1655 2291 y Fj(\003)1679 2282 y Fd(c)1697 2284 y Fr(\))13 b Fg(:)1602 2426 y Fr(\(2.26\))949 2613 y(10)p eop %%Page: 11 11 11 10 bop 224 286 a Fs(De\014nition)19 b(2.27)k Ff(A)17 b(p)n(air)h Fr(\()p Fg(\015)s(;)8 b(T)e Fr(\))p Ff(,)14 b(wher)n(e)k Fg(\015)h Ff(is)e(a)h(sp)n(e)n(ci\014c)n(ation)d(and)i Fg(T)23 b Ff(a)18 b(pr)n(o)n(duct)224 343 y(r)n(enormalization)12 b(tr)n(ansformation,)h(is)f(a)g Fs(monotonicit)o(y-preserving)h(pair)g Ff(if)f(the)224 399 y(family)k Fg(\015)d Fm(\012)d Fg(T)22 b Ff(is)16 b(a)g(monotonicity-pr)n(eserving)g(sp)n(e)n(ci\014c)n (ation.)224 493 y Fr(It)f(do)q(es)h(not)e(seem)i(to)e(b)q(e)i(so)e (simple)j(to)d(construct)h(suc)o(h)h(monotonicit)o(y-preserving)224 550 y(pairs.)k(The)c(only)f(examples)h(w)o(e)f(kno)o(w)g(of)f(are)h (pairs)h(for)e(whic)o(h)i Fg(\015)c Fm(\012)f Fg(T)21 b Fr(is)15 b(Gibbsian)224 606 y(for)f(a)h(FK)o(G)f(in)o(teraction.)20 b(This)15 b(happ)q(ens,)h(for)e(instance,)h(for)f(noisy)i(pro)s (jections)e(\(in)224 663 y(particular)i(noisy)g(decimations\))g(of)e (the)i(Ising)g(measure.)224 806 y Fn(3)67 b(Results)224 907 y Fr(The)16 b(follo)o(wing)g(result)f(follo)o(ws)g(immediately)i (from)e(De\014nitions)h(2.12)e(and)h(2.22.)224 1013 y Fs(Lemma)j(3.1)k Ff(If)17 b Fg(\026)f Fm(2)f(M)691 994 y Fj(+)691 1026 y(1)720 1013 y Fr(\(\012\))i Ff(is)h(asymptotic)n(al)r (ly)f(de)n(c)n(ouple)n(d,)h(then)g(so)f(is)g Fg(\026T)24 b Ff(for)224 1070 y(every)16 b(blo)n(ck-spin)f(tr)n(ansformation)i Fg(T)6 b Ff(.)224 1176 y Fr(F)l(rom)15 b(the)g(results)g(of)g (P\014ster,)g(w)o(e)g(can)g(then)h(conclude)g(the)g(follo)o(wing:)224 1282 y Fs(Theorem)h(3.2)23 b Ff(L)n(et)18 b Fg(\026)g Fm(2)f(M)764 1263 y Fj(+)764 1296 y(1)p Fh(;inv)845 1282 y Fr(\(\012\))h Ff(b)n(e)h(asymptotic)n(al)r(ly)g(de)n(c)n(ouple)n(d)f (and)h Fg(T)25 b Ff(b)n(e)18 b(a)224 1339 y(blo)n(ck-spin)g(tr)n (ansformation)i(such)f(that)i Fg(\026T)k Ff(is)19 b(tr)n (anslation-invariant.)30 b(Then)18 b(the)224 1395 y(r)n(enormalize)n(d) f(me)n(asur)n(e)g Fg(\026T)24 b Ff(satis\014es)16 b(the)i(sp)n(e)n (ci\014c)n(ation-indep)n(endent)c(variational)224 1452 y(principle)i(of)g(De\014nition)g(2.17.)224 1558 y Fr(In)f([24)o(,)f (Section)i(3.4])d(it)h(is)h(sho)o(w)o(ed)f(that)g(the)h(relativ)o(e)g (en)o(trop)o(y)e(densit)o(y)i Fg(h)p Fr(\()8 b Fm(\001)g(j)p Fg(\026T)e Fr(\))13 b(is)224 1615 y(the)i(large)g(deviation)h(rate)e (function)h(of)f(the)h(empirical)i(measure)e Fg(L)1415 1622 y Fj(\003)1454 1615 y Fr(=)1502 1580 y Fc(P)1550 1628 y Fh(x)p Fe(2)p Fj(\003)1627 1615 y Fg(\016)1647 1622 y Fh(\034)1663 1626 y Fb(x)1682 1622 y Fh(\033)1706 1615 y Fr(.)295 1727 y(The)k(next)g(theorem)g(states)f(the)i(criterion) g(used)f(in)h(this)g(pap)q(er)f(to)g(pro)o(v)o(e)f(the)224 1784 y(implication)g(to)c(the)h(righ)o(t)g(in)h(\(2.21\))e(for)g (non-quasilo)q(cal)j(measures)f Fg(\027)s Fr(.)224 1878 y Fs(Theorem)h(3.3)23 b Ff(L)n(et)f Fg(\015)i Ff(b)n(e)f(a)g(sp)n(e)n (ci\014c)n(ation)e(that)i(is)f(quasilo)n(c)n(al)g(in)g(the)h(dir)n(e)n (ction)224 1934 y Fg(\022)i Fm(2)e Fr(\012)f Ff(and)g Fg(\027)k Fm(2)d(G)600 1941 y Fj(in)o(v)650 1934 y Fr(\()p Fg(\015)s Fr(\))p Ff(.)36 b(F)m(or)22 b(e)n(ach)g Fr(\003)g Fm(2)h(S)s Ff(,)h Fg(M)j Fm(2)d Fi(N)p Ff(,)e Fr(\003)g Fm(\032)i Fr(\003)1478 1941 y Fh(M)1539 1934 y Ff(and)e(e)n(ach)224 1995 y(lo)n(c)n(al)e Fg(f)5 b Ff(,)21 b(let)f Fg(\015)488 1973 y Fh(M)r(;\022)485 2009 y Fj(\003)552 1995 y Fr(\()p Fg(f)5 b Fr(\))20 b Ff(denote)g(the)h(function)f Fg(!)i Fm(!)f Fg(\015)1187 2002 y Fj(\003)1213 1995 y Fr(\()p Fg(f)5 b Fm(j)p Fg(!)1299 2002 y Fj(\003)1323 2008 y Fb(M)1367 1995 y Fg(\022)1388 2004 y Fk(L)n Fe(n)p Fj(\003)1455 2010 y Fb(M)1488 1995 y Fr(\))p Ff(.)33 b(Then,)21 b(if)224 2056 y Fg(\026)13 b Fm(2)g(M)362 2037 y Fj(+)362 2070 y(1)p Fh(;)p Fj(in)o(v)439 2056 y Fr(\(\012\))i Ff(is)h(such)g(that)h Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))c(=)g(0)p Ff(,)347 2193 y Fg(\026)g Fm(2)g(G)457 2200 y Fj(in)o(v)506 2193 y Fr(\()p Fg(\015)s Fr(\))28 b Fm(\()-8 b(\))30 b Fg(\027)734 2142 y Fc(h)768 2157 y Fg(d\026)819 2167 y Fj(\003)843 2173 y Fb(M)877 2167 y Fe(n)p Fj(\003)p 768 2182 154 2 v 770 2224 a Fg(d\027)816 2233 y Fj(\003)840 2239 y Fb(M)875 2233 y Fe(n)p Fj(\003)934 2142 y Fc(\020)961 2193 y Fg(\015)988 2171 y Fh(M)r(;\022)985 2207 y Fj(\003)1052 2193 y Fr(\()p Fg(f)5 b Fr(\))10 b Fm(\000)g Fg(\015)1194 2200 y Fj(\003)1220 2193 y Fr(\()p Fg(f)5 b Fr(\))1283 2142 y Fc(\021i)1369 2193 y Fm(\000)-7 b(!)1352 2223 y Fh(M)t Fe(!1)1480 2193 y Fr(0)122 b(\(3.4\))224 2323 y Ff(for)17 b(al)r(l)f Fr(\003)c Fm(2)h(S)19 b Ff(and)d Fg(f)i Fm(2)13 b(F)700 2330 y Fj(lo)q(c)746 2323 y Ff(.)224 2417 y Fr(The)j(righ)o(t-hand-side)h(of)e(\(3.4\))f(sho)o(ws)h(that)f (consistency)j(requires)f(the)g(concen)o(tra-)224 2473 y(tion)c(prop)q(erties)f(of)g Fg(d\026)622 2482 y Fj(\003)646 2488 y Fb(M)680 2482 y Fe(n)p Fj(\003)724 2473 y Fg(=d\027)793 2482 y Fj(\003)817 2488 y Fb(M)851 2482 y Fe(n)p Fj(\003)907 2473 y Fr(to)f(b)q(eat)h(asymptotic)g(div)o(ergences)h(due)g(to)e(the) 949 2613 y(11)p eop %%Page: 12 12 12 11 bop 224 286 a Fr(lac)o(k)14 b(of)f(con)o(tin)o(uit)o(y)h(of)f Fg(\015)653 293 y Fj(\003)679 286 y Fr(.)19 b(This)14 b(imp)q(oses)g(some)g(conditions)g(on)g Fg(\026)f Fr(whic)o(h)i(are)e (remi-)224 343 y(niscen)o(t)h(of)e(what)g(happ)q(ens)i(for)e(un)o(b)q (ounded)i(spin-systems.)20 b(This)13 b(analogy)f(b)q(et)o(w)o(een)224 399 y(un)o(b)q(ounded)25 b(spin)e(systems)g(and)g(non-Gibbsian)h (measures)f(is)g(an)g(early)g(remark)224 456 y(from)18 b(Dobrushin.)31 b(Within)20 b(approac)o(hes)e(based)h(on)g(p)q(oten)o (tials)g(\(w)o(eak)f(Gibbsian-)224 512 y(ness\))11 b(these)h (conditions)g(are)f(de\014ned)i(and)e(handled)i(b)o(y)e (cluster-expansion)i(metho)q(ds)224 569 y([22)o(,)h(19)o(].)19 b(As)14 b(w)o(e)g(discuss)h(b)q(elo)o(w,)f(in)h(fa)o(v)o(orable)e (cases)h(mon)o(tonicit)o(y)g(argumen)o(ts)f(can)224 625 y(b)q(e)j(used)g(instead.)295 731 y(W)l(e)h(presen)o(t)h(t)o(w)o(o)e (applications)j(of)e(the)h(previous)g(theorem.)27 b(First)17 b(w)o(e)g(discuss)224 788 y(systems)e(with)g(monotonicit)o (y-preserving)i(sp)q(eci\014cations.)224 882 y Fs(Corollary)h(3.5)k Ff(Consider)d(a)h(sp)n(e)n(ci\014c)n(ation)e Fg(\015)k Ff(that)e(is)g(monotonicity)g(pr)n(eserving)224 938 y(and)c(tr)n (anslation)g(invariant.)k(Then,)c(with)g(the)h(notation)f(of)g(L)n (emma)g(2.23,)255 1032 y(\(a\))22 b Fg(h)p Fr(\()p Fg(\026)409 1015 y Fe(\000)439 1032 y Fm(j)p Fg(\026)479 1015 y Fj(+)508 1032 y Fr(\))12 b(=)h(0)f Ff(implies)f(that)h Fg(\026)887 1015 y Fe(\000)930 1032 y Fm(2)h(G)s Fr(\()p Fg(\015)1048 1015 y Fj(\(+\))1103 1032 y Fr(\))e Ff(and)h Fg(\026)1243 1015 y Fe(\000)1273 1032 y Fr(\(\012)1324 1045 y Fh(\015)1344 1035 y Fd(\(+\))1395 1032 y Fr(\))g(=)h Fg(\026)1500 1015 y Fe(\000)1530 1032 y Fr(\(\012)1581 1045 y Fh(\015)1601 1035 y Fd(\()p Fa(\000)p Fd(\))1653 1032 y Fr(\))f(=)338 1088 y(1)k Ff(\(henc)n(e)f Fg(\026)548 1072 y Fe(\000)594 1088 y Ff(is)h(almost)g(quasilo)n(c)n(al\).)257 1182 y(\(b\))22 b(F)m(or)16 b Fg(\026)e Fm(2)h(M)564 1163 y Fj(+)564 1196 y(1)p Fh(;)p Fj(in)o(v)640 1182 y Fr(\(\012\))p Ff(,)h Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)848 1166 y Fj(+)878 1182 y Fr(\))e(=)g(0)i Ff(and)h Fg(\026)p Fr(\(\012)1165 1189 y Fe(\006)1195 1182 y Fr(\))c(=)h(1)j Ff(implies)f Fg(\026)e Fm(2)h(G)s Fr(\()p Fg(\015)1632 1166 y Fj(\(+\))1687 1182 y Fr(\))p Ff(,)338 1239 y(and)h(thus)h Fg(\026)f Ff(almost)h(quasilo)n(c)n(al.)224 1333 y(A)o(nalo)n(gous)e(r) n(esults)h(ar)n(e)g(valid)g(inter)n(changing)f(\\)p Fr(+)p Ff(")i(with)g(\\)p Fm(\000)p Ff(".)224 1426 y Fr(By)f(part)f(\(d\))g (of)g(Lemma)g(2.23,)f(and)h(the)h(commen)o(ts)f(immediately)i (thereafter,)d(the)224 1483 y(preceding)25 b(results)e(apply)g(when)h Fg(\026)875 1466 y Fe(\006)928 1483 y Fr(are)e(the)h(pro)s(jections)g (\(p)q(ossibly)h(noisy\))f(of)224 1539 y(the)d(\\plus")h(and)f(\\min)o (us")g(phases)g(of)g(the)g(Ising)h(mo)q(del.)35 b(More)19 b(generally)l(,)j(they)224 1596 y(can)d(b)q(e)g(the)f(renormalized)i (measures)e(of)g(the)h(\\plus")g(and)f(\\min)o(us")h(measures)f(of)224 1652 y(a)f(monotonicit)o(y-preserving)h(sp)q(eci\014cation)i(whenev)o (er)d(the)h(sp)q(eci\014cation)h(and)e(the)224 1709 y(transformation)d (form)g(a)h(monotonicit)o(y-preserving)i(pair)e(\(De\014nition)h (2.27\).)295 1765 y(A)o(t)g(lo)o(w)g(temp)q(erature,)g(the)h (decimations)g(\(p)q(ossibly)h(noisy\))f Fg(\026)1407 1749 y Fj(+)1437 1765 y Fg(T)22 b Fr(and)17 b Fg(\026)1603 1749 y Fe(\000)1633 1765 y Fg(T)22 b Fr(of)224 1822 y(the)15 b(\\plus")h(and)f(\\min)o(us")g(phases)g(of)g(the)g(Ising)h(mo)q(del)g (are)e(non-Gibbsian)j([15)o(,)e(7)o(],)224 1878 y(that)21 b(is,)j(all)f(sp)q(eci\014cations)g(with)f(whic)o(h)h(these)f(measures) g(are)f(consisten)o(t)h(sho)o(w)224 1934 y(essen)o(tial)c(discon)o(tin) o(uities.)28 b(The)18 b(preceding)g(corollary)g(sho)o(ws)e(that,)h(nev) o(ertheless,)224 1991 y(in)f(these)e(cases)h(the)f(implication)j(to)d (the)h(righ)o(t)f(of)g(the)h(v)m(ariational)g(principle)i(\(2.21\))224 2047 y(can)e(b)q(e)h(reco)o(v)o(ered)f(up)h(to)f(a)f(p)q(oin)o(t.)295 2104 y(If)g Fg(\015)i Fr(is)f(a)e(quasilo)q(cal)j(translation-in)o(v)m (arian)o(t)f(sp)q(eci\014cation)h(and)e Fg(T)20 b Fr(a)14 b(blo)q(c)o(k-spin)224 2160 y(transformation,)19 b(then)h Fg(h)p Fr(\()p Fg(\026T)6 b Fm(j)p Fg(\027)s(T)g Fr(\))19 b(=)h(0)f(for)g(eac)o(h)g Fg(\026;)8 b(\027)23 b Fm(2)d(G)1319 2167 y Fj(in)o(v)1368 2160 y Fr(\()p Fg(\015)s Fr(\))e(suc)o(h)i(that)e Fg(\026T)224 2217 y Fr(and)g Fg(\027)s(T)24 b Fr(are)17 b(translation)h(in)o(v)m(arian)o(t)g([7)o(,)g(form)o(ula)f(\(3.28\)].) 25 b(Hence,)19 b(from)d(part)h(\(a\))224 2273 y(of)e(the)g(previous)h (corollary)g(w)o(e)e(conclude)j(the)f(follo)o(wing.)224 2367 y Fs(Corollary)i(3.6)k Ff(L)n(et)16 b Fg(\015)j Ff(b)n(e)e(a)g(quasilo)n(c)n(al,)f(monotonicity-pr)n(eserving,)g(tr)n (anslation-)224 2423 y(invariant)23 b(sp)n(e)n(ci\014c)n(ation,)h(and)f Fg(T)30 b Ff(a)24 b(blo)n(ck-spin)e(tr)n(ansformation)h(that)i(pr)n (eserves)224 2480 y(tr)n(anslation)17 b(invarianc)n(e)f(such)i(that)g (the)g(p)n(air)f Fr(\()p Fg(\015)s(;)8 b(T)e Fr(\))15 b Ff(is)i(monotonicity-pr)n(eserving.)949 2613 y Fr(12)p eop %%Page: 13 13 13 12 bop 224 286 a Ff(L)n(et)14 b Fg(\026)328 270 y Fe(\006)373 286 y Ff(b)n(e)g(the)h(extr)n(emal)g(me)n(asur)n(es)f(for)i Fg(\015)h Ff([p)n(art)e(\(b\))f(of)h(L)n(emma)f(2.23])i(and)f Fg(\031)1608 270 y Fj(\()p Fe(\006)p Fj(\))1679 286 y Ff(b)n(e)224 343 y(the)i(right\(left\)-c)n(ontinuous)e(sp)n(e)n (ci\014c)n(ations)f(such)i(that)h Fg(\026)1232 326 y Fe(\006)1262 343 y Fg(T)i Fm(2)12 b(G)1377 350 y Fj(in)o(v)1427 343 y Fr(\()p Fg(\031)1473 326 y Fe(\006)1502 343 y Fr(\))p Ff(.)295 399 y(Then)460 494 y Fg(\026)487 475 y Fe(\000)517 494 y Fg(T)18 b Fm(2)13 b(G)s Fr(\()p Fg(\031)681 475 y Fj(+)710 494 y Fr(\))62 b Ff(and)h Fg(\026)952 475 y Fe(\000)982 494 y Fg(T)6 b Fr(\(\012)1066 503 y Fh(\031)1087 494 y Fd(+)t Fr(\))12 b(=)h Fg(\026)1219 475 y Fe(\000)1249 494 y Fg(T)6 b Fr(\(\012)1333 503 y Fh(\031)1354 494 y Fa(\000)t Fr(\))12 b(=)h(1)142 b(\(3.7\))224 588 y Ff(\(henc)n(e)15 b Fg(\026)395 572 y Fe(\000)425 588 y Fg(T)22 b Ff(is)15 b(almost)h(quasilo)n(c)n(al\).)j(A)o(nalo)n(gous)c (r)n(esults)g(ar)n(e)h(valid)g(inter)n(changing)224 645 y(\\)p Fr(+)p Ff(")i(with)e(\\)p Fm(\000)p Ff(".)295 732 y Fr(This)11 b(corollary)g(applies)h(in)g(particular)f(for)f (decimations)i(\(p)q(ossibly)f(noisy\))g(of)f(the)224 788 y(Ising)15 b(mo)q(del.)20 b(A)o(t)13 b(lo)o(w)g(temp)q(erature,)h (the)f(renormalized)i(measures)f Fg(\026)1466 772 y Fj(+)1496 788 y Fg(T)19 b Fr(and)14 b Fg(\026)1656 772 y Fe(\000)1686 788 y Fg(T)224 845 y Fr(are)c(in)h(general)g(non-Gibbsian)g([15,)e(7],) h(that)g(is,)h(the)f(sp)q(eci\014cations)i Fg(\031)1431 828 y Fj(+)1471 845 y Fr(and)e Fg(\031)1582 828 y Fe(\000)1621 845 y Fr(sho)o(w)224 901 y(essen)o(tial)i(discon)o(tin)o(uities.)21 b(Nev)o(ertheless,)12 b(the)f(preceding)h(corollary)l(,)g(together)e (with)224 958 y(part)g(\(b\))h(of)f(Corollary)g(3.5)g(sho)o(ws)g(that)g (in)i(these)f(cases)f(the)h(implication)i(to)d(the)h(righ)o(t)224 1014 y(of)19 b(the)h(v)m(ariational)h(principle)i(\(2.21\))17 b(can)j(b)q(e)h(reco)o(v)o(ered,)f(together)f(with)h(almost)224 1071 y(quasilo)q(calit)o(y)l(.)295 1127 y(Sev)o(eral)15 b(remarks)g(are)g(in)h(order.)224 1214 y Fs(Remark)h(3.8)22 b Fr(The)e(preceding)g(corollary)f(strengthens,)h(for)e(\(noisy\))h (decimation)224 1271 y(transformations,)10 b(the)g(results)g(of)g([2)o (,)g(23])f(where)i(only)f(w)o(eak-Gibbsianness)i(is)e(pro)o(v)o(en.)224 1327 y(Our)h(argumen)o(t)e(is)i(apparen)o(tly)f(simpler)i(than)e(the)g (renormalization)h(and)f(expansion-)224 1384 y(based)16 b(pro)q(cedures)g(set)g(up)g(in)g(these)g(references,)g(but,)f(of)g (course,)g(it)h(do)q(es)g(not)f(giv)o(e)224 1440 y(suc)o(h)i(a)f (complete)i(description)g(of)e(the)h(supp)q(ort)f(of)h(the)f(decimated) i(measures)e(and)224 1497 y(it)g(is)f(only)h(restricted)g(to)e(mo)q (dels)i(with)g(monotonicit)o(y)f(prop)q(erties.)224 1595 y Fs(Remark)i(3.9)22 b Fr(F)l(or)16 b Fg(d)d Fr(=)i(2,)g(the)h (corollary)h(implies)h(that)d Ff(al)r(l)20 b Fr(the)d(decimated)g(mea-) 224 1652 y(sures)e(of)f(the)g(Ising)h(mo)q(del)h(are)e(consisten)o(t)g (with)h Fg(\031)1126 1635 y Fj(+)1169 1652 y Fr(and)g(almost)f(quasilo) q(cal.)21 b(This)224 1708 y(follo)o(ws)c(from)f(the)g(results)h(of)g (Aizenman)g([1])f(and)h(Higuc)o(hi)h([14)o(])e(sho)o(wing)h(that)f Fg(\026)1689 1692 y Fj(+)224 1765 y Fr(and)g Fg(\026)340 1748 y Fe(\000)385 1765 y Fr(are)e(the)i(only)g(extremal)f(measures)g (in)h Fm(G)s Fr(\()p Fg(\015)s Fr(\).)224 1863 y Fs(Remark)h(3.10)22 b Fr(Lefev)o(ere)17 b(pro)o(v)o(es)f(in)h([19)o(])f(the)g(implication)j (to)d(the)g(left)h(in)g(\(2.21\),)224 1920 y(for)i Fg(\027)j Fr(a)d(blo)q(c)o(k-transformed)g(measure)g(and)g Fg(\026)h Fr(concen)o(trated)e(on)h(an)h(appropriate)224 1976 y(set)299 1965 y Fc(e)295 1976 y Fr(\012)13 b Fm(\032)g Fr(\012)i(of)f Fg(\027)s Fr(-measure)i(1.)295 2075 y(Our)10 b(second)h(application)h (of)d(Theorem)h(3.3)g(do)q(es)g(not)g(in)o(v)o(olv)o(e)h(an)o(y)e (monotonicit)o(y)224 2131 y(h)o(yp)q(othesis.)34 b(T)l(o)20 b(form)o(ulate)f(it)h(w)o(e)f(need)i(some)e(notation.)33 b(F)l(or)20 b(\003)f(a)g(\014xed)i(\014nite)224 2187 y(v)o(olume,)15 b(and)h Fg(f)k Fr(a)15 b(lo)q(cal)h(function,)g(put)674 2291 y Fg(\016)696 2272 y Fh(\022)694 2302 y Fj(\003)p Fh(;M)768 2291 y Fr(\()p Fg(f)5 b Fr(\))25 b(=)917 2238 y Fc(\014)917 2266 y(\014)917 2293 y(\014)932 2291 y Fg(\015)959 2269 y Fh(M)r(;\022)956 2305 y Fj(\003)1023 2291 y Fr(\()p Fg(f)5 b Fr(\))k Fm(\000)i Fg(\015)1165 2298 y Fj(\003)1191 2291 y Fr(\()p Fg(f)5 b Fr(\))1254 2238 y Fc(\014)1254 2266 y(\014)1254 2293 y(\014)1602 2291 y Fr(\(3.11\))224 2394 y(and)16 b(in)o(tro)q(duce)g(for)e Fg(\017)f(>)g Fr(0)i(the)g(sets)552 2488 y Fg(A)p Fr(\()p Fg(\022)q(;)8 b Fr(\003)p Fg(;)g(f)r(;)g(\017;)g(M)d Fr(\))23 b(=)i Fm(f)p Fg(\021)14 b Fm(2)f Fr(\012)f(:)g Fg(\016)1129 2469 y Fh(\022)1127 2499 y Fj(\003)p Fh(;M)1201 2488 y Fr(\()p Fg(f)5 b Fr(\))12 b Fg(>)h(\017)p Fm(g)g Fg(:)211 b Fr(\(3.12\))949 2613 y(13)p eop %%Page: 14 14 14 13 bop 224 286 a Fr(If)15 b Fg(\015)i Fr(is)f(con)o(tin)o(uous)f(in) h(the)f(direction)h Fg(\022)q Fr(,)f(then)g Fg(\016)1078 270 y Fh(\022)1076 300 y Fj(\003)p Fh(;M)1165 286 y Fr(tends)g(to)g (zero)g(as)f Fg(M)20 b Fr(tends)15 b(to)224 343 y(in\014nit)o(y)l(,)20 b(and)e(hence)g(for)f(an)o(y)h(probabilit)o(y)h(measure)e Fg(\026)p Fr(,)h Fg(\026)p Fr([)p Fg(A)p Fr(\()p Fg(\022)q(;)8 b Fr(\003)p Fg(;)g(f)r(;)g(\017;)g(M)d Fr(\)])15 b(tends)224 399 y(to)g(zero)g(as)g Fg(M)20 b Fr(tends)15 b(to)g(in\014nit)o(y)l(.) 224 493 y Fs(De\014nition)k(3.13)k Ff(L)n(et)c Fg(\013)695 500 y Fh(M)753 493 y Fm(")f(1)i Ff(b)n(e)f(an)g(incr)n(e)n(asing)f(se)n (quenc)n(e)g(of)h(p)n(ositive)g(num-)224 549 y(b)n(ers.)25 b(L)n(et)16 b Fg(\026)i Ff(b)n(e)g(a)g(pr)n(ob)n(ability)f(me)n(asur)n (e)g(on)h Fr(\012)p Ff(.)25 b(We)18 b(say)f(that)i(the)f(sp)n(e)n (ci\014c)n(ation)d Fg(\015)224 606 y Ff(admits)j Fg(\013)402 613 y Fh(M)460 606 y Ff(as)g(a)g Fg(\026)p Ff(-r)n(ate)h(of)f(c)n (ontinuity)g(in)f(the)h(dir)n(e)n(ction)g Fg(\022)h Ff(if)f(for)h(al)r (l)f Fg(\017)e(>)g Fr(0)p Ff(,)i(for)224 662 y(al)r(l)e Fg(f)22 b Ff(lo)n(c)n(al,)15 b(and)h(for)h(al)r(l)f Fr(\003)p Ff(:)580 780 y Fr(lim)8 b(sup)605 820 y Fh(M)t Fe("1)755 749 y Fr(1)p 732 770 69 2 v 732 811 a Fg(\013)761 818 y Fh(M)813 780 y Fr(log)g Fg(\026)p Fr([)p Fg(A)p Fr(\()p Fg(\022)q(;)g Fr(\003)p Fg(;)g(f)r(;)g(\017;)g(M)d Fr(\)])23 b Fg(<)38 b Fr(0)p Fg(:)239 b Fr(\(3.14\))224 911 y(The)23 b(follo)o(wing)g(prop)q(osition)h(sho)o(ws)e(that)f(for)h(a)h(giv)o(en) g(rate)f(of)g(con)o(tin)o(uit)o(y)l(,)i(the)224 968 y(condition)c(of)d (Theorem)i(3.3)e(will)j(b)q(e)f(satis\014ed)f(if)h(the)f(relativ)o(e)h (en)o(tropies)g(tend)f(to)224 1024 y(zero)d(at)g(the)g(same)g(rate.)224 1130 y Fs(Prop)q(osition)k(3.15)j Ff(L)n(et)g Fg(\027)27 b Fm(2)d(G)s Fr(\()p Fg(\015)s Fr(\))c Ff(and)j(supp)n(ose)f(that)h Fg(\013)1318 1137 y Fh(M)1380 1130 y Ff(is)f(a)g Fg(\027)s Ff(-r)n(ate)h(of)g Fg(\022)q Ff(-)224 1186 y(c)n(ontinuity.)d(Supp)n (ose)d(furthermor)n(e)g(that)737 1304 y Fr(lim)723 1335 y Fh(M)t Fe("1)849 1274 y Fr(1)p 826 1294 V 826 1336 a Fg(\013)855 1343 y Fh(M)900 1304 y Fg(H)938 1311 y Fj(\003)962 1317 y Fb(M)998 1304 y Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))25 b(=)g(0)p Fg(:)382 b Fr(\(3.16\))224 1429 y Ff(Then)16 b Fg(\026)d Fm(2)g(G)s Fr(\()p Fg(\015)s Fr(\))p Ff(.)295 1535 y Fr(This)h(prop)q(osition)g(applies,)h(for)d (instance,)i(to)f(Sc)o(honmann's)g(example.)20 b(Indeed,)224 1592 y(if)c Fg(\027)291 1575 y Fj(+)337 1592 y Fr(is)g(the)g(pro)s (jection)g(on)g(a)f(\(one-dimensional\))j(la)o(y)o(er)d(of)h(the)g(lo)o (w-temp)q(erature)224 1648 y(plus-phase)21 b(of)e(the)g(t)o(w)o (o-dimensional)h(Ising)h(mo)q(del,)g(then)e(the)h(estimates)f(in)h([21) o(])224 1704 y(imply)c(that)e(the)h(monotone)g(righ)o(t-con)o(tin)o (uous)g(sp)q(eci\014cation)h Fg(\015)1359 1688 y Fj(+)1403 1704 y Fr(\(suc)o(h)f(that)f Fg(\027)1646 1688 y Fj(+)1688 1704 y Fm(2)224 1761 y(G)s Fr(\()p Fg(\015)299 1744 y Fj(+)327 1761 y Fr(\)\))e(admits)h Fg(\013)551 1768 y Fh(M)603 1761 y Fr(=)g Fg(M)k Fr(as)c Fg(\027)791 1744 y Fj(+)820 1761 y Fr(-rate)f(of)h(righ)o(t-con)o(tin)o(uit)o(y)l(.)19 b(Hence)14 b(for)e(this)h(exam-)224 1817 y(ple)i(w)o(e)f(can)h (conclude)h(that)d(for)h(an)o(y)g(other)g(measure)g Fg(\026)g Fr(on)h(the)f(la)o(y)o(er,)g Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)1588 1801 y Fj(+)1618 1817 y Fr(\))e(=)h(0)224 1874 y(implies)22 b Fg(\026)g Fm(2)f(G)s Fr(\()p Fg(\015)558 1857 y Fj(+)586 1874 y Fr(\).)34 b(This)21 b(is)g(a)e(strengthening)i (of)f(part)f(\(b\))h(of)g(Corollary)g(3.5.)224 1930 y(W)l(e)h (emphasize)h(that)e(suc)o(h)h(a)f Fg(\026)h Fr(can)g Ff(not)k Fr(b)q(e)c(the)g(pro)s(jection)f Fg(\027)1409 1914 y Fe(\000)1460 1930 y Fr(of)g(the)h(min)o(us)224 1987 y(Ising)16 b(phase.)k(Indeed,)c(while)g(at)e(presen)o(t)h(the)g (existence)h(of)e Fg(h)p Fr(\()p Fg(\027)1360 1970 y Fe(\000)1390 1987 y Fm(j)p Fg(\027)1428 1970 y Fj(+)1457 1987 y Fr(\))g(has)h(not)f(rig-)224 2043 y(orously)k(b)q(een)g (established,)h(Sc)o(honmann's)f(original)g(argumen)o(t)f([25)o(])g (implies)i(that)224 2100 y Fg(h)p Fr(\()p Fg(\027)293 2083 y Fe(\000)323 2100 y Fm(j)p Fg(\027)361 2083 y Fj(+)390 2100 y Fr(\))13 b Fg(>)g Fr(0)i(if)g(it)h(exists.)224 2243 y Fn(4)67 b(Pro)r(ofs)224 2346 y Fl(4.1)56 b(Pro)r(of)18 b(of)h(Theorem)e(3.3)224 2432 y Fr(The)g(h)o(yp)q(othesis)g Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))e(=)g(0)h(implies,)i(b) o(y)f(\(2.15\){\(2.1)o(6\),)c(that)j(for)g Fg(n)g Fr(su\016cien)o(tly) 224 2488 y(large)i(the)h Fm(F)453 2495 y Fj(\003)477 2499 y Fb(n)500 2488 y Fr(-measurable)g(function)g Fg(d\026)988 2495 y Fj(\003)1012 2499 y Fb(n)1036 2488 y Fg(=d\027)1105 2495 y Fj(\003)1129 2499 y Fb(n)1171 2488 y Fr(exists.)29 b(Let's)18 b(denote)h(it)f Fg(g)1658 2495 y Fj(\003)1682 2499 y Fb(n)1706 2488 y Fr(.)949 2613 y(14)p eop %%Page: 15 15 15 14 bop 224 286 a Fr(F)l(or)15 b Fg(f)20 b Fr(lo)q(cal)c(and)g(\003)c Fm(2)h(S)s Fr(,)i(pic)o(k)h Fg(M)k Fr(suc)o(h)15 b(that)g(\003)1083 293 y Fh(M)1134 286 y Fm(\033)e Fr(\003)i(and)h Fg(g)1340 293 y Fj(\003)1364 299 y Fb(M)1415 286 y Fr(exist)f(and)h(write)652 385 y Fg(\026)p Fr(\()p Fg(\015)721 392 y Fj(\003)747 385 y Fg(f)f Fm(\000)c Fg(f)5 b Fr(\))25 b(=)g Fg(A)994 392 y Fh(M)1044 385 y Fr(+)10 b Fg(B)1123 392 y Fh(M)1174 385 y Fr(+)g Fg(C)1252 392 y Fh(M)1625 385 y Fr(\(4.1\))224 484 y(with)327 583 y Fg(A)361 590 y Fh(M)426 583 y Fr(=)26 b Fg(\026)514 532 y Fc(h)535 583 y Fg(\015)559 590 y Fj(\003)583 596 y Fb(M)619 583 y Fr(\()p Fg(f)5 b Fr(\))10 b Fm(\000)g Fg(\015)764 561 y Fh(M)r(;\022)761 597 y Fj(\003)785 603 y Fb(M)828 583 y Fr(\()p Fg(f)5 b Fr(\))891 532 y Fc(i)927 583 y Fg(;)23 b(B)997 590 y Fh(M)1062 583 y Fr(=)i Fg(\027)1147 532 y Fc(h)1169 583 y Fr(\()p Fg(g)1209 590 y Fj(\003)1233 596 y Fb(M)1279 583 y Fm(\000)10 b Fg(g)1346 592 y Fj(\003)1370 598 y Fb(M)1404 592 y Fe(n)p Fj(\003)1448 583 y Fr(\))e Fg(f)1501 532 y Fc(i)1625 583 y Fr(\(4.2\))224 693 y(and)14 b Fg(C)344 700 y Fh(M)397 693 y Fr(is)h(the)f(righ)o(t-hand)h(side)g(in)f(\(3.4\).)k(W)l(e)c (shall)i(pro)o(v)o(e)d(that)g Fg(A)1444 700 y Fh(M)1498 693 y Fr(and)h Fg(B)1619 700 y Fh(M)1673 693 y Fr(go)224 749 y(to)h(zero.)295 806 y(Indeed,)j(lim)520 813 y Fh(M)t Fe(!1)638 806 y Fg(A)672 813 y Fh(M)726 806 y Fr(=)e(0)g(follo)o(ws)g (b)o(y)h(dominated)g(con)o(v)o(ergence,)g(b)q(ecause)g Fg(\015)224 862 y Fr(is)f(quasilo)q(cal)h(in)f(the)f(direction)i Fg(\022)f Fr(and)g Fm(j)p Fg(\015)967 846 y Fh(M)964 876 y Fj(\003)1005 862 y Fr(\()p Fg(f)5 b Fr(\))p Fm(j)12 b(\024)h(k)p Fg(f)5 b Fm(k)1214 869 y Fe(1)1251 862 y Fr(.)295 919 y(On)15 b(the)h(other)f(hand,)g(Csisz\023)-23 b(ar's)15 b(inequalit)o(y)h([4])410 1051 y Fg(H)448 1058 y Fj(\001)480 1051 y Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))10 b Fm(\000)g Fg(H)674 1058 y Fj(\001)703 1049 y Fa(0)716 1051 y Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))25 b Fm(\025)908 1020 y Fr(1)p 908 1040 23 2 v 908 1082 a(2)943 987 y Fc(\024)967 989 y(Z)993 1092 y Fj(\012)1020 999 y Fc(\014)1020 1026 y(\014)1020 1053 y(\014)1035 1051 y Fg(g)1057 1058 y Fj(\001)1088 1051 y Fr(\()p Fg(!)r Fr(\))10 b Fm(\000)g Fg(g)1231 1058 y Fj(\001)1260 1049 y Fa(0)1274 1051 y Fr(\()p Fg(!)r Fr(\))1340 999 y Fc(\014)1340 1026 y(\014)1340 1053 y(\014)1354 1051 y Fg(d\027)s Fr(\()p Fg(!)r Fr(\))1469 987 y Fc(\025)1492 998 y Fj(2)1520 1051 y Fg(:)224 1175 y Fr(v)m(alid)17 b(for)e(\001)442 1159 y Fe(0)466 1175 y Fm(\032)e Fr(\001)f Fm(2)h(S)s Fr(,)i(implies)i(that)546 1283 y Fm(j)p Fg(B)593 1290 y Fh(M)632 1283 y Fm(j)25 b(\024)731 1243 y(p)p 769 1243 V 40 x Fr(2)7 b Fm(k)p Fg(f)e Fm(k)872 1290 y Fe(1)916 1233 y Fc(h)938 1283 y Fg(H)976 1290 y Fj(\001)1007 1283 y Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))10 b Fm(\000)g Fg(H)1201 1292 y Fj(\001)p Fe(n)p Fj(\003)1275 1283 y Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))1376 1233 y Fc(i)224 1393 y Fr(for)15 b(an)o(y)f(\001)f Fm(\033)g Fr(\003)510 1400 y Fh(M)549 1393 y Fr(.)20 b(But)15 b(the)g(h)o(yp)q (othesis)h Fg(h)p Fr(\()p Fg(\026)p Fm(j)p Fg(\027)s Fr(\))d(=)g(0)h(implies)k(that)c(the)h(di\013erence)224 1450 y(in)i(en)o(tropies)f(in)h(the)f(righ)o(t-hand)g(side)h(tends)f (to)f(zero)h(as)f(\001)f Fm(")f Fi(L)p Fr(,)h(as)i(sho)o(wn)f(in)i([12) o(])224 1506 y(or)e([24)o(].)k(Hence)d Fg(B)r Fr(\()p Fg(M)5 b Fr(\))13 b Fm(!)697 1513 y Fh(M)750 1506 y Fr(0.)224 1627 y Fl(4.2)56 b(Pro)r(of)18 b(of)h(Corollary)f(3.5)224 1713 y Fr(It)f(is)g(enough)g(to)f(v)o(erify)h(the)f(righ)o(t-hand)h (side)h(of)e(\(3.4\))f(for)h(increasing)i(lo)q(cal)g(func-)224 1770 y(tions)e Fg(f)22 b Fr(since)17 b(linear)g(com)o(binations)g(of)f (these)g(are)g(uniformly)h(dense)g(in)g(the)f(set)g(of)224 1826 y(quasilo)q(cal)h(functions.)224 1946 y Fs(P)o(art)g(\(a\))47 b Fr(By)15 b(Theorem)g(3.3)f(w)o(e)h(only)h(ha)o(v)o(e)f(to)f(sho)o(w)h (that)492 2056 y Fg(C)525 2063 y Fh(M)589 2056 y Fr(=)26 b Fg(\026)677 2037 y Fj(+)707 2005 y Fc(h)728 2056 y Fg(g)750 2065 y Fj(\003)774 2071 y Fb(M)808 2065 y Fe(n)p Fj(\003)860 2005 y Fc(\020)887 2056 y Fg(\015)914 2034 y Fh(M)r(;)p Fj(+)911 2070 y(\003)988 2056 y Fr(\()p Fg(f)5 b Fr(\))k Fm(\000)i Fg(\015)1130 2063 y Fj(\003)1156 2056 y Fr(\()p Fg(f)5 b Fr(\))1219 2005 y Fc(\021i)1305 2056 y Fm(\000)-8 b(!)1287 2086 y Fh(M)t Fe(!1)1403 2056 y Fr(0)12 b Fg(;)174 b Fr(\(4.3\))224 2176 y(where)16 b Fg(g)378 2183 y Fh(D)422 2176 y Fr(=)d Fg(d\026)521 2157 y Fe(\000)521 2191 y Fh(D)553 2176 y Fg(=d\026)627 2157 y Fj(+)627 2191 y Fh(D)674 2176 y Fr(for)h Fg(D)g Fm(\032)f Fi(L)p Fr(.)18 b(W)l(e)d(\014rst)g(p)q(oin)o(t)h(out)f(that) 881 2275 y Fg(C)914 2282 y Fh(M)978 2275 y Fm(\025)26 b Fr(0)563 b(\(4.4\))224 2374 y(b)q(ecause)16 b Fg(\015)i Fr(is)d(monotonicit)o(y)h(preserving,)f(while)588 2481 y Fg(\026)615 2463 y Fj(+)645 2431 y Fc(\020)672 2481 y Fg(g)694 2491 y Fj(\003)718 2497 y Fb(M)752 2491 y Fe(n)p Fj(\003)804 2481 y Fg(\015)831 2459 y Fh(M)r(;)p Fj(+)828 2496 y(\003)905 2481 y Fr(\()p Fg(f)5 b Fr(\))968 2431 y Fc(\021)1019 2481 y Fr(=)26 b Fg(\026)1107 2463 y Fe(\000)1137 2431 y Fc(\020)1164 2481 y Fg(\015)1191 2459 y Fh(M)r(;)p Fj(+)1188 2496 y(\003)1265 2481 y Fr(\()p Fg(f)5 b Fr(\))1328 2431 y Fc(\021)1625 2481 y Fr(\(4.5\))949 2613 y(15)p eop %%Page: 16 16 16 15 bop 224 286 a Fr(b)q(ecause)16 b Fg(\015)419 264 y Fh(M)r(;)p Fj(+)416 301 y(\003)493 286 y Fg(f)k Fr(is)c Fm(F)614 296 y Fj(\003)638 302 y Fb(M)672 296 y Fe(n)p Fj(\003)716 286 y Fr(-measurable.)21 b(On)16 b(the)f(other)g(hand,)459 401 y Fg(\026)486 383 y Fj(+)516 351 y Fc(\020)543 401 y Fg(g)565 411 y Fj(\003)589 417 y Fb(M)623 411 y Fe(n)p Fj(\003)674 401 y Fg(\015)698 408 y Fj(\003)725 401 y Fr(\()p Fg(f)5 b Fr(\))788 351 y Fc(\021)839 401 y Fr(=)26 b Fg(\026)927 383 y Fj(+)957 351 y Fc(h)978 401 y Fg(g)1000 411 y Fj(\003)1024 417 y Fb(M)1058 411 y Fe(n)p Fj(\003)1110 401 y Fg(\026)1137 383 y Fj(+)1167 401 y Fr(\()p Fg(\015)1209 408 y Fj(\003)1235 401 y Fr(\()p Fg(f)5 b Fr(\))p Fm(jF)1344 411 y Fj(\003)1368 417 y Fb(M)1401 411 y Fe(n)p Fj(\003)1445 401 y Fr(\))1463 351 y Fc(i)1625 401 y Fr(\(4.6\))224 513 y(where)13 b Fg(\026)380 496 y Fj(+)410 513 y Fr(\()8 b Fm(\001)g(jF)503 520 y Fh(D)533 513 y Fr(\))k(are)g(the)g (conditional)i(exp)q(ectations)g(of)e(part)g(\(c\))g(of)g(Lemma)g (2.23.)224 569 y(By)j(the)h(last)f(inequalit)o(y)i(there,)291 675 y Fg(\026)318 657 y Fj(+)348 625 y Fc(\020)375 675 y Fg(g)397 685 y Fj(\003)421 691 y Fb(M)455 685 y Fe(n)p Fj(\003)507 675 y Fg(\015)531 682 y Fj(\003)557 675 y Fr(\()p Fg(f)5 b Fr(\))620 625 y Fc(\021)672 675 y Fm(\025)26 b Fg(\026)760 657 y Fj(+)789 625 y Fc(h)811 675 y Fg(g)833 685 y Fj(\003)857 691 y Fb(M)891 685 y Fe(n)p Fj(\003)943 675 y Fg(\026)970 657 y Fe(\000)999 675 y Fr(\()p Fg(\015)1041 682 y Fj(\003)1067 675 y Fr(\()p Fg(f)5 b Fr(\))p Fm(jF)1176 685 y Fj(\003)1200 691 y Fb(M)1233 685 y Fe(n)p Fj(\003)1277 675 y Fr(\))1295 625 y Fc(i)1342 675 y Fr(=)25 b Fg(\026)1429 657 y Fe(\000)1459 625 y Fc(\020)1486 675 y Fg(\015)1510 682 y Fj(\003)1536 675 y Fr(\()p Fg(f)5 b Fr(\))1599 625 y Fc(\021)1639 675 y Fg(:)1625 743 y Fr(\(4.7\))224 800 y(b)q(ecause)20 b(of)f(the)h Fm(F)568 809 y Fj(\003)592 815 y Fb(M)626 809 y Fe(n)p Fj(\003)670 800 y Fr(-measurabilit)o(y)g (of)f Fg(\026)1055 783 y Fe(\000)1085 800 y Fr(\()p Fg(\015)1127 807 y Fj(\003)1153 800 y Fr(\()p Fg(f)5 b Fr(\))p Fm(jF)1262 809 y Fj(\003)1286 815 y Fb(M)1319 809 y Fe(n)p Fj(\003)1363 800 y Fr(\)\()j Fm(\001)g Fr(\).)30 b(F)l(rom)18 b(\(4.4\),)224 856 y(\(4.5\))c(and)h(\(4.7\),)604 965 y(0)25 b Fm(\024)h Fg(C)746 972 y Fh(M)810 965 y Fm(\024)g Fg(\026)898 946 y Fe(\000)928 914 y Fc(\020)955 965 y Fm(j)o Fg(\015)994 943 y Fh(M)r(;)p Fj(+)991 979 y(\003)1068 965 y Fr(\()p Fg(f)5 b Fr(\))10 b Fm(\000)g Fg(\015)1210 972 y Fj(\003)1236 965 y Fr(\()p Fg(f)5 b Fr(\))p Fm(j)1311 914 y Fc(\021)224 1073 y Fr(and)14 b(\(4.3\))e(follo)o(ws)h(from)g(the)h(righ)o(t-con)o (tin)o(uit)o(y)f(of)g Fg(\015)j Fr(and)e(dominated)g(con)o(v)o (ergence.)224 1273 y Fs(P)o(art)j(\(b\))46 b Fr(By)16 b(monotonicit)o(y)554 1381 y(0)54 b Fm(\024)h Fg(\026)748 1363 y Fj(+)777 1331 y Fc(h)799 1381 y Fg(g)821 1391 y Fj(\003)845 1397 y Fb(M)879 1391 y Fe(n)p Fj(\003)931 1331 y Fc(\020)958 1381 y Fg(\015)985 1359 y Fh(M)r(;)p Fj(+)982 1396 y(\003)1059 1381 y Fr(\()p Fg(f)5 b Fr(\))k Fm(\000)i Fg(\015)1201 1388 y Fj(\003)1227 1381 y Fr(\()p Fg(f)5 b Fr(\))1290 1331 y Fc(\021)o(i)631 1480 y Fm(\024)55 b Fg(\026)748 1461 y Fj(+)777 1429 y Fc(h)799 1480 y Fg(g)821 1489 y Fj(\003)845 1495 y Fb(M)879 1489 y Fe(n)p Fj(\003)931 1429 y Fc(\020)958 1480 y Fg(\015)985 1458 y Fh(M)r(;)p Fj(+)982 1494 y(\003)1059 1480 y Fr(\()p Fg(f)5 b Fr(\))k Fm(\000)i Fg(\015)1204 1458 y Fh(M)r(;)p Fe(\000)1201 1494 y Fj(\003)1277 1480 y Fr(\()p Fg(f)5 b Fr(\))1340 1429 y Fc(\021i)631 1578 y Fr(=)55 b Fg(\026)748 1528 y Fc(\020)775 1578 y Fg(\015)802 1556 y Fh(M)r(;)p Fj(+)799 1593 y(\003)876 1578 y Fr(\()p Fg(f)5 b Fr(\))10 b Fm(\000)g Fg(\015)1021 1556 y Fh(M)r(;)p Fe(\000)1018 1593 y Fj(\003)1095 1578 y Fr(\()p Fg(f)5 b Fr(\))1158 1528 y Fc(\021)224 1692 y Fr(where)17 b Fg(g)379 1699 y Fh(D)427 1692 y Fr(indicates)h(the)f(Radon-Nik)o(o)q(dym)h(densit)o (y)f(of)f Fg(\026)1278 1699 y Fh(D)1327 1692 y Fr(with)h(resp)q(ect)h (to)e Fg(\026)1674 1673 y Fj(+)1674 1706 y Fh(D)1706 1692 y Fr(,)224 1755 y(and)k(the)f(last)g(equalit)o(y)h(follo)o(ws)g (from)e(the)i Fm(F)1047 1765 y Fj(\003)1071 1771 y Fb(M)1104 1765 y Fe(n)p Fj(\003)1149 1755 y Fr(-measurabilit)o(y)g(of)f Fg(\015)1534 1733 y Fh(M)r(;)p Fj(+)1531 1770 y(\003)1608 1755 y Fr(\()p Fg(f)5 b Fr(\))12 b Fm(\000)224 1822 y Fg(\015)251 1800 y Fh(M)r(;)p Fe(\000)248 1836 y Fj(\003)325 1822 y Fr(\()p Fg(f)5 b Fr(\).)32 b(The)20 b(last)f(line)i(tends)f(to)f (zero)g(with)h Fg(M)k Fr(b)o(y)c(dominated)g(con)o(v)o(ergence,)224 1878 y(b)q(ecause)c Fg(\026)p Fr(\(\012)470 1885 y Fe(\006)500 1878 y Fr(\))c(=)h(1.)20 b(This)15 b(concludes)i(the)e(pro)q(of)g(b)q (ecause)h(of)f(Theorem)g(3.3.)224 1999 y Fl(4.3)56 b(Pro)r(of)18 b(of)h(Prop)r(osition)f(3.15)224 2085 y Fr(Let)e(us)f(\014x)g(a)g(lo)q (cal)i(function)f Fg(f)5 b Fr(,)14 b(a)h(\014nite)i(set)d(\003)h(and)h (some)e Fg(\017)g(>)f Fr(0.)19 b(W)l(e)c(ha)o(v)o(e)420 2206 y Fg(\027)452 2142 y Fc(\024)481 2171 y Fg(d\026)532 2180 y Fj(\003)556 2186 y Fb(M)591 2180 y Fe(n)p Fj(\003)p 481 2195 154 2 v 483 2237 a Fg(d\027)529 2246 y Fj(\003)553 2252 y Fb(M)588 2246 y Fe(n)p Fj(\003)647 2155 y Fc(\020)675 2206 y Fg(\015)702 2184 y Fh(M)r(;\022)699 2220 y Fj(\003)765 2206 y Fr(\()p Fg(f)5 b Fr(\))10 b Fm(\000)g Fg(\015)907 2213 y Fj(\003)934 2206 y Fr(\()p Fg(f)5 b Fr(\))997 2155 y Fc(\021)1023 2142 y(\025)1072 2206 y Fm(\024)26 b Fg(\017)11 b Fr(+)f(2)p Fm(k)p Fg(f)5 b Fm(k)1303 2213 y Fe(1)1350 2206 y Fc(e)-28 b Fg(\026)1374 2213 y Fh(M)1414 2206 y Fr(\()p Fg(A)1466 2187 y Fh(M)1466 2217 y(\017)1505 2206 y Fr(\))102 b(\(4.8\))224 2339 y(where)16 b Fg(A)390 2322 y Fh(M)390 2350 y(\017)444 2339 y Fr(denotes)g(the)f(set)g (\(3.12\))e(and)j(w)o(e)f(abbreviated)649 2462 y Fc(e)-28 b Fg(\026)673 2469 y Fh(M)713 2462 y Fr(\()p Fg(A)765 2444 y Fh(M)765 2474 y(\017)804 2462 y Fr(\))25 b(=)h Fg(\027)941 2398 y Fc(\022)979 2427 y Fg(d\026)1030 2436 y Fj(\003)1054 2442 y Fb(M)1089 2436 y Fe(n)p Fj(\003)p 979 2452 V 981 2493 a Fg(d\027)1027 2503 y Fj(\003)1051 2509 y Fb(M)1086 2503 y Fe(n)p Fj(\003)1145 2462 y Fr(1)1168 2473 y Fh(A)1194 2464 y Fb(M)1194 2481 y(\017)1230 2398 y Fc(\023)1284 2462 y Fg(:)328 b Fr(\(4.9\))949 2613 y(16)p eop %%Page: 17 17 17 16 bop 224 286 a Fr(By)15 b(\(3.14\))f(there)h(exists)h Fg(c)c(>)h Fr(0)i(suc)o(h)g(that)g(for)f Fg(M)21 b Fr(large)15 b(enough,)785 382 y Fg(\027)s Fr(\()p Fg(A)862 363 y Fh(M)862 393 y(\017)901 382 y Fr(\))25 b Fm(\024)h Fg(e)1026 363 y Fe(\000)p Fh(c)6 b(\013)1097 369 y Fb(M)1146 382 y Fg(;)443 b Fr(\(4.10\))224 477 y(hence,)16 b(for)f(0)d Fg(<)26 b(\016)14 b(<)26 b(c)p Fr(,)14 b(and)i(w)o(e)e(can)i(write)f (the)g(follo)o(wing)h(inequalities:)328 590 y Fc(e)-28 b Fg(\026)352 597 y Fh(M)392 590 y Fr(\()p Fg(A)444 572 y Fh(M)444 602 y(\017)483 590 y Fr(\))54 b Fm(\024)687 560 y Fr(1)p 650 580 99 2 v 650 621 a Fg(\013)679 628 y Fh(M)726 621 y Fg(\016)768 590 y Fr(log)834 528 y Fc(Z)887 590 y Fr(exp)q(\()p Fg(\016)9 b(\013)1033 597 y Fh(M)1080 590 y Fr(1)1103 601 y Fh(A)1129 592 y Fb(M)1129 609 y(\017)1165 590 y Fr(\))f Fg(d\027)25 b Fr(+)1363 560 y(1)p 1325 580 V 1325 621 a Fg(\013)1354 628 y Fh(M)1401 621 y Fg(\016)1436 590 y(H)t Fr(\()t(~)-27 b Fg(\026)1523 597 y Fh(M)1562 590 y Fm(j)p Fg(\027)s Fr(\))555 706 y Fm(\024)687 675 y Fr(1)p 650 696 V 650 737 a Fg(\013)679 744 y Fh(M)726 737 y Fg(\016)768 706 y Fr(log)834 656 y Fc(\020)861 706 y Fr(1)10 b(+)h Fg(e)961 687 y Fh(\013)984 693 y Fb(M)1023 687 y Fh(\016)1050 706 y Fg(\027)s Fr(\()p Fg(A)1127 687 y Fh(M)1127 717 y(\017)1166 706 y Fr(\))1184 656 y Fc(\021)1234 706 y Fr(+)1335 675 y(1)p 1297 696 V 1297 737 a Fg(\013)1326 744 y Fh(M)1373 737 y Fg(\016)1407 706 y(H)t Fr(\()t(~)-27 b Fg(\026)1494 713 y Fh(M)1534 706 y Fm(j)p Fg(\027)s Fr(\))555 821 y Fm(\024)687 790 y Fr(1)p 650 811 V 650 852 a Fg(\013)679 859 y Fh(M)726 852 y Fg(\016)760 821 y(e)781 802 y Fh(\013)804 808 y Fb(M)838 802 y Fj(\()p Fh(\016)q Fe(\000)p Fh(c)p Fj(\))950 821 y Fr(+)1050 790 y(1)p 1013 811 V 1013 852 a Fg(\013)1042 859 y Fh(M)1089 852 y Fg(\016)1123 821 y(H)t Fr(\()t(~)g Fg(\026)1210 828 y Fh(M)1249 821 y Fm(j)p Fg(\027)s Fr(\))13 b Fg(:)271 b Fr(\(4.11\))224 936 y(By)19 b(\(3.16\),)e(the)i(last)f (line)j(tends)e(to)f(zero)g(as)g Fg(M)24 b Fm(!)19 b(1)p Fr(.)30 b(By)19 b(\(4.8\),)f(and)g(the)h(fact)224 993 y(that)d Fg(\017)g(>)f Fr(0)h(is)i(arbitrary)l(,)e(w)o(e)g(conclude)j (that)d(condition)h(\(3.4\))f(of)g(Theorem)g(3.3)g(is)224 1049 y(satis\014ed,)g(whic)o(h)g(implies)h(that)d Fg(\026)f Fm(2)g(G)s Fr(\()p Fg(\015)s Fr(\).)224 1191 y Fn(Ac)n(kno)n(wledgemen) n(ts)224 1293 y Fr(W)l(e)22 b(thank)g(Charles-Edouard)h(P\014ster)f (for)f(giving)i(us)f(an)g(early)h(v)o(ersion)f(of)g([24)o(])224 1349 y(and)g(Aernout)h(v)m(an)f(En)o(ter)g(for)f(commen)o(ts)h(and)g (suggestions.)41 b(W)l(e)22 b(are)g(grateful)224 1406 y(to)e(Eurandom)g(\(RF)h(and)f(ALN\))h(and)g(The)g(Univ)o(ersit)o(y)g (of)f(Groningen)h(\(RF\))f(for)224 1462 y(hospitalit)o(y)c(during)g (the)g(completion)g(of)f(this)g(w)o(ork.)224 1604 y Fn(References)247 1706 y Fr([1])22 b(M.)17 b(Aizenman.)29 b(T)l(ranslation)19 b(in)o(v)m(ariance)g(and)g(instabilit)o(y)g(of)f(phase)g(co)q(exis-)318 1762 y(tence)e(in)g(the)g(t)o(w)o(o)e(dimensional)k(Ising)e(system.)21 b Ff(Commun.)c(Math.)f(Phys.)g Fs(73)p Fr(:)318 1818 y(83{94,)d(1980.)247 1910 y([2])22 b(J.)28 b(Bricmon)o(t,)k(A.)d (Kupiainen,)34 b(R.)29 b(Lefev)o(ere.)61 b(Renormalization)31 b(group)318 1966 y(pathologies)f(and)f(the)h(de\014nition)h(of)e(Gibbs) h(states.)f Ff(Comm.Math.Phys.)318 2023 y Fs(194)p Fr(:359{388,)12 b(1998.)247 2114 y([3])22 b(R.L.)j(Dobrushin)i(and)f(S.B.)f(Shlosman.) 52 b(Gibbsian)27 b(description)g(of)f('non)318 2171 y(Gibbsian')20 b(\014eld.)36 b Ff(R)o(ussian)20 b(Math)h(Surveys)p Fr(,)f Fs(52)p Fr(:285-297,)f(1997.)g(Also)h('Non)318 2227 y(Gibbsian')12 b(states)e(and)i(their)g(description.)g Ff(Comm.Math.Phys.)g Fs(200)p Fr(:125{179,)318 2284 y(1999.)247 2375 y([4])22 b(I.)12 b(Csisz\023)-23 b(ar.)15 b(Information-t)o(yp)q(e)e(measures)f (of)g(di\013erence)i(of)e(probabilit)o(y)h(distri-)318 2432 y(butions)18 b(and)g(indirect)h(observ)m(ations.)27 b Ff(Studia)19 b(Sci.)e(Math.)i(Hungar.)p Fr(,)f Fs(2)p Fr(:299{)318 2488 y(318,)c(1967.)949 2613 y(17)p eop %%Page: 18 18 18 17 bop 247 286 a Fr([5])22 b(A.)16 b(C.)g(D.)g(v)m(an)h(En)o(ter.)23 b(On)17 b(the)g(p)q(ossible)h(failure)g(of)e(the)h(Gibbs)g(prop)q(ert)o (y)f(for)318 343 y(measures)d(on)g(lattice)h(systems.)j Ff(Markov)e(Pr)n(o)n(c.)f(R)n(el.)g(Fields)t Fr(,)e Fs(2)p Fr(:209{24,)f(1996.)247 437 y([6])22 b(A.)11 b(C.)h(D.)f(v)m(an)h(En)o (ter.)i(The)f(renormalization-group)f(p)q(eculiarities)j(of)d (Gri\016ths)318 493 y(and)17 b(Pearce:)25 b(What)17 b(ha)o(v)o(e)h(w)o (e)f(learned?,)j(in)e Ff(Mathematic)n(al)h(R)n(esults)e(in)h(Sta-)318 550 y(tistic)n(al)c(Me)n(chanics)e Fr(\(Pro)q(ceedings)j(of)e(the)h (collo)q(quium)i(with)e(the)f(same)h(name,)318 606 y(Marseille-Lumin)o (y)l(,)19 b(July)e(1998\),)e(S.)h(Miracle-Sol)o(\023)-21 b(e,)17 b(J.)f(Ruiz)i(and)e(V.)g(Zagreb-)318 663 y(no)o(v,)e(W)l(orld)i (Scien)o(ti\014c,)g(Singap)q(ore,)g(1999.)247 756 y([7])22 b(A.C.D.)d(v)m(an)j(En)o(ter,)f(R.)g(F)l(ern\023)-23 b(andez,)23 b(and)e(A.D.)g(Sok)m(al.)38 b(Regularit)o(y)21 b(prop-)318 813 y(erties)g(and)g(pathologies)h(of)f(p)q(osition-space)h (renormalization-group)g(trans-)318 869 y(formations:)j(Scop)q(e)19 b(and)g(limitations)h(of)e(Gibbsian)h(theory)l(.)30 b Ff(J.)19 b(Stat.)g(Phys.)318 926 y Fs(72)p Fr(:879{1167,)12 b(1993.)247 1020 y([8])22 b(R.)16 b(F)l(ern\023)-23 b(andez.)26 b(Random)17 b(\014elds)h(in)g(lattices.)f(The)g(Gibbsianness)h(issue.) 26 b Ff(R)n(e-)318 1076 y(senhas)15 b(do)i(IME-USP)5 b Fr(,)13 b Fs(3)p Fr(:)20 b(391{421,)13 b(1998.)247 1170 y([9])22 b(R.)15 b(F)l(ern\023)-23 b(andez.)20 b(Measures)15 b(for)f(lattice)i(systems.)j Ff(Physic)n(a)d(A)p Fr(,)f Fs(263)p Fr(:)20 b(117{130,)318 1226 y(1999.)224 1320 y([10])i(C.)11 b(M.)h(F)l(ortuin,)h(J.)f(Ginibre)h(and)g(P)l(.)f(W.)g (Kasteleyn.)k(Correlation)c(inequalities)318 1377 y(on)21 b(some)f(partially)i(ordered)g(sets.)37 b Ff(Commun.)21 b(Math.)h(Phys.)f Fs(22)p Fr(:)32 b(89{103,)318 1433 y(1971.)224 1527 y([11])22 b(R.)33 b(F)l(ern\023)-23 b(andez)35 b(and)f(C.-E.)e(P\014ster.)41 b(Global)35 b(sp)q(eci\014cations)g(and)f(non-)318 1583 y(quasilo)q(calit)o(y)23 b(of)f(pro)s(jections)g(of)f(Gibbs)i(measures.)18 b Ff(A)o(nn.Pr)n(ob)n (ab.)p Fs(25)p Fr(:1284{)318 1640 y(1315,)13 b(1997.)224 1733 y([12])22 b(H.O.)i(Georgii.)48 b(Gibbs)26 b(Measures)e(and)h (Phase)g(T)l(ransitions.)48 b(W)l(alter)25 b(de)318 1790 y(Gruyter)11 b(\(de)h(Gruyter)f(Studies)i(in)g(Mathematics,)f(V)l(ol.)g (9\),)f(Berlin-New)j(Y)l(ork,)318 1846 y(1988.)224 1940 y([13])22 b(R.B.)16 b(Gri\016ths)h(and)g(P)l(.A.)f(P)o(earce.)24 b(Mathematicals)17 b(prop)q(erties)g(of)f(p)q(osition-)318 1997 y(space)24 b(renormalization-group)h(transformations.)d Ff(J.Stat.Phys.)i Fs(20)p Fr(:)38 b(499{)318 2053 y(545,)14 b(1979.)224 2147 y([14])22 b(Y.)10 b(Higuc)o(hi.)j(On)e(the)g(absence)g (of)f(non-translationally)i(in)o(v)m(arian)o(t)f(Gibbs)g(states)318 2203 y(for)21 b(the)i(t)o(w)o(o-dimensional)g(Ising)g(system.)41 b(In)23 b Ff(R)n(andom)g(Fields:)32 b(R)o(igor)n(ous)318 2260 y(R)n(esults)11 b(in)i(Statistic)n(al)f(Me)n(chanics)f(and)i (Quantum)g(Field)g(The)n(ory)e Fr(\(Esztergom)318 2316 y(1979\),)21 b(J.)h(F)l(ritz,)h(J.)f(L.)f(Leb)q(o)o(witz)i(and)f(D.)f (Szasz)h(\(eds.\),)g(North)g(Holland,)318 2373 y(Amsterdam,)14 b(1981.)949 2613 y(18)p eop %%Page: 19 19 19 18 bop 224 286 a Fr([15])22 b(R.B.)d(Israel.)32 b(Banac)o(h)20 b(algebras)f(and)g(Kadano\013)g(transformations,)g(in)h Ff(R)n(an-)318 343 y(dom)e(Fields)e(\(Eszter)n(gom,)h(1979\))h Fr(J.F)l(ritz,)f(J.L.Leb)q(o)o(witz,)g(and)g(D.Sz\023)-23 b(asz,)16 b(eds)318 399 y(\(North-Holland,)f(Amsterdam,)f(1981\),)g(V)l (ol)h(I)q(I,)h(pp.593{608,)d(1981.)224 493 y([16])22 b(O.K.)13 b(Kozlo)o(v.)k(Gibbs)c(description)i(of)e(a)g(system)g(of)f (random)h(v)m(ariables.)h Ff(Pr)n(obl.)318 550 y(Inform.)h(T)m(r)n (ansmission.)p Fs(10)p Fr(:258{)o(265,)c(1974.)224 643 y([17])22 b(A.)14 b(Le)h(Ny)l(,)f(Mesures)h(de)g(Gibbs)g(sur)f(un)h(r)o (\023)-21 b(eseau)14 b(et)g(non-gibbsiannit)o(\023)-21 b(e:)21 b(restau-)318 700 y(ration)h(du)h(formalisme)g(gibbsien.)h(Th)o (\022)-21 b(ese)22 b(de)h(do)q(ctorat)e(de)i(l'univ)o(ersit)o(\023)-21 b(e)23 b(de)318 756 y(Rennes)16 b(1,)f(juillet)i(2000:)h(h)o (ttp://www.maths.univ-rennes1.fr/)t(~)-14 b(alen)o(y)l(.)224 850 y([18])22 b(A.)17 b(Le)h(Ny)l(,)h(F)l(ractal)e(F)l(ailure)i(of)e (Quasilo)q(calit)o(y)j(for)d(a)g(Ma)s(jorit)o(y)f(Rule)k(T)l(rans-)318 907 y(formation)14 b(on)h(a)g(T)l(ree,)g Ff(L)n(ett.Math.Phys.)p Fr(,)p Fs(54)p Fr(:11{24,)d(2000.)224 1000 y([19])22 b(R.)k(Lefev)o(ere.)i(V)l(ariational)f(principle)i(for)d(some)g (renormalized)i(measures.)318 1057 y Ff(J.Stat.Phys.)p Fr(,)p Fs(95)p Fr(:785{803,)11 b(1999.)224 1151 y([20])22 b(R.)29 b(Lefev)o(ere.)h(Almost)g(and)g(w)o(eak)f(Gibbsianness:)51 b(A)30 b(long-range)g(pair-)318 1207 y(in)o(teraction)15 b(example.)h Ff(J.Stat.Phys.)p Fr(,)p Fs(96)p Fr(:109{113,)c(1999.)224 1301 y([21])22 b(C.)9 b(Maes,)i(F.)e(Redig)j(and)e(A.)g(V)l(an)g (Mo\013aert.)g(Almost)h(Gibbsian)g(v)o(ersus)f(W)l(eakly)318 1357 y(Gibbsian.)21 b Ff(Sto)n(ch.Pr)n(o)n(c.Appl.)14 b Fs(79)p Fr(:1{15,)f(1999.)224 1451 y([22])22 b(C.)17 b(Maes,)h(F.)f(Redig)j(and)e(A.)g(V)l(an)g(Mo\013aert.)27 b(The)18 b(restriction)h(of)e(the)i(Ising)318 1508 y(mo)q(del)d(to)e(a) h(La)o(y)o(er.)20 b Ff(J.Stat.Phys)14 b Fs(96)p Fr(:69{107,)f(1999.)224 1601 y([23])22 b(C.)f(Maes,)h(F.)f(Redig,)i(S.)f(Shlosman)g(and)g(A.)f (v)m(an)h(Mo\013aert.)37 b(P)o(ercolation,)318 1658 y(path)14 b(large)i(deviations)g(and)f(w)o(eak)f(Gibbsianit)o(y)l(.)21 b Ff(Comm.)16 b(Math.)h(Phys.)e Fs(209)p Fr(:)318 1714 y(517{45,)e(2000.)224 1808 y([24])22 b(C.-E.)13 b(P\014ster.)5 b(Thermo)q(dynamical)15 b(asp)q(ects)g(of)g(classical)h(lattice)f (systems.)k(T)l(o)318 1865 y(app)q(ear)i(in)h(the)f(pro)q(ceedings)i (of)e(the)g(4th)g(brazilian)i(sc)o(ho)q(ol)f(of)e(probabilit)o(y)l(,) 318 1921 y(Mam)o(bucaba,)14 b(RJ,)h(2000.)224 2015 y([25])22 b(R.)11 b(H.)f(Sc)o(honmann.)k(Pro)s(jections)c(of)h(Gibbs)g(measures)g (ma)o(y)g(b)q(e)g(non-Gibbsian.)318 2071 y Ff(Commun.)16 b(Math.)g(Phys.)p Fr(,)f Fs(124)p Fr(:1{7,)f(1989.)224 2165 y([26])22 b(W.G.)16 b(Sulliv)m(an.)31 b(P)o(oten)o(tials)18 b(for)f(almost)g(Mark)o(o)o(vian)g(random)h(\014elds.)g Ff(Com-)318 2222 y(mun.)e(Math.)g(Phys.)f Fs(33)p Fr(:61{74,)f(1973.) 949 2613 y(19)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0107110806230--