Content-Type: multipart/mixed; boundary="-------------9905110844830" This is a multi-part message in MIME format. ---------------9905110844830 Content-Type: text/plain; name="99-162.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-162.keywords" massless Gaussian, random walks, Green functions, large deviations, Gibbs' variational principle ---------------9905110844830 Content-Type: application/postscript; name="variald.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="variald.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: variald.dvi %%Pages: 35 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips variald %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.05.11:1426 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IE 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IE N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IE 256 array N 2 string 0 1 255{IE S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Ft(3.)47 b(F)-8 b(or)33 b(this)118 4973 y(mo)s(del)26 b(and)h(generally)g(when)f Fr(p)h Ft(is)f(the)i(transition)e(of)h(a)h(\014nite)e(range)i(transien)m(t)f (random)f(w)m(alk)h(large)118 5080 y(deviations)35 b(estimates)i(w)m (ere)f(obtained)g(in)f([1].)58 b(In)36 b(con)m(trast)h(to)g([1)q(],)g (w)m(e)g(consider)e(here)h(the)g(more)118 5188 y(delicate)30 b(case)i(of)e(non-lo)s(cal)g(in)m(teraction)g(as)h(w)m(ell.)118 5372 y(If)e Fr(\021)g Fl(6\021)c Ft(1,)30 b(the)f(mo)s(del)g(is)f(not)i (ferromagnetic)g(and)e(monotonicit)m(y)i(to)s(ols)f(suc)m(h)g(as)h(FK)m (G)g(inequalities)118 5480 y(are)36 b(no)f(longer)g(a)m(v)-5 b(ailable.)55 b(Moreo)m(v)m(er,)39 b(there)c(is)g(no)g(more)g(global)g (shift)f(symmetry)h(to)h(b)s(e)e(brok)m(en)p eop %%Page: 3 3 3 2 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)287 b(3)118 370 y Ft(but)29 b(still)e(phase)i(transition)e (can)j(o)s(ccur,)f(if)g(the)g(F)-8 b(ourier)29 b(transform)2550 347 y(^)2528 370 y Fr(J)39 b Ft(of)29 b Fr(J)38 b Ft(has)29 b(zeros.)41 b(In)29 b(order)g(to)118 478 y(obtain)23 b(a)h(massless)f(regime)g(w)m(e)h(require)e(the)i(constan)m(ts)g Fr(\021)j Ft(to)d(ha)m(v)m(e)h(a)f(suitable)e(oscillating)g(structure.) 118 690 y(W)-8 b(e)41 b(no)m(w)f(turn)f(to)i(a)f(brief)e(description)g (of)i(the)g(main)f(results)g(of)h(this)f(pap)s(er.)68 b(W)-8 b(e)41 b(refer)f(to)g(the)118 798 y(relev)-5 b(an)m(t)40 b(sections)g(for)g(precise)f(statemen)m(ts)i(and)e(pro)s(ofs.)69 b(In)39 b(section)h(2,)i(after)f(in)m(tro)s(ducing)c(the)118 906 y(mo)s(dels)20 b(and)h(reviewing)g(some)h(kno)m(wn)f(material,)i(w) m(e)f(pro)m(v)m(e)g(a)g(new)f(result)g(concerning)g(the)h(structure)118 1014 y(of)k(translation)f(in)m(v)-5 b(arian)m(t)24 b(Gibbs)h(measures.) 38 b(It)26 b(states)h(that)f(for)g(ferromagnetic)g(mo)s(dels)e (generated)118 1121 y(b)m(y)35 b(an)f(irreducible)e(random)i(w)m(alk,)h (translation)f(in)m(v)-5 b(arian)m(t)34 b(Gibbs)f(measures)i(cannot)g (b)s(e)f(realized)118 1229 y(as)26 b(mixtures)e(of)i(non-translation)f (in)m(v)-5 b(arian)m(t)25 b(extremal)g(Gibbs)f(measures.)39 b(Or,)26 b(in)f(other)h(w)m(ords,)g(the)118 1337 y(set)31 b(of)f(translation)g(in)m(v)-5 b(arian)m(t)29 b(Gibbs)g(measures)h(is)f (a)i(face)g(of)f(the)h(con)m(v)m(ex)g(set)g(of)g(Gibbs)d(measures.)118 1445 y(See)j(remark)f(3)h(for)f(further)f(discussion)f(of)i(this)f (topic.)118 1657 y(In)d(section)h(3)g(w)m(e)g(in)m(tro)s(duce)f(thermo) s(dynamics.)38 b(W)-8 b(e)28 b(pro)m(v)m(e)g(existence)f(of)g(relativ)m (e)g(en)m(trop)m(y)g(densit)m(y)118 1765 y(and)33 b(establish)f(the)i (corresp)s(onding)d(v)-5 b(ariational)32 b(principle.)47 b(In)33 b(particular,)g(w)m(e)h(sho)m(w)f(that)h(trans-)118 1873 y(lation)40 b(in)m(v)-5 b(arian)m(t)39 b(temp)s(ered)h(Gibbs)f (states)i(are)g(precisely)e(the)h(minima)f(\(zeros\))i(of)g(the)f (relativ)m(e)118 1981 y(en)m(trop)m(y)30 b(densit)m(y)-8 b(.)40 b(In)29 b(this)f(w)m(a)m(y)i(w)m(e)g(extend)g(previous)e (results)g(of)h(K)s(\177)-48 b(unsc)m(h)29 b([22)q(,)g(23)q(].)41 b(T)-8 b(o)30 b(pro)m(v)m(e)g(that)118 2089 y(an)m(y)k(minimizer)d(m)m (ust)j(b)s(e)f(Gibbsian)e(w)m(e)j(mak)m(e)h(use)f(of)f(a)i(new)e(appro) m(ximation)g(whic)m(h)f(enables)h(us)118 2197 y(to)g(go)g(b)s(ey)m(ond) f(the)g(standard)g(quasilo)s(calit)m(y)e(assumption)h(\(see)i([30)q(,)g (20)q(]\).)46 b(The)32 b(main)f(to)s(ol)i(in)d(this)118 2305 y(section)i(is)f(an)g(expansion)g(in)f(terms)i(of)g(random)f(w)m (alks)g(for)g(the)h(Green)g(function)e Fr(G)i Ft(asso)s(ciated)h(to)118 2413 y Fr(J)9 b Ft(,)31 b(closely)f(related)g(to)h(the)g(expansions)e (in)m(tro)s(duced)g(in)g([4].)118 2625 y(Our)h(results)h(concerning)g (large)g(deviations)g(are)h(discussed)d(in)i(section)g(4.)45 b(The)31 b Fi(empiric)-5 b(al)35 b(\014eld)d Ft(on)118 2733 y(a)f(cubic)e(b)s(o)m(x)h(\003)668 2747 y Fm(N)766 2733 y Ft(of)h(side)e Fr(N)40 b Ft(is)30 b(the)g(random)g(probabilit)m (y)e(measure)i(de\014ned)f(b)m(y)1170 2908 y(~)1150 2931 y Fr(R)1219 2945 y Fm(N)1287 2931 y Ft(\()p Fr(\036)p Ft(\))51 b(=)1632 2870 y(1)p 1593 2910 124 4 v 1593 2994 a Fr(N)1676 2967 y Fm(d)1788 2845 y Fk(X)1757 3042 y Fm(x)p Fj(2)p Fo(\003)1893 3053 y Fu(N)1981 2931 y Fr(\016)2021 2946 y Fm(\022)2054 2954 y Fu(x)2105 2946 y Fm(\036)2166 2931 y Fr(;)167 b(\036)25 b Fl(2)g Ft(\012)15 b Fr(:)846 b Ft(\(1.4\))118 3214 y(where)42 b Fr(\016)433 3229 y Fm(\030)513 3214 y Ft(stands)g(for)f(the)i(Dirac)f(measure)g(at)h Fr(\030)49 b Fl(2)44 b Ft(\012)h Fl(\021)f Fn(R)2355 3181 y Fh(Z)2411 3157 y Fu(d)2445 3214 y Ft(,)i(and)41 b Fr(\022)2747 3228 y Fm(x)2791 3214 y Fr(\036)h Ft(denotes)g(the)h (shifted)118 3322 y(con\014guration)29 b Fr(\036)722 3336 y Fj(\001)p Fo(+)p Fm(x)840 3322 y Ft(.)40 b(Let)30 b Fr(P)42 b Ft(denote)30 b(the)f(cen)m(tered)h(Gaussian)f(\014eld)e (with)h(co)m(v)-5 b(ariance)31 b(giv)m(en)e(b)m(y)g Fr(G)p Ft(.)118 3429 y(Since)g Fr(P)44 b Ft(is)29 b(ergo)s(dic)h(with)f(resp)s (ect)i(to)g(lattice)f(shifts,)1216 3568 y(~)1196 3591 y Fr(R)1265 3605 y Fm(N)1378 3591 y Fl(\))45 b Fr(P)28 b(;)91 b(P)34 b Fl(\000)20 b Ft(a)p Fr(:)p Ft(s)p Fr(:)66 b Ft(as)51 b Fr(N)35 b Fl(!)25 b(1)15 b Fr(;)118 3751 y Ft(in)24 b(the)i(sense)g(of)f(w)m(eak)i(con)m(v)m(ergence.)41 b(In)25 b(particular,)g(for)h(an)m(y)f(op)s(en)g(set)h(\000)g(of)f (probabilit)m(y)f(measures)118 3861 y(on)41 b(\012)g(suc)m(h)h(that)g Fr(P)66 b(=)-55 b Fl(2)44 b Ft(\000,)g(w)m(e)e(ha)m(v)m(e)g Fr(P)13 b Ft(\()1623 3838 y(~)1603 3861 y Fr(R)1672 3875 y Fm(N)1784 3861 y Fl(2)43 b Ft(\000\))h Fl(!)f Ft(0,)i(as)d Fr(N)54 b Fl(!)43 b(1)p Ft(.)74 b(W)-8 b(e)43 b(sho)m(w)e(that)h(these) 118 3971 y(probabilities)25 b(are)k(exp)s(onen)m(tially)d(deca)m(ying)j (in)e(the)i(v)m(olume)f Fr(N)2370 3938 y Fm(d)2410 3971 y Ft(,)h(with)e(rate)i(giv)m(en)f(b)m(y)h(the)f(relativ)m(e)118 4079 y(en)m(trop)m(y)k(densit)m(y)-8 b(.)44 b(Since)30 b(w)m(e)i(are)g(in)m(terested)f(in)f(the)i(massless)f(regime,)g (classical)g(large)g(deviations)118 4187 y(results)i(for)h(stationary)h (Gaussian)f(pro)s(cesses)g(based)g(on)g(the)h(con)m(tin)m(uit)m(y)f(of) h(the)f(sp)s(ectral)g(densit)m(y)118 4295 y(\(cf.)67 b(e.g.)g([14)q(,)39 b(3)q(]\))g(cannot)h(b)s(e)e(used.)66 b(F)-8 b(ollo)m(wing)38 b(an)h(idea)f(in)m(tro)s(duced)g(in)f([1)q(],)k (the)e(exp)s(onen)m(tial)118 4403 y(b)s(ounds)26 b(are)j(\014rst)e (established)g(for)g(the)i(h)m(yp)s(ercon)m(tractiv)m(e)g(regime)f(and) g(then)f(lifted)g(to)i(the)f(general)118 4511 y(case.)41 b(In)26 b(con)m(trast)j(to)f(the)f(h)m(yp)s(ercon)m(tractiv)m(e)h (regime,)g(in)e(the)h(phase)g(transition)f(regime)h(the)g(upp)s(er)118 4619 y(estimates)38 b(m)m(ust)f(b)s(e)g(restricted)g(to)i(certain)e (compact)i(sets)e(of)h(w)m(ell)e(b)s(eha)m(v)m(ed)i(measures.)62 b(In)36 b(this)118 4727 y(sense)30 b(w)m(e)h(sp)s(eak)f(of)h(a)g Fi(we)-5 b(ak)31 b Ft(large)f(deviation)g(principle.)118 4939 y(In)e(view)f(of)i(the)g(v)-5 b(ariational)27 b(principle)e(one)k (exp)s(ects)f Fi(critic)-5 b(al)30 b Ft(large)e(deviations)f(in)h(the)g (phase)g(tran-)118 5047 y(sition)d(regime.)40 b(In)26 b(particular,)g(if)g(\000)g(is)g(an)h(op)s(en)f(set)h(suc)m(h)f(that)i Fr(P)48 b(=)-55 b Fl(2)25 b Ft(\000)h(but)g(\000)h(con)m(tains)g(some)g (other)118 5157 y(translation)35 b(in)m(v)-5 b(arian)m(t)35 b(Gibbs)g(measure,)i(then)f Fr(P)13 b Ft(\()1966 5134 y(~)1946 5157 y Fr(R)2015 5171 y Fm(N)2117 5157 y Fl(2)34 b Ft(\000\))i(deca)m(ys)h(slo)m(w)m(er)f(than)g(exp)s(onen)m(tially)118 5264 y(in)i(the)h(v)m(olume.)67 b(This)38 b(is)g(indeed)g(what)h(has)g (b)s(een)f(describ)s(ed)g(in)f(detail)i(in)f([1])i(for)f(\014nite)f (range)118 5372 y(harmonic)29 b(crystals.)41 b(W)-8 b(e)32 b(discuss)d(critical)g(large)i(deviations)e(for)h(a)h(class)f(of)h (long)f(range)h(mo)s(dels)e(of)118 5480 y(the)i(form)f(\(1.3\))i(in)d (a)i(forthcoming)e(pap)s(er,)h(see)h(also)f(our)g(remarks)g(after)h (Theorem)f(4.1.)p eop %%Page: 4 4 4 3 bop 118 45 a Fo(4)1138 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(Finally)-8 b(,)24 b(the)h(app)s(endix)d(con)m (tains)i(an)g(expansion)f(for)h(Green)h(functions)d(in)h(terms)i(of)f (random)f(w)m(alks,)118 474 y(and)30 b(an)g(estimate)h(relev)-5 b(an)m(t)31 b(for)f(the)h(h)m(yp)s(ercon)m(tractiv)m(e)g(regime.)118 679 y Fy(Akno)m(wledgemen)m(ts)p Ft(.)46 b(The)31 b(authors)h(thank)g (Yv)-5 b(an)32 b(V)-8 b(elenik)32 b(and)f(Aernout)i(v)-5 b(an)32 b(En)m(ter)g(for)g(stim-)118 787 y(ulating)d(con)m(v)m (ersations.)1551 1000 y(2.)51 b Fs(The)35 b(Models)118 1162 y Ft(2.1.)54 b Fy(Preliminaries.)45 b Ft(Let)27 b Fn(Z)1217 1129 y Fm(d)1253 1162 y Ft(,)g(b)s(e)f(the)g Fr(d)p Ft(-dimensional)e(in)m(teger)i(lattice.)40 b(F)-8 b(or)27 b Fr(x)e Fl(2)g Fn(Z)3146 1129 y Fm(d)3182 1162 y Ft(,)i Fl(j)p Fr(x)p Fl(j)g Ft(denotes)118 1284 y(the)k(euclidean)e (norm.)41 b(Let)31 b(\012)25 b(=)g Fn(R)1358 1251 y Fh(Z)1414 1228 y Fu(d)1449 1284 y Ft(,)30 b(equipp)s(ed)e(with)i(the)g(pro)s (duct)g(top)s(ology)-8 b(,)31 b(and)f(denote)i(b)m(y)e Fl(F)118 1392 y Ft(the)24 b(corresp)s(onding)d(Borel)j(\014eld.)37 b(Probabilit)m(y)21 b(measures)i(on)g(\(\012)p Fr(;)15 b Fl(F)9 b Ft(\))25 b(are)f(denoted)f(b)m(y)h Fl(M)3266 1406 y Fo(1)3305 1392 y Ft(\(\012\).)39 b(The)118 1500 y(set)31 b Fl(M)369 1514 y Fo(1)409 1500 y Ft(\(\012\))f(is)g(equipp)s (ed)d(with)i(the)i(w)m(eak)g(top)s(ology)-8 b(,)32 b(i.e.)40 b(the)31 b(top)s(ology)f(induced)f(b)m(y)h(con)m(v)m(ergence)118 1610 y(against)44 b(b)s(ounded)e(con)m(tin)m(uous)i(functions)f(on)h (\012.)81 b(F)-8 b(or)45 b(eac)m(h)g(\014nite)e(subset)g(\003)h(of)h Fn(Z)3240 1577 y Fm(d)3276 1610 y Ft(,)i(denoted)118 1718 y(\003)36 b Fl(\032\032)g Fn(Z)460 1685 y Fm(d)496 1718 y Ft(,)j Fl(F)625 1732 y Fo(\003)715 1718 y Ft(is)d(the)h Fr(\033)s Ft(-\014eld)f(generated)i(b)m(y)f(the)g(canonical)g(pro)5 b(jections)36 b Fr(\031)2914 1732 y Fm(x)2958 1718 y Ft(\()p Fr(\036)p Ft(\))h(=)f Fr(\036)3280 1732 y Fm(x)3324 1718 y Ft(,)j Fr(x)d Fl(2)g Ft(\003,)118 1829 y Fr(\036)44 b Fl(2)g Ft(\012,)g(and)d Fr(\036)698 1843 y Fo(\003)793 1829 y Ft(denotes)h(the)g(restriction)f(of)g Fr(\036)k Fl(2)e Ft(\012)e(to)i(\003.)74 b(W)-8 b(e)43 b(also)f(set)g(\003)3035 1796 y Fm(c)3114 1829 y Ft(=)i Fn(Z)3294 1796 y Fm(d)3357 1829 y Fl(n)29 b Ft(\003.)74 b(A)118 1939 y(function)28 b Fr(f)34 b Ft(:)26 b(\012)e Fl(!)i Fn(R)37 b Ft(is)28 b(called)g Fi(lo)-5 b(c)g(al)30 b Ft(if)e(there)h(exists)f(a)h(\003)d Fl(\032\032)e Fn(Z)2406 1906 y Fm(d)2471 1939 y Ft(suc)m(h)k(that)i Fr(f)37 b Ft(is)28 b Fl(F)3107 1953 y Fo(\003)3161 1939 y Ft(-measurable.)118 2047 y(The)j(smallest)g(suc)m(h)g(\003)h(is)e (called)h(the)h(supp)s(ort)e(of)h Fr(f)41 b Ft(and)31 b(is)g(denoted)g Fr(S)2666 2062 y Fm(f)2712 2047 y Ft(.)44 b(Lattice)32 b(shifts)e Fl(f)p Fr(\022)3421 2061 y Fm(y)3463 2047 y Fl(g)i Ft(are)118 2162 y(de\014ned)c(b)m(y)g Fr(\031)606 2176 y Fm(x)666 2162 y Fl(\016)18 b Fr(\022)772 2176 y Fm(y)838 2162 y Ft(=)25 b Fr(\031)986 2176 y Fm(x)p Fo(+)p Fm(y)1122 2162 y Ft(,)k Fr(x;)15 b(y)28 b Fl(2)d Fn(Z)1492 2129 y Fm(d)1528 2162 y Ft(.)40 b(Elemen)m(ts)29 b(of)g Fl(M)2196 2176 y Fo(1)2235 2162 y Ft(\(\012\))g(in)m(v)-5 b(arian)m(t)28 b(under)f(transformations)118 2275 y(induced)35 b(b)m(y)i(shifts)f(are)h(denoted)h Fl(M)1462 2242 y Fm(S)1462 2299 y Fo(1)1513 2275 y Ft(\(\012\).)61 b(The)36 b(set)i(of)f(shift)f (in)m(v)-5 b(arian)m(t)37 b(measures)f Fl(M)3293 2242 y Fm(S)3293 2299 y Fo(1)3344 2275 y Ft(\(\012\))i(is)e(a)118 2386 y(closed)30 b(con)m(v)m(ex)j(subset)c(of)i Fl(M)1176 2400 y Fo(1)1216 2386 y Ft(\(\012\).)41 b(A)31 b(measure)f Fr(\026)c Fl(2)f(M)2147 2353 y Fm(S)2147 2410 y Fo(1)2198 2386 y Ft(\(\012\))30 b(is)g(called)g(ergo)s(dic)g(if)f Fr(\026)p Ft(\()p Fr(A)p Ft(\))d(=)g(0)k(or)h(1)118 2498 y(for)f(an)m(y)h(ev)m(en)m(t)h Fr(A)25 b Fl(2)g(F)40 b Ft(suc)m(h)30 b(that)h Fr(\022)1398 2512 y Fm(x)1441 2498 y Fr(A)26 b Ft(=)f Fr(A)30 b Ft(for)g(all)g Fr(x)25 b Fl(2)g Fn(Z)2223 2465 y Fm(d)2259 2498 y Ft(.)118 2681 y(2.2.)54 b Fy(The)34 b(in)m(teraction.)46 b Ft(Let)31 b Fr(p)25 b Ft(:)g Fn(Z)1442 2648 y Fm(d)1499 2681 y Fl(\002)20 b Fn(Z)1655 2648 y Fm(d)1716 2681 y Fl(!)25 b Ft([0)p Fr(;)15 b Ft(1])32 b(b)s(e)e(a)h(symmetric,)f(homogeneous)h (function)993 2844 y Fr(p)p Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)f Fr(p)p Ft(\(0)p Fr(;)15 b(y)24 b Fl(\000)c Fr(x)p Ft(\))51 b(=)f Fr(p)p Ft(\()p Fr(y)s(;)15 b(x)p Ft(\))g Fr(;)92 b(x;)15 b(y)28 b Fl(2)d Fn(Z)2710 2806 y Fm(d)2762 2844 y Fr(:)118 3000 y Ft(W)-8 b(e)32 b(mak)m(e)f(the)g(follo)m(wing)d (basic)i(assumptions.)243 3180 y Fl(\017)42 b Fr(p)30 b Ft(is)f(the)i(transition)e(function)g(of)h(a)h(random)f(w)m(alk)g(on) g Fn(Z)2341 3147 y Fm(d)2377 3180 y Ft(,)h(i.e.)1329 3264 y Fk(X)1312 3469 y Fm(y)r Fj(2)p Fh(Z)1447 3450 y Fu(d)1507 3350 y Fr(p)p Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)g(1)15 b Fr(;)142 b(x)25 b Fl(2)g Fn(Z)2391 3313 y Fm(d)2442 3350 y Fr(:)1008 b Ft(\(2.1\))243 3655 y Fl(\017)42 b Fr(p)30 b Ft(is)f Fi(tr)-5 b(ansient)p Ft(,)33 b(i.e.)1368 3881 y Fr(\033)1423 3843 y Fo(2)1514 3881 y Ft(=)1666 3767 y Fj(1)1636 3795 y Fk(X)1635 3989 y Fm(n)p Fo(=0)1798 3881 y Fr(p)1844 3843 y Fm(n)1891 3881 y Ft(\(0)p Fr(;)15 b Ft(0\))52 b Fr(<)f Fl(1)15 b Fr(;)1079 b Ft(\(2.2\))330 4118 y(where)30 b Fr(p)639 4085 y Fm(n)716 4118 y Ft(denotes)h(the)f Fr(n)p Fl(\000)p Ft(th)g(iterate)h(of)f Fr(p)p Ft(.)118 4298 y(The)g(mo)s(del)f(is)h(sp)s(eci\014ed)e(b)m(y)j (the)f Fi(p)-5 b(otential)1297 4454 y Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)f Fr(\016)s Ft(\()p Fr(x;)15 b(y)s Ft(\))43 b Fl(\000)d Fr(Q)p Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(;)1008 b Ft(\(2.3\))118 4616 y(where)30 b Fr(\016)s Ft(\()p Fr(x;)15 b(y)s Ft(\))27 b(=)e(1)31 b(if)e Fr(x)c Ft(=)g Fr(y)s Ft(,)31 b(and)e Fr(\016)s Ft(\()p Fr(x;)15 b(y)s Ft(\))27 b(=)e(0)31 b(otherwise,)f(and)g(w)m(e)h (ha)m(v)m(e)g(de\014ned)1345 4772 y Fr(Q)p Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)g Fr(\021)1845 4786 y Fm(x)p Fj(\000)p Fm(y)1996 4772 y Fr(p)p Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(;)1183 b Ft(\(2.4\))118 4933 y(with)29 b Fr(\021)34 b Ft(an)c(arbitrary)f(symmetric)h(v)m(ector)i(with)d(v)-5 b(alues)30 b(in)f Fl(f\000)p Ft(1)p Fr(;)15 b Ft(+1)p Fl(g)1108 5108 y Fr(\021)28 b Fl(2)d(f\000)p Ft(1)p Fr(;)15 b Ft(+1)p Fl(g)1629 5071 y Fh(Z)1681 5047 y Fu(d)1730 5108 y Fr(;)117 b(\021)1917 5122 y Fm(x)1986 5108 y Ft(=)25 b Fr(\021)2127 5122 y Fj(\000)p Fm(x)2317 5108 y Fl(8)p Fr(x)g Fl(2)f Fn(Z)2595 5071 y Fm(d)2647 5108 y Fr(:)118 5264 y Ft(W)-8 b(e)37 b(also)e(use)h(the)g(notation)f Fr(J)44 b Ft(=)33 b(1)-23 b(I)24 b Fl(\000)g Fr(Q)p Ft(.)56 b(W)-8 b(e)37 b(will)c(b)s(e)i(w)m(orking)g(in)f(this)h(setting)g(for)h (the)f(rest)h(of)118 5372 y(the)g(pap)s(er,)h(with)d(the)i(assumptions) f(\(2.1\))i(and)f(\(2.2\).)59 b(In)35 b(section)h(4,)i(the)e(follo)m (wing)f(mild)e(deca)m(y)118 5480 y(condition)c(will)f(b)s(e)i (considered:)p eop %%Page: 5 5 5 4 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)287 b(5)243 366 y Fl(\017)42 b Ft(there)31 b(exists)f Fr(\014)g(>)25 b Ft(0,)31 b(suc)m(h)f(that)1428 516 y(lim)1381 581 y Fj(j)p Fm(x)p Fj(j!1)1631 516 y Fl(j)p Fr(x)p Fl(j)1733 478 y Fm(d)p Fo(+)p Fm(\014)1887 516 y Fr(p)p Ft(\(0)p Fr(;)15 b(x)p Ft(\))52 b(=)e(0)15 b Fr(:)1077 b Ft(\(2.5\))118 718 y(W)-8 b(e)33 b(emphasize)e(that)h (our)f(results)f(up)h(to)h(section)g(4)f(hold)g(without)f(an)m(y)i(h)m (yp)s(othesis)e(on)h(the)h(deca)m(y)118 826 y(of)f Fr(p)p Ft(.)40 b(Thanks)29 b(to)i(the)g(transience)f(condition)f(\(2.2\),)j (the)f(Green)g(function)1424 1048 y Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))52 b(=)1909 934 y Fj(1)1879 962 y Fk(X)1878 1156 y Fm(n)p Fo(=0)2026 1048 y Fr(Q)2098 1010 y Fm(n)2145 1048 y Ft(\()p Fr(x;)15 b(y)s Ft(\))1120 b(\(2.6\))118 1299 y(is)30 b(w)m(ell)f(de\014ned,)g(cf.)41 b([32)q(].)g(Let)1258 1276 y(^)1236 1299 y Fr(J)e Ft(denote)31 b(the)g(F)-8 b(ourier)30 b(transform)g(of)g Fr(J)9 b Ft(,)1081 1442 y(^)1059 1465 y Fr(J)g Ft(\()p Fr(\022)s Ft(\))50 b(=)1423 1379 y Fk(X)1405 1584 y Fm(x)p Fj(2)p Fh(Z)1543 1565 y Fu(d)1588 1465 y Fr(J)9 b Ft(\(0)p Fr(;)15 b(x)p Ft(\))g(e)1909 1428 y Fm(i)f(x)p Fj(\001)p Fm(\022)2060 1465 y Fr(;)117 b(\022)27 b Fl(2)e Ft(\()p Fl(\000)p Fr(\031)s(;)15 b(\031)s Ft(])2639 1428 y Fm(d)2696 1465 y Fr(:)118 1732 y Ft(It)30 b(is)g(easily)f(c)m(hec)m(k)m(ed)j(that)f(\(2.1\))h(implies)1636 1709 y(^)1614 1732 y Fr(J)i Fq(>)25 b Ft(0,)31 b(and)f(\(2.2\))i (implies)27 b(that)k Fr(f)k Fl(\021)2992 1709 y Ft(^)2970 1732 y Fr(J)3029 1699 y Fj(\000)p Fo(1)3153 1732 y Ft(is)29 b(in)m(tegrable,)118 1840 y(with)1048 2011 y Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)1595 1950 y(1)p 1512 1990 212 4 v 1512 2074 a(\(2)p Fr(\031)s Ft(\))1682 2047 y Fm(d)1748 1888 y Fk(Z)1799 2094 y Fo(\()p Fj(\000)p Fm(\031)r(;\031)r Fo(])2007 2075 y Fu(d)2077 2011 y Fr(f)10 b Ft(\()p Fr(\022)s Ft(\))15 b(e)2303 1974 y Fm(i)c Fo(\()p Fm(y)r Fj(\000)p Fm(x)p Fo(\))p Fj(\001)p Fm(\022)2599 2011 y Fr(d\022)18 b(:)743 b Ft(\(2.7\))118 2301 y Fi(R)-5 b(emark)34 b(1)p Ft(.)41 b(Let)31 b(us)f(de\014ne)1231 2449 y Fl(N)38 b Ft(=)25 b Fl(f)p Fr(\022)j Fl(2)d Ft(\()p Fl(\000)p Fr(\031)s(;)15 b(\031)s Ft(])1923 2411 y Fm(d)2005 2449 y Ft(:)2128 2426 y(^)2106 2449 y Fr(J)9 b Ft(\()p Fr(\022)s Ft(\))25 b(=)g(0)p Fl(g)15 b Fr(;)943 b Ft(\(2.8\))118 2620 y(the)36 b(set)h(of)f(zeros)g(of)902 2597 y(^)880 2620 y Fr(J)9 b Ft(.)57 b(If)36 b Fr(\021)i Fl(\021)c Ft(1)i(in)f(\(2.4\))j(then)d(clearly)h(0)f Fl(2)f(N)13 b Ft(.)57 b(Moreo)m(v)m(er,)40 b(it)35 b(is)g(kno)m(wn)h(that)118 2728 y Fl(N)46 b Ft(=)33 b Fl(f)p Ft(0)p Fl(g)k Ft(if)d(and)h(only)f (if)g Fr(p)h Ft(is)f Fi(irr)-5 b(e)g(ducible)p Ft(,)38 b(cf.)55 b([32)r(].)g(By)35 b(irreducibilit)m(y)c(w)m(e)36 b(mean)f(that)h(for)f(an)m(y)118 2838 y Fr(x;)15 b(y)33 b Fl(2)d Fn(Z)444 2805 y Fm(d)513 2838 y Ft(there)k(exists)f(an)g(in)m (teger)h Fr(n)29 b(>)h Ft(0)k(suc)m(h)f(that)h Fr(p)2154 2805 y Fm(n)2201 2838 y Ft(\()p Fr(x;)15 b(y)s Ft(\))31 b Fr(>)e Ft(0.)50 b(In)33 b(the)g(non)g(ferromagnetic)118 2946 y(case)k Fr(\021)h Fl(6\021)d Ft(1)h(it)g(could)f(happ)s(en)g (that)h Fl(N)48 b Ft(=)34 b Fl(;)p Ft(.)59 b(As)36 b(discussed)e(in)h (the)h(in)m(tro)s(duction,)g(this)f(reduces)118 3054 y(to)g(an)f(h)m(yp)s(ercon)m(tractiv)m(e)h(regime,)g(with)e Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))36 b(absolutely)d(summable.)50 b(One)34 b(should)e(therefore)118 3162 y(k)m(eep)j(in)d(mind)g(that)j (the)f(in)m(teresting)f(cases)i(are)f(the)g(massless)f(mo)s(dels,)h (obtained)f(b)m(y)h(c)m(ho)s(osing)g Fr(\021)118 3269 y Ft(in)29 b(suc)m(h)h(a)h(w)m(a)m(y)g(that)g Fl(N)39 b(6)p Ft(=)24 b Fl(;)p Ft(.)42 b(In)29 b(section)i(2.5)g(w)m(e)g (discuss)e(some)h(examples.)118 3457 y(Let)39 b(\003)f Fl(\032\032)g Fn(Z)635 3424 y Fm(d)671 3457 y Ft(.)64 b(W)-8 b(e)39 b(denote)g(b)m(y)f Fr(J)1409 3471 y Fo(\003)1500 3457 y Ft(and)g Fr(Q)1757 3471 y Fo(\003)1848 3457 y Ft(the)g(restriction)f(to)i(\003)f(of)h Fr(J)47 b Ft(and)37 b Fr(Q)h Ft(resp)s(ectiv)m(ely)-8 b(.)118 3565 y(Then)29 b Fr(J)405 3579 y Fo(\003)484 3565 y Ft(=)c(1)-23 b(I)635 3579 y Fo(\003)709 3565 y Fl(\000)20 b Fr(Q)872 3579 y Fo(\003)955 3565 y Ft(is)30 b(a)g(p)s(ositiv)m(e)g(de\014nite)f (matrix)h(and)f(\000)2311 3579 y Fo(\003)2390 3565 y Fl(\021)c Fr(J)2545 3527 y Fj(\000)p Fo(1)2536 3594 y(\003)2669 3565 y Ft(is)30 b(giv)m(en)g(b)m(y)1146 3793 y(\000)1203 3807 y Fo(\003)1256 3793 y Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)1670 3679 y Fj(1)1639 3707 y Fk(X)1638 3901 y Fm(n)p Fo(=0)1786 3793 y Fr(Q)1858 3755 y Fm(n)1858 3815 y Fo(\003)1911 3793 y Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(;)118 b(x;)15 b(y)28 b Fl(2)d Ft(\003)15 b Fr(:)842 b Ft(\(2.9\))118 4021 y(Let)33 b Fr(G)354 4035 y Fo(\003)440 4021 y Ft(b)s(e)f(the)h (restriction)f(of)h Fr(G)g Ft(to)g(\003.)48 b(Note)34 b(that)f Fr(G)2114 4035 y Fo(\003)2200 4021 y Ft(is)f(p)s(ositiv)m(e)f (de\014nite.)47 b(In)32 b(the)h(app)s(endix,)118 4129 y(cf.)41 b(Lemma)29 b(A.1,)i(w)m(e)e(sho)m(w)h(that)g Fr(G)1393 4143 y Fo(\003)1475 4129 y Ft(has)f(an)h(expansion)e (analogous)i(to)g(\(2.9\))h(where)e Fr(Q)3262 4143 y Fo(\003)3344 4129 y Ft(m)m(ust)g(b)s(e)118 4244 y(replaced)h(b)m(y)g (an)g(e\013ectiv)m(e)i(coupling)d(matrix)1761 4221 y(~)1740 4244 y Fr(Q)1812 4258 y Fo(\003)1865 4244 y Ft(.)40 b(This)29 b(giv)m(es)1533 4393 y Fr(G)1604 4355 y Fj(\000)p Fo(1)1604 4422 y(\003)1750 4393 y Ft(=)50 b(1)-23 b(I)1926 4407 y Fo(\003)2015 4393 y Fl(\000)2142 4370 y Ft(~)2121 4393 y Fr(Q)2193 4407 y Fo(\003)3429 4393 y Ft(\(2.10\))118 4541 y(The)24 b(matrix)f Fr(G)659 4555 y Fo(\003)736 4541 y Ft(can)i(b)s(e)e(seen)h(as)g(the)h(Green)f(function)f(of)h Fr(J)33 b Ft(on)24 b(\003)g(with)f(free)h(b)s(oundary)e(conditions,)118 4649 y(while)h(\000)407 4663 y Fo(\003)485 4649 y Ft(is)h(asso)s (ciated)h(to)h(Diric)m(hlet)d(b)s(oundary)g(conditions.)38 b(W)-8 b(e)26 b(call)e Fr(P)2716 4663 y Fo(\003)2794 4649 y Ft(the)h(cen)m(tered)h(Gaussian)118 4757 y(measure)k(on)h Fn(R)658 4724 y Fo(\003)748 4757 y Ft(with)e(co)m(v)-5 b(ariance)31 b Fr(G)1465 4771 y Fo(\003)1519 4757 y Ft(,)g(so)f(that) 778 4886 y Fr(dP)883 4900 y Fo(\003)p 778 4927 159 4 v 780 5010 a Fr(d\036)881 5024 y Fo(\003)947 4947 y Ft(\()p Fr(\036)p Ft(\))51 b(=)f(\(2)p Fr(\031)s Ft(\))1413 4910 y Fj(\000j)p Fo(\003)p Fj(j)p Fm(=)p Fo(2)1648 4947 y Ft(\(det)15 b Fr(G)1895 4961 y Fo(\003)1949 4947 y Ft(\))1984 4910 y Fj(\000)p Fo(1)p Fm(=)p Fo(2)2179 4947 y Ft(exp)g(\()p Fl(\000)2449 4886 y Ft(1)p 2449 4927 46 4 v 2449 5010 a(2)2535 4874 y Fk(\012)2578 4947 y Fr(\036;)g(G)2743 4909 y Fj(\000)p Fo(1)2743 4976 y(\003)2839 4947 y Fr(\036)2893 4874 y Fk(\013)2936 4947 y Ft(\))g Fr(:)118 5144 y Ft(Here)30 b Fr(d\036)433 5158 y Fo(\003)512 5144 y Ft(=)608 5076 y Fk(Q)694 5171 y Fm(x)p Fj(2)p Fo(\003)849 5144 y Fr(d\036)950 5158 y Fm(x)1024 5144 y Ft(denotes)g(Leb)s(esgue)g(measure,)g Fl(h)p Fr(;)15 b Fl(i)30 b Ft(is)f(the)h(usual)e(scalar)i(pro)s(duct,)f (and)g Fl(j)p Ft(\003)p Fl(j)118 5262 y Ft(is)d(the)h(cardinalit)m(y)e (of)i(the)g(set)g(\003.)39 b(The)26 b(measures)h Fl(f)p Fr(P)2001 5276 y Fo(\003)2055 5262 y Fr(;)j Ft(\003)c Fl(\032\032)e Fn(Z)2430 5229 y Fm(d)2467 5262 y Fl(g)j Ft(form)f(a)h(consisten)m(t)g(family)e(and)118 5370 y(there)30 b(exist)f(a)h(unique)e(measure)h Fr(P)38 b Fl(2)25 b(M)1578 5384 y Fo(1)1618 5370 y Ft(\(\012\))30 b(suc)m(h)f(that)h(its)f (marginal)f(on)h(\003)h(is)e(giv)m(en)i(b)m(y)f Fr(P)3416 5384 y Fo(\003)3470 5370 y Ft(.)40 b(W)-8 b(e)118 5480 y(also)33 b(refer)g(to)h Fr(P)46 b Ft(as)34 b(the)f(cen)m(tered)h (Gaussian)f(\014eld)f(on)h Fn(Z)2160 5447 y Fm(d)2196 5480 y Ft(,)h(with)e(co)m(v)-5 b(ariance)34 b Fr(G)p Ft(.)50 b(F)-8 b(rom)34 b(\(2.7\))h(and)p eop %%Page: 6 6 6 5 bop 118 45 a Fo(6)1138 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(Riemann-Leb)s(esgue)d(lemma,)i(w)m(e)g(ha)m(v)m (e)h Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))26 b Fl(!)g Ft(0,)i(as)g Fl(j)p Fr(x)14 b Fl(\000)g Fr(y)s Fl(j)26 b(!)f(1)p Ft(.)39 b(This)26 b(implies)f Fr(P)40 b Ft(is)26 b(trivial)118 474 y(on)k(tail)g(ev)m(en)m(ts,)i(cf.)41 b(e.g.)h([20)q(],)31 b(c)m(hapter)g(7.)41 b(In)29 b(particular,)g Fr(P)44 b Ft(is)29 b(ergo)s(dic)h(w.r.t.)41 b(lattice)30 b(shifts.)118 645 y(2.3.)54 b Fy(Gibbs)35 b(Measures.)46 b Ft(Let)780 800 y(\012)846 814 y Fm(J)946 800 y Ft(=)1067 700 y Fk(n)1127 800 y Fr(\036)26 b Fl(2)f Ft(\012)g(:)1476 714 y Fk(X)1459 919 y Fm(y)r Fj(2)p Fh(Z)1594 900 y Fu(d)1639 800 y Fl(j)p Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fr(\036)1987 814 y Fm(y)2030 800 y Fl(j)41 b Fr(<)f Fl(1)15 b Fr(;)41 b Ft(for)25 b(all)f Fr(x)h Fl(2)g Fn(Z)2862 763 y Fm(d)2898 700 y Fk(o)2974 800 y Fr(:)430 b Ft(\(2.11\))118 1071 y(Note)43 b(that)f(\012)623 1085 y Fm(J)712 1071 y Ft(is)f(a)g(tail)g(ev)m(en)m(t.)75 b(Fix)41 b(\003)i Fl(\032\032)g Fn(Z)1912 1038 y Fm(d)1990 1071 y Ft(and)d(a)i(b)s (oundary)d(condition)h Fr(\034)54 b Fl(2)43 b Ft(\012)3358 1085 y Fm(J)3406 1071 y Ft(.)74 b(The)118 1179 y(Hamiltonian)29 b(is)g(de\014ned)g(b)m(y)824 1367 y Fr(H)907 1330 y Fm(\034)900 1390 y Fo(\003)952 1367 y Ft(\()p Fr(\036)p Ft(\))52 b(=)1259 1306 y(1)p 1259 1346 46 4 v 1259 1429 a(2)1360 1281 y Fk(X)1329 1478 y Fm(x;y)r Fj(2)p Fo(\003)1537 1367 y Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fr(\036)1860 1381 y Fm(x)1905 1367 y Fr(\036)1959 1381 y Fm(y)2046 1367 y Ft(+)2241 1281 y Fk(X)2162 1482 y Fm(x)p Fj(2)p Fo(\003)p Fm(;y)10 b(=)-43 b Fj(2)p Fo(\003)2466 1367 y Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fr(\036)2789 1381 y Fm(x)2834 1367 y Fr(\034)2874 1381 y Fm(y)2931 1367 y Fr(:)473 b Ft(\(2.12\))118 1623 y(W)-8 b(e)32 b(de\014ne)d Fi(sp)-5 b(e)g(ci\014c)g(ations)33 b Fr(\015)1133 1590 y Fm(\034)1128 1650 y Fo(\003)1206 1623 y Fl(2)25 b(M)1401 1637 y Fo(1)1441 1623 y Ft(\(\012\))31 b(b)m(y)857 1859 y Fr(\015)909 1822 y Fm(\034)904 1882 y Fo(\003)957 1859 y Ft(\()p Fr(d\036)p Ft(\))c(=)1251 1704 y Fk(\()1324 1798 y Ft(\()p Fr(Z)1428 1765 y Fm(\034)1421 1825 y Fo(\003)1474 1798 y Ft(\))1509 1765 y Fj(\000)p Fo(1)1634 1798 y Ft(exp)15 b(\()p Fl(\000)p 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Fl(H)900 5387 y Fm(J)999 5373 y Ft(=)50 b Fl(f)p Fr(\036)26 b Fl(2)f Ft(\012)g(:)1540 5286 y Fk(X)1523 5492 y Fm(y)r Fj(2)p Fh(Z)1658 5473 y Fu(d)1703 5373 y Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fr(\036)2026 5387 y Fm(y)2119 5373 y Ft(=)50 b(0)p Fr(;)41 b Ft(for)25 b(all)g Fr(x)g Fl(2)g Fn(Z)2835 5335 y Fm(d)2871 5373 y Fl(g)15 b Fr(:)p eop %%Page: 7 7 7 6 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)287 b(7)118 366 y Ft(Let)34 b Fl(G)338 380 y Fo(ext)472 366 y Ft(denote)g(the)g(set)g(of)f(extremal)h(p)s(oin)m (ts)e(of)i(the)g(con)m(v)m(ex)h(set)f Fl(G)5 b Ft(.)50 b(A)34 b(result)e(of)i(Rozano)m(v)h([31)q(])118 474 y(and)30 b(Dobrushin)e([12)q(])j(\(see)g(also)f(Theorems)g(\(13.24\))j(and)d (\(13.36\))j(of)e([20)q(]\))g(sho)m(ws)f(that)1420 636 y Fl(G)1474 650 y Fo(ext)1625 636 y Ft(=)50 b Fl(f)p Fr(\015)1843 598 y Fm(\033)1891 636 y Fr(;)65 b(\033)29 b Fl(2)c(H)2225 650 y Fm(J)2274 636 y Fl(g)15 b Fr(:)1070 b Ft(\(2.19\))118 910 y Fi(R)-5 b(emark)37 b(2)p Ft(.)50 b(The)33 b(transience)g(assumption)f(\(2.2\))k(guaran)m(tees)f (existence)e(of)h(Gibbs)e(measures,)i(i.e.)118 1018 y Fl(G)48 b(6)p Ft(=)43 b Fl(;)p Ft(.)72 b(W)-8 b(e)43 b(remark)d(that)i(if)e Fr(p)g Ft(has)h(\014nite)f(range,)k(the)d (transience)f(h)m(yp)s(othesis)g(forces)h Fr(d)i Fq(>)g Ft(3,)118 1126 y(cf.)62 b(e.g.)h([32)r(].)f(Moreo)m(v)m(er,)42 b(condition)36 b(\(2.17\))j(is)e(automatically)g(satis\014ed.)62 b(It)37 b(is)g(then)g(useful)f(to)118 1234 y(restrict)41 b(the)h(atten)m(tion)g(to)g(measures)g(on)f(smaller)f(subsets)g(of)i (\012.)73 b(One)41 b(can)h(in)m(tro)s(duce)e(sets)i(of)118 1342 y(temp)s(ered)32 b(con\014gurations)p 1110 1269 66 4 v 31 w(\012,)h(and)f(de\014ne)f Fi(temp)-5 b(er)g(e)g(d)37 b(Gibbs)d(me)-5 b(asur)g(es)34 b Ft(as)f(the)f(set)h(of)g Fr(\026)28 b Fl(2)g(G)38 b Ft(suc)m(h)118 1449 y(that)e Fr(\026)p Ft(\()p 410 1376 V(\012\))d(=)h(1.)55 b(A)36 b(natural)e(c)m(hoice)i(is)e(e.g.)p 1764 1376 V 57 w(\012)f(=)g Fl(S)2029 1416 y Fj(0)2052 1449 y Ft(\()p Fn(Z)2152 1416 y Fm(d)2188 1449 y Ft(\),)k(where)e Fl(S)2615 1416 y Fj(0)2638 1449 y Ft(\()p Fn(Z)2738 1416 y Fm(d)2775 1449 y Ft(\))g(denotes)h(the)f(space)h(of)118 1557 y(con\014gurations)30 b(whic)m(h)f(increase)h(at)h(most)g(p)s(olynomially)c(at)k(in\014nit)m (y)-8 b(.)118 1773 y Fi(R)j(emark)32 b(3)p Ft(.)40 b(A)29 b(general)f(result)f(\(see)i([20)q(],)g(Theorem)f(\(14.17\)\))j(states) f(that)e(a)h(shift)e(in)m(v)-5 b(arian)m(t)27 b(Gibbs)118 1880 y(measure)j(has)g(a)g(unique)f(decomp)s(osition)f(in)h(terms)h(of) g(ergo)s(dic)g(Gibbs)e(measures.)41 b(But)30 b(an)g(ergo)s(dic)118 1988 y(Gibbs)24 b(measure)i(need)f(not)h(b)s(e)f(extremal)g(in)g Fl(G)31 b Ft(in)24 b(general)i(\(an)f(ergo)s(dic)g(measure)h(is)f(not)g (necessarily)118 2096 y(tail)41 b(trivial\).)73 b(Indeed)41 b(there)g(are)h(man)m(y)g(w)m(ell)f(kno)m(wn)g(examples)g(\(see)i(e.g.) 75 b([16)q(],)45 b(section)d(2\))g(of)118 2204 y(shift)c(in)m(v)-5 b(arian)m(t)37 b(Gibbs)h(measures)g(whic)m(h)g(can)h(b)s(e)f(written)f (as)i(mixtures)f(of)h(non)f(shift)f(in)m(v)-5 b(arian)m(t)118 2312 y(elemen)m(ts)28 b(of)g Fl(G)642 2326 y Fo(ext)742 2312 y Ft(.)40 b(In)27 b(Theorem)h(2.1)g(b)s(elo)m(w)f(w)m(e)h(pro)m(v) m(e)h(that)f(for)g(ferromagnetic)g(mo)s(dels)e(generated)118 2420 y(b)m(y)41 b(an)f Fi(irr)-5 b(e)g(ducible)42 b Ft(random)e(w)m (alk)g(this)g(cannot)h(happ)s(en,)h(at)f(least)g(if)f(w)m(e)h(restrict) g(to)g(temp)s(ered)118 2528 y(measures.)64 b(In)37 b(other)i(w)m(ords)f (all)f(ergo)s(dic)g(Gibbs)g(measures)h(are)g(extremal)h(in)e Fl(G)43 b Ft(and)38 b(\\extremal)118 2636 y(shift)43 b(in)m(v)-5 b(arian)m(t)43 b(Gibbs)f(=)i(shift)e(in)m(v)-5 b(arian)m(t)43 b(extremal)h(Gibbs".)80 b(W)-8 b(e)46 b(also)d(giv)m(e)i(an)f(example)f(of)118 2744 y(mo)s(dels)c(where)h (this)g(is)g(violated.)70 b(In)40 b(particular)f(these)i(examples)f (imply)f(that)i(in)e(our)h(setting,)118 2852 y(assuming)33 b(a)i(massless)e(regime,)j(all)d(ergo)s(dic)h(Gibbs)f(measures)h(are)h (extremal)f(in)f Fl(G)40 b Ft(if)33 b(and)h(only)f(if)118 2960 y Fr(\021)c Fl(\021)c Ft(1)30 b(and)g Fr(p)g Ft(is)g(irreducible.) 118 3153 y(2.4.)54 b Fy(F)-9 b(erromagnetic)39 b(mo)s(dels.)45 b Ft(The)34 b(ferromagnetic)h(case)g(is)e(obtained)h(if)f Fr(\021)j Fl(\021)31 b Ft(1)k(in)e(\(2.4\),)k(i.e.)118 3264 y Fr(Q)42 b Ft(=)g Fr(p)p Ft(,)h(or)e(equiv)-5 b(alen)m(tly)39 b Fr(J)50 b Ft(is)40 b(the)h(generator)g(of)g(a)g(\(con)m(tin)m(uous)g (time\))f(random)g(w)m(alk)h(on)f Fn(Z)3600 3231 y Fm(d)3636 3264 y Ft(.)118 3371 y(Since)353 3303 y Fk(P)449 3398 y Fm(y)506 3371 y Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))26 b(=)f(0)j(for)g(an)m(y)h Fr(x)p Ft(,)f(an)m(y)h(constan)m(t)g (con\014guration)e Fr(\033)2516 3385 y Fm(x)2586 3371 y Fl(\021)d Fr(c)29 b Ft(is)e(in)g Fl(H)3019 3385 y Fm(J)3095 3371 y Ft(and)h(the)g(global)118 3483 y(shift)40 b(symmetry)h Fr(\036)j Fl(!)f Fr(\036)27 b Ft(+)h Fr(c)p Ft(,)44 b(is)d(sp)s(on)m (taneously)f(brok)m(en.)73 b(In)41 b(particular)e(the)j(set)f Fl(G)47 b Ft(con)m(tains)118 3591 y(uncoun)m(tably)29 b(man)m(y)i(measures.)218 3725 y(A)24 b(classical)g(example)g(is)g(the) h Fi(massless)j(fr)-5 b(e)g(e)28 b(\014eld)p Ft(.)39 b(It)25 b(is)f(obtained)g(when)f Fr(p)h Ft(is)g(the)h(simple)d(random) 118 3833 y(w)m(alk)1266 4054 y Fr(p)p Ft(\()p Fr(x;)15 b(y)s Ft(\))26 b(=)1644 3899 y Fk(\()1717 3993 y Ft(1)p Fr(=)p Ft(2)p Fr(d)93 b Ft(if)29 b Fl(j)p Fr(x)21 b Fl(\000)f Fr(y)s Fl(j)25 b Ft(=)g(1)1717 4122 y(0)230 b(otherwise)118 4309 y(i.e.)52 b Fr(J)41 b Ft(=)32 b Fl(\000)p Ft(\001,)j(if)e(\001)h (is)f(the)i(discrete)f(Laplace)g(op)s(erator)h(on)f Fn(Z)2382 4276 y Fm(d)2418 4309 y Ft(.)53 b(Here,)36 b(as)e(in)f(other)i (\014nite)e(range)118 4417 y(mo)s(dels)27 b(one)h(considers)e(temp)s (ered)h(Gibbs)f(measures,)i(de\014ned)f(as)h(in)e(remark)i(2)g(ab)s(o)m (v)m(e,)h([12)r(].)40 b(Large)118 4525 y(deviations)29 b(for)i(\014nite)e(range)i(harmonic)e(crystal)h(w)m(ere)h(considered)e (in)g([1)q(].)218 4660 y(As)c(an)g(example)f(of)i(a)f(non)g(lo)s(cal)f (in)m(teraction)h(w)m(e)h(shall)d(consider)h(the)h(follo)m(wing)f(mo)s (del,)h(to)h(whic)m(h)118 4767 y(w)m(e)31 b(refer)f(as)h(the)f Fr(\013)p Ft(-)p Fi(stable)i Ft(crystal.)41 b(F)-8 b(or)31 b Fr(\013)25 b Fl(2)g Ft(\(0)p Fr(;)15 b Ft(2)22 b Fl(^)e Fr(d)p Ft(\),)31 b(let)g Fr(p)f Ft(b)s(e)f(suc)m(h)h(that)1405 4936 y(lim)1358 5001 y Fj(j)p Fm(x)p Fj(j!1)1608 4936 y Fl(j)p Fr(x)p Fl(j)1710 4898 y Fm(d)p Fo(+)p Fm(\013)1866 4936 y Fr(p)p Ft(\(0)p Fr(;)15 b(x)p Ft(\))52 b(=)e Fr(c)2331 4950 y Fm(\013)2396 4936 y Fr(:)1008 b Ft(\(2.20\))118 5157 y(for)43 b(some)g Fr(c)549 5171 y Fm(\013)645 5157 y Fl(2)i Ft(\(0)p Fr(;)15 b Fl(1)p Ft(\).)80 b(The)42 b(random)g(w)m(alk)h(will)d(then)j(b)s(e)f(in)f(the)j(domain)d(of)i (attraction)h(of)118 5264 y(a)e Fi(stable)i(law)e Ft(\(see)h([27)q(])f (for)f(some)h(in)m(teresting)f(prop)s(erties)f(of)h(stable)h(random)e (w)m(alks\).)75 b(In)40 b(this)118 5372 y(case)f(the)g(condition)d (\(2.17\))41 b(forces)d(all)f(Gibbs)g(measures)h(to)g(b)s(e)g(temp)s (ered,)i(with)c(no)i(additional)118 5480 y(requiremen)m(t.)p eop %%Page: 8 8 8 7 bop 118 45 a Fo(8)1138 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)218 366 y Ft(In)43 b(ferromagnetic)h(mo)s(dels)e(the)i(co)m (v)-5 b(ariance)45 b(matrices)f(\000)2309 380 y Fo(\003)2362 366 y Fr(;)15 b(G)2473 380 y Fo(\003)2527 366 y Ft(,)47 b(\003)h Fl(\032\032)f Fn(Z)2964 333 y Fm(d)3043 366 y Ft(admit)c(a)h(simple)118 474 y(in)m(terpretation)e(in)f(terms)h(of)h (the)g(random)e(w)m(alk.)77 b(W)-8 b(e)44 b(see)f(from)f(\(2.9\))i (that)f(\000)3056 488 y Fo(\003)3151 474 y Ft(is)f(the)g(Green)118 582 y(function)32 b(of)i(the)g(random)e(w)m(alk)h(with)g(transition)f Fr(p)h Ft(whic)m(h)f(is)h(\\killed")f(up)s(on)f(exiting)i(the)h(b)s(o)m (x)f(\003.)118 690 y(On)25 b(the)h(other)f(hand)g Fr(G)943 704 y Fo(\003)1022 690 y Ft(is)f(the)i(Green)g(function)e(of)i(the)g (random)e(w)m(alk)i(\\em)m(b)s(edded")f(in)f(\003,)j(see)f(the)118 798 y(remark)j(after)h(Lemma)f(A.1.)41 b(One)29 b(can)g(use)g(these)g (facts)h(to)g(giv)m(e)g(the)f(follo)m(wing)e(represen)m(tation)j(for) 118 906 y(the)k(mean)g(v)-5 b(alues)33 b(\(2.15\).)52 b(If)34 b Fn(P)1255 920 y Fm(x)1333 906 y Ft(denotes)g(the)g (probabilit)m(y)d(for)i(the)h(discrete)f(time)h(random)f(w)m(alk)118 1014 y Fr(\030)158 1028 y Fo(0)197 1014 y Fr(;)15 b(\030)277 1028 y Fo(1)332 1014 y Fr(:)g(:)g(:)44 b Ft(de\014ned)27 b(b)m(y)g Fr(p)p Ft(,)i(with)d(start)j(in)d Fr(\030)1578 1028 y Fo(0)1643 1014 y Ft(=)f Fr(x)g Fl(2)g Ft(\003,)j(and)g(if)e Fr(t)2306 1028 y Fo(\003)2387 1014 y Ft(denotes)i(the)g(\014rst)g (hitting)e(time)i(for)118 1122 y Fn(Z)183 1089 y Fm(d)239 1122 y Fl(n)21 b Ft(\003,)31 b(w)m(e)g(ha)m(v)m(e)1070 1311 y Fr(\015)1122 1274 y Fm(\034)1117 1334 y Fo(\003)1170 1311 y Ft(\()p Fr(\036)1259 1325 y Fm(x)1303 1311 y Ft(\))51 b(=)1511 1225 y Fk(X)1510 1426 y Fm(y)10 b(=)-43 b Fj(2)q Fo(\003)1674 1311 y Fn(P)1729 1325 y Fm(x)1773 1237 y Fk(\000)1815 1311 y Fr(\030)1855 1325 y Fm(t)1880 1336 y Fg(\003)1955 1311 y Ft(=)25 b Fr(y)2099 1237 y Fk(\001)2156 1311 y Fr(\034)2196 1325 y Fm(y)2252 1311 y Fr(;)117 b(\034)35 b Fl(2)25 b Ft(\012)2621 1325 y Fm(J)2684 1311 y Fr(:)118 1582 y Ft(This)i(t)m(yp)s(e)i(of)g(form)m(ulas)f(represen)m (t)h(only)f(a)h(sp)s(ecial)f(case)h(of)g(the)h(so)f(called)f Fi(r)-5 b(andom)33 b(walk)f(r)-5 b(epr)g(esen-)118 1690 y(tation)p Ft(,)35 b(whic)m(h)c(can)i(b)s(e)f(generalized)g(to)i (anharmonic)d(crystals)i(with)e(con)m(v)m(ex)j(pair)e(in)m(teraction,)h (cf.)118 1798 y(e.g.)42 b([10)q(])31 b(for)f(a)g(recen)m(t)i(accoun)m (t.)118 2031 y(W)-8 b(e)32 b(turn)d(to)i(the)g(result)e(announced)g(in) h(remark)g(3)g(ab)s(o)m(v)m(e.)118 2207 y Fy(Theorem)k(2.1.)46 b Fi(L)-5 b(et)31 b Fr(\026)g Fi(b)-5 b(e)31 b(a)h(shift)f(invariant)h (Gibbs)f(me)-5 b(asur)g(e)33 b(for)f Fr(J)i Ft(=)25 b(1)-23 b(I)17 b Fl(\000)f Fr(p)31 b Fi(with)h Fr(p)f Fi(irr)-5 b(e)g(ducible,)118 2315 y(and)39 b(assume)g Fr(\026)f Fi(has)i(\014nite)e(\014rst)h(moments,)i(i.e.)59 b Fr(\026)p Ft(\()p Fl(j)p Fr(\036)2098 2329 y Fo(0)2138 2315 y Fl(j)p Ft(\))36 b Fr(<)f Fl(1)p Fi(.)59 b(Then)39 b Fr(\026)f Fi(is)g(a)h(mixtur)-5 b(e)39 b(of)g(shift)118 2423 y(invariant)c (elements)g(of)g Fl(G)1052 2437 y Fo(ext)1152 2423 y Fi(.)46 b(In)35 b(p)-5 b(articular,)36 b(ther)-5 b(e)35 b(exists)g(a)f(pr)-5 b(ob)g(ability)37 b(me)-5 b(asur)g(e)35 b Fr(\032)3191 2437 y Fo(0)3265 2423 y Fi(on)g Fn(R)42 b Fi(with)118 2531 y(\014nite)33 b(\014rst)g(moment)h(and)1479 2746 y Fr(\026)50 b Ft(=)1705 2622 y Fk(Z)1796 2648 y Fj(1)1756 2828 y(\0001)1916 2746 y Fr(\015)1963 2760 y Fm(t)2008 2746 y Fr(\032)2055 2760 y Fo(0)2094 2746 y Ft(\()p Fr(dt)p Ft(\))15 b Fr(;)1145 b Ft(\(2.21\))118 2969 y Fi(wher)-5 b(e)34 b Fr(\015)422 2983 y Fm(t)477 2969 y Ft(=)25 b Fr(\015)625 2936 y Fm(\033)704 2969 y Fi(with)34 b Fr(\033)954 2983 y Fm(x)1023 2969 y Fl(\021)25 b Fr(t)g Fl(2)g Fn(R)r Fi(.)118 3145 y(Pr)-5 b(o)g(of.)45 b Ft(Let)32 b Fr(\026)f Ft(b)s(e)f(giv)m(en)i(as)f(in)f(the)h(h)m(yp)s (othesis,)f(and)h(let)g Fr(\032)c Fl(2)f(M)2444 3159 y Fo(1)2483 3145 y Ft(\(\012\))32 b(b)s(e)e(its)h(extremal)g(decomp)s (o-)118 3253 y(sition)e(in)g Fl(G)5 b Ft(,)31 b(i.e.)1521 3466 y Fr(\026)50 b Ft(=)1748 3342 y Fk(Z)1798 3548 y Fo(\012)1884 3466 y Fr(\015)1936 3428 y Fm(\033)1998 3466 y Fr(\032)p Ft(\()p Fr(d\033)s Ft(\))15 b Fr(:)1172 b Ft(\(2.22\))118 3687 y(In)38 b(particular,)i(cf.)66 b(\(2.19\),)43 b Fr(\032)p Ft(\()p Fl(H)1312 3701 y Fm(J)1362 3687 y Ft(\))c(=)g(1.)67 b(Moreo)m(v)m(er,)43 b Fr(\032)c Ft(is)f(shift)f(in)m(v)-5 b(arian)m(t)38 b(and)h(has)f(\014nite)g (\014rst)118 3795 y(momen)m(ts.)53 b(W)-8 b(e)35 b(w)m(an)m(t)g(to)g (sho)m(w)f(that)h Fr(\032)f Ft(is)g(actually)g(concen)m(trated)i(on)e (the)g(constan)m(t)i(elemen)m(ts)e(of)118 3903 y Fl(H)195 3917 y Fm(J)244 3903 y Ft(.)40 b(This)29 b(w)m(ould)g(reduce)h(the)h (in)m(tegral)f(on)g(\012)g(in)f(\(2.22\))k(to)e(the)f(in)m(tegral)g(on) g Fn(R)39 b Ft(in)29 b(\(2.21\),)k(thereb)m(y)118 4013 y(pro)m(ving)d(the)g(theorem.)41 b(It)31 b(is)e(su\016cien)m(t)h(to)h (sho)m(w)f(that)h(for)g(an)m(y)f Fr(x;)15 b(y)29 b Fl(2)c Fn(Z)2725 3980 y Fm(d)2761 4013 y Ft(,)30 b(for)g(an)m(y)h Fr(\014)g(>)24 b Ft(0,)1418 4184 y Fr(\032)1465 4110 y Fk(\010)1518 4184 y Fl(j)p Fr(\036)1597 4198 y Fm(x)1662 4184 y Fl(\000)c Fr(\036)1807 4198 y Fm(y)1848 4184 y Fl(j)26 b Fr(>)f(\014)2051 4110 y Fk(\011)2154 4184 y Ft(=)51 b(0)15 b Fr(:)1068 b Ft(\(2.23\))118 4360 y(In)29 b(order)g(to)h(pro)m(v)m(e)h(\(2.23\))h(w)m(e)e(ma)m(y)g(pro)s(ceed)f (as)h(follo)m(ws.)40 b(Observ)m(e)29 b(that)h Fr(\036)c Fl(2)f(H)2981 4374 y Fm(J)3059 4360 y Ft(is)k(equiv)-5 b(alen)m(t)29 b(to)118 4468 y Fr(p)164 4482 y Fm(t)193 4468 y Fr(\036)d Ft(=)f Fr(\036)30 b Ft(for)g(all)g Fr(t)25 b Fq(>)g Ft(0,)31 b(where)1295 4708 y Fr(p)1341 4722 y Fm(t)1370 4708 y Ft(\()p Fr(x;)15 b(y)s Ft(\))41 b(=)g(e)1773 4670 y Fj(\000)p Fm(t)1919 4594 y Fj(1)1889 4621 y Fk(X)1888 4815 y Fm(n)p Fo(=0)2061 4646 y Fr(t)2094 4613 y Fm(n)p 2061 4687 80 4 v 2061 4770 a Fr(n)p Ft(!)2181 4708 y Fr(p)2227 4670 y Fm(n)2274 4708 y Ft(\()p Fr(x;)15 b(y)s Ft(\))118 4982 y(is)33 b(the)i(con)m(tin)m(uous)f(time)g(k)m(ernel)g (on)g Fn(Z)1511 4949 y Fm(d)1581 4982 y Ft(with)f(F)-8 b(ourier)34 b(transform)f(exp)15 b(\()p Fl(\000)p Fr(t)2847 4959 y Ft(^)2825 4982 y Fr(J)9 b Ft(\))q(.)52 b(Since)33 b Fr(\032)p Ft(\()p Fl(H)3397 4996 y Fm(J)3446 4982 y Ft(\))g(=)e(1)118 5090 y(w)m(e)f(can)g(replace)g Fr(\036)f Ft(in)g(\(2.23\))j(b)m(y)d Fr(p)1347 5104 y Fm(t)1376 5090 y Fr(\036)p Ft(,)i(and)e(b)m(y)g(Cheb)m(yshev's)g(inequalit)m(y)f (and)h(the)h(shift)e(in)m(v)-5 b(ariance)118 5197 y(of)31 b Fr(\032)p Ft(,)759 5382 y Fr(\032)806 5308 y Fk(\010)859 5382 y Fl(j)p Fr(\036)938 5396 y Fm(x)1003 5382 y Fl(\000)20 b Fr(\036)1148 5396 y Fm(y)1189 5382 y Fl(j)26 b Fr(>)f(\014)1392 5308 y Fk(\011)1496 5382 y Fq(6)50 b Fr(\014)1673 5344 y Fj(\000)p Fo(1)1782 5382 y Fr(\032)p Ft(\()p Fl(j)p Fr(\036)1943 5396 y Fo(0)1984 5382 y Fl(j)p Ft(\))2075 5295 y Fk(X)2059 5500 y Fm(z)s Fj(2)p Fh(Z)2192 5481 y Fu(d)2253 5382 y Fl(j)p Fr(p)2324 5396 y Fm(t)2353 5382 y Ft(\()p Fr(x;)15 b(z)t Ft(\))22 b Fl(\000)e Fr(p)2720 5396 y Fm(t)2749 5382 y Ft(\()p Fr(y)s(;)15 b(z)t Ft(\))p Fl(j)g Fr(:)p eop %%Page: 9 9 9 8 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)287 b(9)118 366 y Ft(Since)29 b Fr(t)i Ft(is)e(arbitrary)-8 b(,)30 b(\(2.23\))i(follo)m(ws)e(from)1223 544 y(lim)1202 600 y Fm(t)p Fj(!1)1415 457 y Fk(X)1399 662 y Fm(z)s Fj(2)p Fh(Z)1532 643 y Fu(d)1593 544 y Fl(j)p Fr(p)1664 558 y Fm(t)1693 544 y Ft(\()p Fr(x;)15 b(z)t Ft(\))22 b Fl(\000)e Fr(p)2060 558 y Fm(t)2089 544 y Ft(\()p Fr(y)s(;)15 b(z)t Ft(\))p Fl(j)52 b Ft(=)e(0)15 b Fr(:)853 b Ft(\(2.24\))118 803 y(If)25 b Fr(p)g Ft(has)g(\014nite)f(range)h(and)g(is)f (irreducible,)f(\(2.24\))28 b(follo)m(ws)c(form)h(the)g(standard)f(lo)s (cal)h(limit)e(theorem)118 911 y(estimates,)29 b(just)e(as)h(in)e(the)i (case)g(of)g(the)g(simple)d(random)i(w)m(alk,)h(cf.)40 b(e.g.)h([25)q(],)29 b(Corollary)d(1.2.3.)42 b(F)-8 b(or)118 1018 y(a)30 b(general)f(irreducible)d(random)j(w)m(alk)g(one)g(can)h (use)f(the)g(follo)m(wing)f(argumen)m(t,)i(cf.)41 b([8)q(].)f(Let)30 b Fr(K)i(>)25 b Ft(0)118 1126 y(b)s(e)30 b(a)h(large)f(n)m(um)m(b)s(er) f(suc)m(h)h(that)h Fr(\016)e Fl(2)c Ft(\(0)p Fr(;)15 b Ft(1\))32 b(if)1465 1304 y Fr(\016)55 b Ft(=)1769 1217 y Fk(X)1681 1418 y Fm(x)p Fo(:)11 b Fj(j)p Fm(x)p Fj(j)19 b Ff(6)g Fm(K)2020 1304 y Fr(p)p Ft(\(0)p Fr(;)c(x)p Ft(\))g Fr(:)118 1571 y Ft(De\014ne)31 b(t)m(w)m(o)g(homogeneous)g (transitions)e Fr(r)s Ft(\()p Fr(x;)15 b(y)s Ft(\))31 b(and)f Fr(s)p Ft(\()p Fr(x;)15 b(y)s Ft(\),)31 b(b)m(y)397 1733 y Fr(r)s Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)f Fr(\016)866 1696 y Fj(\000)p Fo(1)976 1733 y Fr(p)p Ft(\()p Fr(x;)15 b(y)s Ft(\))g(1)1292 1752 y Fj(fj)p Fm(x)p Fj(\000)p Fm(y)r Fj(j)21 b Ff(6)e Fm(K)5 b Fj(g)1713 1733 y Fr(;)116 b(s)p Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)g(\(1)21 b Fl(\000)e Fr(\016)s Ft(\))2549 1696 y Fj(\000)p Fo(1)2660 1733 y Fr(p)p Ft(\()p Fr(x;)c(y)s Ft(\))g(1)2976 1752 y Fj(fj)p Fm(x)p Fj(\000)p Fm(y)r Fj(j)p Fm(>K)5 b Fj(g)3357 1733 y Fr(:)118 1892 y Ft(Notice)39 b(that)f(since)e Fr(p)h Ft(is)g(irreducible,)f Fr(K)44 b Ft(can)38 b(b)s(e)f(c)m(hosen)h (in)e(suc)m(h)h(a)h(w)m(a)m(y)g(that)g Fr(r)i Ft(is)c(irreducible.)118 2000 y(If)c Fr(r)252 2014 y Fm(t)314 2000 y Ft(and)g Fr(s)536 2014 y Fm(t)598 2000 y Ft(are)h(the)g(corresp)s(onding)d(con)m (tin)m(uous)i(time)h(k)m(ernels,)f(then)h(it)f(is)f(easy)-8 b(,)35 b(using)c(e.g.)48 b(the)118 2108 y(F)-8 b(ourier)30 b(transform,)g(to)h(see)g(that)1118 2280 y Fr(p)1164 2294 y Fm(t)1194 2280 y Ft(\()p Fr(x;)15 b(z)t Ft(\))52 b(=)1602 2193 y Fk(X)1575 2398 y Fm(z)1611 2379 y Fe(0)1633 2398 y Fj(2)p Fh(Z)1730 2379 y Fu(d)1791 2280 y Fr(r)1832 2295 y Fm(\016)r(t)1895 2280 y Ft(\()p Fr(x;)15 b(z)2068 2242 y Fj(0)2092 2280 y Ft(\))g Fr(s)2185 2298 y Fo(\(1)p Fj(\000)p Fm(\016)r Fo(\))p Fm(t)2394 2280 y Ft(\()p Fr(z)2475 2242 y Fj(0)2499 2280 y Fr(;)g(z)t Ft(\))g Fr(:)118 2534 y Ft(W)-8 b(e)32 b(ha)m(v)m(e)567 2606 y Fk(X)551 2811 y Fm(z)s Fj(2)p Fh(Z)684 2792 y Fu(d)744 2692 y Fl(j)p Fr(p)815 2706 y Fm(t)845 2692 y Ft(\()p Fr(x;)15 b(z)t Ft(\))22 b Fl(\000)d Fr(p)1211 2706 y Fm(t)1241 2692 y Ft(\()p Fr(y)s(;)c(z)t Ft(\))p Fl(j)52 b Fq(6)1688 2606 y Fk(X)1633 2811 y Fm(z)s(;z)1725 2792 y Fe(0)1746 2811 y Fj(2)p Fh(Z)1843 2792 y Fu(d)1904 2692 y Fl(j)p Fr(r)1970 2707 y Fm(\016)r(t)2034 2692 y Ft(\()p Fr(x;)15 b(z)2207 2655 y Fj(0)2231 2692 y Ft(\))20 b Fl(\000)g Fr(r)2418 2707 y Fm(\016)r(t)2482 2692 y Ft(\()p Fr(y)s(;)15 b(z)2651 2655 y Fj(0)2675 2692 y Ft(\))p Fl(j)g Fr(s)2793 2711 y Fo(\(1)p Fj(\000)p Fm(\016)r Fo(\))p Fm(t)3002 2692 y Ft(\()p Fr(z)3083 2655 y Fj(0)3107 2692 y Fr(;)g(z)t Ft(\))1522 2964 y(=)1670 2877 y Fk(X)1643 3082 y Fm(z)1679 3063 y Fe(0)1701 3082 y Fj(2)p Fh(Z)1798 3063 y Fu(d)1859 2964 y Fl(j)p Fr(r)1925 2979 y Fm(\016)r(t)1988 2964 y Ft(\()p Fr(x;)g(z)2161 2926 y Fj(0)2186 2964 y Ft(\))20 b Fl(\000)g Fr(r)2373 2979 y Fm(\016)r(t)2437 2964 y Ft(\()p Fr(y)s(;)15 b(z)2606 2926 y Fj(0)2630 2964 y Ft(\))p Fl(j)739 b Ft(\(2.25\))118 3223 y(No)m(w)31 b(\(2.24\))i(follo)m(ws)c(from)h(\(2.25\))j(since)d Fr(r)i Ft(has)f(\014nite)e(range.)p 3595 3223 4 62 v 3599 3165 55 4 v 3599 3223 V 3653 3223 4 62 v 118 3410 a(2.5.)54 b Fy(Non)33 b(ferromagnetic)f(mo)s(dels.)45 b Ft(Here)29 b(w)m(e)g(consider)f(some)h(examples)f(of)h(p)s(oten)m(tials)f Fr(J)37 b Ft(sat-)118 3518 y(isfying)28 b(conditions)h(\(2.1\))k(and)c (\(2.2\))j(but)e(suc)m(h)g(that)1574 3609 y Fk(X)1556 3814 y Fm(x)p Fj(2)p Fh(Z)1693 3795 y Fu(d)1739 3695 y Fr(J)9 b Ft(\(0)p Fr(;)15 b(x)p Ft(\))52 b Fl(6)p Ft(=)e(0)1206 b(\(2.26\))118 3954 y(Supp)s(ose)39 b Fr(\021)45 b Ft(in)39 b(\(2.4\))k(is)d(suc)m(h)h(that)h Fl(N)56 b Ft(=)43 b Fl(;)p Ft(,)h(cf.)73 b(\(2.8\).)h(Then)40 b(one)i(can)f(sho)m(w)g (\(see)h(e.g.)74 b([20)q(],)118 4062 y(Corollary)28 b(\(13.40\)\))33 b(that)e(the)e(unique)f(elemen)m(t)i(of)g Fl(G)35 b Ft(with)29 b(uniformly)e(b)s(ounded)g(\014rst)i(momen)m(ts)h(is)118 4170 y Fr(P)13 b Ft(.)46 b(Moreo)m(v)m(er)35 b Fr(P)45 b Ft(is)31 b(h)m(yp)s(ercon)m(tractiv)m(e)i(in)e(this)g(case,)j(cf.)47 b(app)s(endix)30 b(B.)i(If)g Fr(p)g Ft(has)g(long)g(range,)h(e.g.)118 4278 y Fr(p)d Ft(satis\014es)g(\(2.20\))j(for)d(some)h Fr(\013)26 b(>)g Ft(0,)31 b(one)g(can)g(actually)f(sho)m(w)g(that)h Fl(G)g Ft(=)26 b Fl(f)p Fr(P)13 b Fl(g)p Ft(.)42 b(This)28 b(is)i(due)g(to)h(the)118 4386 y(fact)g(that)g(condition)e(\(2.17\))k (forces)e(all)e(Gibbs)g(measures)h(to)h(b)s(e)f(strongly)g(temp)s (ered,)g(cf.)40 b([12)r(,)30 b(20)q(].)218 4595 y(More)i(in)m (teresting)e(is)h(the)g(case)i Fl(N)40 b(6)p Ft(=)26 b Fl(;)p Ft(.)44 b(Let)32 b Fr(\022)1879 4609 y Fo(0)1945 4595 y Fl(2)26 b(N)45 b Ft(\(b)m(y)31 b(symmetry)-8 b(,)32 b(this)e(implies)f Fl(\000)p Fr(\022)3360 4609 y Fo(0)3425 4595 y Fl(2)e(N)13 b Ft(\).)118 4705 y(Then)36 b(it)g(is)g(easy)h(to)g (see)h(that)f(there)g(exists)f Fr(m)g Fl(2)f Fn(Z)2010 4672 y Fm(d)2083 4705 y Ft(suc)m(h)h(that)h Fr(\022)2540 4719 y Fo(0)2615 4705 y Ft(=)e Fr(\031)s Ft(\()p Fr(m)2891 4719 y Fo(1)2931 4705 y Fr(;)15 b(:)g(:)g(:)i(;)e(m)3213 4720 y Fm(d)3253 4705 y Ft(\),)39 b(and)d(the)118 4813 y(co)s(e\016cien)m(ts)31 b Fr(\021)623 4827 y Fm(x)698 4813 y Ft(ha)m(v)m(e)g(the)g(form)1444 5059 y Fr(\021)1489 5073 y Fm(x)1559 5059 y Ft(=)25 b(cos)1791 4985 y Fk(\000)1833 5059 y Fr(\031)1951 4945 y Fm(d)1903 4972 y Fk(X)1912 5168 y Fm(i)p Fo(=1)2050 5059 y Fr(m)2130 5073 y Fm(i)2173 5059 y Fr(x)2225 5073 y Fm(i)2253 4985 y Fk(\001)2310 5059 y Fr(;)118 5315 y Ft(for)30 b(an)m(y)h Fr(x)25 b Fl(2)g Fn(Z)657 5282 y Fm(d)723 5315 y Ft(suc)m(h)30 b(that)h Fr(p)p Ft(\(0)p Fr(;)15 b(x)p Ft(\))27 b Fr(>)e Ft(0.)41 b(Moreo)m(v)m(er,)33 b(all)c(oscillating)g(con\014gurations) 1157 5480 y Fr(\033)1209 5494 y Fm(x)1278 5480 y Ft(=)c Fr(\013)31 b Ft(cos)16 b(\()p Fr(\022)1678 5494 y Fo(0)1738 5480 y Fl(\001)k Fr(x)p Ft(\))c Fr(;)116 b(x)25 b Fl(2)g Fn(Z)2255 5443 y Fm(d)2306 5480 y Fr(;)15 b(\013)26 b Fl(2)f Fn(R)18 b Fr(;)p eop %%Page: 10 10 10 9 bop 118 45 a Fo(10)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(are)40 b(elemen)m(ts)g(of)g Fl(H)850 380 y Fm(J)899 366 y Ft(.)69 b(W)-8 b(e)41 b(no)m(w)e(giv)m(e)i(an)e (example)h(of)g(translation)e(in)m(v)-5 b(arian)m(t)39 b(Gibbs)g(measures)118 474 y(whic)m(h)c(are)i(not)g(mixtures)e(of)i (translation)e(in)m(v)-5 b(arian)m(t)36 b(extremal)h(Gibbs)d(states.)61 b(Suc)m(h)35 b(cases)j(w)m(ere)118 582 y(\014rst)25 b(p)s(oin)m(ted)f (out)i(b)m(y)f(Rozano)m(v,)k([31)q(].)39 b(Since)24 b(\(2.26\))k (implies)23 b(that)j Fr(P)38 b Ft(is)25 b(the)g(unique)f(shift)g(in)m (v)-5 b(arian)m(t)118 690 y(elemen)m(t)39 b(of)f Fl(G)627 704 y Fo(ext)727 690 y Ft(,)i(it)e(will)d(b)s(e)j(enough)f(to)i (exhibit)e(a)h(shift)f(in)m(v)-5 b(arian)m(t)37 b Fr(\026)h Fl(2)f(G)44 b Ft(with)37 b Fr(\026)g Fl(6)p Ft(=)h Fr(P)13 b Ft(.)64 b(The)118 798 y(follo)m(wing)37 b(example)g(sho)m(ws)h(m)m (uc)m(h)g(more,)j(namely)c(that)i(for)f(an)m(y)g Fr(L)g(>)g Ft(0,)i(w)m(e)f(can)f(can)g(\014nd)f Fr(\026)h Fl(2)118 906 y(G)29 b(\\)23 b(M)394 873 y Fm(S)394 930 y Fo(1)445 906 y Ft(\(\012\))35 b(with)f Fr(\026)p Ft(\()p Fr(\036)972 873 y Fo(2)972 930 y(0)1012 906 y Ft(\))f Fr(>)g(L)p Ft(.)55 b(Let)36 b Fr(T)46 b(>)33 b Ft(0)i(and)g(let)g Fr(\032)2142 920 y Fm(T)2232 906 y Ft(b)s(e)f(the)i(cen)m(tered)g (Gaussian)e(\014eld)g(with)118 1014 y(sp)s(ectral)c(densit)m(y)1470 1213 y(^)-53 b Fr(\032)1509 1227 y Fm(T)1604 1213 y Ft(=)1725 1152 y Fr(T)p 1725 1192 66 4 v 1735 1275 a Ft(2)1816 1213 y(\()p Fr(\016)1891 1228 y Fm(\022)1924 1237 y Fg(0)1984 1213 y Ft(+)20 b Fr(\016)2115 1228 y Fj(\000)p Fm(\022)2203 1237 y Fg(0)2242 1213 y Ft(\))15 b Fr(;)118 1396 y Ft(or,)31 b(in)e(other)h(w)m(ords)g Fr(\032)906 1410 y Fm(T)992 1396 y Ft(is)f(the)i(cen)m(tered)g(Gaussian)e(\014eld)g(with)h(co)m(v) -5 b(ariance)1297 1554 y Fr(\032)1344 1568 y Fm(T)1399 1554 y Ft(\()p Fr(\033)1486 1568 y Fm(x)1530 1554 y Fr(;)15 b(\033)1622 1568 y Fm(y)1664 1554 y Ft(\))25 b(=)g Fr(T)j Ft(cos)16 b Fr(\022)2081 1568 y Fo(0)2140 1554 y Fl(\001)21 b Ft(\()p Fr(x)f Fl(\000)g Fr(y)s Ft(\))118 1711 y(\(cf.)41 b(e.g.)h([20)q(],)31 b(Prop)s(osition)d(\(13.A7\))33 b(to)e(c)m(hec)m(k)h(that)f Fr(\032)2081 1725 y Fm(T)2167 1711 y Ft(is)e(w)m(ell)g(de\014ned\).)40 b(Then,)30 b(for)g(an)m(y)h Fr(T)38 b(>)25 b Ft(0,)1519 1915 y Fr(\026)1574 1929 y Fm(T)1669 1915 y Ft(=)1780 1792 y Fk(Z)1886 1915 y Fr(\015)1938 1878 y Fm(\033)1985 1915 y Fr(\032)2032 1929 y Fm(T)2087 1915 y Ft(\()p Fr(d\033)s Ft(\))1170 b(\(2.27\))118 2114 y(is)44 b(a)h(cen)m(tered)h(Gaussian)f(\014eld,)i (since)d(it)h(is)f(a)h(con)m(v)m(olution)g(of)g(cen)m(tered)h(Gaussian) e(measures.)118 2222 y(Moreo)m(v)m(er)33 b(it)d(is)f(translation)g(in)m (v)-5 b(arian)m(t,)30 b(since)g(its)g(co)m(v)-5 b(ariance)1112 2380 y Fr(G)1183 2394 y Fm(T)1239 2380 y Ft(\()p Fr(x;)15 b(y)s Ft(\))26 b(=)f Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))22 b(+)d Fr(T)44 b Ft(cos)15 b Fr(\022)2240 2394 y Fo(0)2300 2380 y Fl(\001)20 b Ft(\()p Fr(x)h Fl(\000)f Fr(y)s Ft(\))15 b Fr(;)118 2539 y Ft(is)38 b(homogeneous)h(on)f Fn(Z)981 2506 y Fm(d)1017 2539 y Ft(.)65 b(T)-8 b(o)39 b(sho)m(w)f(that)h Fr(\026)1742 2553 y Fm(T)1835 2539 y Fl(2)g(G)k Ft(w)m(e)c(ha)m(v)m(e)h (to)f(sho)m(w)f Fr(\032)2791 2553 y Fm(T)2846 2539 y Ft(\()p Fl(H)2958 2553 y Fm(J)3007 2539 y Ft(\))h(=)g(1.)65 b(But)38 b(this)118 2647 y(follo)m(ws)30 b(from)321 2805 y Fr(\032)368 2819 y Fm(T)423 2731 y Fk(\000)465 2805 y Fl(j)522 2718 y Fk(X)505 2923 y Fm(y)r Fj(2)p Fh(Z)640 2904 y Fu(d)685 2805 y Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fr(\033)1006 2819 y Fm(y)1049 2805 y Fl(j)1074 2767 y Fo(2)1113 2731 y Fk(\001)1196 2805 y Ft(=)40 b Fr(T)1432 2718 y Fk(X)1388 2923 y Fm(y)r(;z)s Fj(2)p Fh(Z)1577 2904 y Fu(d)1623 2805 y Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fr(J)9 b Ft(\()p Fr(x;)15 b(z)t Ft(\))g(cos)j Fr(\022)2356 2819 y Fo(0)2416 2805 y Fl(\001)i Ft(\()p Fr(y)j Fl(\000)d Fr(z)t Ft(\))41 b(=)f Fr(T)2991 2782 y Ft(^)2969 2805 y Fr(J)10 b Ft(\()p Fr(\022)3107 2819 y Fo(0)3146 2805 y Ft(\))3181 2767 y Fo(2)3261 2805 y Ft(=)40 b(0)15 b Fr(;)118 3082 y Ft(for)24 b(an)m(y)h Fr(x)g Fl(2)g Fn(Z)645 3049 y Fm(d)681 3082 y Ft(.)39 b(W)-8 b(e)26 b(remark)e(that)h(in)e(the)i(case)h(of)e(Theorem)g(2.1)i (the)f(measure)f Fr(\026)3000 3096 y Fm(T)3079 3082 y Ft(can)h(b)s(e)f(realized)118 3190 y(only)31 b(for)g Fr(\022)502 3204 y Fo(0)568 3190 y Ft(=)c(0,)32 b(cf.)44 b(remark)31 b(1,)i(and)d Fr(\032)1544 3204 y Fm(T)1631 3190 y Ft(is)g(concen)m(trated)j(on)f(the)f(constan)m(t)i (con\014gurations.)43 b(On)118 3298 y(the)32 b(other)g(hand)f(if)f Fr(\021)h Fl(\021)d Ft(1)k(but)f Fr(p)g Ft(is)g(not)h(irreducible,)d (there)j(exists)f(0)d Fl(6)p Ft(=)g Fr(\022)2759 3312 y Fo(0)2825 3298 y Fl(2)f(N)45 b Ft(and)31 b(w)m(e)h(see)h(that)118 3406 y(the)e(measure)f(\(2.27\))j(violates)d(the)g(claim)g(of)g (Theorem)h(2.1.)817 3647 y(3.)51 b Fs(Thermod)n(ynamics)32 b(and)i(V)-11 b(aria)-6 b(tional)33 b(Principle)218 3809 y Ft(In)42 b(this)f(section)i(w)m(e)g(pro)m(v)m(e)h(a)f(v)m(ersion)f (of)h(Gibbs')e(v)-5 b(ariational)42 b(principle.)74 b(W)-8 b(e)44 b(\014rst)d(pro)m(v)m(e)j(a)118 3917 y(basic)c(thermo)s(dynamic) f(iden)m(tit)m(y)h(\(cf.)72 b(Prop)s(osition)38 b(3.1\))k(expressing)d (relativ)m(e)i(en)m(trop)m(y)g(densit)m(y)118 4025 y(as)g(a)f (di\013erence)g(of)h(free)f(energies.)71 b(In)39 b(the)i(case)g(of)g(b) s(ounded)d(in)m(teractions,)43 b(this)c(is)g(a)i(classical)118 4133 y(result)34 b(whic)m(h)f(go)s(es)i(bac)m(k)h(to)f(F\177)-45 b(ollmer's)34 b(pap)s(er)f([17)q(].)54 b(F)-8 b(or)35 b(Gaussian)f(random)g(\014elds)f(the)i(iden)m(tit)m(y)118 4241 y(w)m(as)d(pro)m(v)m(en)g(b)m(y)g(K)s(\177)-48 b(unsc)m(h)30 b(under)g(a)i(strong)g(assumption)e(on)i(the)g(deca)m(y)h(of)e(the)h(p) s(oten)m(tial,)g(cf.)45 b([22)q(],)118 4349 y(Theorem)32 b(2.11.)46 b(W)-8 b(e)33 b(sho)m(w)f(that)h(in)d(our)i(setting)f(the)i (result)d(is)h(true)h(with)e(no)i(assumption)e(on)i(the)118 4457 y(deca)m(y)g(of)e Fr(J)9 b Ft(,)31 b(i.e.)40 b(only)30 b(assuming)f(\(2.1\))j(and)e(\(2.2\).)118 4642 y(3.1.)54 b Fy(Thermo)s(dynamics.)44 b Ft(F)-8 b(or)32 b(an)m(y)e Fr(N)35 b Fl(2)25 b Fn(Z)1770 4656 y Fo(+)1855 4642 y Ft(let)1370 4804 y(\003)1433 4818 y Fm(N)1526 4804 y Ft(=)g([)p Fl(\000)p Fr(N)r(=)p Ft(2)p Fr(;)15 b(N)r(=)p Ft(2\))2123 4767 y Fm(d)2187 4804 y Fl(\\)20 b Fn(Z)2333 4767 y Fm(d)2384 4804 y Fr(:)118 4964 y Ft(W)-8 b(e)33 b(will)c(alw)m(a)m(ys)j(assume)f Fr(N)41 b Ft(is)31 b(an)g(ev)m(en)h (in)m(teger,)h(so)e(that)h(\003)2299 4978 y Fm(N)2398 4964 y Ft(has)f(side)g Fr(N)41 b Ft(and)31 b Fl(j)p Ft(\003)3125 4978 y Fm(N)3193 4964 y Fl(j)c Ft(=)g Fr(N)3426 4931 y Fm(d)3466 4964 y Ft(.)44 b(W)-8 b(e)118 5072 y(de\014ne)30 b(the)g(sets)h(of)g(translation)e(in)m(v)-5 b(arian)m(t)30 b(measures)776 5229 y Fl(K)845 5243 y Fm(L)948 5229 y Ft(=)1069 5155 y Fk(\010)1122 5229 y Fr(\026)25 b Fl(2)g(M)1397 5191 y Fm(S)1397 5251 y Fo(1)1448 5229 y Ft(\(\012\))p Fr(;)92 b(\026)p Ft(\()p Fr(!)1851 5191 y Fo(2)1848 5251 y(0)1890 5229 y Ft(\))26 b Fq(6)f Fr(L)2109 5155 y Fk(\011)2162 5229 y Fr(;)167 b Fl(K)2423 5243 y Fj(1)2543 5229 y Ft(=)2697 5143 y Fk([)2659 5340 y Fm(L)20 b Ff(>)f Fo(0)2867 5229 y Fl(K)2936 5243 y Fm(L)3475 5229 y Ft(\(3.1\))118 5480 y(A)31 b(tigh)m(tness)f(argumen)m(t)h(sho)m(ws)f(that)h Fl(K)1533 5494 y Fm(L)1616 5480 y Ft(is)e(compact)j(in)d(the)h(w)m(eak) h(top)s(ology)g(for)f(an)m(y)h Fr(L)25 b Fq(>)g Ft(0.)p eop %%Page: 11 11 11 10 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(11)218 366 y Ft(Giv)m(en)24 b(t)m(w)m(o)j(measures)d Fr(\026;)15 b(\027)6 b Ft(,)26 b(let)f Fr(\026)1404 380 y Fm(N)1471 366 y Fr(;)15 b(\027)1556 380 y Fm(N)1648 366 y Ft(denote)26 b(their)e(marginals)f(on)i(\003)2741 380 y Fm(N)2808 366 y Ft(.)39 b(The)24 b Fi(r)-5 b(elative)29 b(entr)-5 b(opy)118 474 y Ft(of)31 b Fr(\026)f Ft(with)f(resp)s(ect)h (to)h Fr(\027)36 b Ft(on)31 b(\003)1207 488 y Fm(N)1304 474 y Ft(is)f(de\014ned)f(b)m(y)1016 729 y Fr(H)1092 743 y Fm(N)1159 729 y Ft(\()p Fr(\026)p Fl(j)p Fr(\027)6 b Ft(\))51 b(=)1532 542 y Fk(8)1532 624 y(<)1532 788 y(:)1613 592 y(R)1704 665 y Fr(d\026)1806 679 y Fm(N)1888 665 y Ft(log)2031 604 y Fr(d\026)2133 618 y Fm(N)p 2031 644 170 4 v 2036 728 a Fr(d\027)2128 742 y Fm(N)2301 665 y Ft(if)29 b Fr(\026)2439 679 y Fm(N)2506 665 y Fr(<)-37 b(<\027)2656 679 y Fm(N)1613 832 y Ft(+)p Fl(1)526 b Ft(otherwise)3475 729 y(\(3.2\))118 988 y(Let)28 b(also)e Fr(H)532 1002 y Fm(N)599 988 y Ft(\()p Fr(\026)p Ft(\))h(denote)h(the)f (relativ)m(e)g(en)m(trop)m(y)h(of)f Fr(\026)f Ft(with)g(resp)s(ect)h (to)g(Leb)s(esgue)g(measure)g(on)g(\003)3569 1002 y Fm(N)3636 988 y Ft(.)118 1098 y(The)j Fi(entr)-5 b(opy)34 b(density)d Ft(of)g Fr(\026)25 b Fl(2)g(M)1319 1065 y Fm(S)1319 1122 y Fo(1)1370 1098 y Ft(\(\012\))31 b(is)e(de\014ned)g(as)1344 1291 y Fr(s)p Ft(\()p Fr(\026)p Ft(\))50 b(=)h Fl(\000)i Ft(lim)1770 1351 y Fm(N)7 b Fj(!1)2053 1229 y Ft(1)p 2014 1270 124 4 v 2014 1353 a Fr(N)2097 1327 y Fm(d)2167 1291 y Fr(H)2243 1305 y Fm(N)2310 1291 y Ft(\()p Fr(\026)p Ft(\))1040 b(\(3.3\))118 1472 y(A)30 b(general)g(subadditivit)m(y)c (argumen)m(t)31 b(sho)m(ws)e(that)h(this)f(limit)e(alw)m(a)m(ys)k (exists,)e(although)g(it)h(ma)m(y)g(b)s(e)118 1580 y(equal)39 b(to)h Fl(\0001)p Ft(.)67 b(Moreo)m(v)m(er,)44 b Fr(s)39 b Ft(is)f Fi(a\016ne)i Ft(\(see)h(e.g.)68 b(c)m(hapter)40 b(15)g(of)g([20)q(]\).)68 b(The)39 b Fi(internal)j(ener)-5 b(gy)118 1688 y(density)31 b Ft(of)g Fr(\026)25 b Fl(2)g(K)766 1702 y Fj(1)871 1688 y Ft(is)k(de\014ned)g(as)1018 1874 y Fr(e)p Ft(\()p Fr(\026)p Ft(\))51 b(=)1368 1813 y(1)p 1368 1853 46 4 v 1368 1937 a(2)1492 1874 y(lim)1453 1935 y Fm(N)7 b Fj(!1)1737 1813 y Ft(1)p 1698 1853 124 4 v 1698 1937 a Fr(N)1781 1910 y Fm(d)1926 1788 y Fk(X)1866 1985 y Fm(x;y)r Fj(2)p Fo(\003)2059 1996 y Fu(N)2132 1874 y Fr(J)i Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fr(\026)p Ft(\()p Fr(\036)2545 1888 y Fm(x)2590 1874 y Fr(\036)2644 1888 y Fm(y)2686 1874 y Ft(\))g Fr(:)714 b Ft(\(3.4\))118 2124 y(The)30 b(ergo)s(dic)g(theorem)g(implies)e(\(cf.)41 b([22)q(],)31 b(Theorem)f(2.2\))i(that)e(for)g Fr(\026)25 b Fl(2)g(K)2785 2138 y Fj(1)2891 2124 y Ft(the)30 b(limit)e(exists)i (and)118 2232 y(equals)1280 2418 y Fr(e)p Ft(\()p Fr(\026)p Ft(\))52 b(=)1630 2357 y(1)p 1630 2397 46 4 v 1630 2480 a(2)1717 2332 y Fk(X)1701 2537 y Fm(y)r Fj(2)p Fh(Z)1835 2518 y Fu(d)1880 2418 y Fr(J)9 b Ft(\(0)p Fr(;)15 b(y)s Ft(\))p Fr(\026)p Ft(\()p Fr(\036)2286 2432 y Fo(0)2328 2418 y Fr(\036)2382 2432 y Fm(y)2423 2418 y Ft(\))g Fr(:)977 b Ft(\(3.5\))118 2671 y(The)30 b Fi(pr)-5 b(essur)g(e)32 b Ft(is)d(de\014ned)g(b)m(y)1386 2857 y Fr(p)51 b Ft(=)89 b(lim)1604 2918 y Fm(N)7 b Fj(!1)1887 2795 y Ft(1)p 1848 2836 124 4 v 1848 2919 a Fr(N)1931 2893 y Fm(d)2032 2857 y Ft(log)17 b Fr(Z)2234 2819 y Fo(0)2227 2879 y(\003)2276 2890 y Fu(N)2353 2857 y Fr(;)1097 b Ft(\(3.6\))118 3051 y(where)33 b Fr(Z)453 3018 y Fo(0)446 3078 y(\003)495 3089 y Fu(N)590 3051 y Ft(is)g(giv)m(en)h(b)m(y)f(\(2.14\))j(with)c Fr(\034)40 b Ft(=)31 b(0.)50 b(By)34 b(Theorem)f(2.5)i(in)d([22)q(])i (the)g(ab)s(o)m(v)m(e)g(limit)e(exists)118 3161 y(and)e(equals)1005 3351 y Fr(p)40 b Ft(=)1212 3289 y(1)p 1212 3330 46 4 v 1212 3413 a(2)1283 3351 y(log)17 b(\(2)p Fr(\031)s Ft(\))41 b(+)1748 3289 y(1)p 1748 3330 V 1748 3413 a(2)1897 3289 y(1)p 1814 3330 212 4 v 1814 3413 a(\(2)p Fr(\031)s Ft(\))1984 3387 y Fm(d)2051 3227 y Fk(Z)2101 3433 y Fo(\()p Fj(\000)p Fm(\031)r(;\031)r Fo(])2309 3414 y Fu(d)2364 3351 y Ft(log)16 b Fr(f)10 b Ft(\()p Fr(\022)s Ft(\))p Fr(d\022)718 b Ft(\(3.7\))118 3584 y(where)38 b Fr(f)47 b Ft(=)613 3561 y(^)591 3584 y Fr(J)650 3551 y Fj(\000)p Fo(1)782 3584 y Ft(is)37 b(the)i(sp)s(ectral)e(densit)m(y)h(of)g Fr(P)51 b Ft(\(cf.)65 b(\(2.7\)\).)i(W)-8 b(e)39 b(de\014ne)e(the)i Fi(r)-5 b(elative)41 b(entr)-5 b(opy)118 3692 y(density)31 b Ft(of)g Fr(\026)f Ft(w.r.t.)41 b Fr(\027)36 b Ft(as)1286 3879 y Fr(h)p Ft(\()p Fr(\026)p Fl(j)p Fr(\027)6 b Ft(\))51 b(=)89 b(lim)1711 3939 y Fm(N)7 b Fj(!1)1995 3817 y Ft(1)p 1956 3858 124 4 v 1956 3941 a Fr(N)2039 3915 y Fm(d)2109 3879 y Fr(H)2185 3893 y Fm(N)2252 3879 y Ft(\()p Fr(\026)p Fl(j)p Fr(\027)f Ft(\))15 b Fr(:)982 b Ft(\(3.8\))118 4091 y Fy(Prop)s(osition)36 b(3.1.)46 b Fi(The)33 b(function)g Fr(h)p Ft(\()15 b Fl(\001)g(j)p Fr(P)e Ft(\))34 b Fi(is)f(wel)5 b(l)33 b(de\014ne)-5 b(d)34 b(on)f Fl(M)2572 4058 y Fm(S)2572 4115 y Fo(1)2623 4091 y Ft(\(\012\))p Fi(,)f(and)849 4323 y Fr(h)p Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))51 b(=)1295 4167 y Fk(\()1368 4261 y Ft(+)p Fl(1)595 b Fi(if)33 b Fr(\026)25 b Fl(2)f(M)2488 4228 y Fm(S)2488 4285 y Fo(1)2539 4261 y Ft(\(\012\))d Fl(n)g(K)2831 4275 y Fj(1)1368 4391 y Fr(e)p Ft(\()p Fr(\026)p Ft(\))36 b Fl(\000)f Fr(s)p Ft(\()p Fr(\026)p Ft(\))h(+)f Fr(p)92 b Fi(if)33 b Fr(\026)25 b Fl(2)f(K)2448 4405 y Fj(1)3475 4323 y Ft(\(3.9\))118 4553 y Fi(Mor)-5 b(e)g(over,)34 b Fr(h)p Ft(\()15 b Fl(\001)g(j)p Fr(P)e Ft(\))34 b Fi(is)f(a\016ne)g(on)g Fl(K)1402 4567 y Fj(1)1477 4553 y Fi(.)118 4717 y(Pr)-5 b(o)g(of.)43 b Ft(Let)30 b Fr(\026)25 b Fl(2)f(M)835 4731 y Fo(1)875 4717 y Ft(\(\012\).)41 b(W)-8 b(e)30 b(c)m(hec)m(k)h(the)e(\014rst)g(line)e(of)i(\(3.9\).)42 b(An)29 b(alternativ)m(e)h(expression)e(for)h Fr(H)3595 4731 y Fm(N)118 4824 y Ft(is)h(giv)m(en)g(b)m(y)g(\(cf.)41 b(e.g.)h(Lemma)31 b(\(3.2.13\))i(of)d([11)r(]\))1144 4979 y Fr(H)1220 4993 y Fm(N)1287 4979 y Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))51 b(=)f(sup)1725 5058 y Fm( )1833 4906 y Fk(\010)1886 4979 y Fr(\026)p Ft(\()p Fr( )s Ft(\))36 b Fl(\000)f Ft(log)16 b Fr(P)d Ft(\(e)2493 4942 y Fm( )2547 4979 y Ft(\))2582 4906 y Fk(\011)3429 4979 y Ft(\(3.10\))118 5205 y(where)22 b(the)h(suprem)m(um)e(is)g(tak)m(en)j(o)m(v)m(er)g(b)s (ounded)c Fl(F)1895 5219 y Fo(\003)1944 5230 y Fu(N)2006 5205 y Fl(\000)p Ft(measurable)i(functions.)36 b(Let)23 b Fr(\013)j Fl(2)f Ft(\(0)p Fr(;)15 b(\033)3469 5172 y Fo(2)3510 5205 y Fr(=)p Ft(2\),)118 5327 y(so)29 b(that)g(log)16 b Fr(P)d Ft(\(e)701 5294 y Fm(\013\036)788 5271 y Fg(2)788 5315 y(0)828 5327 y Ft(\))26 b Fl(\021)f Fr(c)p Ft(\()p Fr(\013)p Ft(\))h Fr(<)f Fl(1)p Ft(.)40 b(Then)28 b(c)m(ho)s(osing)g Fr( )2092 5341 y Fm(T)2147 5327 y Ft(\()p Fr(\036)p Ft(\))e(=)f Fr(\013)p Ft(\()p Fr(\036)2540 5294 y Fo(2)2540 5351 y(0)2597 5327 y Fl(^)15 b Fr(T)e Ft(\),)45 b Fr(T)38 b(>)25 b Ft(0)j(in)f(\(3.10\))k(giv)m(es)1224 5480 y Fr(H)1300 5494 y Fm(N)1367 5480 y Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))51 b Fq(>)g Fr(\013\026)p Ft(\()p Fr(\036)1963 5443 y Fo(2)1963 5503 y(0)2023 5480 y Fl(^)20 b Fr(T)13 b Ft(\))35 b Fl(\000)g Fr(c)p Ft(\()p Fr(\013)p Ft(\))15 b Fr(:)p eop %%Page: 12 12 12 11 bop 118 45 a Fo(12)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(Letting)i Fr(T)38 b Fl(!)25 b(1)p Ft(,)30 b(this)f(implies)f Fr(H)1353 380 y Fm(N)1420 366 y Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))26 b Fq(>)f Fr(\013\026)p Ft(\()p Fr(\036)1965 333 y Fo(2)1965 391 y(0)2005 366 y Ft(\))c Fl(\000)e Fr(c)p Ft(\()p Fr(\013)p Ft(\).)43 b(Hence,)31 b Fr(h)p Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))26 b(=)f Fl(1)31 b Ft(if)e Fr(\026)35 b(=)-55 b Fl(2)25 b(K)3516 380 y Fj(1)3591 366 y Ft(.)218 580 y(W)-8 b(e)35 b(turn)d(to)j(the)f(pro)s(of)f(of)h(the)g(iden)m(tit) m(y)f(in)g(the)h(second)g(line)e(of)i(\(3.9\).)53 b(W)-8 b(e)35 b(ma)m(y)f(assume)g(that)118 688 y Fr(\026)173 702 y Fm(N)269 688 y Ft(has)28 b(a)h(densit)m(y)-8 b(,)29 b(otherwise)f(the)h(iden)m(tit)m(y)f(holds)f(trivially)-8 b(,)27 b(since)h Fr(s)p Ft(\()p Fr(\026)p Ft(\))e(=)f Fl(\0001)j Ft(in)f(this)h(case.)41 b(Let)118 796 y Fr( )177 810 y Fm(N)275 796 y Ft(b)s(e)30 b(the)g(densit)m(y)g(of)g Fr(P)1030 810 y Fm(N)1098 796 y Ft(,)h(and)e(call)h Fr( )1559 763 y Fo(0)1556 823 y Fm(N)1654 796 y Ft(the)g(densit)m(y)g(of)g Fr(\015)2279 763 y Fo(0)2274 823 y(\003)2323 834 y Fu(N)2385 796 y Ft(,)h(cf.)41 b(\(2.13\).)i(W)-8 b(e)31 b(ha)m(v)m(e)883 976 y Fr(H)959 990 y Fm(N)1026 976 y Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))51 b(=)f Fr(H)1495 990 y Fm(N)1562 976 y Ft(\()p Fr(\026)p Ft(\))36 b Fl(\000)f Fr(\026)p Ft(\(log)17 b Fr( )2114 939 y Fo(0)2111 999 y Fm(N)2178 976 y Ft(\))36 b(+)f Fr(\026)p Ft(\(log)17 b Fr( )2640 939 y Fo(0)2637 999 y Fm(N)2704 976 y Fr( )2766 938 y Fj(\000)p Fo(1)2763 1005 y Fm(N)2861 976 y Ft(\))118 1137 y(Since)1211 1319 y(lim)1172 1379 y Fm(N)7 b Fj(!1)1455 1257 y Ft(1)p 1416 1298 124 4 v 1416 1381 a Fr(N)1499 1355 y Fm(d)1570 1319 y Fr(\026)p Ft(\(log)17 b Fr( )1855 1281 y Fo(0)1852 1341 y Fm(N)1919 1319 y Ft(\))51 b(=)f Fl(\000)15 b Fr(p)35 b Fl(\000)g Fr(e)p Ft(\()p Fr(\026)p Ft(\))15 b Fr(;)118 1513 y Ft(w)m(e)31 b(need)f(to)h(pro)m(v)m(e)1330 1712 y(lim)1291 1772 y Fm(N)7 b Fj(!1)1575 1650 y Ft(1)p 1536 1691 V 1536 1774 a Fr(N)1619 1748 y Fm(d)1689 1712 y Fr(\026)p Ft(\(log)17 b Fr( )1974 1674 y Fo(0)1971 1734 y Fm(N)2039 1712 y Fr( )2101 1673 y Fj(\000)p Fo(1)2098 1740 y Fm(N)2195 1712 y Ft(\))51 b(=)f(0)15 b Fr(:)942 b Ft(\(3.11\))118 1906 y(Recall)30 b(that)643 2087 y Fr( )705 2049 y Fo(0)702 2109 y Fm(N)770 2087 y Fr( )832 2049 y Fj(\000)p Fo(1)829 2116 y Fm(N)926 2087 y Ft(\()p Fr(\036)p Ft(\))52 b(=)e(det)1364 2013 y Fk(\000)1406 2087 y Fr(G)1477 2101 y Fm(N)1545 2087 y Ft(\000)1602 2049 y Fj(\000)p Fo(1)1602 2116 y Fm(N)1696 2013 y Fk(\001)1738 2032 y Fo(1)p Fm(=)p Fo(2)1878 2087 y Ft(exp)2032 1986 y Fk(\020)2107 2087 y Fl(\000)2207 2025 y Ft(1)p 2207 2066 46 4 v 2207 2149 a(2)2293 2013 y Fk(\012)2336 2087 y Fr(\036;)15 b Ft(\(\000)2522 2049 y Fj(\000)p Fo(1)2522 2116 y Fm(N)2637 2087 y Fl(\000)20 b Fr(G)2799 2049 y Fj(\000)p Fo(1)2799 2116 y Fm(N)2894 2087 y Ft(\))p Fr(\036)2983 2013 y Fk(\013)3042 1986 y(\021)3111 2087 y Fr(;)118 2279 y Ft(where)30 b Fr(G)452 2293 y Fm(N)545 2279 y Fl(\021)25 b Fr(G)712 2293 y Fo(\003)761 2304 y Fu(N)853 2279 y Ft(and)30 b(\000)1087 2293 y Fm(N)1180 2279 y Fl(\021)25 b Ft(\000)1333 2293 y Fo(\003)1382 2304 y Fu(N)1443 2279 y Ft(.)41 b(The)30 b(argumen)m(t)g(of)h(Theorem)f(2.5)i (in)d([22)q(],)i(sho)m(ws)f(that)1270 2485 y(lim)1231 2546 y Fm(N)7 b Fj(!1)1514 2424 y Ft(1)p 1475 2464 124 4 v 1475 2548 a Fr(N)1558 2521 y Fm(d)1644 2485 y Ft(log)17 b(det)1918 2412 y Fk(\000)1960 2485 y Fr(G)2031 2499 y Fm(N)2098 2485 y Ft(\000)2155 2447 y Fj(\000)p Fo(1)2155 2514 y Fm(N)2250 2412 y Fk(\001)2342 2485 y Ft(=)50 b(0)15 b Fr(:)118 2679 y Ft(W)-8 b(e)32 b(are)e(left)h(with)1154 2861 y(lim)1115 2921 y Fm(N)7 b Fj(!1)1398 2799 y Ft(1)p 1359 2840 V 1359 2923 a Fr(N)1442 2897 y Fm(d)1513 2861 y Fr(\026)1568 2787 y Fk(\002)15 b(\012)1664 2861 y Fr(\036;)g Ft(\(\000)1850 2822 y Fj(\000)p Fo(1)1850 2889 y Fm(N)1965 2861 y Fl(\000)20 b Fr(G)2127 2822 y Fj(\000)p Fo(1)2127 2889 y Fm(N)2222 2861 y Ft(\))p Fr(\036)2311 2787 y Fk(\013)15 b(\003)2458 2861 y Ft(=)50 b(0)15 b Fr(:)765 b Ft(\(3.12\))118 3055 y(Let)31 b Fr(x;)15 b(y)28 b Fl(2)d Ft(\003)595 3069 y Fm(N)663 3055 y Ft(.)40 b(Using)30 b(the)g(expansion)g(of)g (Lemma)h(A.1,)g(w)m(e)g(ha)m(v)m(e)587 3285 y(\000)644 3247 y Fj(\000)p Fo(1)644 3313 y Fm(N)738 3285 y Ft(\()p Fr(x;)15 b(y)s Ft(\))37 b Fl(\000)e Fr(G)1162 3247 y Fj(\000)p Fo(1)1162 3313 y Fm(N)1257 3285 y Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)1670 3171 y Fj(1)1640 3198 y Fk(X)1639 3393 y Fm(n)p Fo(=1)1847 3198 y Fk(X)1802 3400 y Fm(z)1835 3409 y Fg(1)1878 3400 y Fm(=)-43 b Fj(2)p Fo(\003)1966 3411 y Fu(N)2039 3285 y Fl(\001)15 b(\001)g(\001)2209 3198 y Fk(X)2160 3400 y Fm(z)2193 3408 y Fu(n)2243 3400 y Fm(=)-43 b Fj(2)p Fo(\003)2331 3411 y Fu(N)2419 3285 y Fr(Q)p Ft(\()p Fr(x;)15 b(z)2660 3299 y Fo(1)2700 3285 y Ft(\))g Fl(\001)g(\001)g(\001)i Fr(Q)p Ft(\()p Fr(z)3021 3299 y Fm(n)3069 3285 y Fr(;)e(y)s Ft(\))118 3544 y(Since)29 b Fr(\026)d Fl(2)e(K)590 3558 y Fj(1)665 3544 y Ft(,)31 b(b)m(y)f(Sc)m(h)m(w)m(arz')i(inequalit)m(y)-8 b(,)539 3641 y Fk(\014)539 3696 y(\014)569 3719 y Fr(\026)624 3645 y Fk(\002)15 b(\012)720 3719 y Fr(\036;)g Ft(\(\000)906 3680 y Fj(\000)p Fo(1)906 3747 y Fm(N)1021 3719 y Fl(\000)20 b Fr(G)1183 3680 y Fj(\000)p Fo(1)1183 3747 y Fm(N)1278 3719 y Ft(\))p Fr(\036)1367 3645 y Fk(\013)15 b(\003)1463 3641 y(\014)1463 3696 y(\014)1544 3719 y Fq(6)1740 3632 y Fk(X)1680 3829 y Fm(x;y)r Fj(2)p Fo(\003)1873 3840 y Fu(N)1946 3641 y Fk(\014)1946 3696 y(\014)1976 3719 y Ft(\000)2033 3680 y Fj(\000)p Fo(1)2033 3747 y Fm(N)2127 3719 y Ft(\()p Fr(x;)g(y)s Ft(\))21 b Fl(\000)f Fr(G)2520 3680 y Fj(\000)p Fo(1)2520 3747 y Fm(N)2615 3719 y Ft(\()p Fr(x;)15 b(y)s Ft(\))2825 3641 y Fk(\014)2825 3696 y(\014)2871 3719 y Fl(j)p Fr(\026)p Ft(\()p Fr(\036)3040 3733 y Fm(x)3085 3719 y Fr(\036)3139 3733 y Fm(y)3180 3719 y Ft(\))p Fl(j)778 4038 y Fq(6)50 b Fr(\026)p Ft(\()p Fr(\036)1043 4000 y Fo(2)1043 4060 y(0)1083 4038 y Ft(\))1193 3951 y Fk(X)1133 4148 y Fm(x;y)r 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Fn(P)2822 4976 y Fm(x)2866 4889 y Fk(\000)2908 4962 y Fr(\030)2948 4976 y Fo(1)3022 4962 y Fr(=)-55 b Fl(2)25 b Ft(\003)3161 4976 y Fm(N)3229 4889 y Fk(\001)3285 4962 y Fr(:)118 5217 y Ft(W)-8 b(e)32 b(then)e(ha)m(v)m(e)848 5301 y Fk(\014)848 5356 y(\014)878 5378 y Fr(\026)933 5305 y Fk(\002)15 b(\012)1029 5378 y Fr(\036;)g Ft(\(\000)1215 5340 y Fj(\000)p Fo(1)1215 5407 y Fm(N)1330 5378 y Fl(\000)20 b Fr(G)1492 5340 y Fj(\000)p Fo(1)1492 5407 y Fm(N)1587 5378 y Ft(\))p Fr(\036)1676 5305 y Fk(\013)15 b(\003)1772 5301 y(\014)1772 5356 y(\014)1853 5378 y Fq(6)50 b Fr(\026)p Ft(\()p Fr(\036)2118 5341 y Fo(2)2118 5401 y(0)2158 5378 y Ft(\))2239 5292 y Fk(X)2208 5489 y Fm(x)p Fj(2)p Fo(\003)2344 5500 y Fu(N)2446 5292 y Fk(X)2417 5493 y Fm(z)11 b(=)-43 b Fj(2)p Fo(\003)2549 5504 y Fu(N)2636 5378 y Fr(p)p Ft(\()p Fr(x;)15 b(z)t Ft(\))g Fr(:)p eop %%Page: 13 13 13 12 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 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b(ener)-5 b(gy)1468 1017 y Fr(f)10 b Ft(\()p Fr(\026)p Ft(\))50 b(=)g Fr(e)p Ft(\()p Fr(\026)p Ft(\))36 b Fl(\000)f Fr(s)p Ft(\()p Fr(\026)p Ft(\))118 1168 y(if)30 b(and)g(only)g(if)g Fr(\026)g Ft(is)g(a)h(translation)f (in)m(v)-5 b(arian)m(t)30 b(Gibbs)f(state.)44 b(In)30 b(view)g(of)h(Prop)s(osition)d(3.1)k(this)e(can)118 1276 y(b)s(e)j(turned)f(in)m(to)i(a)g(statemen)m(t)h(ab)s(out)e(relativ)m(e) h(en)m(trop)m(y)g(densit)m(y)-8 b(.)50 b(Let)34 b Fl(G)2733 1243 y Fm(S)2728 1300 y Fo(2)2814 1276 y Ft(=)c Fl(G)e(\\)22 b(K)3149 1290 y Fj(1)3224 1276 y Ft(,)34 b(the)g(set)g(of)118 1384 y(translation)29 b(in)m(v)-5 b(arian)m(t)30 b(Gibbs)f(states)i (with)e(\014nite)h(second)g(momen)m(t.)118 1549 y Fy(Theorem)k(3.2.)46 b Fi(L)-5 b(et)33 b Fr(\026)25 b Fl(2)g(M)1202 1516 y Fm(S)1202 1573 y Fo(1)1253 1549 y Ft(\(\012\))p Fi(.)42 b(Then,)1322 1708 y Fr(\026)25 b Fl(2)g(G)1547 1670 y Fm(S)1542 1730 y Fo(2)1699 1708 y Fl(\()-15 b(\))101 b Fr(h)p Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))51 b(=)f(0)972 b(\(3.14\))118 1888 y Fi(Pr)-5 b(o)g(of.)43 b Ft(Let)28 b Fr(\026)d Fl(2)g(G)784 1855 y Fm(S)779 1913 y Fo(2)835 1888 y Ft(,)k(and)e(denote)i(b)m(y)f Fr(\032)d Fl(2)g(M)1745 1902 y Fo(1)1784 1888 y Ft(\(\012\))k(its)e (extremal)h(decomp)s(osition,)g(i.e.)39 b(\(cf.)i(\(2.19\)\))1233 2088 y Fr(\026)50 b Ft(=)1459 1964 y Fk(Z)1580 2088 y Fr(\015)1632 2050 y Fm(\033)1694 2088 y Fr(\032)p Ft(\()p Fr(d\033)s Ft(\))15 b Fr(;)143 b(\032)p Ft(\()p Fl(H)2255 2102 y Fm(J)2305 2088 y Ft(\))25 b(=)g(1)15 b Fr(:)118 2281 y Ft(Note)32 b(that)f Fr(\026)25 b Fl(2)g(K)770 2295 y Fj(1)875 2281 y Ft(implies)j Fr(\032)d Fl(2)g(K)1411 2295 y Fj(1)1486 2281 y Ft(.)40 b(By)31 b(con)m(v)m(exit)m(y)1211 2474 y Fr(H)1287 2488 y Fm(N)1354 2474 y Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))51 b Fq(6)1763 2350 y Fk(Z)1884 2474 y Fr(H)1960 2488 y Fm(N)2027 2474 y Ft(\()p Fr(\015)2114 2436 y Fm(\033)2161 2474 y Fl(j)p Fr(P)13 b Ft(\))i Fr(\032)p Ft(\()p Fr(d\033)s Ft(\))g Fr(:)863 b Ft(\(3.15\))118 2667 y(Moreo)m(v)m(er,)33 b(b)m(y)d(explicit)f(computation,)1325 2856 y Fr(H)1401 2870 y Fm(N)1468 2856 y Ft(\()p Fr(\015)1555 2819 y Fm(\033)1602 2856 y Fl(j)p Fr(P)13 b Ft(\))51 b(=)1916 2795 y(1)p 1916 2835 46 4 v 1916 2919 a(2)2001 2783 y Fk(\012)2044 2856 y Fr(\033)n(;)15 b(G)2205 2818 y Fj(\000)p Fo(1)2205 2885 y Fm(N)2301 2856 y Fr(\033)2356 2783 y Fk(\013)2429 2856 y Fr(:)118 3032 y Ft(F)-8 b(or)31 b Fr(\033)e Fl(2)c(H)525 3046 y Fm(J)604 3032 y Ft(w)m(e)30 b(can)h(write)729 3120 y Fk(\012)772 3194 y Fr(\033)n(;)15 b(G)933 3155 y Fj(\000)p Fo(1)933 3222 y Fm(N)1028 3194 y Fr(\033)1083 3120 y Fk(\013)1177 3194 y Ft(=)1298 3120 y Fk(\012)1341 3194 y Fr(\033)n(;)g Ft(\()p Fr(G)1537 3155 y Fj(\000)p Fo(1)1537 3222 y Fm(N)1653 3194 y Fl(\000)20 b Fr(J)1794 3208 y Fm(N)1861 3194 y Ft(\))p Fr(\033)1951 3120 y Fk(\013)2030 3194 y Fl(\000)2167 3107 y Fk(X)2136 3304 y Fm(x)p Fj(2)p Fo(\003)2272 3315 y Fu(N)2375 3107 y Fk(X)2345 3309 y Fm(y)10 b(=)-43 b Fj(2)p Fo(\003)2478 3320 y Fu(N)2551 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Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fl(j)g(j)p Fr(\036)1346 5035 y Fm(y)1390 5021 y Fl(j)1415 4947 y Fk(\001)1497 5021 y Ft(=)40 b Fr(\026)p Ft(\()p Fl(j)p Fr(\036)1777 5035 y Fo(0)1817 5021 y Fl(j)p Ft(\))1925 4935 y Fk(X)1908 5140 y Fm(y)r Fj(2)p Fh(Z)2043 5121 y Fu(d)2088 5021 y Fl(j)p Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fl(j)42 b Fr(<)e Fl(1)15 b Fr(;)142 b(x)25 b Fl(2)g Fn(Z)3061 4984 y Fm(d)3112 5021 y Fr(:)118 5293 y Ft(Let)31 b Fr(g)j Ft(b)s(e)c(a)g(lo)s(cal,)g(b)s(ounded)f(con)m(tin)m(uous)h(function,)f (and)h(let)g(\003)25 b Fl(\032\032)g Fn(Z)2633 5260 y Fm(d)2669 5293 y Ft(.)41 b(W)-8 b(e)31 b(will)d(sho)m(w)1526 5444 y Fr(\026)p Ft(\()p Fr(\015)1663 5458 y Fo(\003)1716 5444 y Ft(\()p Fr(g)s Ft(\)\))53 b(=)d Fr(\026)p Ft(\()p Fr(g)s Ft(\))15 b Fr(:)1177 b Ft(\(3.17\))p eop %%Page: 14 14 14 13 bop 118 45 a Fo(14)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(Let)i Fr(V)45 b Fl(\032\032)25 b Fn(Z)611 333 y Fm(d)678 366 y Ft(b)s(e)k(a)i(large)f(set)h(con)m (taining)f(b)s(oth)g(\003)g(and)g(the)h(supp)s(ort)d(of)j Fr(g)s Ft(.)41 b(W)-8 b(e)32 b(write)1658 570 y Fr(f)1703 584 y Fm(V)1814 570 y Ft(=)1947 508 y Fr(d\026)2049 522 y Fm(V)p 1945 549 167 4 v 1945 632 a Fr(dP)2050 646 y Fm(V)118 776 y Ft(for)e(the)h(densit)m(y)e(of)i Fr(\026)f Ft(w.r.t.)41 b Fr(P)i Ft(on)30 b Fr(V)21 b Ft(.)40 b(Then,)30 b(since)g Fr(\026)p Ft(\()p Fr(g)s Ft(\))c(=)f Fr(P)13 b Ft(\()p Fr(f)2465 790 y Fm(V)2526 776 y Fr(g)s Ft(\),)31 b(w)m(e)g(ha)m(v)m(e)268 936 y Fl(j)p Fr(\026)p Ft(\()p Fr(\015)430 950 y Fo(\003)484 936 y Ft(\()p Fr(g)s Ft(\)\))37 b Fl(\000)e Fr(\026)p Ft(\()p Fr(g)s Ft(\))p Fl(j)52 b Fq(6)e Fl(j)p Fr(\026)p Ft(\()p Fr(\015)1309 950 y Fo(\003)1363 936 y Ft(\()p Fr(g)s Ft(\)\))37 b Fl(\000)e Fr(P)13 b Ft(\()p Fr(f)1808 950 y Fm(V)1869 936 y Fr(\015)1916 950 y Fo(\003)1969 936 y Ft(\()p Fr(g)s Ft(\)\))p Fl(j)37 b Ft(+)e Fl(j)p Fr(P)13 b Ft(\()p Fr(f)2464 950 y Fm(V)2525 936 y Fr(\015)2572 950 y Fo(\003)2625 936 y Ft(\()p Fr(g)s Ft(\)\))37 b Fl(\000)e Fr(P)13 b Ft(\()p Fr(f)3070 950 y Fm(V)3131 936 y Fr(g)s Ft(\))p Fl(j)i Fr(:)152 b Ft(\(3.18\))118 1102 y(T)-8 b(o)35 b(pro)m(v)m(e)g(\(3.17\))i(w)m(e)d(sho)m(w)g(that)h (b)s(oth)f(terms)g(in)f(the)i(r.h.s.)51 b(of)35 b(\(3.18\))i(can)d(b)s (e)g(made)g(arbitrarily)118 1210 y(small)26 b(b)m(y)h(c)m(ho)s(osing)f Fr(V)48 b Ft(large)27 b(enough.)39 b(T)-8 b(o)27 b(b)s(e)g(more)g (precise,)g(\(3.17\))j(is)c(a)h(consequence)h(of)f(the)g(next)118 1318 y(t)m(w)m(o)32 b(lemmas.)40 b(Note)32 b(that)f(Lemma)f(3.3)i(b)s (ecomes)e(trivial)f(if)g Fr(p)h Ft(has)g(\014nite)f(range.)118 1488 y Fy(Lemma)k(3.3.)46 b Fi(F)-7 b(or)34 b(any)f Fr(")25 b(>)g Ft(0)p Fi(,)33 b(ther)-5 b(e)34 b(exists)f Fr(N)i Ft(=)25 b Fr(N)10 b Ft(\()p Fr(")p Ft(\))26 b Fl(2)f Fn(Z)2362 1502 y Fo(+)2449 1488 y Fi(such)33 b(that)h(for)f(any)g Fr(V)46 b Fl(\033)25 b Ft(\003)3419 1502 y Fm(N)1283 1649 y Fl(j)p Fr(\026)p Ft(\()p Fr(\015)1445 1663 y Fo(\003)1498 1649 y Ft(\()p Fr(g)s Ft(\)\))38 b Fl(\000)d Fr(P)13 b Ft(\()p Fr(f)1944 1663 y Fm(V)2004 1649 y Fr(\015)2051 1663 y Fo(\003)2105 1649 y Ft(\()p Fr(g)s Ft(\)\))p Fl(j)52 b Fr(<)e(")933 b Ft(\(3.19\))118 1836 y Fy(Lemma)33 b(3.4.)46 b Fi(F)-7 b(or)35 b(any)g Fr(")29 b(>)g Ft(0)35 b Fi(and)g(for)g(al)5 b(l)35 b Fr(N)j Fl(2)29 b Fn(Z)2018 1850 y Fo(+)2072 1836 y Fi(,)35 b(ther)-5 b(e)35 b(exists)g Fr(V)49 b Ft(=)28 b Fr(V)20 b Ft(\()p Fr(";)15 b(N)10 b Ft(\))30 b Fl(\032\032)e Fn(Z)3388 1803 y Fm(d)3424 1836 y Fi(,)35 b(such)118 1944 y(that)f Fr(V)45 b Fl(\033)25 b Ft(\003)561 1958 y Fm(N)629 1944 y Fi(,)32 b(and)1308 2104 y Fl(j)p Fr(P)13 b Ft(\()p Fr(f)1484 2118 y Fm(V)1545 2104 y Fr(\015)1592 2118 y Fo(\003)1645 2104 y Ft(\()p Fr(g)s Ft(\)\))37 b Fl(\000)e Fr(P)13 b Ft(\()p Fr(f)2090 2118 y Fm(V)2151 2104 y Fr(g)s Ft(\))p Fl(j)51 b Fr(<)g(")957 b Ft(\(3.20\))118 2380 y Fi(Pr)-5 b(o)g(of)34 b(of)f(L)-5 b(emma)34 b(3.3)p Ft(.)41 b(Notice)32 b(that)769 2541 y Fr(P)13 b Ft(\()p Fr(f)920 2555 y Fm(V)980 2541 y Fr(\015)1027 2555 y Fo(\003)1080 2541 y Ft(\()p Fr(g)s Ft(\)\))53 b(=)d Fr(P)13 b Ft(\()p Fr(f)1556 2555 y Fm(V)1617 2541 y Fr(P)g Ft(\()p Fr(\015)1770 2555 y Fo(\003)1823 2541 y Ft(\()p Fr(g)s Ft(\))p Fl(jF)2029 2555 y Fm(V)2092 2541 y Ft(\)\))51 b(=)f Fr(\026)p Ft(\()p Fr(P)13 b Ft(\()p Fr(\015)2577 2555 y Fo(\003)2631 2541 y Ft(\()p Fr(g)s Ft(\))p Fl(jF)2837 2555 y Fm(V)2900 2541 y Ft(\)\))i Fr(:)419 b Ft(\(3.21\))118 2702 y(De\014ne)1473 2862 y Fr(\015)1525 2825 y Fm(\034)1520 2885 y Fo(\003)p Fm(;V)1649 2862 y Ft(\()p Fr(g)s Ft(\))27 b(=)e Fr(\015)1940 2818 y Fm(\034)1971 2829 y Fu(V)2024 2818 y Fo(0)2059 2831 y Fu(V)2109 2817 y(c)1935 2891 y Fo(\003)2149 2862 y Ft(\()p Fr(g)s Ft(\))15 b Fr(;)118 3027 y Ft(i.e.)38 b(w)m(e)24 b(put)f(0-b)s(oundary)f(conditions)g(outside)g(of)i Fr(V)c Ft(.)38 b(Note)25 b(that)f Fr(\015)2454 3041 y Fo(\003)p Fm(;V)2584 3027 y Ft(\()p Fr(g)s Ft(\))g(is)f(no)m(w)g(a)h Fl(F)3125 3041 y Fm(V)3186 3027 y Ft(-measurable)118 3135 y(function.)40 b(Let)30 b(us)g(\014x)g(a)h(sequence)f Fl(f)p Fr(V)1468 3149 y Fm(N)1536 3135 y Fl(g)p Ft(,)h(with)e Fr(V)1897 3149 y Fm(N)1990 3135 y Fl(\033)c Ft(\003)2149 3149 y Fm(N)2216 3135 y Ft(,)31 b Fr(N)k Fl(2)25 b Fn(Z)2531 3149 y Fo(+)2585 3135 y Ft(.)41 b(W)-8 b(e)32 b(de\014ne)1413 3296 y Fr(')1472 3310 y Fm(N)1565 3296 y Ft(=)25 b Fr(\015)1708 3310 y Fo(\003)1761 3296 y Ft(\()p Fr(g)s Ft(\))d Fl(\000)e Fr(\015)2037 3310 y Fo(\003)p Fm(;V)2147 3321 y Fu(N)2208 3296 y Ft(\()p Fr(g)s Ft(\))15 b Fr(:)118 3456 y Ft(Using)30 b(\(3.21\))i(w)m(e)f(see)g(that)g(for)f(an)m(y)h Fr(N)770 3617 y Fl(j)p Fr(\026)p Ft(\()p Fr(\015)932 3631 y Fo(\003)985 3617 y Ft(\()p Fr(g)s Ft(\)\))22 b Fl(\000)e Fr(P)13 b Ft(\()p Fr(f)1400 3631 y Fm(V)1441 3642 y Fu(N)1503 3617 y Fr(\015)1550 3631 y Fo(\003)1603 3617 y Ft(\()p Fr(g)s Ft(\)\))p Fl(j)27 b Fq(6)e Fr(\026)p Ft(\()p Fl(j)p Fr(')2076 3631 y Fm(N)2144 3617 y Fl(j)p Ft(\))c(+)f Fr(\026)p Ft(\()p Fr(P)13 b Ft(\()p Fl(j)p Fr(')2596 3631 y Fm(N)2665 3617 y Fl(j)i(jF)2795 3631 y Fm(V)2836 3642 y Fu(N)2899 3617 y Ft(\)\))g Fr(:)118 3778 y Ft(T)-8 b(o)31 b(pro)m(v)m(e)g(\(3.19\))i(it)d(is)f(then)h(su\016cien)m(t)g(to) h(sho)m(w)1577 3939 y(lim)1538 3999 y Fm(N)7 b Fj(!1)1772 3939 y Fr(\026)p Ft(\()p Fl(j)p Fr(')1946 3953 y Fm(N)2014 3939 y Fl(j)p Ft(\))26 b(=)f(0)1188 b(\(3.22\))118 4132 y(and)1381 4293 y(lim)1342 4354 y Fm(N)7 b Fj(!1)1577 4293 y Fr(\026)p Ft(\()p Fr(P)13 b Ft(\()p Fl(j)p Fr(')1857 4307 y Fm(N)1925 4293 y Fl(j)i(jF)2055 4307 y Fm(V)2096 4318 y Fu(N)2159 4293 y Ft(\)\))26 b(=)f(0)15 b Fr(:)993 b Ft(\(3.23\))118 4487 y(W)-8 b(e)25 b(ma)m(y)f(assume)g Fr(g)j Ft(to)d(b)s(e)f(a)h(b)s(ounded)e(Lipshitz-con)m(tin)m(uous)f(lo) s(cal)i(function.)38 b(F)-8 b(or)24 b(suc)m(h)f(a)h(function,)118 4595 y(de\014ne)592 4786 y Fl(k)p Fr(g)s Fl(k)728 4800 y Fo(L)829 4786 y Ft(=)65 b(sup)950 4872 y Fm(x)p Fj(2)p Fh(Z)1087 4853 y Fu(d)1157 4786 y Ft(sup)1325 4685 y Fk(n)1410 4724 y Fl(j)p Fr(g)s Ft(\()p Fr(\033)s Ft(\))22 b Fl(\000)e Fr(g)s Ft(\()p Fr(\021)s Ft(\))p Fl(j)p 1410 4765 500 4 v 1487 4848 a(j)p Fr(\033)1564 4862 y Fm(x)1628 4848 y Fl(\000)g Fr(\021)1764 4862 y Fm(x)1808 4848 y Fl(j)1935 4786 y Fr(;)66 b(\033)n(;)15 b(\021)29 b Fl(2)c Ft(\012)40 b(:)51 b Fr(\033)2510 4800 y Fm(y)2577 4786 y Ft(=)25 b Fr(\021)2718 4800 y Fm(y)2759 4786 y Fr(;)41 b Fl(8)15 b Fr(y)28 b Fl(6)p Ft(=)d Fr(x)3127 4685 y Fk(o)118 5063 y Ft(F)-8 b(rom)29 b(\(2.15\))h(and)e(\(2.16\))i(w)m(e)e (kno)m(w)g(that)h(the)f(measure)g Fr(\015)2169 4989 y Fm(\034)2200 5000 y Fu(V)2236 5017 y(N)2299 4989 y Fo(0)2334 5005 y Fu(V)2384 4988 y(c)2370 5031 y(N)2164 5091 y Fo(\003)2465 5063 y Ft(can)g(b)s(e)f(seen)i(as)f(the)g(measure)g Fr(\015)3613 5030 y Fm(\034)3608 5090 y Fo(\003)118 5178 y Ft(shifted)h(b)m(y)h(the) h(mean)f(v)-5 b(alues)30 b(v)m(ector)i Fr(\030)1527 5145 y Fm(N)1594 5178 y Ft(\()p Fr(\034)10 b Ft(\),)31 b(giv)m(en)g(b)m(y) 648 5378 y Fr(\030)692 5341 y Fm(N)688 5401 y(x)759 5378 y Ft(\()p Fr(\034)10 b Ft(\))51 b Fl(\021)g Fr(\015)1104 5305 y Fm(\034)1135 5316 y Fu(V)1171 5333 y(N)1233 5305 y Fo(0)1268 5320 y Fu(V)1318 5303 y(c)1304 5347 y(N)1099 5407 y Fo(\003)1371 5378 y Ft(\()p Fr(\036)1460 5392 y Fm(x)1504 5378 y Ft(\))36 b Fl(\000)f Fr(\015)1733 5341 y Fm(\034)1728 5401 y Fo(\003)1781 5378 y Ft(\()p Fr(\036)1870 5392 y Fm(x)1915 5378 y Ft(\))51 b(=)2123 5292 y Fk(X)2122 5489 y Fm(y)r Fj(2)p Fo(\003)2295 5292 y Fk(X)2270 5493 y Fm(z)11 b(=)-43 b Fj(2)p Fm(V)2394 5504 y Fu(N)2467 5378 y Ft(\000)2524 5392 y Fo(\003)2577 5378 y Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fr(J)9 b Ft(\()p Fr(y)s(;)15 b(z)t Ft(\))p Fr(\034)3090 5392 y Fm(z)p eop %%Page: 15 15 15 14 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(15)118 366 y Ft(W)-8 b(e)32 b(then)e(ha)m(v)m(e)199 547 y Fl(j)p Fr(')283 561 y Fm(N)351 547 y Ft(\()p Fr(\034)10 b Ft(\))p Fl(j)52 b Fq(6)684 424 y Fk(Z)790 547 y Fr(\015)842 510 y Fm(\034)837 570 y Fo(\003)890 547 y Ft(\()p Fr(d\036)p Ft(\))15 b Fl(j)p Fr(g)s Ft(\()p Fr(\036)23 b Ft(+)d Fr(\030)1394 510 y Fm(N)1461 547 y Ft(\()p Fr(\034)10 b Ft(\)\))22 b Fl(\000)e Fr(g)s Ft(\()p Fr(\036)p Ft(\))p Fl(j)548 756 y Fq(6)50 b Fl(k)p Fr(g)s Fl(k)805 770 y Fo(L)887 670 y Fk(X)885 867 y Fm(x)p Fj(2)p Fo(\003)1036 756 y Fl(j)p Fr(\030)1105 719 y Fm(N)1101 779 y(x)1172 756 y Ft(\()p Fr(\034)10 b Ft(\))p Fl(j)52 b Fq(6)e Fl(j)p Ft(\003)p Fl(j)1603 719 y Fo(2)1658 756 y Fl(k)p Fr(g)s Fl(k)1794 770 y Fo(L)1902 756 y Ft(sup)1874 835 y Fm(x;y)r Fj(2)p Fo(\003)2082 756 y Fl(j)p Ft(\000)2164 770 y Fo(\003)2217 756 y Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fl(j)32 b Ft(sup)2485 835 y Fm(y)r Fj(2)p Fo(\003)2661 670 y Fk(X)2636 871 y Fm(z)11 b(=)-43 b Fj(2)o Fm(V)2759 882 y Fu(N)2832 756 y Fl(j)p Fr(J)9 b Ft(\()p Fr(y)s(;)15 b(z)t Ft(\))p Fl(j)g(j)p Fr(\034)3225 770 y Fm(z)3267 756 y Fl(j)31 b Fr(:)81 b Ft(\(3.24\))118 1012 y(Let)31 b(us)f(\014x)1181 1169 y Fr(C)7 b Ft(\()p Fr(g)s(;)15 b Ft(\003\))27 b(=)e Fl(j)p Ft(\003)p Fl(j)1708 1131 y Fo(2)1763 1169 y Fl(k)p Fr(g)s Fl(k)1899 1183 y Fo(L)2007 1169 y Ft(sup)1979 1247 y Fm(x;y)r Fj(2)p Fo(\003)2187 1169 y Fl(j)p Ft(\000)2269 1183 y Fo(\003)2322 1169 y Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fl(j)g Fr(:)118 1390 y Ft(Using)30 b(the)g(shift)f(in)m(v)-5 b(ariance)30 b(of)g Fr(\026)g Ft(and)g(the)h(fact)g(that)g Fr(\026)25 b Fl(2)g(K)2298 1404 y Fj(1)2373 1390 y Ft(,)30 b(w)m(e)h(ha)m(v)m(e)997 1560 y Fr(\026)p Ft(\()p Fl(j)p Fr(')1171 1574 y Fm(N)1239 1560 y Fl(j)p Ft(\))51 b Fq(6)f Fr(C)7 b Ft(\()p Fr(g)s(;)15 b Ft(\003\))32 b(sup)1796 1638 y Fm(y)r Fj(2)p Fo(\003)1971 1474 y Fk(X)1946 1675 y Fm(z)11 b(=)-43 b Fj(2)p Fm(V)2070 1686 y Fu(N)2143 1560 y Fl(j)p Fr(J)9 b Ft(\()p Fr(y)s(;)15 b(z)t Ft(\))p Fl(j)g Fr(\026)p Ft(\()p Fl(j)p Fr(\036)2640 1574 y Fm(z)2682 1560 y Fl(j)p Ft(\))1350 1826 y(=)50 b Fr(C)7 b Ft(\()p Fr(g)s(;)15 b Ft(\003\))g Fr(\026)p Ft(\()p Fl(j)p Fr(\036)1946 1840 y Fo(0)1988 1826 y Fl(j)p Ft(\))31 b(sup)2081 1904 y Fm(y)r Fj(2)p Fo(\003)2256 1739 y Fk(X)2231 1941 y Fm(z)11 b(=)-43 b Fj(2)o Fm(V)2354 1952 y Fu(N)2427 1826 y Fl(j)p Fr(J)9 b Ft(\()p Fr(y)s(;)15 b(z)t Ft(\))p Fl(j)g Fr(:)649 b Ft(\(3.25\))118 2086 y(No)m(w,)31 b(the)g(r.h.s.)40 b(of)31 b(\(3.25\))h(v)-5 b(anishes)29 b(as)i Fr(N)k Fl(!)25 b(1)p Ft(,)31 b(and)f(\(3.22\))i(follo)m(ws.)218 2219 y(W)-8 b(e)31 b(turn)f(to)h(the)f(pro)s(of)g(of)g(\(3.23\).)43 b(F)-8 b(rom)31 b(\(3.24\))i(w)m(e)e(ha)m(v)m(e)646 2394 y Fr(\026)p Ft(\()p Fr(P)13 b Ft(\()p Fl(j)p Fr(')926 2408 y Fm(N)995 2394 y Fl(j)i(jF)1125 2408 y Fm(V)1166 2419 y Fu(N)1229 2394 y Ft(\)\))51 b Fq(6)f Fr(C)7 b Ft(\()p Fr(g)s(;)15 b Ft(\003\))32 b(sup)1796 2472 y Fm(y)r Fj(2)p Fo(\003)1971 2308 y Fk(X)1946 2509 y Fm(z)11 b(=)-43 b Fj(2)o Fm(V)2069 2520 y Fu(N)2142 2394 y Fl(j)p Fr(J)9 b Ft(\()p Fr(y)s(;)15 b(z)t Ft(\))p Fl(j)g Fr(\026)p Ft(\()p Fr(P)e Ft(\()p Fl(j)p Fr(\036)2745 2408 y Fm(z)2788 2394 y Fl(j)i(jF)2918 2408 y Fm(V)2959 2419 y Fu(N)3022 2394 y Ft(\)\))g Fr(:)297 b Ft(\(3.26\))118 2655 y(Let)31 b(us)f(\014x)f Fr(\034)36 b Fl(2)25 b 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y Fj(0)660 3287 y Fm(z)700 3265 y Fl(j)p Fr(\034)765 3279 y Fm(V)806 3290 y Fu(N)868 3265 y Ft(\))21 b Fl(\000)e Fr(P)13 b Ft(\()p Fr(\036)1174 3279 y Fm(z)1215 3265 y Fl(j)p Fr(\034)1280 3279 y Fm(V)1321 3290 y Fu(N)1383 3265 y Ft(\))p Fr(P)g Ft(\()p Fr(\036)1578 3227 y Fj(0)1578 3287 y Fm(z)1618 3265 y Fl(j)p Fr(\034)1683 3279 y Fm(V)1724 3290 y Fu(N)1786 3265 y Ft(\))26 b(=)f(\000)2000 3279 y Fm(V)2056 3256 y Fu(c)2041 3302 y(N)2103 3265 y Ft(\()p Fr(z)t(;)15 b(z)2270 3227 y Fj(0)2294 3265 y Ft(\))g Fr(;)142 b(z)t(;)15 b(z)2643 3227 y Fj(0)2693 3265 y Fl(2)25 b Fn(Z)2844 3227 y Fm(d)2900 3265 y Fl(n)c Fr(V)3019 3279 y Fm(N)3101 3265 y Fr(;)303 b Ft(\(3.28\))118 3422 y(where)1333 3631 y(\000)1390 3645 y Fm(V)1447 3622 y Fu(c)1431 3668 y(N)1493 3631 y Ft(\()p Fr(z)t(;)15 b(z)1660 3593 y Fj(0)1684 3631 y Ft(\))26 b(=)1872 3517 y Fj(1)1842 3545 y Fk(X)1841 3739 y Fm(n)p Fo(=0)2004 3631 y Fr(Q)2076 3593 y Fm(n)2076 3653 y(V)2133 3630 y Fu(c)2117 3676 y(N)2179 3631 y Ft(\()p Fr(z)t(;)15 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b(y)s Ft(\))38 b(b)m(y)g(its)f(absolute)g(v)-5 b(alue)37 b Fr(p)p Ft(\()p Fr(x;)15 b(y)s Ft(\))38 b(w)m(e)g(can)g(reduce)g (ourselv)m(es)f(to)h(the)g(ferromagnetic)g(case.)118 4795 y(Let)26 b Fn(P)331 4809 y Fm(z)371 4795 y Ft(,)h Fr(z)i Fl(2)c Fn(Z)645 4762 y Fm(d)692 4795 y Fl(n)10 b Fr(V)800 4809 y Fm(N)893 4795 y Ft(denote)26 b(the)g(probabilit)m(y)c (asso)s(ciated)k(to)h(the)e(random)g(w)m(alk)g Fr(\030)3054 4809 y Fo(0)3118 4795 y Ft(=)g Fr(z)t(;)15 b(\030)3340 4809 y Fo(1)3380 4795 y Fr(;)g(\030)3460 4809 y Fo(2)3500 4795 y Fr(;)g(:)g(:)g(:)118 4903 y Ft(with)29 b(transition)g Fr(p)p Ft(.)40 b(Let)31 b Fr(t)1047 4917 y Fm(N)1145 4903 y Ft(b)s(e)e(the)i(\014rst)f(hitting)f(time)h(of)g(the)h(set)g Fr(V)2573 4917 y Fm(N)2640 4903 y Ft(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)740 5073 y Fr(P)13 b Ft(\()p Fr(\036)900 5036 y Fo(2)900 5096 y Fm(z)955 5073 y Fl(j)p Fr(\034)1020 5087 y Fm(V)1061 5098 y Fu(N)1123 5073 y Ft(\))41 b(=)f(\000)1367 5087 y Fm(V)1423 5064 y Fu(c)1408 5110 y(N)1470 5073 y Ft(\()p Fr(z)t(;)15 b(z)t Ft(\))21 b(+)1784 4999 y Fk(\000)1867 4987 y(X)1841 5184 y Fm(y)r Fj(2)p Fm(V)1982 5161 y Fu(c)1966 5207 y(N)2067 4987 y Fk(X)2039 5184 y Fm(x)p Fj(2)p Fm(V)2167 5195 y Fu(N)2240 5073 y Ft(\000)2297 5087 y Fm(V)2354 5064 y Fu(c)2338 5110 y(N)2400 5073 y Ft(\()p Fr(z)t(;)15 b(y)s Ft(\))p Fr(J)9 b Ft(\()p Fr(y)s(;)15 b(x)p Ft(\))p Fr(\034)2913 5087 y Fm(x)2959 4999 y Fk(\001)3001 5022 y Fo(2)1184 5347 y Fq(6)40 b Fr(\033)1350 5309 y Fo(2)1410 5347 y Ft(+)1568 5260 y Fk(X)1500 5458 y Fm(x;x)1600 5439 y Fe(0)1622 5458 y Fj(2)p Fm(V)1710 5469 y Fu(N)1783 5347 y Fn(P)1838 5361 y Fm(z)1878 5347 y Ft(\()p Fr(\030)1953 5361 y Fm(t)1978 5372 y Fu(N)2066 5347 y Ft(=)25 b Fr(x)p Ft(\))p Fn(P)2304 5361 y Fm(z)2345 5347 y Ft(\()p Fr(\030)2420 5361 y Fm(t)2445 5372 y Fu(N)2532 5347 y Ft(=)g Fr(x)2680 5309 y Fj(0)2703 5347 y Ft(\))p Fl(j)p Fr(\034)2803 5361 y Fm(x)2848 5347 y Fr(\034)2888 5362 y Fm(x)2928 5343 y Fe(0)2954 5347 y Fl(j)15 b Fr(;)p eop %%Page: 16 16 16 15 bop 118 45 a Fo(16)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(where)35 b(w)m(e)i(ha)m(v)m(e)g(used)e(\(2.2\))j (to)e(b)s(ound)e(\000)1636 380 y Fm(V)1692 358 y Fu(c)1677 404 y(N)1739 366 y Ft(\()p Fr(z)t(;)15 b(z)t Ft(\).)58 b(W)-8 b(e)37 b(no)m(w)f(use)g(Sc)m(h)m(w)m(arz')h(inequalit)m(y)d(and) h(the)118 475 y(shift)29 b(in)m(v)-5 b(ariance)30 b(of)g Fr(\026)g Ft(to)h(write)312 635 y Fr(\026)367 562 y Fk(\000)408 635 y Fr(P)13 b Ft(\()p Fr(\036)568 598 y Fo(2)568 658 y Fm(z)624 635 y Fl(jF)714 649 y Fm(V)755 660 y Fu(N)817 635 y Ft(\))852 562 y Fk(\001)934 635 y Fq(6)41 b Fr(\033)1101 598 y Fo(2)1160 635 y Ft(+)1319 549 y Fk(X)1251 747 y Fm(x;x)1351 728 y Fe(0)1373 747 y Fj(2)p Fm(V)1461 758 y Fu(N)1534 635 y Fn(P)1589 649 y Fm(z)1629 635 y Ft(\()p Fr(\030)1704 649 y Fm(t)1729 660 y Fu(N)1817 635 y Ft(=)25 b Fr(x)p Ft(\))p Fn(P)2055 649 y Fm(z)2095 635 y Ft(\()p Fr(\030)2170 649 y Fm(t)2195 660 y Fu(N)2283 635 y Ft(=)g Fr(x)2431 598 y Fj(0)2454 635 y Ft(\))p Fr(\026)p Ft(\()p Fl(j)p Fr(\034)2644 649 y Fm(x)2689 635 y Fr(\034)2729 650 y Fm(x)2769 632 y Fe(0)2795 635 y Fl(j)p Ft(\))41 b Fq(6)f Fr(\033)3062 598 y Fo(2)3122 635 y Ft(+)20 b Fr(\026)p Ft(\()p Fr(\034)3353 598 y Fo(2)3343 658 y(0)3392 635 y Ft(\))15 b Fr(:)118 879 y Ft(Therefore,)30 b(b)m(y)h(Sc)m(h)m(w)m (arz')g(inequalit)m(y)779 1044 y Fr(\026)834 971 y Fk(\000)875 1044 y Fr(P)13 b Ft(\()p Fl(j)p Fr(\036)1060 1058 y Fm(z)1101 1044 y Fl(j)i(jF)1231 1058 y Fm(V)1272 1069 y Fu(N)1335 1044 y Ft(\))1370 971 y Fk(\001)1437 1044 y Fq(6)25 b Fr(\026)1588 971 y Fk(\000)1629 1044 y Fr(P)13 b Ft(\()p Fr(\036)1789 1007 y Fo(2)1789 1067 y Fm(z)1845 1044 y Fl(jF)1935 1058 y Fm(V)1976 1069 y Fu(N)2038 1044 y Ft(\))2073 971 y Fk(\001)2115 993 y Fo(1)p Fm(=)p Fo(2)2250 1044 y Fq(6)2346 971 y Fk(\000)2388 1044 y Fr(\033)2443 1007 y Fo(2)2503 1044 y Ft(+)20 b Fr(\026)p Ft(\()p Fr(\034)2734 1007 y Fo(2)2724 1067 y(0)2773 1044 y Ft(\))2808 971 y Fk(\001)2850 993 y Fo(1)p Fm(=)p Fo(2)2976 1044 y 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b Fj(n)p Fo(\003)1445 1797 y Ft(=)27 b Fr(d\026)1645 1815 y Fm(V)16 b Fj(n)p Fo(\003)1791 1797 y Fr(=dP)1941 1815 y Fm(V)h Fj(n)p Fo(\003)2119 1797 y Ft(is)31 b Fl(F)2277 1811 y Fo(\003)2326 1792 y Fu(c)2363 1797 y Fl(\000)p Ft(measurable,)g(the)i(DLR)f(prop)s(ert)m(y)118 1905 y(of)f Fr(P)43 b Ft(implies)1320 2051 y Fr(P)13 b Ft(\()p Fr(f)1471 2070 y Fm(V)j Fj(n)p Fo(\003)1616 2051 y Fr(\015)1663 2065 y Fo(\003)1716 2051 y Ft(\()p Fr(g)s Ft(\)\))53 b(=)d Fr(P)13 b Ft(\()p Fr(f)2192 2070 y Fm(V)i Fj(n)p Fo(\003)2337 2051 y Fr(g)s Ft(\))g Fr(;)118 2198 y Ft(and)30 b(w)m(e)h(can)f(write)850 2344 y Fl(j)p Fr(P)13 b Ft(\()p Fr(f)1026 2358 y Fm(V)1087 2344 y Fr(\015)1134 2358 y Fo(\003)1187 2344 y Ft(\()p Fr(g)s Ft(\)\))37 b Fl(\000)e Fr(P)13 b Ft(\()p Fr(f)1632 2358 y Fm(V)1693 2344 y Fr(g)s Ft(\))p Fl(j)52 b Fq(6)e Ft(2)15 b Fl(k)p Fr(g)s Fl(k)2168 2358 y Fj(1)2260 2344 y Fr(P)e Ft(\()p Fl(j)p Fr(f)2436 2363 y Fm(V)j Fj(n)p Fo(\003)2617 2344 y Fl(\000)35 b Fr(f)2768 2358 y Fm(V)2828 2344 y Fl(j)p Ft(\))15 b Fr(:)118 2495 y Ft(The)29 b(Lemma)g(no)m(w)h(follo)m(ws)e(from)h(a)g(classical)g (argumen)m(t)h(\(cf.)40 b([30)r(,)29 b(20)q(]\).)41 b(Precisely)-8 b(,)29 b(follo)m(wing)f([20)q(],)118 2602 y(Step)k(2)g(in)e(the)i(pro)s (of)f(of)h(Theorem)f(\(15.37\),)k(one)d(sho)m(ws)g(that)g(\(3.16\))i (implies)29 b(that)j(for)g(an)m(y)g Fr(")c(>)f Ft(0)118 2713 y(and)j(for)g(all)f Fr(N)35 b Fl(2)25 b Fn(Z)819 2727 y Fo(+)874 2713 y Ft(,)31 b(there)f(exists)g Fr(V)46 b Ft(=)25 b Fr(V)20 b Ft(\()p Fr(";)15 b(N)10 b Ft(\))27 b Fl(\032\032)d Fn(Z)2174 2680 y Fm(d)2210 2713 y Ft(,)31 b(suc)m(h)f(that)h Fr(V)45 b Fl(\033)25 b Ft(\003)2925 2727 y Fm(N)2993 2713 y Ft(,)30 b(and)1468 2859 y Fr(P)13 b Ft(\()p Fl(j)p Fr(f)1644 2873 y Fm(V)1741 2859 y Fl(\000)35 b Fr(f)1892 2878 y Fm(V)15 b Fj(n)p Fo(\003)2036 2859 y Fl(j)p Ft(\))52 b Fr(<)e(")1118 b Ft(\(3.30\))118 3006 y(This)29 b(completes)h(the)h(pro)s(of)f(of)g(Lemma)h(3.4.)p 3595 3006 V 3599 2948 55 4 v 3599 3006 V 3653 3006 4 62 v 1425 3192 a(4.)51 b Fs(Lar)n(ge)34 b(Devia)-6 b(tions)218 3354 y Ft(In)43 b(this)g(section)h(w)m(e)g(discuss)f(v)m(olume)g(order) h(large)g(deviations)f(of)h(the)g(empirical)e(\014eld)h(with)118 3461 y(resp)s(ect)33 b(to)h(the)f(measure)f Fr(P)13 b Ft(.)49 b(F)-8 b(rom)33 b(no)m(w)g(on,)g(in)f(addition)f(to)j(the)f (usual)e(assumptions)g(\(2.1\))k(and)118 3569 y(\(2.2\),)f(w)m(e)f (assume)e(that)i(the)f(coupling)e(co)s(e\016cien)m(ts)j(satisfy)e(the)h (deca)m(y)h(condition)d(\(2.5\).)47 b(W)-8 b(e)33 b(also)118 3677 y(assume)d Fl(N)39 b(6)p Ft(=)25 b Fl(;)p Ft(,)31 b(i.e.)41 b(w)m(e)30 b(fo)s(cus)g(on)h(the)f(massless)g(regime.)41 b(The)30 b(\(p)s(erio)s(dized\))e Fi(empiric)-5 b(al)34 b(\014eld)d Fr(R)3594 3691 y Fm(N)118 3785 y Ft(is)f(de\014ned)f(b)m(y) 1333 3970 y Fr(R)1402 3984 y Fm(N)1469 3970 y Ft(\()p Fr(\036)p Ft(\))52 b(=)1815 3908 y(1)p 1776 3949 124 4 v 1776 4032 a Fr(N)1859 4006 y Fm(d)1971 3884 y Fk(X)1939 4080 y Fm(x)p Fj(2)p Fo(\003)2075 4091 y Fu(N)2163 3970 y Fr(\016)2203 3985 y Fm(\022)2236 3993 y Fu(x)2287 3985 y Fm(\036)2329 3996 y Fu(N)2406 3970 y Fr(;)1044 b Ft(\(4.1\))118 4211 y(where)38 b Fr(\016)429 4226 y Fm(\030)505 4211 y Ft(stands)g(for)g(the)h(Dirac)f(measure)g(at)h Fr(\030)k Fl(2)38 b Ft(\012,)i Fr(\022)2200 4225 y Fm(x)2243 4211 y Fr(\036)f Ft(denotes)g(the)f(shifted)f(con\014guration)118 4326 y Fr(\036)172 4340 y Fj(\001)p Fo(+)p Fm(x)291 4326 y Ft(,)d(and)e Fr(\036)583 4340 y Fm(N)683 4326 y Ft(denotes)i(the)f (con\014guration)g(on)g(\003)1920 4340 y Fm(N)2020 4326 y Ft(p)s(erio)s(dized)d(to)k(the)g(whole)e(of)h Fn(Z)3167 4293 y Fm(d)3203 4326 y Ft(.)49 b(With)32 b(this)118 4434 y(de\014nition)27 b Fr(R)590 4448 y Fm(N)657 4434 y Ft(\()p Fr(\036)p Ft(\))f Fl(2)f(M)1002 4401 y Fm(S)1002 4458 y Fo(1)1053 4434 y Ft(\(\012\))30 b(for)f(all)f Fr(\036)e Fl(2)e Ft(\012,)30 b(and)f(the)g(map)g Fr(\036)c Fl(!)h Fr(R)2565 4448 y Fm(N)2632 4434 y Ft(\()p Fr(\036)p Ft(\))k(is)e Fl(F)2941 4448 y Fo(\003)2990 4459 y Fu(N)3052 4434 y Ft(-measurable.)40 b(It)118 4542 y(is)29 b(w)m(ell)f(kno)m(wn,)h (cf.)41 b(e.g.)h([21)q(],)30 b(that)g(as)g(far)f(as)h(the)f(large)h (deviation)e(b)s(eha)m(viour)g(is)h(concerned)h(\(4.1\))118 4650 y(is)g(equiv)-5 b(alen)m(t)29 b(to)i(\(1.4\).)43 b(Let)1065 4878 y Fl(I)7 b Ft(\()p Fr(\026)p Ft(\))50 b(=)1418 4723 y Fk(\()1492 4817 y Ft(+)15 b Fl(1)188 b Fr(\026)25 b Fl(2)f(M)2131 4831 y Fo(1)2171 4817 y Ft(\(\012\))d Fl(n)f(M)2502 4784 y Fm(S)2502 4841 y Fo(1)2553 4817 y Ft(\(\012\))1492 4946 y Fr(h)p Ft(\()p Fr(\026)p Fl(j)p Fr(P)13 b Ft(\))92 b Fr(\026)25 b Fl(2)f(M)2131 4913 y Fm(S)2131 4971 y Fo(1)2182 4946 y Ft(\(\012\))3475 4878 y(\(4.2\))118 5102 y(The)35 b(main)f(result)g(in)g(this)g(section) h(states)h(that)g Fr(R)1935 5116 y Fm(N)2037 5102 y Ft(satis\014es)f(a) g(w)m(eak)h(large)g(deviation)e(principle)118 5210 y(with)29 b(rate)i(function)e Fl(I)7 b Ft(.)40 b(Recall)30 b(the)h(de\014nition)d (\(3.1\))k(of)f(the)f(compact)i(sets)f Fl(K)2913 5224 y Fm(L)2990 5210 y Fl(\032)25 b(M)3195 5177 y Fm(S)3195 5234 y Fo(1)3246 5210 y Ft(\(\012\).)118 5372 y Fy(Theorem)34 b(4.1.)46 b Fi(F)-7 b(or)47 b(any)g Fr(L)j(>)g Ft(0)p Fi(,)g(the)c(r)-5 b(estriction)48 b(of)e Fl(I)53 b Fi(to)47 b Fl(K)2529 5386 y Fm(L)2627 5372 y Fi(is)f(lower)i(semi-c)-5 b(ontinuous.)118 5480 y(Mor)g(e)g(over,)34 b(the)f(fol)5 b(lowing)33 b(b)-5 b(ounds)34 b(hold:)p eop %%Page: 17 17 17 16 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(17)243 366 y Fl(\017)42 b Fi(F)-7 b(or)34 b(any)f(close)-5 b(d)34 b(set)e Fr(F)39 b Fl(\032)25 b(M)1382 380 y Fo(1)1421 366 y Ft(\(\012\))p Fi(,)33 b(for)g(any)h Fr(L)25 b(>)g Ft(0)p Fi(,)827 561 y Ft(lim)15 b(sup)864 639 y Fm(N)7 b Fj(!1)1184 500 y Ft(1)p 1145 540 124 4 v 1145 624 a Fr(N)1228 597 y Fm(d)1314 561 y Ft(log)17 b Fr(P)1518 488 y Fk(\000)1559 561 y Fr(R)1628 575 y Fm(N)1721 561 y Fl(2)25 b Fr(F)33 b Fl(\\)20 b(K)2048 575 y Fm(L)2100 488 y Fk(\001)2167 561 y Fq(6)70 b Fl(\000)136 b Ft(inf)2425 622 y Fm(\026)p Fj(2)p Fm(F)10 b Fj(\\K)2670 633 y Fu(L)2731 561 y Fl(I)d Ft(\()p Fr(\026)p Ft(\))15 b Fr(:)522 b Ft(\(4.3\))243 768 y Fl(\017)42 b Fi(F)-7 b(or)34 b(any)f(op)-5 b(en)34 b(set)e Fr(G)26 b Fl(\032)f(M)1331 782 y Fo(1)1371 768 y Ft(\(\012\))p Fi(,)999 963 y Ft(lim)15 b(inf)1023 1024 y Fm(N)7 b Fj(!1)1330 902 y Ft(1)p 1291 942 V 1291 1026 a Fr(N)1374 999 y Fm(d)1460 963 y Ft(log)16 b Fr(P)1663 889 y Fk(\000)1705 963 y Fr(R)1774 977 y Fm(N)1867 963 y Fl(2)24 b Fr(G)2023 889 y Fk(\001)2116 963 y Fq(>)71 b Fl(\000)h Ft(inf)2384 1024 y Fm(\026)p Fj(2)p Fm(G)2559 963 y Fl(I)7 b Ft(\()p Fr(\026)p Ft(\))15 b Fr(:)694 b Ft(\(4.4\))118 1174 y(Before)31 b(starting)g(the)f(pro)s (of)g(of)g(Theorem)g(4.1)i(w)m(e)f(mak)m(e)g(some)g(remarks.)118 1369 y Fi(R)-5 b(emark)36 b(1)p Ft(.)48 b(Since)32 b(the)h(upp)s(er)d (b)s(ound)h(\(4.3\))k(is)c(restricted)i(to)g(compact)h(sets)f(w)m(e)g (sp)s(eak)g(of)f(a)i Fi(we)-5 b(ak)118 1477 y Ft(large)32 b(deviation)e(principle.)40 b(The)31 b(standard)g(w)m(a)m(y)h(to)g (remo)m(v)m(e)h(this)d(restriction)h(w)m(ould)f(b)s(e)g(to)j(pro)m(v)m (e)118 1585 y(exp)s(onen)m(tial)c(tigh)m(tness)i(in)e(the)h(form)1053 1775 y(lim)1022 1835 y Fm(L)p Fj(!1)1241 1775 y Ft(lim)15 b(sup)1278 1853 y Fm(N)7 b Fj(!1)1599 1713 y Ft(1)p 1560 1754 V 1560 1837 a Fr(N)1643 1811 y Fm(d)1728 1775 y Ft(log)17 b Fr(P)1932 1701 y Fk(\000)1974 1775 y Fr(R)2043 1789 y Fm(N)2145 1775 y Fr(=)-55 b Fl(2)25 b(K)2290 1789 y Fm(L)2342 1701 y Fk(\001)2435 1775 y Ft(=)50 b Fl(\0001)15 b Fr(:)118 1977 y Ft(But)33 b(this)e(is)g(false)h(in)f(the)h(massless)g (regime.)46 b(Indeed,)32 b(for)g(\014xed)g Fr(L)c(>)g Ft(0,)34 b(the)e(measure)h Fr(\026)3292 1991 y Fm(T)3379 1977 y Ft(de\014ned)118 2085 y(b)m(y)e(\(2.27\))i(is)d(not)i(in)d Fl(K)938 2099 y Fm(L)991 2085 y Ft(,)i(for)g(an)m(y)g Fr(T)40 b(>)26 b(L)20 b Fl(\000)h Fr(\033)1778 2052 y Fo(2)1817 2085 y Ft(.)43 b(But)31 b Fr(\026)2121 2099 y Fm(T)2202 2085 y Fl(2)26 b(G)2348 2052 y Fm(S)2343 2109 y Fo(2)2400 2085 y Ft(,)31 b(and)f(therefore)i(the)f(lo)m(w)m(er)g (b)s(ound)118 2193 y(\(4.4\))h(together)g(with)d(Theorem)h(3.2)i (implies)899 2388 y(lim)15 b(inf)923 2448 y Fm(N)7 b Fj(!1)1230 2326 y Ft(1)p 1191 2367 V 1191 2450 a Fr(N)1274 2424 y Fm(d)1360 2388 y Ft(log)17 b Fr(P)1564 2314 y Fk(\000)1605 2388 y Fr(R)1674 2402 y Fm(N)1777 2388 y Fr(=)-55 b Fl(2)25 b(K)1922 2402 y Fm(L)1974 2314 y Fk(\001)2066 2388 y Fq(>)71 b Fl(\000)35 b Fr(h)p Ft(\()p Fr(\026)2456 2402 y Fm(T)2511 2388 y Fl(j)p Fr(P)13 b Ft(\))41 b(=)g(0)15 b Fr(:)118 2572 y Ft(It)32 b(is)f(also)h(easy)h(to)g(see)f(that,)i(in)c (con)m(trast)k(to)e(the)h(h)m(yp)s(ercon)m(tractiv)m(e)g(case)g (discussed)d(in)h(Theorem)118 2680 y(4.2)36 b(b)s(elo)m(w,)f Fl(I)41 b Ft(cannot)35 b(b)s(e)f(lo)m(w)m(er)h(semi-con)m(tin)m(uous)f (on)g(the)h(whole)f(of)h Fl(M)2757 2694 y Fo(1)2797 2680 y Ft(\(\012\).)53 b(Indeed,)35 b(consider)118 2788 y(e.g.)65 b(mixtures)36 b(of)j(the)f(t)m(yp)s(e)g(\(2.21\).)66 b(Let)39 b Fr(\026)1737 2802 y Fm(n)1783 2788 y Ft(,)h Fr(n)e Fl(2)g Fn(Z)2105 2802 y Fo(+)2197 2788 y Ft(b)s(e)g(the)g (measures)g(obtained)f(with)g(suc)m(h)118 2896 y(mixtures)i(when)g Fr(\032)799 2863 y Fm(n)799 2920 y Fo(0)846 2896 y Ft(\()p Fr(dt)p Ft(\))j(=)e Fr( )1208 2910 y Fm(n)1256 2896 y Ft(\()p Fr(t)p Ft(\))p Fr(dt)g Ft(with)f(densities)f Fr( )2134 2910 y Fm(n)2222 2896 y Fl(2)j Fr(L)2386 2863 y Fo(2)2425 2896 y Ft(\()p Fn(R)s Ft(\))47 b(suc)m(h)39 b(that)i Fr( )3082 2910 y Fm(n)3169 2896 y Ft(con)m(v)m(erges)h(in)118 3004 y Fr(L)180 2971 y Fo(1)219 3004 y Ft(\()p Fn(R)s Ft(\))e(to)34 b(some)f Fr( )h Fl(2)29 b Fr(L)978 2971 y Fo(1)1018 3004 y Ft(\()p Fn(R)s Ft(\))f Fl(n)23 b Fr(L)1306 2971 y Fo(2)1345 3004 y Ft(\()p Fn(R)s Ft(\).)56 b(Clearly)31 b Fr(\026)1929 3018 y Fm(n)2006 3004 y Fl(!)f Fr(\026)j Ft(w)m(eakly)-8 b(,)35 b(as)e Fr(n)d Fl(!)f(1)p Ft(,)34 b(where)f Fr(\026)g Ft(is)f(giv)m(en)118 3112 y(b)m(y)f(\(2.21\))h (with)d Fr(\032)761 3126 y Fo(0)801 3112 y Ft(\()p Fr(dt)p Ft(\))d(=)f Fr( )s Ft(\()p Fr(t)p Ft(\))p Fr(dt)p Ft(.)43 b(On)29 b(the)i(other)g(hand)e Fl(I)7 b Ft(\()p Fr(\026)2306 3126 y Fm(n)2353 3112 y Ft(\))25 b(=)h(0)k(for)h(all)e Fr(n)h Ft(but)g(since)g Fr(\026)35 b(=)-55 b Fl(2)25 b(K)3561 3126 y Fj(1)3636 3112 y Ft(,)118 3220 y(w)m(e)31 b(ha)m(v)m(e)h Fl(I)7 b Ft(\()p Fr(\026)p Ft(\))24 b(=)h Fl(1)p Ft(.)118 3415 y Fi(R)-5 b(emark)34 b(2)p Ft(.)41 b(In)30 b(view)f(of)i(the)g(v)-5 b(ariational)29 b(principle)e(pro)m(v) m(en)k(in)e(Theorem)h(3.2,)h(w)m(e)g(ha)m(v)m(e)1322 3568 y Fl(I)7 b Ft(\()p Fr(\026)p Ft(\))51 b(=)f(0)127 b Fl(\()-15 b(\))126 b Fr(\026)25 b Fl(2)f(G)2365 3530 y Fm(S)2360 3590 y Fo(2)2432 3568 y Fr(:)118 3721 y Ft(In)40 b(particular)f(w)m(e)j(exp)s(ect)f(deviations)f(of)h(order)f Fr(o)p Ft(\()p Fr(N)2082 3688 y Fm(d)2123 3721 y Ft(\))h(in)f(the)h (class)f(of)h(shift)f(in)m(v)-5 b(arian)m(t)40 b(Gibbs)118 3829 y(states.)57 b(W)-8 b(e)37 b(then)e(sp)s(eak)g(of)h Fi(critic)-5 b(al)36 b Ft(large)g(deviations.)55 b(One)35 b(can)h(in)m(v)m(estigate)g(this)f(issue)f(at)i(the)118 3937 y(lev)m(el)30 b(of)h(the)f(empirical)e(mean)1396 4127 y Fr(M)1484 4141 y Fm(N)1551 4127 y Ft(\()p Fr(\036)p Ft(\))52 b(=)1897 4065 y(1)p 1858 4106 V 1858 4189 a Fr(N)1941 4163 y Fm(d)2053 4041 y Fk(X)2022 4237 y Fm(x)p Fj(2)p Fo(\003)2158 4248 y Fu(N)2245 4127 y Fr(\036)2299 4141 y Fm(x)2358 4127 y Fr(:)118 4373 y Ft(With)30 b(resp)s(ect)g(to)h Fr(P)13 b Ft(,)31 b Fr(M)990 4387 y Fm(N)1088 4373 y Ft(is)e(a)i(Gaussian)e(v)-5 b(ariable)30 b(with)f(mean)h(zero)h(and)f (v)-5 b(ariance)1124 4562 y Fr(\033)1179 4525 y Fo(2)1176 4585 y Fm(N)1294 4562 y Ft(=)50 b Fr(P)13 b Ft(\()p Fr(M)1619 4525 y Fo(2)1609 4585 y Fm(N)1677 4562 y Ft(\))50 b(=)1950 4501 y(1)p 1894 4541 159 4 v 1894 4625 a Fr(N)1977 4598 y Fo(2)p Fm(d)2152 4476 y Fk(X)2093 4673 y Fm(x;y)r Fj(2)p Fo(\003)2286 4684 y Fu(N)2373 4562 y Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))118 4810 y(It)31 b(is)e(easy)i(to)g(c)m(hec)m(k)1063 4975 y Fr(N)1146 4937 y Fm(d)1187 4975 y Fr(\033)1242 4937 y Fo(2)1239 4997 y Fm(N)1331 4975 y Fl(!)25 b Fr(f)10 b Ft(\(0\))26 b(=)1757 4888 y Fk(X)1739 5093 y Fm(x)p Fj(2)p Fh(Z)1876 5074 y Fu(d)1921 4975 y Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(;)118 b(N)36 b Fl(!)25 b(1)15 b Fr(:)118 5222 y Ft(This)29 b(implies)e(that,)k(for)g(an)m(y)f Fr(\014)h(>)25 b Ft(0)1145 5428 y(lim)1106 5489 y Fm(N)7 b Fj(!1)1390 5367 y Ft(1)p 1351 5408 124 4 v 1351 5491 a Fr(N)1434 5465 y Fm(d)1535 5428 y Ft(log)16 b Fr(P)1738 5355 y Fk(\000)1780 5428 y Fr(M)1868 5442 y Fm(N)1961 5428 y Fq(>)25 b Fr(\014)2113 5355 y Fk(\001)2205 5428 y Ft(=)50 b Fl(\000)2467 5367 y Fr(\014)2523 5334 y Fo(2)p 2407 5408 216 4 v 2407 5491 a Ft(2)p Fr(f)10 b Ft(\(0\))2648 5428 y Fr(:)p eop %%Page: 18 18 18 17 bop 118 45 a Fo(18)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(In)24 b(other)g(w)m(ords,)h(if)e(the)i(mo)s(del) e(is)g(not)i(ferromagnetic,)h(i.e.)38 b Fr(f)10 b Ft(\(0\))26 b Fr(<)f Fl(1)p Ft(,)h(the)e(ev)m(en)m(t)i Fr(M)3125 380 y Fm(N)3217 366 y Fq(>)f Fr(\014)30 b Ft(cannot)118 476 y(b)s(e)e(realized)g(in)f(the)h(class)g Fl(G)1100 443 y Fm(S)1095 500 y Fo(2)1152 476 y Ft(.)40 b(F)-8 b(or)29 b(ferromagnetic)g(mo)s(dels,)e Fr(f)10 b Ft(\(0\))26 b(=)f(+)p Fl(1)j Ft(and)g(the)h(ev)m(en)m(t)h Fr(M)3417 490 y Fm(N)3509 476 y Fq(>)25 b Fr(\014)118 587 y Ft(can)30 b(alw)m(a)m(ys)h(b)s(e)e(pro)s(duced)f(within)g(the)i(class)g Fl(G)1805 554 y Fm(S)1800 611 y Fo(2)1886 587 y Ft(\(the)g(latter)h (con)m(tains)f(all)f(the)h(\014elds)e Fr(\015)3247 554 y Fm(\033)3324 587 y Ft(with)h Fr(\033)k Ft(a)118 695 y(constan)m(t\).)50 b(F)-8 b(or)34 b(\014nite)e(range)h(ferromagnetic)g (mo)s(dels)f(with)f(irreducible)f(random)i(w)m(alks,)i(the)f(full)118 803 y(picture)j(of)i(critical)e(large)h(deviations)g(has)g(b)s(een)f (obtained)h(in)f(the)i(pap)s(er)e([1)q(].)61 b(Since)37 b(a)g(classical)118 913 y(in)m(v)-5 b(ariance)27 b(principle)d (applies,)j(one)h(has)g(here)f(the)h(order)g Fr(N)2234 880 y Fm(d)p Fj(\000)p Fo(2)2364 913 y Ft(,)h(for)e Fr(d)f Fq(>)f Ft(3.)40 b(In)27 b(particular)f(one)i(can)118 1021 y(pro)m(v)m(e)k(that)f(exp)s(onen)m(tial)f(tigh)m(tness)g(holds)f (at)j(this)d(order.)41 b(It)31 b(is)f(p)s(ossible)e(to)j(extend)g(this) f(analysis)118 1129 y(of)h(critical)g(large)g(deviations)f(to)i(the)f Fr(\013)p Fl(\000)p Ft(stable)h(mo)s(dels,)e(cf.)43 b(\(2.20\),)34 b(for)d(an)m(y)h Fr(d)27 b Fq(>)f Ft(1.)43 b(The)31 b(correct)118 1239 y(order)37 b(here)g(is)f Fr(N)748 1206 y Fm(d)p Fj(\000)p Fm(\013)888 1239 y Ft(,)j(due)d(to)i(the)f(di\013eren)m(t)g (scaling.)60 b(This)35 b(will)f(b)s(e)j(the)g(sub)5 b(ject)37 b(of)g(a)g(separate)118 1347 y(pap)s(er,)30 b([5].)118 1541 y Fi(R)-5 b(emark)40 b(3)p Ft(.)62 b(The)37 b(pro)s(of)f(of)i(the) f(upp)s(er)e(b)s(ound)g(in)h(theorem)i(4.1)g(is)e(based)h(on)g(the)h (argumen)m(t)f(in-)118 1648 y(tro)s(duced)g(in)f([1)q(])i(for)f(the)h (case)h(of)f(\014nite)f(range)h(ferromagnetic)g(in)m(teraction.)62 b(The)37 b(main)g(to)s(ol)h(in)118 1756 y(extending)25 b(this)f(to)i(our)e(general)h(setting)h(is)e(the)h(expansion)f(pro)m(v) m(ed)i(in)e(the)h(app)s(endix,)f(Lemma)h(A.1,)118 1864 y(whic)m(h)34 b(is)h(hea)m(vily)f(used)h(for)g(comparisons)g(of)g (di\013eren)m(t)g(Gaussian)g(\014elds.)54 b(The)35 b(main)g(line)f(of)h (the)118 1972 y(pro)s(of)24 b(can)g(b)s(e)g(roughly)f(describ)s(ed)f (as)i(follo)m(ws.)38 b(W)-8 b(e)25 b(p)s(erturb)d(the)j(original)d (measure)i Fr(P)37 b Ft(b)m(y)25 b(remo)m(ving)118 2080 y(the)38 b(singularit)m(y)d(of)j(its)f(sp)s(ectral)g(densit)m(y)-8 b(,)39 b(cf.)62 b(\(4.5\).)i(This)36 b(de\014nes)h(a)h(new)f(Gaussian)f (\014eld,)i Fr(P)3599 2047 y Fm(")3636 2080 y Ft(,)118 2188 y(with)e(b)s(ounded)f(sp)s(ectral)h(densit)m(y)-8 b(.)60 b(The)36 b(new)h(\014eld)e(is)h(h)m(yp)s(ercon)m(tractiv)m(e)i (and)e(satis\014es)h(a)g(strong)118 2296 y(large)32 b(deviation)e (principle,)f(cf.)44 b(Theorem)31 b(4.2.)46 b(This)29 b(is)i(a)h(m)m(ultidimensional)27 b(v)m(ersion)k(of)g(the)h(sort)118 2404 y(of)e(results)e(pro)m(v)m(en)i(in)e(section)h(5.4)i(of)e([11)q (],)h(see)g(also)g([6)q(].)40 b(Hyp)s(ercon)m(tractivit)m(y)30 b(is)e(obtained)h(via)g(the)118 2512 y(classical)h(theorem)i(of)f (Nelson)g([28)q(],)h(cf.)43 b(Lemma)32 b(B.2)g(in)e(the)h(app)s(endix.) 41 b(W)-8 b(e)32 b(then)f(sho)m(w)g(that)h(the)118 2620 y(b)s(ounds)f(satis\014ed)h(b)m(y)g Fr(P)988 2587 y Fm(")1058 2620 y Ft(are)h(stable)g(\(on)g(the)g(v)m(olume)f(scale\))i(when)d(the) i(singularit)m(y)e(is)h(restored,)118 2728 y(cf.)40 b(Theorem)28 b(4.3.)41 b(This)27 b(is)g(the)i(most)g(delicate)f(part)g(of)h(the)f (pro)s(of.)40 b(In)27 b(particular,)h(the)g(argumen)m(ts)118 2836 y(of)j([1])g(do)f(not)h(apply)e(and)h(\014ner)f(estimates)i(are)g (needed,)f(cf.)41 b(Lemma)30 b(4.6.)118 3029 y Fi(R)-5 b(emark)36 b(4)p Ft(.)49 b(The)33 b(lo)m(w)m(er)g(b)s(ound)e(in)h (theorem)h(4.1)h(can)g(b)s(e)e(pro)m(v)m(en)h(along)g(the)h(line)d(of)i (the)g(classical)118 3137 y(Donsk)m(er-V)-8 b(aradhan)36 b(argumen)m(t)g([13)q(].)56 b(This)34 b(is)g(done)h(at)h(the)g(end)e (of)i(this)e(section.)56 b(Our)34 b(pro)s(of)g(of)118 3245 y(Theorem)d(4.1)h(also)e(sho)m(ws,)h(ho)m(w)m(ev)m(er,)i(ho)m(w)e (the)g(lo)m(w)m(er)g(b)s(ound)e(for)h Fr(P)44 b Ft(can)31 b(b)s(e)f(in)g(principle)d(deriv)m(ed)118 3353 y(from)j(a)h(lo)m(w)m (er)f(b)s(ound)f(on)h Fr(P)1124 3320 y Fm(")1161 3353 y Ft(.)118 3547 y Fi(R)-5 b(emark)39 b(5)p Ft(.)56 b(As)35 b(a)h(\014nal)e(remark)i(w)m(e)f(observ)m(e)h(that)g(the)g(pro)s(of)f (of)g(Theorem)h(4.1)g(carries)f(through)118 3655 y(with)d(minor)g(mo)s (di\014cations)g(if)h(w)m(e)h(replace)f(the)h(in\014nite)d(v)m(olume)j (measure)f Fr(P)47 b Ft(b)m(y)33 b Fr(P)3141 3622 y Fo(0)3128 3682 y Fm(N)3227 3655 y Ft(=)d Fr(\015)3380 3622 y Fo(0)3375 3682 y(\003)3424 3693 y Fu(N)3486 3655 y Ft(,)k(i.e.)118 3764 y(the)g(Gibbs)f(measure)g(on)h(\003)1096 3778 y Fm(N)1197 3764 y Ft(with)f(zero)i(b)s(oundary)d(condition.)50 b(In)33 b(particular)f(the)i(statemen)m(t)i(of)118 3872 y(Theorem)h(4.1)h(holds)d(as)j(it)e(is)g(when)g Fr(P)50 b Ft(is)36 b(replaced)h(b)m(y)f Fr(P)2239 3839 y Fo(0)2226 3899 y Fm(N)2294 3872 y Ft(.)61 b(Indeed,)37 b(this)f(amoun)m(ts)i(to)f (replace)118 3980 y(the)i(co)m(v)-5 b(ariance)41 b Fr(G)803 3994 y Fm(N)909 3980 y Ft(b)m(y)e(\000)1101 3994 y Fm(N)1168 3980 y Ft(.)67 b(As)39 b(\(the)g(pro)s(of)f(of)7 b(\))40 b(Prop)s(osition)d(3.1)j(sho)m(ws,)h(this)d(cannot)i(a\013ect)118 4088 y(thermo)s(dynamics,)34 b(and)f(in)g(general)h(will)e(not)i (a\013ect)i(an)m(y)e(v)m(olume)g(order)f(asymptotic.)53 b(Of)33 b(course)118 4196 y(one)27 b(cannot)g(exp)s(ect)h(this)d(to)j (b)s(e)e(true)g(for)h(critical)e(large)i(deviations,)g(where)f(b)s (oundary)f(e\013ects)j(pla)m(y)118 4304 y(an)i(imp)s(ortan)m(t)g(role,) g(cf.)41 b([5)q(].)118 4476 y(4.1.)54 b Fy(Pro)s(of)31 b(of)g(Theorem)e(4.1.)46 b Ft(F)-8 b(or)27 b(eac)m(h)h Fr(")e(>)f Ft(0,)j(w)m(e)e(de\014ne)g(the)h(measure)f Fr(P)2942 4443 y Fm(")3004 4476 y Fl(2)f(M)3199 4490 y Fo(1)3239 4476 y 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Fr(\026)p Fl(j)p Fr(P)1726 5337 y Fm(")1764 5370 y Ft(\))91 b Fr(\026)25 b Fl(2)g(M)2165 5337 y Fm(S)2165 5395 y Fo(1)2216 5370 y Ft(\(\012\))3475 5302 y(\(4.6\))p eop %%Page: 19 19 19 18 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(19)118 366 y Fy(Theorem)34 b(4.2.)46 b Fi(The)34 b(function)f Fl(I)1365 380 y Fm(")1428 366 y Ft(:)27 b Fl(M)1589 380 y Fo(1)1629 366 y Ft(\(\012\))g Fl(!)f Ft([0)p Fr(;)15 b Fl(1)p Ft(])35 b Fi(is)e(lower)h(semi-c)-5 b(ontinuous)35 b(and)f(its)f(level)118 474 y(sets)1171 563 y Fk(\010)1224 636 y Fr(\026)25 b Fl(2)g(M)1499 650 y Fo(1)1538 636 y Ft(\(\012\))h(:)51 b Fl(I)7 b Ft(\()p Fr(\026)p Ft(\))40 b Fq(6)g Fr(L)2171 563 y Fk(\011)2239 636 y Fr(;)116 b(L)25 b Fq(>)g Ft(0)118 798 y Fi(ar)-5 b(e)34 b(c)-5 b(omp)g(act.)44 b(F)-7 b(or)33 b(e)-5 b(ach)34 b Fr(\026)25 b Fl(2)g(K)1270 812 y Fj(1)1377 798 y Fi(it)33 b(satis\014es)979 979 y Fl(I)1029 993 y Fm(")1065 979 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Fq(6)70 b Fl(\000)37 b Ft(inf)2422 5035 y Fm(\026)p Fj(2)p Fm(F)2581 4975 y Fl(I)2631 4989 y Fm(")2667 4975 y Ft(\()p Fr(\026)p Ft(\))15 b Fr(;)597 b Ft(\(4.13\))118 5193 y Fi(and,)34 b(for)f(e)-5 b(ach)33 b(op)-5 b(en)34 b(set)f Fr(G)25 b Fl(\032)g(M)1329 5207 y Fo(1)1369 5193 y Ft(\(\012\))p Fi(,)951 5398 y Ft(lim)15 b(inf)975 5458 y Fm(N)7 b Fj(!1)1283 5336 y Ft(1)p 1244 5377 V 1244 5460 a Fr(N)1327 5434 y Fm(d)1412 5398 y Ft(log)17 b Fr(P)1616 5360 y Fm(N)s(;")1731 5324 y Fk(\000)1773 5398 y Fr(R)1842 5412 y Fm(N)1935 5398 y Fl(2)24 b Fr(G)2091 5324 y Fk(\001)2184 5398 y Fq(>)71 b Fl(\000)36 b Ft(inf)2416 5458 y Fm(\026)p Fj(2)p Fm(G)2576 5398 y Fl(I)2626 5412 y Fm(")2662 5398 y Ft(\()p Fr(\026)p Ft(\))15 b Fr(:)602 b Ft(\(4.14\))p eop %%Page: 20 20 20 19 bop 118 45 a Fo(20)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(Let)k(us)f(assume,)i(in)d(fact,)k(the)d(v)-5 b(alidit)m(y)32 b(of)g(Theorem)h(4.3.)49 b(Let)33 b Fr(L)c(>)g Ft(0)k(b)s(e)f(\014xed.)47 b(F)-8 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1123 y Fl(\001)15 b(\001)g(\001)2091 1037 y Fk(X)2080 1238 y Fm(z)2113 1250 y Fu(l)2145 1238 y Fm(=)-43 b Fj(2)p Fo(\003)2263 1123 y Fr(Q)2335 1086 y Fm(")2372 1123 y Ft(\()p Fr(x;)15 b(z)2541 1137 y Fo(1)2581 1123 y Ft(\))g Fl(\001)g(\001)g(\001)i Fr(Q)2825 1086 y Fm(")2862 1123 y Ft(\()p Fr(z)2939 1138 y Fm(l)2965 1123 y Fr(;)e(y)s Ft(\))g Fr(:)301 b Ft(\(4.22\))118 1386 y Fi(Pr)-5 b(o)g(of.)43 b Ft(The)30 b(co)m(v)-5 b(ariance)32 b Fr(G)1097 1353 y Fm(")1164 1386 y Ft(is,)e(b)m(y)g (de\014nition,)1011 1577 y Fr(G)1082 1540 y Fm(")1119 1577 y Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)1595 1516 y(1)p 1512 1556 212 4 v 1512 1639 a(\(2)p Fr(\031)s Ft(\))1682 1613 y Fm(d)1748 1453 y Fk(Z)1799 1660 y Fo(\()p Fj(\000)p Fm(\031)r(;\031)r Fo(])2007 1641 y Fu(d)2077 1577 y Fr(f)2132 1540 y Fm(")2168 1577 y Ft(\()p Fr(\022)s Ft(\))15 b(e)2339 1540 y Fm(i)d Fo(\()p Fm(y)r Fj(\000)p Fm(x)p Fo(\))p Fj(\001)p Fm(\022)2636 1577 y Fr(d\022)17 b(:)118 1801 y Ft(F)-8 b(rom)34 b(\(4.5\))h(w)m(e)e(see)h(that)g Fr(f)1121 1768 y Fm(")1190 1801 y Ft(is)e(the)h(F)-8 b(ourier)33 b(transform)f(of)h(\(1)23 b(+)f Fr(")p Ft(\))2560 1768 y Fj(\000)p Fo(1)2655 1801 y Ft(\(1)-23 b(I)23 b Fl(\000)f Fr(Q)2933 1768 y Fm(")2969 1801 y Ft(\))3004 1768 y Fj(\000)p Fo(1)3099 1801 y Ft(.)49 b(This)31 b(implies)118 1909 y(\(4.20\).)43 b(The)30 b(rest)g(is)g(a)h(straigh)m(tforw)m(ard)f (consequence)h(of)f(Lemma)h(A.1)g(applied)d(to)j(\(4.20\).)p 3595 1909 4 62 v 3599 1851 55 4 v 3599 1909 V 3653 1909 4 62 v 118 2075 a Fy(Lemma)i(4.5.)46 b Fi(L)-5 b(et)41 b Ft(\003)f Fl(\032)f Ft(\003)1138 2089 y Fm(N)1205 2075 y Fi(.)66 b(Then)41 b(the)g(mar)-5 b(ginal)42 b(of)f Fr(P)2274 2042 y Fm(N)s(;")2430 2075 y Fi(on)g Ft(\003)g Fi(is)f(the)h(c)-5 b(enter)g(e)g(d)42 b(Gaussian)118 2182 y(me)-5 b(asur)g(e)34 b(with)g(c)-5 b(ovarianc)g(e)1263 2383 y Fr(G)1334 2339 y Fm(";N)1334 2412 y Fo(\003)1504 2383 y Ft(=)50 b(\(1)21 b(+)f Fr(")p Ft(\))1894 2346 y Fj(\000)p Fo(1)2036 2270 y Fj(1)2005 2297 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y(random)28 b(w)m(alk)g(on)h Fn(Z)850 4494 y Fm(d)914 4527 y Ft(with)f(constan)m(t)i Fi(kil)5 b(ling)p Ft(.)39 b(The)28 b(probabilit)m(y)e(to)k(die)d(at)j(eac)m(h)f(step)g(is)f Fr("=)p Ft(\(1)18 b(+)e Fr(")p Ft(\).)118 4636 y(Similarly)-8 b(,)36 b(the)h(couplings)f Fr(Q)1173 4603 y Fm(";N)1329 4636 y Ft(de\014ne)g(a)i(random)e(w)m(alk)h(for)g(whic)m(h)f(killing)e (o)s(ccurs)j(only)f Fi(inside)118 4744 y Ft(\003)181 4758 y Fm(N)248 4744 y Ft(.)66 b(Moreo)m(v)m(er,)44 b(the)39 b(co)m(v)-5 b(ariance)40 b Fr(G)1459 4711 y Fm(")1535 4744 y Ft(can)f(b)s(e)f(seen)h(as)g(the)g(Green)g(function)e(asso)s (ciated)j(to)f(the)118 4852 y(random)34 b(w)m(alk)h(with)e(constan)m(t) j(killing)c(probabilit)m(y)g Fr("=)p Ft(\(1)25 b(+)e Fr(")p Ft(\),)37 b(with)c(killing)f(starting)j(at)g(time)g(0,)118 4960 y(cf.)41 b(\(4.20\).)218 5150 y(W)-8 b(e)44 b(turn)f(to)h(the)g (pro)s(of)f(of)g(the)h(upp)s(er)d(b)s(ound)h(\(4.13\).)82 b(W)-8 b(e)45 b(use)e(here)g(the)h Fi(pr)-5 b(oje)g(ctive)46 b(limit)118 5258 y 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Fl(2)34 b Fn(Z)833 993 y Fo(+)923 1026 y Ft(and)g(denote)i(b) m(y)g Fr(\032)1581 1040 y Fm(M)1695 1026 y Ft(a)g(metric)f(on)g Fl(M)2304 1040 y Fo(1)2344 1026 y Ft(\(\012)2445 1040 y Fm(M)2524 1026 y Ft(\),)i(compatible)e(with)f(the)h(w)m(eak)118 1134 y(top)s(ology)29 b(and)f(suc)m(h)h(that)g Fr(\032)1111 1148 y Fm(M)1219 1134 y Ft(is)f(dominated)g(b)m(y)g(the)h(total)h(v)-5 b(ariation)28 b(norm)g(\(cf.)40 b(e.g.)i([15)q(]\).)e(F)-8 b(or)30 b(a)118 1242 y(giv)m(en)g Fr(\016)g Fl(2)24 b Ft(\(0)p Fr(;)15 b Ft(1\),)33 b(de\014ne)1261 1435 y Fr(R)1330 1450 y Fm(N)s(;\016)1446 1435 y Ft(\()p Fr(\036)p Ft(\))52 b(=)1845 1373 y(1)p 1753 1414 231 4 v 1753 1497 a Fl(j)p Ft(\003)1841 1512 y Fm(N)s(;\016)1958 1497 y Fl(j)2062 1348 y Fk(X)2008 1545 y Fm(x)p Fj(2)p Fo(\003)2144 1557 y Fu(N)r(;\016)2262 1435 y Fr(\016)2302 1450 y Fm(\022)2335 1458 y Fu(x)2375 1450 y Fm(\036)2417 1461 y Fu(N)2494 1435 y Fr(;)118 1689 y Ft(where)1062 1838 y(\003)1125 1853 y Fm(N)s(;\016)1292 1838 y Fl(\021)1413 1764 y Fk(\002)1451 1838 y Ft(\(1)21 b Fl(\000)f Fr(\016)s Ft(\))p Fr(N)r(=)p Ft(2)p Fr(;)15 b Ft(\(1)24 b Fl(\000)c Fr(\016)s Ft(\))p Fr(N)r(=)p Ft(2)2364 1764 y Fk(\003)2404 1787 y Fm(d)2480 1838 y Fl(\\)35 b Fn(Z)2641 1800 y Fm(d)2692 1838 y Fr(:)118 1988 y Ft(It)d(is)e(easy)i(to)g(c)m(hec)m(k)h(that)f(there)g(exist)f (constan)m(ts)h Fr(\016)1960 2002 y Fo(0)2000 1988 y Ft(\()p Fr(d)p Ft(\))c Fr(>)f Ft(0)32 b(and)e Fr(c)p Ft(\()p Fr(d)p Ft(\))f Fr(<)d Fl(1)p Ft(,)32 b(only)f(dep)s(ending)d (on)118 2096 y Fr(d)p Ft(,)j(suc)m(h)f(that)h(for)f(an)m(y)h(b)s (ounded)d(measurable)i(function)f Fr(f)10 b Ft(,)29 b(for)i(an)m(y)f Fr(N)36 b Fl(2)24 b Fn(Z)2816 2063 y Fo(+)2871 2096 y Ft(,)947 2245 y(sup)946 2324 y Fm(\036)p Fj(2)p Fo(\012)1116 2168 y Fk(\014)1116 2222 y(\014)1162 2245 y Fr(R)1231 2259 y Fm(N)1298 2245 y Ft(\()p Fr(\036)p Ft(\))15 b(\()p Fr(f)10 b Ft(\))41 b Fl(\000)f Fr(R)1783 2260 y Fm(N)s(;\016)1900 2245 y Ft(\()p Fr(\036)p Ft(\))15 b(\()p Fr(f)10 b Ft(\))2179 2168 y Fk(\014)2179 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Fr(\031)2559 4386 y Fj(\000)p Fo(1)2556 4452 y Fm(M)2678 4424 y Fl(2)k Fr(F)2835 4386 y Fo(\()p Fm(\016)r Fo(\))2943 4350 y Fk(\001)3000 4424 y Fr(:)118 4578 y Ft(F)-8 b(or)31 b(eac)m(h)h(1)25 b Fr(<)g(p)g(<)g Fl(1)p Ft(,)31 b Fr(q)d Ft(=)d Fr(p=)p Ft(\()p Fr(p)20 b Fl(\000)g Ft(1\),)32 b(H\177)-45 b(older's)30 b(inequalit)m(y)e (yields)250 4771 y(log)17 b Fr(P)454 4733 y Fm(N)s(;")569 4697 y Fk(\000)611 4771 y Fr(R)680 4785 y Fm(N)767 4771 y Fl(\016)k Fr(\031)888 4733 y Fj(\000)p Fo(1)885 4800 y Fm(M)1007 4771 y Fl(2)k Fr(F)1151 4785 y Fm(M)1231 4697 y Fk(\001)1313 4771 y Fq(6)1434 4709 y Ft(1)p 1434 4750 46 4 v 1434 4833 a Fr(p)1505 4771 y Ft(log)16 b Fr(P)1708 4733 y Fm(")1745 4697 y Fk(\000)1787 4771 y Fr(\036)1841 4727 y Fm(p)1841 4800 y(N)s(;\016)1958 4697 y Fk(\001)2035 4771 y Ft(+)2151 4709 y(1)p 2151 4750 V 2152 4833 a Fr(q)2221 4771 y Ft(log)h Fr(P)2425 4733 y Fm(")2461 4697 y Fk(\000)2503 4771 y Fr(R)2572 4785 y Fm(N)2660 4771 y Fl(\016)j Fr(\031)2780 4733 y Fj(\000)p Fo(1)2777 4800 y Fm(M)2900 4771 y Fl(2)25 b Fr(F)3057 4733 y Fo(\(2)p Fm(\016)r Fo(\))3200 4697 y Fk(\001)3257 4771 y Fr(;)147 b Ft(\(4.30\))118 4969 y(where)30 b(w)m(e)h(ha)m(v)m(e) g(used)f(\(4.28\))j(again.)40 b(W)-8 b(e)32 b(shall)d(need)h(the)g (follo)m(wing)f(k)m(ey)i(appro)m(ximation)f(result.)118 5133 y Fy(Lemma)j(4.6.)46 b Fi(L)-5 b(et)36 b Fr(\036)911 5148 y Fm(N)s(;\016)1063 5133 y Fi(b)-5 b(e)35 b(the)h(density)g (de\014ne)-5 b(d)36 b(in)f(\(4.29\).)52 b(F)-7 b(or)36 b(al)5 b(l)36 b Fr(p)30 b Fq(>)g Ft(1)p Fi(,)36 b Fr(")31 b(>)f Ft(0)36 b Fi(and)g Fr(\016)e(>)c Ft(0)p Fi(,)118 5241 y(we)j(have)775 5411 y Ft(lim)736 5471 y Fm(N)7 b Fj(!1)1019 5349 y Ft(1)p 980 5390 124 4 v 980 5473 a Fr(N)1063 5447 y Fm(d)1149 5411 y Ft(log)16 b Fr(P)1352 5373 y Fm(")1389 5337 y Fk(\000)1431 5411 y Fr(\036)1485 5366 y Fm(p)1485 5440 y(N)s(;\016)1602 5337 y Fk(\001)1694 5411 y Ft(=)89 b(lim)1815 5471 y Fm(N)7 b Fj(!1)2098 5349 y Ft(1)p 2060 5390 V 2060 5473 a Fr(N)2143 5447 y Fm(d)2228 5411 y Ft(log)17 b Fr(P)2432 5373 y Fm(N)s(;")2547 5337 y Fk(\000)2589 5411 y Fr(\036)2643 5366 y Fj(\000)p Fm(p)2643 5440 y(N)s(;\016)2760 5337 y Fk(\001)2852 5411 y Ft(=)50 b(0)p Fr(:)386 b Ft(\(4.31\))p eop %%Page: 23 23 23 22 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(23)218 366 y Ft(Let)25 b(us)f(sho)m(w)h(that)h(Lemma) f(4.6)h(allo)m(ws)e(to)h(pro)m(v)m(e)h(the)f(desired)f(estimate.)39 b(Supp)s(ose)23 b(\(4.31\))28 b(holds.)118 474 y(Then)h(\(4.30\))k (together)f(with)d(the)i(upp)s(er)d(b)s(ound)g(of)j(Theorem)f(\(4.2\))i (imply)568 674 y(lim)15 b(sup)605 752 y Fm(N)7 b Fj(!1)926 613 y Ft(1)p 887 653 124 4 v 887 736 a Fr(N)970 710 y Fm(d)1055 674 y Ft(log)17 b Fr(P)1259 637 y Fm(N)s(;")1374 600 y Fk(\000)1416 674 y Fr(R)1485 688 y Fm(N)1573 674 y Fl(\016)j Fr(\031)1693 636 y Fj(\000)p Fo(1)1690 703 y Fm(M)1813 674 y Fl(2)25 b Fr(F)1957 688 y Fm(M)2036 600 y Fk(\001)1065 917 y Fq(6)61 b Fl(\000)1313 855 y Ft(1)p 1313 896 46 4 v 1314 979 a Fr(q)1399 917 y Ft(inf)1539 843 y Fk(\010)1592 917 y Fl(I)1642 931 y Fm(")1679 917 y Ft(\()p Fr(\026)p Ft(\))15 b Fr(;)66 b(\026)25 b Fl(2)g(M)2185 931 y Fo(1)2225 917 y Ft(\(\012\))41 b(:)76 b Fr(\026)20 b Fl(\016)g Fr(\031)2698 878 y Fj(\000)p Fo(1)2695 945 y Fm(M)2818 917 y Fl(2)25 b Fr(F)2975 879 y Fo(\(2)p Fm(\016)r Fo(\))3118 843 y Fk(\011)3186 917 y Fr(:)118 1122 y Ft(This)32 b(implies)e(\(4.27\))36 b(since)c Fr(q)k Ft(and)d Fr(\016)k Ft(are)d(arbitrarily)c(close)k(to)g(1)g(and)e(0)i (resp)s(ectiv)m(ely)-8 b(.)49 b(The)33 b(pro)s(of)118 1230 y(of)e(\(4.13\))h(is)e(complete.)218 1362 y(Before)37 b(pro)m(ving)e(Lemma)h(4.6,)i(w)m(e)e(deriv)m(e)g(the)g(lo)m(w)m(er)g (b)s(ound)e(\(4.14\).)59 b(Let)36 b(us)g(\014x)f(an)h(op)s(en)f(set)118 1470 y Fr(G)189 1484 y Fm(M)294 1470 y Fl(\032)25 b(M)499 1484 y Fo(1)538 1470 y Ft(\(\012)639 1484 y Fm(M)718 1470 y Ft(\),)31 b(for)f(a)h(giv)m(en)g Fr(M)k Fl(2)25 b Fn(Z)1536 1437 y Fo(+)1590 1470 y Ft(.)41 b(Giv)m(en)30 b Fr(\016)f(>)c Ft(0)31 b(de\014ne)933 1637 y Fr(G)1004 1599 y Fo(\()p Fm(\016)r Fo(\))1148 1637 y Ft(=)1269 1563 y Fk(\010)1322 1637 y Fr(\026)25 b Fl(2)g(M)1597 1651 y Fo(1)1637 1637 y Ft(\(\012)1738 1651 y Fm(M)1817 1637 y Ft(\))91 b Fr(\032)1990 1651 y Fm(M)2069 1637 y Ft(\()p Fr(\026;)15 b(G)2270 1599 y Fm(c)2270 1659 y(M)2350 1637 y Ft(\))41 b Fq(>)f Fr(c)p Ft(\()p Fr(d)p Ft(\))p Fr(\016)2753 1563 y Fk(\011)2821 1637 y Fr(:)118 1794 y Ft(Then,)30 b(b)m(y)g(\(4.28\))748 1960 y Fr(P)819 1923 y Fm(N)s(;")934 1887 y Fk(\000)976 1960 y Fr(R)1045 1974 y Fm(N)1132 1960 y Fl(\016)21 b Fr(\031)1253 1922 y Fj(\000)p Fo(1)1250 1989 y Fm(M)1373 1960 y Fl(2)j Fr(G)1529 1974 y Fm(M)1609 1887 y Fk(\001)1701 1960 y Fq(>)50 b Fr(P)1893 1923 y Fm(N)s(;")2009 1887 y Fk(\000)2050 1960 y Fr(R)2119 1975 y Fm(N)s(;\016)2256 1960 y Fl(\016)21 b Fr(\031)2377 1922 y Fj(\000)p Fo(1)2374 1989 y Fm(M)2497 1960 y Fl(2)j Fr(G)2653 1923 y Fo(\()p Fm(\016)r Fo(\))2762 1887 y Fk(\001)1701 2120 y Ft(=)50 b Fr(P)1893 2082 y Fm(")1930 2046 y Fk(\000)1987 2120 y Fr(\036)2041 2135 y Fm(N)s(;\016)2173 2120 y Ft(;)40 b Fr(R)2307 2135 y Fm(N)s(;\016)2444 2120 y Fl(\016)21 b Fr(\031)2565 2081 y Fj(\000)p Fo(1)2562 2148 y Fm(M)2685 2120 y Fl(2)j Fr(G)2841 2082 y Fo(\()p Fm(\016)r Fo(\))2950 2046 y Fk(\001)3007 2120 y Fr(:)118 2281 y Ft(F)-8 b(or)31 b(an)m(y)g(1)26 b Fr(<)f(p)g(<)g Fl(1)p Ft(,)30 b Fr(p=q)e Ft(=)d Fr(p)20 b Fl(\000)g Ft(1,)31 b(H\177)-45 b(older's)30 b(inequalit)m(y)f(yields) 749 2444 y(log)r Fr(P)938 2406 y Fm(N)s(;")1053 2370 y Fk(\000)1095 2444 y Fr(R)1164 2458 y Fm(N)1251 2444 y Fl(\016)21 b Fr(\031)1372 2405 y Fj(\000)p Fo(1)1369 2472 y Fm(M)1492 2444 y Fl(2)k Fr(G)1649 2458 y Fm(M)1728 2370 y Fk(\001)892 2616 y Fq(>)60 b Fl(\000)1140 2555 y Fr(q)p 1139 2595 V 1139 2678 a(p)1210 2616 y Ft(log)17 b Fr(P)1414 2578 y Fm(N)s(;")1529 2542 y Fk(\000)1571 2616 y Fr(\036)1625 2572 y Fj(\000)p Fm(p)1625 2645 y(N)s(;\016)1742 2542 y Fk(\001)1819 2616 y Ft(+)35 b Fr(q)18 b Ft(log)e Fr(P)2187 2578 y Fm(")2224 2542 y Fk(\000)2266 2616 y Fr(R)2335 2630 y Fm(N)2422 2616 y Fl(\016)21 b Fr(\031)2543 2578 y Fj(\000)p Fo(1)2540 2645 y Fm(M)2663 2616 y Fl(2)k Fr(G)2820 2578 y Fo(\(2)p Fm(\016)r Fo(\))2948 2542 y Fk(\001)3005 2616 y Fr(:)399 b Ft(\(4.32\))118 2816 y(F)-8 b(rom)29 b(Lemma)g(4.6,)i(the)e(arbitrariness)d(of)j Fr(q)j Ft(and)c Fr(\016)s Ft(,)i(and)e(from)h(the)g(lo)m(w)m(er)g(b)s (ound)e(of)h(Theorem)h(4.2,)118 2924 y(w)m(e)i(ha)m(v)m(e)866 3101 y(lim)15 b(inf)890 3161 y Fm(N)7 b Fj(!1)1198 3039 y Ft(1)p 1159 3080 124 4 v 1159 3163 a Fr(N)1242 3137 y Fm(d)1327 3101 y Ft(log)17 b Fr(P)1531 3063 y Fm(N)s(;")1647 3027 y Fk(\000)1688 3101 y Fr(R)1757 3115 y Fm(N)1845 3101 y Fl(\016)j Fr(\031)1965 3063 y Fj(\000)p Fo(1)1962 3129 y Fm(M)2085 3101 y Fl(2)25 b Fr(G)2242 3115 y Fm(M)2321 3027 y Fk(\001)2413 3101 y Fq(>)71 b Fl(\000)35 b(I)2711 3115 y Fm(")2747 3101 y Ft(\()p Fr(\026)p Ft(\))15 b Fr(;)118 3295 y Ft(for)32 b(an)m(y)h Fr(\026)28 b Fl(2)g(M)714 3309 y Fo(1)754 3295 y Ft(\(\012\))k(suc)m(h)g(that)h Fr(\026)1383 3309 y Fm(M)1491 3295 y Fl(2)28 b Fr(G)1651 3309 y Fm(M)1730 3295 y Ft(.)47 b(This)30 b(implies)g(\(4.14\).)48 b(The)32 b(pro)s(of)g(of)g(Theorem)g(4.3)118 3403 y(is)e(completed.)p 3595 3403 4 62 v 3599 3345 55 4 v 3599 3403 V 3653 3403 4 62 v 118 3684 a(4.3.)54 b Fy(Pro)s(of)32 b(of)f(Lemma)e(4.6.)46 b Ft(W)-8 b(e)29 b(\014x)d Fr(")g(>)f Ft(0)j(and)e Fr(\016)j(>)c Ft(0.)40 b(Let)28 b Fr(G)2482 3651 y Fm(N)s(;")2625 3684 y Ft(b)s(e)f(the)g(co)m(v)-5 b(ariance)29 b(function)118 3792 y(of)i(the)f(\014eld)f Fr(P)646 3759 y Fm(N)s(;")762 3792 y Ft(,)h(cf.)41 b(\(4.25\).)i(Let)31 b(us)f(de\014ne)f(the)i (matrices)643 3954 y Fr(C)708 3968 y Fm(N)825 3954 y Ft(=)947 3881 y Fk(\010)1000 3954 y Fr(G)1071 3917 y Fm(N)s(;")1187 3954 y Ft(\()p Fr(x;)15 b(y)s Ft(\))1397 3881 y Fk(\011)1451 3990 y Fm(x;y)r Fj(2)p Fo(\003)1644 4004 y Fu(N)r(;\016)r(=)p Fg(2)1828 3954 y Fr(;)142 b(D)2070 3968 y Fm(N)2188 3954 y Ft(=)2309 3881 y Fk(\010)2362 3954 y Fr(G)2433 3917 y Fm(")2470 3954 y Ft(\()p Fr(x;)15 b(y)s Ft(\))2680 3881 y Fk(\011)2734 3990 y Fm(x;y)r Fj(2)p Fo(\003)2927 4004 y Fu(N)r(;\016)r(=)p Fg(2)3112 3954 y Fr(:)118 4142 y Ft(Then)29 b(\(4.29\))k(can)e(b)s(e)f(written) 684 4342 y Fr(\036)738 4357 y Fm(N)s(;\016)855 4342 y Ft(\()p Fr(\033)s Ft(\))52 b(=)e(det)15 b(\()p Fr(D)1404 4356 y Fm(N)1487 4342 y Fr(C)1559 4303 y Fj(\000)p Fo(1)1552 4370 y Fm(N)1653 4342 y Ft(\))1698 4260 y Fg(1)p 1698 4272 31 3 v 1698 4313 a(2)1788 4342 y Ft(exp)1942 4241 y Fk(\020)2017 4342 y Fl(\000)2118 4280 y Ft(1)p 2118 4321 46 4 v 2118 4404 a(2)2204 4268 y Fk(\012)2246 4342 y Fr(\033)n(;)g Ft(\()p Fr(C)2443 4303 y Fj(\000)p Fo(1)2436 4370 y Fm(N)2574 4342 y Fl(\000)35 b Fr(D)2758 4303 y Fj(\000)p Fo(1)2755 4370 y Fm(N)2852 4342 y Ft(\))p Fr(\033)2942 4268 y Fk(\013)3000 4241 y(\021)3070 4342 y Fr(:)334 b Ft(\(4.33\))118 4524 y(An)30 b(explicit)f(computation)h(sho)m(ws)g (that)852 4700 y Fr(P)923 4662 y Fm(")959 4626 y Fk(\000)1001 4700 y Fr(\036)1055 4655 y Fm(p)1055 4729 y(N)s(;\016)1172 4626 y Fk(\001)1264 4700 y Ft(=)50 b(det)16 b(\(1)-23 b(I)1617 4714 y Fm(N)1705 4700 y Ft(+)20 b Fr(B)1865 4714 y Fm(N)1932 4700 y Ft(\))1967 4658 y Fm(p=)p Fo(2)2108 4700 y Ft(det)15 b(\(1)-23 b(I)2339 4714 y Fm(N)2428 4700 y Ft(+)20 b Fr(p)15 b(B)2649 4714 y Fm(N)2716 4700 y Ft(\))2751 4658 y Fj(\000)2816 4631 y Fg(1)p 2816 4643 31 3 v 2816 4684 a(2)2887 4700 y Fr(;)517 b Ft(\(4.34\))118 4862 y(where)30 b(w)m(e)h(ha)m(v)m(e)g(de\014ned)1422 5018 y Fr(B)1491 5032 y Fm(N)1608 5018 y Ft(=)51 b Fr(D)1805 5032 y Fm(N)1887 5018 y Fr(C)1959 4980 y Fj(\000)p Fo(1)1952 5047 y Fm(N)2088 5018 y Fl(\000)36 b Ft(1)-23 b(I)2250 5032 y Fm(N)2332 5018 y Fr(;)118 5175 y Ft(with)29 b(1)-23 b(I)380 5189 y Fm(N)478 5175 y Ft(denoting)30 b(the)h(unit)d(diagonal)i (matrix)g(on)g(\003)2049 5193 y Fm(N)s(;\016)r(=)p Fo(2)2236 5175 y Ft(.)41 b(Moreo)m(v)m(er,)33 b(setting)d Fr(q)e Ft(=)d Fr(p=)p Ft(\()p Fr(p)c Fl(\000)f Ft(1\),)750 5400 y Fr(P)821 5363 y Fm(N)s(;")936 5327 y Fk(\000)978 5400 y Fr(\036)1032 5356 y Fj(\000)p Fm(p)1032 5430 y(N)s(;\016)1149 5327 y Fk(\001)1241 5400 y Ft(=)50 b(det)15 b(\(1)-23 b(I)1593 5414 y Fm(N)1682 5400 y Ft(+)20 b Fr(B)1842 5414 y Fm(N)1909 5400 y Ft(\))1944 5358 y Fj(\000)p Fm(p=)p Fo(2)p Fm(q)2174 5400 y Ft(det)15 b(\(1)-23 b(I)2405 5414 y Fm(N)2493 5400 y Fl(\000)2616 5339 y Fr(p)p 2594 5379 90 4 v 2594 5463 a Ft(2)p Fr(q)2709 5400 y(B)2778 5414 y Fm(N)2845 5400 y Ft(\))2880 5326 y Fj(\000)2945 5299 y Fg(1)p 2945 5311 31 3 v 2945 5352 a(2)3005 5400 y Fr(:)399 b Ft(\(4.35\))p eop %%Page: 24 24 24 23 bop 118 45 a Fo(24)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(Let)i(us)f(denote)h(b)m(y)1349 576 y Fl(f)p Fr(\025)1447 590 y Fm(j)1484 576 y Ft(\()p Fr(N)10 b Ft(\))15 b Fr(;)67 b(j)30 b Ft(=)25 b(1)p Fr(;)15 b(:)g(:)g(:)j(;)d Fl(j)p Ft(\003)2243 591 y Fm(N)s(;\016)2360 576 y Fl(jg)118 785 y Ft(the)29 b(set)h(of)f(eigen)m(v)-5 b(alues)29 b(of)g(the)g(matrix)g Fr(B)1606 799 y Fm(N)1673 785 y Ft(.)40 b(Observ)m(e)29 b(that,)h(b)m(y)f(elemen)m(tary)g(matrix) g(calculus,)f(w)m(e)118 893 y(ha)m(v)m(e)887 1102 y(sup)939 1179 y Fm(j)1054 1102 y Fl(j)p Fr(\025)1132 1116 y Fm(j)1169 1102 y Ft(\()p Fr(N)10 b Ft(\))p Fl(j)51 b Fq(6)g Fl(k)p Fr(B)1634 1116 y Fm(N)1701 1102 y Fl(k)g(\021)101 b Ft(sup)1918 1180 y Fm(x)p Fj(2)p Fo(\003)2054 1192 y Fu(N)r(;\016)2240 1016 y Fk(X)2188 1213 y Fm(y)r Fj(2)p Fo(\003)2321 1225 y Fu(N)r(;\016)2455 1102 y Fl(j)p Fr(B)2549 1116 y Fm(N)2616 1102 y Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fl(j)g Fr(:)538 b Ft(\(4.36\))118 1421 y(F)-8 b(rom)31 b(\(4.34\),)716 1673 y(log)q Fr(P)904 1636 y Fm(")941 1600 y Fk(\000)983 1673 y Fr(\036)1037 1629 y Fm(p)1037 1703 y(N)s(;\016)1153 1600 y Fk(\001)1246 1673 y Ft(=)1377 1612 y Fr(p)p 1377 1652 46 4 v 1377 1736 a Ft(2)1463 1587 y Fk(X)1516 1782 y Fm(i)1624 1673 y Ft(log)17 b(\(1)k(+)f Fr(\025)2002 1687 y Fm(i)2030 1673 y Ft(\))36 b Fl(\000)2217 1612 y Ft(1)p 2217 1652 V 2217 1736 a(2)2303 1587 y Fk(X)2356 1782 y Fm(i)2464 1673 y Ft(log)17 b(\(1)k(+)f Fr(p)15 b(\025)2903 1687 y Fm(i)2931 1673 y Ft(\))859 1947 y Fq(6)990 1885 y Fr(p)p 990 1926 V 990 2009 a Ft(2)1045 1947 y Fl(j)p Ft(\003)1133 1962 y Fm(N)s(;\016)1250 1947 y Fl(j)31 b Ft(log)17 b(\(1)k(+)e Fl(k)p Fr(B)1744 1961 y Fm(N)1812 1947 y Fl(k)p Ft(\))36 b Fl(\000)2059 1885 y Ft(1)p 2059 1926 V 2059 2009 a(2)2115 1947 y Fl(j)p Ft(\003)2203 1962 y Fm(N)s(;\016)2320 1947 y Fl(j)30 b Ft(log)17 b(\(1)k Fl(\000)f Fr(p)15 b Fl(k)p Fr(B)2875 1961 y Fm(N)2942 1947 y Fl(k)p Ft(\))h Fr(:)118 2187 y Ft(A)31 b(similar)c(computation)k(for)f(\(4.35\))i(sho)m(ws)612 2404 y(log)q Fr(P)800 2366 y Fm(N)s(;")916 2330 y Fk(\000)957 2404 y Fr(\036)1011 2359 y Fj(\000)p Fm(p)1011 2433 y(N)s(;\016)1128 2330 y Fk(\001)754 2601 y Fq(6)61 b Fl(\000)1024 2540 y Fr(p)p 1002 2580 90 4 v 1002 2664 a Ft(2)p Fr(q)1101 2601 y Fl(j)p Ft(\003)1189 2616 y Fm(N)s(;\016)1306 2601 y Fl(j)31 b Ft(log)16 b(\(1)21 b(+)f Fl(k)p Fr(B)1800 2615 y Fm(N)1868 2601 y Fl(k)p Ft(\))36 b Fl(\000)2100 2540 y Ft(1)p 2100 2580 46 4 v 2100 2664 a(2)2155 2601 y Fl(j)p Ft(\003)2243 2616 y Fm(N)s(;\016)2360 2601 y Fl(j)31 b Ft(log)17 b(\(1)k(+)2772 2540 y Fr(p)p 2750 2580 90 4 v 2750 2664 a Ft(2)p Fr(q)2865 2601 y Fl(k)p Fr(B)2979 2615 y Fm(N)3046 2601 y Fl(k)p Ft(\))16 b Fr(:)118 2859 y Ft(In)30 b(order)g(to)h(pro)m(v)m(e)g(\(4.31\))i(it)c(is)h(then) g(su\016cien)m(t)g(to)h(sho)m(w)1569 3068 y(lim)1530 3129 y Fm(N)7 b Fj(!1)1765 3068 y Fl(k)p Fr(B)1879 3082 y Fm(N)1946 3068 y Fl(k)51 b Ft(=)g(0)15 b Fr(:)1180 b Ft(\(4.37\))118 3311 y(W)-8 b(e)32 b(ha)m(v)m(e)1261 3520 y Fl(k)p Fr(B)1375 3534 y Fm(N)1442 3520 y Fl(k)51 b Fq(6)g Fl(k)p Fr(D)1780 3534 y Fm(N)1848 3520 y Fl(k)15 b(k)p Fr(C)2025 3482 y Fj(\000)p Fo(1)2018 3548 y Fm(N)2155 3520 y Fl(\000)35 b Fr(D)2339 3482 y Fj(\000)p Fo(1)2336 3548 y Fm(N)2433 3520 y Fl(k)15 b Fr(;)118 3729 y Ft(where)38 b(the)g(norm)f Fl(k)26 b(\001)g(k)38 b Ft(is)f(de\014ned)g(as)h(in)f 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Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)g(\(1)21 b(+)e Fr(")p Ft(\))1932 4931 y Fk(h)1997 5009 y Ft(~)1976 5032 y Fr(Q)2048 4988 y Fm(";N)2048 5061 y Fo(\003)2097 5075 y Fu(N)r(;\016)r(=)p Fg(2)2267 5032 y Ft(\()p Fr(x;)c(y)s Ft(\))36 b Fl(\000)2640 5009 y Ft(~)2619 5032 y Fr(Q)2691 4995 y Fm(")2691 5055 y Fo(\003)2740 5069 y Fu(N)r(;\016)r(=)p Fg(2)2910 5032 y Ft(\()p Fr(x;)15 b(y)s Ft(\))3120 4931 y Fk(i)3179 5032 y Fr(:)118 5269 y Ft(W)-8 b(e)32 b(no)m(w)e(use)g(explicitly)e (the)j(expressions)e(\(4.19\),)k(\(4.22\))f(and)e(\(4.24\).)43 b(Giv)m(en)30 b(sites)1302 5478 y Fr(z)1344 5492 y Fo(1)1419 5478 y Fr(=)-55 b Fl(2)25 b Ft(\003)1558 5497 y Fm(N)s(;\016)r(=)p Fo(2)1760 5478 y Fr(;)31 b(:)15 b(:)g(:)31 b(;)g(z)2050 5493 y Fm(l)2111 5478 y Fr(=)-55 b Fl(2)25 b Ft(\003)2250 5497 y Fm(N)s(;\016)r(=)p Fo(2)2452 5478 y Fr(;)p eop %%Page: 25 25 25 24 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 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b(pro)s(cess)f(with)f(killing)e(o)s(ccurring)i(only)h(inside)e (\003)3569 1772 y Fm(N)3636 1758 y Ft(,)118 1866 y(cf.)45 b(the)32 b(remark)f(after)h(Lemma)g(4.5.)46 b(W)-8 b(e)32 b(observ)m(e)h(that)f(the)g(quan)m(tit)m(y)g(app)s(earing)e(in)g(the)i (r.h.s.)44 b(of)118 1974 y(\(4.39\),)38 b(apart)d(from)f(the)h(factor)g (2,)i(is)c(the)i(probabilit)m(y)d(that)j(the)g(w)m(alk)m(er)g(started)g (in)e Fr(x)f Fl(2)g Ft(\003)3449 1993 y Fm(N)s(;\016)r(=)p Fo(2)3636 1974 y Ft(,)118 2082 y(arriv)m(es)c(ev)m(en)m(tually)h(in)e Fr(y)h Fl(2)d Ft(\003)1169 2100 y Fm(N)s(;\016)r(=)p Fo(2)1385 2082 y Ft(taking)j(only)g(jumps)f(out)i(of)f(\003)2454 2100 y Fm(N)s(;\016)r(=)p Fo(2)2670 2082 y Ft(and)g(visiting)f(at)i (least)g(once)118 2204 y(the)k(set)h Fn(Z)487 2171 y Fm(d)545 2204 y Fl(n)22 b Ft(\003)675 2218 y Fm(N)743 2204 y Ft(.)48 b(Let)34 b Fl(f)p Fr(\030)1067 2218 y Fm(n)1114 2204 y Fl(g)p Ft(,)g Fr(n)c Ft(=)f(0)p Fr(;)15 b Ft(1)g Fr(:)g(:)g(:)j Ft(,)33 b Fr(\030)1769 2218 y Fo(0)1838 2204 y Ft(=)d Fr(x)p Ft(,)k(denote)f(the)g(ab)s(o)m(v)m(e)i (describ)s(ed)30 b(pro)s(cess,)k(and)118 2312 y(let)f Fn(P)307 2326 y Fm(x)384 2312 y Ft(b)s(e)g(the)g(asso)s(ciated)g (probabilit)m(y)e(measure.)48 b(Let)34 b Fr(\034)2176 2326 y Fm(")2245 2312 y Ft(b)s(e)f(the)g(time)f(at)i(whic)m(h)e Fl(f)p Fr(\030)3202 2326 y Fm(n)3249 2312 y Fl(g)h Ft(is)g(killed.)118 2423 y(Let)e(also)f Fr(\034)503 2437 y Fm(N)601 2423 y Ft(b)s(e)f(the)i(hitting)e(time)h(for)g Fn(Z)1590 2390 y Fm(d)1646 2423 y Fl(n)21 b Ft(\003)1775 2437 y Fm(N)1842 2423 y Ft(,)31 b(i.e.)1294 2577 y Fr(\034)1334 2591 y Fm(N)1452 2577 y Ft(=)50 b(inf)21 b Fl(f)p Fr(n)k Fq(>)g Ft(1)15 b Fr(;)66 b(\030)2111 2591 y Fm(n)2193 2577 y Fr(=)-55 b Fl(2)25 b Ft(\003)2332 2591 y Fm(N)2400 2577 y Fl(g)15 b Fr(:)118 2731 y Ft(Summing)28 b(o)m(v)m(er)k Fr(y)c Fl(2)d Ft(\003)939 2750 y Fm(N)s(;\016)r(=)p Fo(2)1156 2731 y Ft(in)k(\(4.39\))k(w)m(e)e(can)g(estimate)1037 2902 y Fl(k)p Fr(D)1160 2864 y Fj(\000)p Fo(1)1157 2931 y Fm(N)1290 2902 y Fl(\000)k Fr(C)1468 2864 y Fj(\000)p Fo(1)1461 2931 y Fm(N)1561 2902 y Fl(k)51 b Fq(6)g Ft(2)113 b(sup)1854 2981 y Fm(x)p Fj(2)p Fo(\003)1990 2995 y Fu(N)r(;\016)r(=)p Fg(2)2186 2902 y Fn(P)2241 2916 y Fm(x)2301 2829 y Fk(\000)2357 2902 y Fr(\034)2397 2916 y Fm(")2474 2902 y Fr(>)40 b(\034)2625 2916 y Fm(N)2707 2902 y Ft(\))687 b(\(4.40\))118 3140 y(Observ)m(e)35 b(that)h(due)f(to)h(the)g(constan)m(t)g(killing)c (inside)i(\003)2107 3154 y Fm(N)2174 3140 y Ft(,)j(there)e(exists)g Fr(m)p Ft(\()p Fr(")p Ft(\))g Fr(>)e Ft(0)j(suc)m(h)f(that)h(for)118 3248 y(an)m(y)31 b Fr(N)10 b Ft(,)30 b Fr(x)c Fl(2)e Ft(\003)654 3266 y Fm(N)s(;\016)r(=)p Fo(2)872 3248 y Ft(for)30 b(an)m(y)h Fr(l)c(>)e Ft(0)31 b(su\016cien)m(tly)e(large)1107 3415 y Fn(P)1162 3429 y Fm(x)1221 3341 y Fk(\000)1278 3415 y Fr(\034)1318 3429 y Fm(")1395 3415 y Fr(>)40 b(l)17 b Ft(;)31 b Fr(\034)1646 3429 y Fm(N)1738 3415 y Ft(=)25 b Fr(l)17 b Ft(\))51 b Fq(6)65 b Ft(exp)15 b(\()p Fl(\000)p Fr(m)p Ft(\()p Fr(")p Ft(\))g Fr(l)r Ft(\))i Fr(:)118 3569 y Ft(W)-8 b(e)32 b(then)e(estimate,)h(for)f(an)m(y)h(su\016cien)m (tly)e(large)i(in)m(teger)f Fr(n)p Ft(,)638 3793 y Fn(P)693 3807 y Fm(x)752 3720 y Fk(\000)809 3793 y Fr(\034)849 3807 y Fm(")926 3793 y Fr(>)40 b(\034)1077 3807 y Fm(N)1159 3793 y Ft(\))51 b Fq(6)1426 3680 y Fm(n)1381 3707 y Fk(X)1391 3904 y Fm(l)q Fo(=0)1543 3793 y Fn(P)1598 3807 y Fm(x)1658 3720 y Fk(\000)1699 3793 y Fr(\034)1739 3807 y Fm(N)1831 3793 y Ft(=)25 b Fr(l)1956 3720 y Fk(\001)2033 3793 y Ft(+)2209 3680 y Fj(1)2179 3707 y Fk(X)2139 3904 y Fm(l)q Fo(=)p Fm(n)p Fo(+1)2379 3793 y Fn(P)2434 3807 y Fm(x)2494 3720 y Fk(\000)2536 3793 y Fr(\034)2576 3807 y Fm(")2637 3793 y Fr(>)g(l)17 b Ft(;)41 b Fr(\034)2883 3807 y Fm(N)2975 3793 y Ft(=)25 b Fr(l)3100 3720 y Fk(\001)1245 4033 y Fq(6)50 b Fr(n)15 b Fn(P)1491 4047 y Fm(x)1551 3959 y Fk(\000)1592 4033 y Fr(\034)1632 4047 y Fm(N)1724 4033 y Fq(6)25 b Fr(n)1875 3959 y Fk(\001)1952 4033 y Ft(+)35 b(exp)15 b(\()p Fl(\000)p Fr(m)p Ft(\()p Fr(")p Ft(\))g Fr(n=)p Ft(2\))i Fr(:)682 b Ft(\(4.41\))118 4188 y(On)30 b(the)g(other)h(hand,)e(for)h(eac)m(h)i(\014xed)e Fr(n)25 b(>)g Ft(0,)1129 4356 y(sup)1046 4434 y Fm(x)p Fj(2)p Fo(\003)1182 4448 y Fu(N)r(;\016)r(=)p Fg(2)1378 4356 y Fn(P)1433 4370 y Fm(x)1492 4282 y Fk(\000)1534 4356 y Fr(\034)1574 4370 y Fm(N)1666 4356 y Fq(6)g Fr(n)1817 4282 y Fk(\001)1909 4356 y Fq(6)50 b Fr(n)2199 4269 y Fk(X)2115 4470 y Fj(j)p Fm(z)s Fj(j)p Fm(>\016)r(N)q(=n)2444 4356 y Fr(p)p Ft(\(0)p Fr(;)15 b(z)t Ft(\))g Fr(;)118 4619 y Ft(is)34 b(v)-5 b(anishing)33 b(in)h(the)i(limit)d Fr(N)43 b Fl(!)33 b(1)p Ft(.)55 b(Since)34 b Fr(n)h Ft(is)f(arbitrary)g (in)g(\(4.41\),)39 b(this)34 b(implies)e(\(4.38\))38 b(and)118 4727 y(the)31 b(Lemma)f(follo)m(ws.)p 3595 4727 4 62 v 3599 4669 55 4 v 3599 4727 V 3653 4727 4 62 v 118 4907 a(4.4.)54 b Fy(The)28 b(h)m(yp)s(ercon)m(tractiv)m(e)h (regime.)45 b Ft(This)23 b(subsection)h(is)g(dev)m(oted)i(to)f(the)g (pro)s(of)f(of)h(theorem)118 5014 y(4.2.)62 b(Let)37 b(the)h(rate)f(functions)f Fl(I)43 b Ft(and)36 b Fl(I)1571 5028 y Fm(")1645 5014 y Ft(b)s(e)g(de\014ned)g(as)h(in)f(\(4.2\))i(and) f(\(4.6\).)62 b(Let)38 b(also)f Fr(\036)p Ft(\()p Fr(")p Ft(\))h(b)s(e)118 5122 y(giv)m(en)30 b(b)m(y)h(\(4.8\).)118 5289 y Fy(Lemma)i(4.7.)46 b Fi(F)-7 b(or)34 b(e)-5 b(ach)33 b Fr(")26 b(>)f Ft(0)p Fi(,)32 b Fl(I)1390 5303 y Fm(")1459 5289 y Fi(satis\014es)i(the)f(variational)i(principle)1043 5444 y Fl(I)1093 5458 y Fm(")1129 5444 y Ft(\()p Fr(\026)p Ft(\))51 b(=)f(0)77 b Fl(\()-15 b(\))75 b Fr(\026)40 b Ft(=)g Fr(P)2067 5406 y Fm(")2119 5444 y Fr(;)142 b(\026)25 b Fl(2)g(M)2561 5458 y Fo(1)2600 5444 y Ft(\(\012\))693 b(\(4.42\))p eop %%Page: 26 26 26 25 bop 118 45 a Fo(26)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Fi(and)34 b(the)f(identity)986 538 y Fl(I)1036 552 y Fm(")1073 538 y Ft(\()p Fr(\026)p Ft(\))51 b(=)f Fl(I)7 b Ft(\()p Fr(\026)p Ft(\))35 b(+)1705 476 y Fr(")p 1703 517 46 4 v 1703 600 a Ft(2)1758 538 y Fr(\026)p Ft(\()p Fr(\036)1902 500 y Fo(2)1902 560 y(0)1942 538 y Ft(\))h(+)f Fr(\036)p Ft(\()p Fr(")p Ft(\))15 b Fr(;)118 b(\026)25 b Fl(2)g(K)2678 552 y Fj(1)2768 538 y Fr(:)636 b Ft(\(4.43\))118 730 y Fi(Pr)-5 b(o)g(of.)43 b Ft(F)-8 b(rom)31 b(Prop)s(osition)e(3.1)i(w)m(e)g(ha)m(v)m(e)1269 883 y Fr(h)p Ft(\()p Fr(\026)p Fl(j)p Fr(P)1507 845 y Fm(")1545 883 y Ft(\))51 b(=)f Fr(e)1794 845 y Fm(")1831 883 y Ft(\()p Fr(\026)p Ft(\))41 b Fl(\000)f Fr(s)p Ft(\()p Fr(\026)p Ft(\))h(+)f Fr(p)2474 845 y Fm(")3429 883 y Ft(\(4.44\))118 1041 y(where)26 b(the)g(en)m(trop)m(y)h Fr(s)f Ft(is)f(giv)m(en)i(b)m(y)f(\(3.3\))i(\(indep)s(enden)m(t)c(of)i Fr(")p Ft(\),)i(and)e Fr(e)2571 1008 y Fm(")2608 1041 y Ft(,)h Fr(p)2706 1008 y Fm(")2769 1041 y Ft(are)g(the)f(in)m(ternal)f (energy)118 1148 y(and)32 b(the)g(pressure)f(asso)s(ciated)i(to)g(the)f (p)s(oten)m(tial)g Fr(J)e Ft(+)21 b Fr(")p Ft(1)-23 b(I.)48 b(According)31 b(to)i(\(3.7\))h(and)e(\(4.5\))i(w)m(e)f(see)118 1256 y(that)1555 1409 y Fr(p)1601 1372 y Fm(")1683 1409 y Fl(\000)45 b Fr(p)50 b Ft(=)h Fr(\036)p Ft(\()p Fr(")p Ft(\))15 b Fr(:)118 1562 y Ft(Then)29 b(\(4.43\))k(follo)m(ws)d(from)f (the)i(ob)m(vious)f(iden)m(tit)m(y)g(\(cf.)41 b(\(3.5\)\))1374 1739 y Fr(e)1416 1701 y Fm(")1454 1739 y Ft(\()p Fr(\026)p Ft(\))k Fl(\000)h Fr(e)p Ft(\()p Fr(\026)p Ft(\))51 b(=)2092 1677 y Fr(")p 2090 1718 V 2090 1801 a Ft(2)2146 1739 y Fr(\026)p Ft(\()p Fr(\036)2290 1701 y Fo(2)2290 1761 y(0)2329 1739 y Ft(\))15 b Fr(:)118 1917 y Ft(T)-8 b(o)27 b(pro)m(v)m(e)g(the)f(v)-5 b(ariational)25 b(principle)e(w)m(e)k(ma)m (y)f(use)g(theorem)h(3.2)g(and)f(the)g(fact)h(that)g Fr(P)3190 1884 y Fm(")3253 1917 y Ft(is)e(the)h(only)118 2025 y(shift)i(in)m(v)-5 b(arian)m(t)28 b(Gibbs)g(measure)h(with)f (\014nite)g(second)h(momen)m(t)h(for)f(the)h(p)s(oten)m(tial)e Fr(J)f Ft(+)18 b Fr(")p Ft(1)-23 b(I,)31 b(cf.)40 b(e.g.)118 2133 y(Corollary)29 b(\(13.40\))k(in)c([20)q(].)p 3595 2133 4 62 v 3599 2075 55 4 v 3599 2133 V 3653 2133 4 62 v 118 2331 a Fy(Upp)s(er)42 b(b)s(ound)p Ft(.)59 b(Once)36 b(the)g(Hyp)s(ercon)m(tractivit)m(y)h(of)f Fr(P)2188 2298 y Fm(")2261 2331 y Ft(is)f(established,)h(cf.)58 b(Lemma)36 b(B.2,)j(the)118 2439 y(pro)s(of)30 b(of)i(the)f(upp)s(er)e (b)s(ound)g(in)h(theorem)h(4.2)i(follo)m(ws)d(the)h(standard)f(line,)g (cf.)43 b(section)31 b(5.4)i(of)e([11)q(].)118 2547 y(W)-8 b(e)32 b(p)s(oin)m(t)d(out)i(the)f(main)g(steps)g(needed.)118 2677 y Fi(Step)j(1)p Ft(.)42 b(Let)31 b Fr(Q)25 b Fl(\032\032)g Fn(Z)925 2644 y Fm(d)991 2677 y Ft(b)s(e)30 b(a)h(\014xed)f(cub)s(e)g (con)m(taining)g(the)g(origin.)40 b(Using)30 b(Lemma)g(B.2)i(one)f(sho) m(ws)118 2785 y(that)g(for)f(all)f(b)s(ounded)g(measurable)g(function)g Fr(f)40 b Ft(suc)m(h)30 b(that)h Fr(S)2326 2800 y Fm(f)2396 2785 y Fl(\032)25 b Fr(Q)p Ft(,)31 b(the)f(limit)1009 2986 y(\003)1072 3000 y Fm(Q)1132 2986 y Ft(\()p Fr(f)10 b Ft(\))51 b(=)89 b(lim)1429 3046 y Fm(N)7 b Fj(!1)1712 2924 y Ft(1)p 1673 2965 124 4 v 1673 3048 a Fr(N)1756 3022 y Fm(d)1842 2986 y Ft(log)17 b Fr(P)2046 2948 y Fm(")2082 2912 y Fk(\000)2139 2986 y Ft(exp)2324 2900 y Fk(X)2293 3096 y Fm(x)p Fj(2)p Fo(\003)2429 3107 y Fu(N)2502 2986 y Fr(f)29 b Fl(\016)21 b Fr(\022)2685 3000 y Fm(x)2729 2912 y Fk(\001)3429 2986 y Ft(\(4.45\))118 3233 y(exists,)42 b(cf.)68 b(app)s(endix.)e(An)39 b(application)e(of)j (H\177)-45 b(older's)39 b(inequalit)m(y)f(sho)m(ws)h(that)h(\003)3109 3247 y Fm(Q)3208 3233 y Ft(is)f(a)h(con)m(v)m(ex)118 3341 y(functional.)f(Its)31 b Fi(L)-5 b(e)g(gendr)g(e)33 b(tr)-5 b(ansform)33 b Ft(is)d(the)g(con)m(v)m(ex)i(function)d(on)i Fl(M)2647 3355 y Fo(1)2686 3341 y Ft(\(\012)2787 3355 y Fm(Q)2847 3341 y Ft(\))g(giv)m(en)f(b)m(y)1223 3498 y(\003)1286 3461 y Fj(\003)1286 3521 y Fm(Q)1346 3498 y Ft(\()p Fr(\026)p Ft(\))51 b(=)f(sup)1691 3577 y Fm(f)1816 3425 y Fk(\010)1869 3498 y Fr(\026)p Ft(\()p Fr(f)10 b Ft(\))40 b Fl(\000)g Ft(\003)2263 3512 y Fm(Q)2323 3498 y Ft(\()p Fr(f)10 b Ft(\))2448 3425 y Fk(\011)2516 3498 y Fr(;)888 b Ft(\(4.46\))118 3716 y(where)30 b(the)h(suprem)m(um)d (is)i(tak)m(en)h(o)m(v)m(er)h(con)m(tin)m(uous)e(b)s(ounded)e Fr(f)40 b Ft(with)29 b Fr(S)2688 3731 y Fm(f)2758 3716 y Fl(\032)c Fr(Q)p Ft(.)118 3846 y Fi(Step)36 b(2)p Ft(.)50 b(F)-8 b(ollo)m(wing)32 b(Lemma)i(5.4.13)h(and)e(Corollary)f(5.1.11)j (in)d([11)r(],)i(one)g(can)f(c)m(hec)m(k)i(that)f(\003)3457 3813 y Fj(\003)3530 3846 y Ft(has)118 3954 y(compact)e(lev)m(el)e(sets) h(on)f Fl(M)1102 3968 y Fo(1)1141 3954 y Ft(\(\012)1242 3968 y Fm(Q)1302 3954 y Ft(\))h(and)f(for)g(eac)m(h)h(closed)f(set)h Fr(F)2356 3968 y Fm(Q)2442 3954 y Fl(\032)25 b(M)2647 3968 y Fo(1)2686 3954 y Ft(\(\012)2787 3968 y Fm(Q)2847 3954 y Ft(\),)805 4154 y(lim)15 b(sup)842 4232 y Fm(N)7 b Fj(!1)1162 4093 y Ft(1)p 1123 4133 V 1123 4217 a Fr(N)1206 4190 y Fm(d)1292 4154 y Ft(log)17 b Fr(P)1496 4117 y Fm(")1533 4081 y Fk(\000)1574 4154 y Fr(R)1643 4168 y Fm(N)1731 4154 y Fl(\016)j Fr(\031)1851 4116 y Fj(\000)p Fo(1)1848 4183 y Fm(Q)1971 4154 y Fl(2)25 b Fr(F)2115 4168 y Fm(Q)2175 4081 y Fk(\001)2267 4154 y Fq(6)71 b Fl(\000)57 b Ft(inf)2499 4215 y Fm(\026)p Fj(2)p Fm(F)2633 4226 y Fu(Q)2701 4154 y Ft(\003)2764 4117 y Fj(\003)2764 4177 y Fm(Q)2824 4154 y Ft(\()p Fr(\026)p Ft(\))p Fr(:)455 b Ft(\(4.47\))118 4383 y Fi(Step)33 b(3)p Ft(.)41 b(Let)1015 4535 y(\003)1078 4498 y Fj(\003)1118 4535 y Ft(\()p Fr(\026)p Ft(\))51 b(=)f(sup)1456 4614 y Fm(Q)1567 4535 y Ft(\003)1630 4498 y Fj(\003)1630 4558 y Fm(Q)1690 4535 y Ft(\()p Fr(\026)20 b Fl(\016)h Fr(\031)1921 4497 y Fj(\000)p Fo(1)1918 4564 y Fm(Q)2015 4535 y Ft(\))15 b Fr(;)168 b(\026)25 b Fl(2)g(M)2533 4549 y Fo(1)2572 4535 y Ft(\(\012\))15 b Fr(;)118 4752 y Ft(where)26 b(the)h(suprem)m(um)d(is)i(tak)m(en)i(o)m(v)m(er)f(all)f (\014nite)f(cub)s(es)h(con)m(taining)g(the)g(origin.)38 b(By)27 b(de\014nition,)e(the)118 4860 y(function)37 b(\003)545 4827 y Fj(\003)622 4860 y Ft(is)f(non-negativ)m(e,)41 b(con)m(v)m(ex)f(and)d(lo)m(w)m(er)h(semicon)m(tin)m(uous.)61 b(Moreo)m(v)m(er,)42 b(the)c(standard)118 4968 y Fi(pr)-5 b(oje)g(ctive)40 b(limit)e Ft(argumen)m(t)f(\(cf.)62 b(e.g.)h([9],)40 b(theorem)d(4.6.1\))j(sho)m(ws)d(that)g(\003)2868 4935 y Fj(\003)2945 4968 y Ft(has)g(compact)i(lev)m(el)118 5076 y(sets,)31 b(and)f(\(4.47\))j(can)d(b)s(e)g(lifted)f(to)977 5272 y(lim)15 b(sup)1014 5350 y Fm(N)7 b Fj(!1)1335 5210 y Ft(1)p 1296 5251 V 1296 5334 a Fr(N)1379 5308 y Fm(d)1465 5272 y Ft(log)16 b Fr(P)1668 5234 y Fm(")1705 5198 y Fk(\000)1747 5272 y Fr(R)1816 5286 y Fm(N)1908 5272 y Fl(2)25 b Fr(F)2065 5198 y Fk(\001)2157 5272 y Fq(6)71 b Fl(\000)36 b Ft(inf)2390 5332 y Fm(\026)p Fj(2)p Fm(F)2549 5272 y Ft(\003)2612 5234 y Fj(\003)2651 5272 y Ft(\()p Fr(\026)p Ft(\))p Fr(:)628 b Ft(\(4.48\))118 5480 y(for)30 b(all)g(closed)g(sets)g Fr(F)39 b Fl(\032)25 b(M)1131 5494 y Fo(1)1170 5480 y Ft(\(\012\).)p eop %%Page: 27 27 27 26 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(27)118 366 y Fi(Step)33 b(4)p Ft(.)41 b(With)30 b(the)h(same)g(argumen)m(ts)f(as)h(in)e([11)q(],)i(Lemma)f (5.4.17,)j(one)e(then)f(pro)m(v)m(es)h(the)g(iden)m(tit)m(y)1253 513 y(\003)1316 476 y Fj(\003)1355 513 y Ft(\()p Fr(\026)p Ft(\))46 b(=)f Fl(I)1692 527 y Fm(")1728 513 y Ft(\()p Fr(\026)p Ft(\))15 b Fr(;)143 b(\026)25 b Fl(2)g(M)2311 527 y Fo(1)2350 513 y Ft(\(\012\))15 b Fr(:)118 660 y Ft(F)-8 b(rom)39 b(step)f(3,)k(this)37 b(implies)e(that)k Fl(I)1433 674 y Fm(")1508 660 y Ft(has)f(compact)i(lev)m(el)e(sets)h (and)f(satis\014es)f(the)i(upp)s(er)d(b)s(ound)118 768 y(\(4.9\))c(of)f(theorem)g(4.2.)118 954 y Fy(Lo)m(w)m(er)k(Bound)p Ft(.)44 b(W)-8 b(e)32 b(remark)f(that)g(a)h(pro)s(of)e(of)h(the)g(lo)m (w)m(er)h(b)s(ound)c(in)i(theorem)i(4.2)g(using)d(h)m(yp)s(er-)118 1062 y(con)m(tractivit)m(y)h(as)e(in)f([6)q(,)h(11)q(])h(w)m(ould)e (require)g(a)h(condition)f(stronger)i(than)f(Lemma)g(B.2)i(\(cf.)40 b(section)118 1170 y(5.4)30 b(of)e([11)q(]\).)41 b(Since)27 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y Fm(i)424 2552 y Fq(>)c Ft(0)30 b(and)f Fr(\026)826 2566 y Fm(i)884 2552 y Ft(ergo)s(dic,)h (for)f(eac)m(h)i Fr(i)26 b Ft(=)f(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(n)p Ft(.)40 b(F)-8 b(or)31 b(eac)m(h)g Fr(N)10 b Ft(,)30 b(let)g(\003)2786 2519 y Fm(i)2814 2552 y Ft(\()p Fr(N)10 b Ft(\))p Fr(;)15 b(i)27 b Ft(=)e(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(n)30 b Ft(b)s(e)f(a)118 2663 y(partition)g(of)i(the)f(unit)f(b)s(o)m(x)h([0) p Fr(;)15 b Ft(1])1303 2630 y Fm(d)1376 2663 y Ft(suc)m(h)30 b(that)809 2901 y(\003)872 2864 y Fm(i)900 2901 y Ft(\()p Fr(N)10 b Ft(\))41 b(=)1245 2788 y Fm(d)1205 2815 y Fk(Y)1208 3013 y Fm(l)q Fo(=1)1337 2901 y Ft([)p Fr(a)1410 2864 y Fm(i)1410 2924 y(l)1438 2901 y Ft(\()p Fr(N)10 b Ft(\))p Fr(;)15 b(b)1670 2864 y Fm(i)1670 2924 y(l)1699 2901 y Ft(\()p Fr(N)10 b Ft(\)])15 b Fr(;)93 b Ft(0)25 b Fq(6)g Fr(a)2224 2864 y Fm(i)2224 2924 y(l)2253 2901 y Ft(\()p Fr(N)10 b Ft(\))26 b Fr(<)e(b)2566 2864 y Fm(i)2566 2924 y(l)2595 2901 y Ft(\()p Fr(N)10 b Ft(\))26 b Fq(6)f Ft(1)15 b 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Fm(n)2446 5309 y Fk(Y)2447 5504 y Fm(i)p Fo(=1)2592 5395 y Fr(\026)2647 5358 y Fm(i)2675 5322 y Fk(\000)2717 5395 y Fr(R)2786 5409 y Fm(N)s(;i)2918 5395 y Fl(2)g Fr(U)3066 5409 y Fm(i)3094 5322 y Fk(\001)3151 5395 y Fr(:)p eop %%Page: 28 28 28 27 bop 118 45 a Fo(28)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(The)h(ergo)s(dic)g(theorem)h(implies)1417 518 y(lim)1378 578 y Fm(N)7 b Fj(!1)1612 518 y Fr(\026)1667 480 y Fm(i)1695 518 y Ft(\()p 1730 445 70 4 v Fr(R)1800 532 y Fm(N)s(;i)1933 518 y Fl(2)24 b Fr(U)2080 532 y Fm(i)2109 518 y Ft(\))51 b(=)f(1)15 b Fr(;)118 702 y Ft(for)30 b(all)g Fr(i)25 b Ft(=)g(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(n)p Ft(,)31 b(and)e(therefore)1383 876 y(lim)1344 936 y Fm(N)7 b Fj(!1)1578 876 y Fr(\027)1629 838 y Fo(\()p Fm(N)g Fo(\))1751 775 y Fk(\020)1805 876 y Fr(R)1874 890 y Fm(N)1967 876 y Fl(2)25 b Fr(G)2124 775 y Fk(\021)2229 876 y Ft(=)50 b(1)15 b Fr(:)118 1065 y Ft(Again)30 b(the)h(argumen)m(t)g(of)f([11)q 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Fm(y)r Fj(2)p Fo(\003)2725 4135 y Fu(N)r(;j)2842 4013 y Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(\036)3180 4027 y Fm(x)3225 4013 y Fr(\036)3279 4027 y Fm(y)3321 3912 y Fk(\021)3390 4013 y Fr(:)118 4282 y Ft(where)30 b(\000)438 4249 y Fm(i)438 4309 y(N)535 4282 y Ft(is)g(the)g(matrix)g(in)m(v)m(erse)g(of)h Fl(f)p Fr(J)9 b Ft(\()p Fr(x;)15 b(y)s Ft(\))p Fl(g)1843 4307 y Fm(x;y)r Fj(2)p Fo(\003)2036 4284 y Fu(i)2036 4330 y(N)2099 4282 y Ft(.)41 b(By)31 b(theorem)f(2.5)i(of)e([22)r(],)695 4501 y(lim)656 4562 y Fm(N)7 b Fj(!1)939 4440 y Ft(1)p 900 4481 124 4 v 900 4564 a Fr(N)983 4538 y Fm(d)1069 4501 y Ft(log)17 b(det)e(\000)1400 4515 y Fm(N)1518 4501 y Ft(=)89 b(lim)1639 4562 y Fm(N)7 b Fj(!1)1936 4440 y Ft(1)p 1868 4481 181 4 v 1868 4567 a Fl(j)p Ft(\003)1956 4535 y Fm(i)1956 4595 y(N)2024 4567 y Fl(j)2074 4501 y Ft(log)17 b(det)e(\000)2405 4464 y Fm(i)2405 4524 y(N)2487 4501 y Fr(;)117 b(i)25 b Ft(=)g(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(n)g(:)118 4715 y Ft(It)31 b(follo)m(ws,)e(using)g(\(4.51\),)k(that)1044 4941 y(lim)1005 5001 y Fm(N)7 b Fj(!1)1288 4879 y Ft(1)p 1249 4920 124 4 v 1249 5003 a Fr(N)1332 4977 y Fm(d)1418 4941 y Ft(log)1551 4840 y Fk(\020)1620 4941 y Ft(det)16 b(\000)1819 4955 y Fm(N)1953 4827 y(n)1916 4854 y Fk(Y)1917 5050 y Fm(i)p Fo(=1)2032 4941 y Ft(\(det)g(\000)2266 4903 y Fm(i)2266 4963 y(N)2333 4941 y Ft(\))2368 4903 y Fj(\000)p Fo(1)2463 4840 y Fk(\021)2567 4941 y Ft(=)51 b(0)15 b Fr(:)118 5179 y Ft(Therefore)30 b(\(4.56\))j(follo)m(ws)c (from)755 5373 y(lim)716 5434 y Fm(N)7 b Fj(!1)1000 5312 y Ft(1)p 961 5352 V 961 5435 a Fr(N)1044 5409 y Fm(d)1179 5287 y Fk(X)1129 5484 y Fm(x)p Fj(2)p Fo(\003)1265 5495 y Fu(N)r(;i)1443 5287 y Fk(X)1391 5484 y Fm(y)r Fj(2)p Fo(\003)1524 5495 y Fu(N)r(;j)1641 5373 y Fr(J)i Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(\027)1976 5336 y Fo(\()p Fm(N)7 b Fo(\))2099 5373 y Ft(\()p Fr(\036)2188 5387 y Fm(x)2233 5373 y Fr(\036)2287 5387 y Fm(y)2328 5373 y Ft(\))51 b(=)f(0)15 b Fr(;)92 b Ft(if)31 b Fr(i)41 b Fl(6)p Ft(=)f Fr(j)21 b(:)p eop %%Page: 29 29 29 28 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(29)118 366 y Ft(This)28 b(is)g(easily)h(c)m(hec)m(k)m (ed)i(as)e(in)f(Prop)s(osition)g(3.1.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)f(sho)m(wn)f(that)h(equalit)m(y)f(holds)f(in)g(\(4.54\).) 118 474 y(The)i(pro)s(of)g(of)g(\(4.49\))j(is)c(complete.)p 3595 474 4 62 v 3599 416 55 4 v 3599 474 V 3653 474 4 62 v 1253 759 a Fs(Appendix)k Ft(A.)51 b Fs(The)34 b(Exp)-6 b(ansion)218 921 y Ft(The)34 b(p)s(oin)m(t)h(of)g(this)f(section)h(is)g (to)h(replace)f(\(2.6\))i(b)m(y)e(an)g(expansion)f(in)m(v)m(olving)g (only)g(paths)h(b)s(e-)118 1031 y(longing)c(to)j(some)f(\014nite)e (region)h(\003)d Fl(\032\032)g Fn(Z)1619 998 y Fm(d)1687 1031 y Ft(rather)k(than)f(the)h(whole)f(lattice.)47 b(W)-8 b(e)34 b(\014x)e(a)h(\014nite)e(b)s(o)m(x)118 1139 y(\003)25 b Fl(\032\032)g Fn(Z)438 1106 y Fm(d)474 1139 y Ft(,)31 b(and)f(for)g(eac)m(h)h(in)m(teger)g Fr(n)25 b Fq(>)g Ft(1,)31 b(w)m(e)g(de\014ne)e(the)i(matrix)722 1303 y Fr(B)791 1317 y Fm(n)838 1303 y Ft(\()p Fr(x;)15 b(y)s Ft(\))52 b(=)1237 1217 y Fk(X)1221 1418 y Fm(z)1254 1427 y Fg(1)1296 1418 y Fm(=)-43 b Fj(2)p Fo(\003)1399 1303 y Fl(\001)15 b(\001)g(\001)1541 1217 y Fk(X)1521 1418 y Fm(z)1554 1426 y Fu(n)1604 1418 y Fm(=)-43 b Fj(2)p Fo(\003)1722 1303 y Fr(Q)p Ft(\()p Fr(x;)15 b(z)1963 1317 y Fo(1)2004 1303 y Ft(\))g Fl(\001)g(\001)g(\001)h Fr(Q)p Ft(\()p Fr(z)2324 1317 y Fm(n)2372 1303 y Fr(;)f(y)s Ft(\))26 b Fr(;)166 b(x)15 b(;)g(y)44 b Fl(2)25 b Ft(\003)118 1556 y(Let)31 b(also)f Fr(B)532 1570 y Fo(0)572 1556 y Ft(\()p Fr(x;)15 b(y)s Ft(\))26 b(=)f Fr(Q)p Ft(\()p Fr(x;)15 b(y)s Ft(\).)41 b(Here)31 b Fr(Q)f Ft(is)g(giv)m(en)g(b)m(y)g (\(2.4\).)118 1721 y Fy(Lemma)j(A.1.)46 b Fi(F)-7 b(or)33 b(e)-5 b(ach)34 b Fr(x;)15 b(y)28 b Fl(2)d Ft(\003)p Fi(,)33 b(we)f(have)1393 1941 y Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))52 b(=)1878 1827 y Fj(1)1848 1855 y Fk(X)1847 2049 y Fm(n)p Fo(=0)2016 1918 y Ft(~)1995 1941 y Fr(Q)2067 1903 y Fm(n)2067 1963 y Fo(\003)2120 1941 y Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(;)1082 b Ft(\(A.1\))118 2193 y Fi(wher)-5 b(e)396 2170 y Ft(~)375 2193 y Fr(Q)447 2207 y Fo(\003)532 2193 y Fi(is)33 b(the)g(matrix)h(given)e(by)1403 2390 y Ft(~)1382 2413 y Fr(Q)1454 2427 y Fo(\003)1507 2413 y Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)1920 2300 y Fj(1)1890 2327 y Fk(X)1899 2525 y Fm(l)q Fo(=0)2051 2413 y Fr(B)2120 2428 y Fm(l)2146 2413 y Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(:)1056 b Ft(\(A.2\))118 2771 y Fi(R)-5 b(emark)p Ft(.)58 b(In)35 b(the)h Fi(ferr)-5 b(omagnetic)38 b Ft(case)f Fr(Q)d Ft(=)g Fr(p)p Ft(,)j(the)f(ab)s(o)m(v)m(e)h(Lemma)e(b)s(ecomes)h (a)h(restatemen)m(t)g(of)118 2881 y(the)28 b(results)f(of)h([32)q(],)h (c)m(hapter)g(3.)40 b(In)27 b(particular,)g(in)g(this)g(case)2332 2858 y(~)2312 2881 y Fr(Q)2384 2895 y Fo(\003)2436 2881 y Ft(\()p Fr(x;)15 b(y)s Ft(\))29 b(can)f(b)s(e)g(in)m(terpreted)f(as)h (the)118 2989 y(transition)h(function)g(of)i(a)f(`renormalized')g(or)g Fi(emb)-5 b(e)g(dde)g(d)33 b Ft(random)c(w)m(alk.)118 3183 y Fi(Pr)-5 b(o)g(of.)43 b Ft(Let)31 b Fr(x;)15 b(y)29 b Fl(2)c Ft(\003)30 b(b)s(e)g(\014xed.)40 b(Then,)29 b(from)h(\(2.6\),)j(w)m(e)e(ha)m(v)m(e)986 3408 y Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))52 b(=)e Fr(\016)s Ft(\()p Fr(x;)15 b(y)s Ft(\))47 b(+)e Fr(Q)p Ft(\()p Fr(x;)15 b(y)s Ft(\))47 b(+)2332 3294 y Fj(1)2302 3322 y Fk(X)2301 3516 y Fm(n)p Fo(=2)2464 3408 y Fr(Q)2536 3371 y Fm(n)2583 3408 y Ft(\()p Fr(x;)15 b(y)s Ft(\))659 b(\(A.3\))118 3639 y(Moreo)m(v)m(er,)760 3745 y Fj(1)730 3772 y Fk(X)729 3967 y Fm(n)p Fo(=2)862 3859 y Fr(Q)934 3821 y Fm(n)981 3859 y Ft(\()p Fr(x;)15 b(y)s Ft(\))51 b(=)1395 3745 y Fj(1)1364 3772 y Fk(X)1364 3967 y Fm(n)p Fo(=1)1559 3772 y Fk(X)1527 3978 y Fm(z)1560 3987 y Fg(1)1594 3978 y Fj(2)p Fh(Z)1692 3959 y Fu(d)1737 3859 y Fl(\001)15 b(\001)g(\001)1894 3772 y Fk(X)1858 3978 y Fm(z)1891 3986 y Fu(n)1933 3978 y Fj(2)p Fh(Z)2031 3959 y 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b(or)27 b(in)m(tegers)f Fr(n)f Fq(>)g Ft(1)118 4550 y(and)30 b Fr(k)e Fq(6)d Fr(n)p Ft(,)30 b(de\014ne)g(the)h(matrix)e(elemen)m(t)i Fl(A)p Ft(\()p Fr(n;)15 b(k)s Ft(\))31 b(b)m(y)718 4720 y Fl(A)p Ft(\()p Fr(n;)15 b(k)s Ft(\))51 b(=)1291 4634 y Fk(X)1188 4822 y Fu(\031)r Fe(2f)p Fg(\003)p Fu(;)p Fg(\003)1401 4808 y Fu(c)1433 4822 y Fe(g)1465 4808 y Fu(n)1508 4822 y Fg(:)1226 4881 y(#)1276 4898 y(\003)1321 4881 y(\()p Fu(\031)r Fg(\)=)p Fu(k)1629 4634 y Fk(X)1571 4835 y Fm(z)1604 4844 y Fg(1)1638 4835 y Fj(2)p Fm(\031)r Fo(\(1\))1833 4720 y Fl(\001)15 b(\001)g(\001)2020 4634 y Fk(X)1955 4835 y Fm(z)1988 4843 y Fu(n)2030 4835 y Fj(2)p Fm(\031)r Fo(\()p Fm(n)p Fo(\))2248 4720 y Fr(Q)p Ft(\()p Fr(x;)g(z)2489 4734 y Fo(1)2529 4720 y Ft(\))g Fl(\001)g(\001)g(\001)i Fr(Q)p Ft(\()p Fr(z)2850 4734 y Fm(n)2897 4720 y Fr(;)e(y)s Ft(\))g Fr(;)118 5027 y Ft(where)30 b(w)m(e)i(only)e(ha)m(v)m(e)i(summed)d(the)i(con)m (tributions)e(of)i(those)h(sequences)f Fr(\031)e Fl(2)c(f)p Ft(\003)p Fr(;)15 b Ft(\003)3130 4994 y Fm(c)3166 5027 y Fl(g)3211 4994 y Fm(n)3289 5027 y Ft(suc)m(h)31 b(that)118 5135 y(\003)g(o)s(ccurs)f(exactly)h Fr(k)i Ft(times)d(\(#)1240 5149 y Fo(\003)1293 5135 y Ft(\()p Fr(\031)s Ft(\))c(=)f Fr(k)s Ft(\).)42 b(Setting)30 b Fl(A)p Ft(\(0)p Fr(;)15 b Ft(0\))27 b(=)e(0,)31 b(w)m(e)f(ha)m(v)m(e)863 5246 y Fj(1)832 5274 y Fk(X)831 5468 y Fm(n)p Fo(=2)995 5360 y Fr(Q)1067 5322 y Fm(n)1113 5360 y Ft(\()p Fr(x;)15 b(y)s Ft(\))52 b(=)1527 5246 y Fj(1)1497 5274 y Fk(X)1496 5468 y Fm(n)p Fo(=1)1709 5246 y Fm(n)1664 5274 y Fk(X)1666 5471 y Fm(k)r Fo(=0)1826 5360 y Fl(A)p Ft(\()p Fr(n;)15 b(k)s Ft(\))51 b(=)2317 5246 y Fj(1)2286 5274 y Fk(X)2287 5471 y Fm(k)r Fo(=0)2486 5246 y Fj(1)2455 5274 y Fk(X)2453 5471 y Fm(n)p Fo(=)p Fm(k)2619 5360 y Fl(A)p Ft(\()p Fr(n;)15 b(k)s Ft(\))g Fr(:)505 b Ft(\(A.4\))p eop %%Page: 30 30 30 29 bop 118 45 a Fo(30)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(No)m(w)i(observ)m(e)g(that)g Fl(A)p Ft(\()p Fr(n;)15 b Ft(0\))26 b(=)f Fr(B)1320 380 y Fm(n)1367 366 y Ft(\()p Fr(x;)15 b(y)s Ft(\),)31 b(for)g(all)e Fr(n)c Fq(>)g Ft(1.)41 b(In)29 b(particular)1206 486 y Fj(1)1176 513 y Fk(X)1175 707 y Fm(n)p Fo(=1)1338 599 y Fl(A)p Ft(\()p Fr(n;)15 b Ft(0\))52 b(=)1815 576 y(~)1794 599 y Fr(Q)1866 613 y Fo(\003)1919 599 y Ft(\()p Fr(x;)15 b(y)s Ft(\))41 b Fl(\000)f Fr(Q)p Ft(\()p Fr(x;)15 b(y)s Ft(\))g Fr(;)118 859 y Ft(where)402 836 y(~)381 859 y Fr(Q)453 873 y Fo(\003)536 859 y Ft(is)30 b(de\014ned)f(b)m(y)h (\(A.2\).)42 b(F)-8 b(rom)31 b(\(A.3\))h(and)e(\(A.4\))i(w)m(e)f(ha)m (v)m(e)880 1092 y Fr(G)p Ft(\()p Fr(x;)15 b(y)s Ft(\))52 b(=)e Fr(\016)s Ft(\()p Fr(x;)15 b(y)s Ft(\))42 b(+)1761 1069 y(~)1740 1092 y Fr(Q)1812 1106 y Fo(\003)1865 1092 y Ft(\()p Fr(x;)15 b(y)s Ft(\))46 b(+)2268 978 y Fj(1)2238 1005 y Fk(X)2239 1203 y Fm(k)r Fo(=1)2437 978 y Fj(1)2407 1005 y Fk(X)2404 1203 y Fm(n)p Fo(=)p Fm(k)2571 1092 y Fl(A)p Ft(\()p Fr(n;)15 b(k)s Ft(\))g Fr(:)118 1334 y Ft(In)30 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Fq(6)g Fr(l)615 4999 y Fj(\000)p Fm(\013)720 5032 y Fi(.)118 5201 y(Pr)-5 b(o)g(of.)43 b Ft(Giv)m(en)31 b(t)m(w)m(o)g(con\014gurations)f Fr(\030)t(;)15 b(\021)29 b Fl(2)c Ft(\012)30 b(of)g(\014nite)g(supp)s (ort,)e(w)m(e)j(de\014ne)1277 5373 y Fl(h)p Fr(\030)t(;)15 b(\021)s Fl(i)1480 5400 y Fm(")1567 5373 y Ft(=)1735 5286 y Fk(X)1689 5492 y Fm(x;y)r Fj(2)p Fh(Z)1882 5473 y Fu(d)1943 5373 y Fr(\030)1983 5387 y Fm(x)2042 5373 y Fr(G)2113 5335 y Fm(")2150 5373 y Ft(\()p Fr(x;)g(y)s Ft(\))g Fr(\021)2420 5387 y Fm(y)2478 5373 y Fr(;)p eop %%Page: 31 31 31 30 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(31)118 366 y Ft(and)30 b(let)g Fl(k)p Fr(\030)t Fl(k)560 333 y Fo(2)560 389 y Fm(")651 366 y Ft(=)50 b Fl(h)q Fr(\030)t(;)15 b(\030)t Fl(i)971 393 y Fm(")1023 366 y Ft(.)41 b(F)-8 b(ollo)m(wing)30 b([6])h(w)m(e)g (de\014ne,)f(for)g(\003)2247 380 y Fo(1)2286 366 y Fr(;)15 b Ft(\003)2389 380 y Fo(2)2455 366 y Fl(\032\032)24 b Fn(Z)2686 333 y Fm(d)902 525 y Fr(\034)10 b Ft(\(\003)1050 539 y Fo(1)1090 525 y Fr(;)15 b Ft(\003)1193 539 y Fo(2)1233 525 y Ft(\))51 b(=)f(sup)1577 451 y Fk(\010)1645 525 y Fl(h)q Fr(\030)t(;)15 b(\021)s Fl(i)1849 552 y Fm(")1916 525 y Fr(;)66 b Fl(k)p Fr(\030)t Fl(k)2141 539 y Fm(")2204 525 y Ft(=)25 b Fl(k)p Fr(\021)s Fl(k)2438 539 y Fm(")2501 525 y Ft(=)g(1)15 b Fr(;)1668 674 y(\030)29 b Ft(=)c(0)46 b(on)41 b(\003)2124 636 y Fm(c)2124 696 y Fo(1)2178 674 y Fr(;)66 b(\021)29 b Ft(=)c(0)46 b(on)40 b(\003)2729 636 y Fm(c)2729 696 y Fo(2)2784 600 y Fk(\011)2852 674 y Fr(;)118 828 y Ft(and)30 b(set,)h(for)f(an)m(y)h Fr(l)c(>)e Ft(0)742 984 y Fr(\034)10 b Ft(\()p Fr(l)r Ft(\))51 b(=)f(sup)1230 910 y Fk(\010)1299 984 y Fr(\034)10 b Ft(\(\003)1447 998 y Fo(1)1486 984 y Fr(;)15 b Ft(\003)1589 998 y Fo(2)1629 984 y Ft(\))g(;)67 b(\003)1834 998 y Fo(1)1873 984 y Fr(;)15 b Ft(\003)1976 998 y Fo(2)2042 984 y Fl(\032\032)24 b Fn(Z)2273 946 y Fm(d)2325 984 y Fr(;)65 b(d)p Ft(\(\003)2560 998 y Fo(1)2601 984 y Fr(;)15 b Ft(\003)2704 998 y Fo(2)2744 984 y Ft(\))25 b Fr(>)g(l)2944 910 y Fk(\011)3012 984 y Fr(:)419 b Ft(\(B.2\))118 1142 y(W)-8 b(e)27 b(ma)m(y)f(use)f(no)m(w) h(Nelson's)f(estimate)i(\(cf.)39 b(e.g.)h(theorem)26 b(5.11)i(and)d(Lemma)g(5.12)i(in)e([6]\),)j(to)e(write)1302 1296 y Fl(k)p Fr(f)e(g)s Fl(k)1507 1310 y Fo(1)1598 1296 y Fq(6)51 b Fl(k)p Fr(f)10 b Fl(k)1865 1315 y Fo(1+)p Fm(\034)e Fo(\()p Fm(l)q Fo(\))2090 1296 y Fl(k)p Fr(g)s Fl(k)2226 1315 y Fo(1+)p Fm(\034)g Fo(\()p Fm(l)q Fo(\))2453 1296 y Fr(;)118 1462 y Ft(for)30 b(an)m(y)h(pair)e(of)i(lo)s(cal)e (functions)g Fr(f)5 b(;)15 b(g)29 b Fl(2)c Fr(L)1636 1429 y Fo(1)1675 1462 y Ft(\()p Fr(P)1781 1429 y Fm(")1818 1462 y Ft(\))31 b(whose)f(supp)s(orts)e(satisfy)i Fr(d)p Ft(\()p Fr(S)2946 1477 y Fm(f)2992 1462 y Fr(;)15 b(S)3088 1476 y Fm(g)3128 1462 y Ft(\))26 b Fr(>)f(l)r Ft(.)118 1577 y(Let)34 b(us)f(c)m(hec)m(k)j(the)e(deca)m(y)h(of)f Fr(\034)10 b Ft(\()p Fr(l)r 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b(is)f(then)g(su\016cien)m(t)h(to)g(sho)m(w)1445 2888 y Fk(X)1421 3089 y Fj(j)p Fm(x)p Fj(j)11 b Fm(>)g(l)1631 2974 y Fl(j)p Fr(G)1727 2937 y Fm(")1764 2974 y Ft(\(0)p Fr(;)k(x)p Ft(\))p Fl(j)52 b Fq(6)f Fr(l)2199 2937 y Fj(\000)p Fm(\013)2318 2974 y Fr(;)1113 b Ft(\(B.4\))118 3237 y(for)37 b(some)h Fr(\013)g(>)f Ft(0)h(and)f(for)g(all)f Fr(l)k Ft(su\016cien)m(tly)c(large.)62 b(W)-8 b(e)39 b(\014rst)e(observ)m(e)h(that)g Fr(G)3022 3204 y Fm(")3059 3237 y Ft(\(0)p Fr(;)15 b(x)p Ft(\))39 b(is)e(indeed)118 3345 y(absolutely)29 b(summable,)g(cf.)41 b(\(4.20\).)i(W)-8 b(e)32 b(can)f(then)f(write)686 3487 y Fk(X)662 3689 y Fj(j)p Fm(x)p Fj(j)11 b Fm(>)h(l)872 3574 y Fl(j)p Fr(G)968 3536 y Fm(")1005 3574 y Ft(\(0)p Fr(;)j(x)p Ft(\))p Fl(j)53 b Fq(6)1450 3487 y Fk(X)1426 3689 y Fj(j)p Fm(x)p Fj(j)11 b Fm(>)g(l)1636 3574 y Fr(p)p Ft(\(0)p Fr(;)k(x)p Ft(\))36 b(+)2062 3460 y Fj(1)2032 3487 y Fk(X)2031 3682 y Fm(n)p Fo(=2)2164 3574 y Ft(\(1)21 b(+)f Fr(")p Ft(\))2433 3536 y Fj(\000)p Fm(n)2590 3487 y Fk(X)2566 3689 y Fj(j)p Fm(x)p Fj(j)11 b Fm(>)g(l)2775 3574 y Fr(p)2821 3536 y Fm(n)2868 3574 y Ft(\(0)p Fr(;)k(x)p Ft(\))g Fr(:)341 b Ft(\(B.5\))118 3837 y(Let)29 b Fr(\014)34 b Ft(b)s(e)28 b(the)g(constan)m(t)i(app)s(earing)d(in)h(\(2.5\).)41 b(It)29 b(is)f(easy)h(to)g(c)m(hec)m(k)h(that)f(the)g(\014rst)f(term)g (ab)s(o)m(v)m(e)i(can)118 3947 y(b)s(e)c(b)s(ounded)e(b)m(y)i Fr(l)756 3914 y Fj(\000)p Fm(\014)884 3947 y Ft(as)h(so)s(on)f(as)g Fr(l)i Ft(is)e(su\016cien)m(tly)f(large.)39 b(Let)27 b Fn(P)2369 3961 y Fo(0)2435 3947 y Ft(denote)g(the)g(probabilit)m(y)c (measure)118 4055 y(for)30 b(the)h(random)e(w)m(alk)i Fl(f)p Fr(\030)1045 4069 y Fo(0)1084 4055 y Fr(;)15 b(\030)1164 4069 y Fo(1)1204 4055 y Fr(;)g(\030)1284 4069 y Fo(2)1324 4055 y Fr(;)g(:)g(:)g(:)h Fl(g)31 b Ft(with)e(transition)g Fr(p)h Ft(and)g(starting)g(p)s(oin)m(t)f Fr(\030)3052 4069 y Fo(0)3117 4055 y Ft(=)c(0.)41 b(F)-8 b(or)31 b(eac)m(h)118 4163 y Fr(n)25 b Fq(>)g Ft(2,)31 b(w)m(e)g(ha)m(v)m(e)618 4244 y 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8929 l 2075 8900 l 2041 8873 l 2010 8847 l 1982 8823 l 1957 8801 l 1935 8781 l 1917 8763 l 1902 8746 l 1890 8732 l 1882 8719 l 1877 8709 l 1875 8700 l 1877 8692 l 1883 8686 l 1893 8682 l 1908 8680 l 1926 8679 l 1949 8681 l 1975 8683 l 2006 8688 l 2040 8693 l 2077 8700 l 2118 8709 l 2161 8718 l 2207 8729 l 2256 8740 l 2306 8752 l 2358 8765 l 2412 8778 l 2466 8792 l 2521 8807 l 2576 8821 l 2631 8836 l 2685 8851 l 2739 8866 l 2792 8881 l 2843 8896 l 2893 8911 l 2941 8925 l 2987 8940 l 3030 8954 l 3071 8968 l 3109 8982 l 3144 8995 l 3177 9008 l 3207 9022 l 3234 9035 l 3258 9048 l 3280 9061 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 1425 5550 m 1440 5567 l 1458 5583 l 1479 5598 l 1503 5612 l 1530 5626 l 1560 5639 l 1593 5652 l 1628 5665 l 1667 5677 l 1708 5689 l 1751 5701 l 1795 5713 l 1842 5725 l 1889 5736 l 1937 5747 l 1985 5757 l 2033 5767 l 2081 5777 l 2127 5786 l 2172 5794 l 2215 5801 l 2256 5807 l 2294 5811 l 2329 5815 l 2361 5817 l 2389 5817 l 2413 5815 l 2434 5812 l 2450 5806 l 2462 5798 l 2471 5788 l 2475 5775 l 2476 5761 l 2473 5746 l 2467 5727 l 2459 5706 l 2447 5683 l 2432 5656 l 2414 5627 l 2394 5596 l 2371 5562 l 2345 5526 l 2318 5487 l 2288 5447 l 2256 5405 l 2223 5362 l 2189 5318 l 2154 5272 l 2118 5227 l 2081 5181 l 2044 5135 l 2008 5090 l 1971 5045 l 1935 5002 l 1900 4960 l 1866 4919 l 1833 4881 l 1801 4845 l 1771 4812 l 1742 4781 l 1715 4753 l 1690 4728 l 1666 4707 l 1645 4689 l 1625 4674 l 1607 4663 l 1590 4655 l 1575 4650 l 1558 4649 l 1543 4652 l 1529 4659 l 1516 4672 l 1504 4688 l 1492 4709 l 1481 4735 l 1471 4763 l 1461 4795 l 1451 4831 l 1442 4868 l 1433 4908 l 1424 4950 l 1416 4993 l 1409 5037 l 1402 5081 l 1395 5125 l 1389 5169 l 1384 5211 l 1381 5253 l 1378 5293 l 1376 5331 l 1376 5366 l 1377 5400 l 1380 5431 l 1385 5459 l 1391 5485 l 1400 5509 l 1411 5530 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 5175 1650 m 5176 1669 l 5175 1690 l 5172 1713 l 5168 1740 l 5164 1769 l 5157 1802 l 5150 1837 l 5142 1875 l 5133 1916 l 5123 1959 l 5112 2004 l 5100 2050 l 5089 2097 l 5076 2146 l 5064 2194 l 5051 2242 l 5038 2289 l 5025 2334 l 5012 2378 l 4999 2420 l 4986 2459 l 4973 2494 l 4961 2526 l 4948 2554 l 4936 2577 l 4924 2597 l 4912 2611 l 4900 2620 l 4887 2625 l 4875 2625 l 4864 2621 l 4852 2613 l 4840 2601 l 4828 2586 l 4815 2566 l 4801 2543 l 4787 2516 l 4772 2486 l 4756 2453 l 4740 2416 l 4724 2377 l 4707 2336 l 4689 2292 l 4672 2247 l 4654 2200 l 4636 2153 l 4618 2105 l 4601 2056 l 4583 2008 l 4566 1961 l 4550 1914 l 4534 1869 l 4519 1825 l 4504 1783 l 4491 1743 l 4479 1705 l 4467 1670 l 4457 1638 l 4448 1608 l 4441 1581 l 4435 1557 l 4430 1535 l 4427 1517 l 4425 1500 l 4425 1477 l 4429 1459 l 4438 1448 l 4449 1441 l 4464 1440 l 4482 1442 l 4503 1448 l 4525 1457 l 4549 1469 l 4574 1481 l 4599 1495 l 4625 1508 l 4650 1521 l 4674 1533 l 4698 1544 l 4720 1553 l 4742 1561 l 4762 1566 l 4781 1571 l 4800 1575 l 4819 1579 l 4838 1582 l 4858 1584 l 4879 1585 l 4901 1585 l 4923 1583 l 4947 1581 l 4971 1577 l 4995 1574 l 5019 1570 l 5043 1567 l 5066 1566 l 5088 1565 l 5108 1567 l 5126 1572 l 5141 1580 l 5155 1591 l 5165 1606 l 5171 1626 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 8850 3375 m 8874 3378 l 8900 3378 l 8929 3378 l 8959 3375 l 8992 3371 l 9028 3365 l 9065 3357 l 9105 3349 l 9147 3339 l 9191 3327 l 9238 3315 l 9286 3301 l 9336 3287 l 9388 3271 l 9441 3255 l 9495 3237 l 9550 3220 l 9607 3201 l 9664 3182 l 9721 3163 l 9778 3143 l 9835 3123 l 9892 3102 l 9949 3082 l 10004 3061 l 10059 3040 l 10112 3019 l 10164 2998 l 10214 2978 l 10263 2957 l 10309 2936 l 10353 2915 l 10395 2894 l 10434 2873 l 10471 2852 l 10505 2831 l 10536 2810 l 10564 2789 l 10590 2767 l 10613 2745 l 10633 2723 l 10650 2700 l 10666 2674 l 10679 2647 l 10690 2618 l 10697 2589 l 10702 2557 l 10705 2525 l 10705 2490 l 10703 2454 l 10699 2417 l 10693 2378 l 10686 2338 l 10677 2296 l 10666 2254 l 10654 2210 l 10641 2166 l 10627 2121 l 10612 2076 l 10596 2031 l 10580 1985 l 10563 1940 l 10545 1895 l 10527 1852 l 10509 1809 l 10490 1767 l 10472 1727 l 10453 1689 l 10434 1652 l 10414 1618 l 10395 1586 l 10375 1556 l 10355 1529 l 10335 1505 l 10315 1484 l 10293 1466 l 10271 1451 l 10249 1439 l 10225 1431 l 10200 1425 l 10176 1423 l 10151 1423 l 10124 1426 l 10096 1431 l 10067 1439 l 10036 1449 l 10003 1461 l 9969 1475 l 9933 1492 l 9896 1510 l 9858 1531 l 9818 1553 l 9777 1577 l 9735 1602 l 9692 1628 l 9648 1656 l 9604 1685 l 9559 1714 l 9513 1745 l 9468 1776 l 9422 1807 l 9376 1839 l 9331 1871 l 9286 1903 l 9241 1935 l 9198 1966 l 9155 1998 l 9113 2029 l 9072 2059 l 9033 2089 l 8995 2119 l 8959 2147 l 8925 2176 l 8892 2203 l 8861 2230 l 8832 2255 l 8805 2281 l 8780 2305 l 8757 2330 l 8736 2353 l 8717 2377 l 8700 2400 l 8682 2429 l 8666 2458 l 8652 2487 l 8641 2517 l 8632 2548 l 8624 2579 l 8618 2611 l 8614 2644 l 8610 2678 l 8608 2712 l 8607 2747 l 8608 2783 l 8609 2819 l 8610 2855 l 8613 2892 l 8616 2928 l 8620 2965 l 8625 3001 l 8631 3036 l 8637 3071 l 8644 3105 l 8652 3138 l 8660 3169 l 8670 3199 l 8681 3227 l 8693 3253 l 8706 3277 l 8721 3299 l 8737 3318 l 8756 3335 l 8776 3349 l 8798 3360 l 8823 3369 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 9000 5850 m 9019 5854 l 9040 5856 l 9064 5857 l 9090 5857 l 9119 5856 l 9151 5854 l 9186 5850 l 9223 5846 l 9264 5840 l 9306 5834 l 9350 5827 l 9397 5819 l 9445 5811 l 9494 5802 l 9544 5792 l 9595 5783 l 9646 5773 l 9696 5763 l 9746 5752 l 9795 5742 l 9843 5732 l 9889 5722 l 9933 5712 l 9974 5703 l 10013 5693 l 10048 5685 l 10081 5676 l 10109 5668 l 10134 5660 l 10155 5652 l 10173 5645 l 10186 5638 l 10195 5632 l 10200 5625 l 10201 5617 l 10198 5610 l 10189 5602 l 10176 5595 l 10157 5587 l 10134 5579 l 10107 5571 l 10076 5563 l 10041 5555 l 10003 5547 l 9962 5539 l 9919 5530 l 9873 5522 l 9827 5514 l 9779 5505 l 9731 5497 l 9684 5489 l 9636 5481 l 9590 5473 l 9545 5465 l 9502 5457 l 9461 5450 l 9423 5443 l 9386 5436 l 9353 5429 l 9322 5423 l 9294 5416 l 9269 5411 l 9246 5405 l 9225 5400 l 9186 5390 l 9154 5380 l 9128 5370 l 9107 5360 l 9091 5349 l 9078 5338 l 9067 5327 l 9057 5318 l 9048 5310 l 9039 5304 l 9029 5302 l 9019 5305 l 9009 5312 l 9000 5325 l 8996 5335 l 8992 5348 l 8987 5364 l 8983 5381 l 8978 5402 l 8973 5424 l 8967 5449 l 8960 5476 l 8954 5504 l 8947 5534 l 8940 5564 l 8933 5595 l 8927 5625 l 8922 5655 l 8918 5684 l 8916 5712 l 8916 5738 l 8919 5762 l 8924 5784 l 8932 5803 l 8943 5819 l 8958 5832 l 8977 5842 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 6450 8025 m 6466 8037 l 6484 8048 l 6504 8058 l 6527 8067 l 6552 8076 l 6579 8085 l 6607 8093 l 6638 8101 l 6671 8109 l 6705 8117 l 6740 8125 l 6776 8133 l 6814 8140 l 6851 8147 l 6889 8154 l 6928 8160 l 6966 8165 l 7003 8170 l 7040 8174 l 7076 8176 l 7111 8177 l 7144 8177 l 7176 8175 l 7206 8171 l 7235 8166 l 7261 8158 l 7286 8147 l 7309 8134 l 7330 8119 l 7350 8100 l 7366 8081 l 7381 8060 l 7395 8036 l 7408 8008 l 7421 7978 l 7433 7946 l 7444 7910 l 7455 7871 l 7465 7830 l 7475 7787 l 7485 7741 l 7494 7693 l 7503 7644 l 7512 7594 l 7520 7542 l 7529 7490 l 7536 7438 l 7544 7386 l 7551 7334 l 7558 7283 l 7564 7233 l 7570 7185 l 7575 7139 l 7579 7095 l 7583 7054 l 7586 7015 l 7589 6979 l 7590 6947 l 7591 6918 l 7590 6892 l 7588 6870 l 7585 6852 l 7581 6837 l 7575 6825 l 7566 6816 l 7556 6811 l 7543 6810 l 7529 6814 l 7512 6822 l 7494 6834 l 7474 6849 l 7452 6868 l 7429 6890 l 7405 6915 l 7379 6943 l 7353 6972 l 7325 7003 l 7297 7036 l 7269 7069 l 7241 7102 l 7212 7135 l 7184 7168 l 7156 7200 l 7129 7231 l 7102 7260 l 7076 7287 l 7051 7313 l 7026 7336 l 7003 7356 l 6981 7375 l 6959 7390 l 6939 7404 l 6919 7415 l 6900 7425 l 6872 7436 l 6846 7444 l 6819 7448 l 6794 7447 l 6768 7443 l 6743 7436 l 6718 7425 l 6693 7413 l 6668 7399 l 6643 7384 l 6619 7369 l 6596 7355 l 6573 7343 l 6551 7333 l 6531 7326 l 6511 7323 l 6493 7323 l 6477 7328 l 6462 7337 l 6450 7350 l 6441 7364 l 6434 7380 l 6427 7400 l 6421 7422 l 6416 7448 l 6411 7475 l 6406 7506 l 6402 7538 l 6398 7572 l 6394 7607 l 6390 7644 l 6387 7680 l 6384 7717 l 6382 7754 l 6381 7790 l 6381 7824 l 6383 7857 l 6386 7888 l 6391 7917 l 6398 7944 l 6407 7968 l 6418 7989 l 6433 8008 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 4050 7950 m 4069 7960 l 4090 7970 l 4113 7979 l 4140 7987 l 4169 7995 l 4200 8002 l 4234 8009 l 4270 8015 l 4309 8021 l 4349 8027 l 4391 8033 l 4435 8038 l 4480 8043 l 4526 8048 l 4573 8052 l 4620 8057 l 4666 8061 l 4713 8065 l 4759 8069 l 4803 8073 l 4847 8076 l 4888 8080 l 4928 8083 l 4966 8086 l 5002 8089 l 5035 8092 l 5065 8094 l 5093 8096 l 5117 8098 l 5139 8099 l 5158 8100 l 5175 8100 l 5189 8100 l 5201 8099 l 5210 8098 l 5218 8096 l 5223 8094 l 5227 8092 l 5229 8089 l 5230 8086 l 5229 8083 l 5228 8080 l 5225 8076 l 5221 8073 l 5216 8069 l 5211 8065 l 5205 8061 l 5198 8057 l 5191 8052 l 5184 8048 l 5176 8043 l 5167 8038 l 5159 8033 l 5149 8027 l 5140 8021 l 5130 8015 l 5120 8009 l 5109 8002 l 5097 7995 l 5085 7987 l 5071 7979 l 5057 7970 l 5042 7960 l 5025 7950 l 5007 7939 l 4986 7927 l 4964 7914 l 4940 7900 l 4915 7885 l 4887 7869 l 4857 7851 l 4826 7833 l 4793 7813 l 4758 7792 l 4722 7771 l 4685 7749 l 4647 7726 l 4609 7703 l 4570 7679 l 4530 7656 l 4491 7633 l 4452 7610 l 4414 7587 l 4376 7566 l 4340 7545 l 4304 7526 l 4270 7508 l 4238 7491 l 4207 7476 l 4179 7463 l 4152 7451 l 4127 7442 l 4105 7434 l 4084 7429 l 4066 7426 l 4050 7425 l 4033 7427 l 4018 7431 l 4005 7439 l 3994 7450 l 3985 7464 l 3978 7481 l 3971 7501 l 3966 7523 l 3961 7547 l 3958 7572 l 3955 7599 l 3953 7627 l 3952 7656 l 3951 7685 l 3951 7713 l 3953 7742 l 3955 7769 l 3958 7796 l 3963 7821 l 3969 7844 l 3977 7866 l 3987 7886 l 3999 7905 l 4013 7921 l 4030 7936 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 3836 5448 m 3849 5463 l 3864 5478 l 3882 5492 l 3902 5507 l 3926 5522 l 3952 5537 l 3980 5553 l 4011 5569 l 4044 5585 l 4079 5601 l 4115 5618 l 4153 5635 l 4192 5651 l 4232 5668 l 4272 5685 l 4312 5701 l 4352 5717 l 4391 5732 l 4430 5747 l 4467 5760 l 4503 5773 l 4537 5785 l 4570 5795 l 4600 5804 l 4628 5812 l 4654 5818 l 4678 5822 l 4700 5824 l 4719 5825 l 4736 5823 l 4755 5818 l 4771 5810 l 4785 5799 l 4797 5785 l 4807 5768 l 4815 5748 l 4822 5726 l 4828 5701 l 4832 5675 l 4836 5647 l 4838 5618 l 4840 5589 l 4842 5559 l 4842 5529 l 4843 5500 l 4842 5472 l 4842 5445 l 4840 5419 l 4838 5394 l 4835 5372 l 4831 5351 l 4826 5332 l 4819 5314 l 4811 5298 l 4799 5282 l 4785 5266 l 4768 5252 l 4749 5240 l 4726 5228 l 4701 5218 l 4674 5208 l 4645 5199 l 4615 5191 l 4584 5183 l 4552 5175 l 4520 5167 l 4489 5160 l 4458 5152 l 4429 5144 l 4402 5136 l 4376 5127 l 4353 5117 l 4333 5107 l 4315 5096 l 4299 5085 l 4286 5073 l 4272 5056 l 4262 5038 l 4256 5019 l 4254 4998 l 4255 4976 l 4259 4953 l 4265 4929 l 4272 4905 l 4279 4882 l 4286 4859 l 4292 4838 l 4295 4819 l 4297 4803 l 4296 4789 l 4292 4779 l 4286 4773 l 4278 4770 l 4268 4770 l 4257 4772 l 4245 4777 l 4231 4783 l 4216 4791 l 4200 4800 l 4184 4809 l 4168 4820 l 4151 4831 l 4135 4843 l 4120 4856 l 4104 4870 l 4089 4886 l 4075 4903 l 4061 4923 l 4051 4939 l 4040 4957 l 4028 4976 l 4015 4997 l 4001 5019 l 3985 5043 l 3969 5068 l 3951 5094 l 3933 5120 l 3914 5147 l 3896 5175 l 3878 5203 l 3861 5230 l 3846 5257 l 3833 5284 l 3822 5310 l 3815 5335 l 3811 5360 l 3811 5383 l 3815 5406 l 3823 5427 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 8850 9750 m 8865 9751 l 8882 9748 l 8901 9743 l 8922 9734 l 8946 9723 l 8973 9708 l 9002 9691 l 9033 9671 l 9066 9648 l 9102 9623 l 9140 9596 l 9180 9567 l 9221 9536 l 9264 9503 l 9309 9469 l 9354 9433 l 9400 9397 l 9446 9360 l 9492 9322 l 9538 9285 l 9584 9247 l 9629 9209 l 9673 9172 l 9716 9136 l 9757 9100 l 9796 9066 l 9833 9033 l 9868 9001 l 9900 8971 l 9930 8942 l 9956 8915 l 9979 8890 l 9999 8867 l 10016 8845 l 10030 8825 l 10040 8807 l 10046 8790 l 10050 8775 l 10050 8758 l 10046 8743 l 10037 8729 l 10023 8715 l 10005 8703 l 9983 8691 l 9957 8680 l 9927 8670 l 9893 8660 l 9856 8651 l 9816 8641 l 9774 8633 l 9729 8624 l 9683 8616 l 9636 8609 l 9588 8602 l 9540 8595 l 9492 8589 l 9444 8584 l 9397 8579 l 9352 8575 l 9308 8572 l 9266 8571 l 9226 8570 l 9189 8571 l 9154 8573 l 9122 8577 l 9092 8582 l 9066 8590 l 9042 8599 l 9020 8611 l 9000 8625 l 8982 8642 l 8966 8661 l 8951 8683 l 8937 8709 l 8924 8738 l 8912 8770 l 8900 8805 l 8889 8844 l 8879 8885 l 8869 8929 l 8860 8975 l 8851 9023 l 8842 9072 l 8834 9123 l 8827 9175 l 8820 9226 l 8813 9278 l 8807 9328 l 8802 9378 l 8797 9426 l 8794 9472 l 8791 9516 l 8790 9557 l 8789 9594 l 8790 9628 l 8793 9658 l 8798 9684 l 8804 9706 l 8812 9724 l 8822 9737 l 8835 9746 l cp gs col7 0.75 shd ef gr gs col0 s gr % Polyline n 1350 2475 m 1351 2499 l 1353 2523 l 1356 2547 l 1361 2572 l 1366 2597 l 1372 2623 l 1379 2649 l 1386 2676 l 1394 2703 l 1402 2731 l 1411 2758 l 1420 2786 l 1430 2813 l 1440 2840 l 1450 2867 l 1461 2893 l 1472 2919 l 1484 2944 l 1497 2968 l 1511 2992 l 1525 3014 l 1540 3035 l 1557 3056 l 1575 3075 l 1592 3091 l 1609 3106 l 1628 3121 l 1648 3137 l 1668 3153 l 1690 3169 l 1712 3186 l 1735 3203 l 1758 3221 l 1782 3239 l 1806 3258 l 1831 3276 l 1856 3295 l 1881 3313 l 1906 3330 l 1932 3347 l 1958 3362 l 1984 3376 l 2010 3389 l 2036 3399 l 2062 3407 l 2088 3412 l 2114 3415 l 2141 3414 l 2168 3410 l 2195 3402 l 2222 3390 l 2250 3375 l 2271 3361 l 2292 3345 l 2314 3326 l 2337 3306 l 2361 3283 l 2385 3257 l 2410 3230 l 2437 3200 l 2464 3168 l 2492 3135 l 2521 3099 l 2551 3062 l 2581 3024 l 2612 2984 l 2643 2943 l 2675 2901 l 2707 2859 l 2739 2816 l 2770 2773 l 2802 2729 l 2833 2686 l 2864 2643 l 2893 2600 l 2922 2558 l 2950 2517 l 2976 2476 l 3002 2437 l 3025 2399 l 3047 2362 l 3067 2327 l 3085 2293 l 3101 2261 l 3115 2230 l 3127 2201 l 3136 2174 l 3143 2148 l 3148 2123 l 3150 2100 l 3149 2072 l 3145 2046 l 3137 2021 l 3125 1998 l 3109 1976 l 3091 1955 l 3069 1935 l 3044 1916 l 3017 1898 l 2987 1880 l 2955 1863 l 2922 1847 l 2887 1831 l 2851 1816 l 2815 1801 l 2778 1787 l 2741 1773 l 2703 1759 l 2667 1746 l 2631 1734 l 2595 1722 l 2561 1710 l 2527 1699 l 2495 1689 l 2464 1680 l 2435 1672 l 2406 1664 l 2378 1658 l 2351 1653 l 2325 1650 l 2297 1648 l 2269 1648 l 2241 1650 l 2213 1654 l 2184 1659 l 2156 1665 l 2127 1673 l 2098 1682 l 2069 1692 l 2040 1703 l 2010 1715 l 1981 1728 l 1951 1741 l 1922 1754 l 1893 1768 l 1864 1782 l 1835 1796 l 1807 1810 l 1780 1824 l 1753 1839 l 1727 1852 l 1702 1866 l 1678 1880 l 1655 1894 l 1633 1908 l 1612 1921 l 1593 1936 l 1575 1950 l 1554 1969 l 1535 1989 l 1517 2010 l 1501 2032 l 1486 2054 l 1473 2077 l 1460 2101 l 1448 2125 l 1437 2149 l 1427 2174 l 1417 2198 l 1407 2223 l 1398 2249 l 1390 2274 l 1382 2299 l 1375 2324 l 1368 2349 l 1362 2374 l 1357 2399 l 1353 2424 l 1351 2450 l cp gs col7 0.75 shd ef gr gs col0 s gr /Times-Roman ff 450.00 scf sf 1575 5400 m gs 1 -1 sc (f) col0 sh gr /Times-Roman ff 285.00 scf sf 1725 5550 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 285.00 scf sf 9150 9450 m gs 1 -1 sc (n) col0 sh gr /Times-Italic ff 450.00 scf sf 1875 11550 m gs 1 -1 sc (L) col0 sh gr /Times-Italic ff 450.00 scf sf 3225 11550 m gs 1 -1 sc (l) col0 sh gr /Times-Italic ff 450.00 scf sf 2250 7725 m gs 1 -1 sc (A) col0 sh gr /Times-Italic ff 450.00 scf sf 2250 9900 m gs 1 -1 sc (C) col0 sh gr /Times-Italic ff 450.00 scf sf 9225 5100 m gs 1 -1 sc (D) col0 sh gr /Times-Roman ff 285.00 scf sf 1950 3300 m gs 1 -1 sc (1) col0 sh gr /Times-Italic ff 450.00 scf sf 2100 2475 m gs 1 -1 sc (A) col0 sh gr /Times-Italic ff 450.00 scf sf 4350 2550 m gs 1 -1 sc (B) col0 sh gr /Times-Italic ff 450.00 scf sf 7050 2625 m gs 1 -1 sc (A) col0 sh gr /Times-Italic ff 450.00 scf sf 9450 2625 m gs 1 -1 sc (B) col0 sh gr /Times-Italic ff 450.00 scf sf 4575 4875 m gs 1 -1 sc (D) col0 sh gr /Times-Italic ff 450.00 scf sf 6825 5025 m gs 1 -1 sc (C) col0 sh gr /Times-Italic ff 450.00 scf sf 6900 9825 m gs 1 -1 sc (C) col0 sh gr /Times-Italic ff 450.00 scf sf 4425 9900 m gs 1 -1 sc (D) col0 sh gr /Times-Italic ff 450.00 scf sf 9150 10200 m gs 1 -1 sc (D) col0 sh gr /Times-Italic ff 450.00 scf sf 6825 7200 m gs 1 -1 sc (A) col0 sh gr /Times-Italic ff 450.00 scf sf 4500 7500 m gs 1 -1 sc (B) col0 sh gr /Times-Italic ff 450.00 scf sf 9375 7575 m gs 1 -1 sc (B) col0 sh gr /Times-Italic ff 450.00 scf sf 2175 4875 m gs 1 -1 sc (C) col0 sh gr /Times-Roman ff 450.00 scf sf 9000 9300 m gs 1 -1 sc (f) col0 sh gr /Times-Roman ff 450.00 scf sf 1800 3150 m gs 1 -1 sc (f) col0 sh gr /Times-Italic ff 180.00 scf sf 3375 4125 m gs 1 -1 sc (2) col0 sh gr /Times-Italic ff 360.00 scf sf 3075 4050 m gs 1 -1 sc (Q) col0 sh gr /Times-Italic ff 180.00 scf sf 3450 1650 m gs 1 -1 sc (1) col0 sh gr /Times-Italic ff 360.00 scf sf 3150 1575 m gs 1 -1 sc (Q) col0 sh gr /Times-Italic ff 180.00 scf sf 10575 8925 m gs 1 -1 sc (n) col0 sh gr /Times-Italic ff 360.00 scf sf 10275 8850 m gs 1 -1 sc (Q) col0 sh gr $F2psEnd rs showpage %%EndDocument %%EndDocument endTexFig 817 1956 a Fw(Figure)28 b(1.)38 b Fx(Geometric)26 b(construction)g(in) g(the)f(pro)r(of)i(of)f(Lemma)e(B.2)j(.)118 2234 y Ft(Using)41 b(the)i(b)s(ound)d Fr(\034)10 b Ft(\()p Fr(l)r Ft(\))45 b Fq(6)f Fr(l)1184 2201 y Fj(\000)p Fm(\013)1289 2234 y Ft(,)h(cf.)76 b(Lemma)42 b(B.1,)k(it)c(is)f(straigh)m(tforw)m(ard)h (to)h(c)m(hec)m(k)g(that)g Fr(\032)p Ft(\()p Fr(l)r Ft(\))118 2342 y(satis\014es)30 b(the)h(desired)d(prop)s(erties.)p 3595 2342 4 62 v 3599 2284 55 4 v 3599 2342 V 3653 2342 4 62 v 218 2589 a(The)22 b(h)m(yp)s(othesis)g(of)i(Lemma)f(B.2)h(are)g (tailored)f(in)e(suc)m(h)i(a)h(w)m(a)m(y)g(that)g(they)g(apply)d(to)j (the)g(follo)m(wing)118 2697 y(situation)i(\(see)h([11)q(],)h(Lemma)f (5.4.13,)j(for)c(the)h(one-dimensional)d(case\).)41 b(Supp)s(ose)25 b(a)i(lo)s(cal)f(b)s(ounded)118 2805 y(function)j Fr(f)40 b Ft(is)29 b(giv)m(en,)i(whose)f(supp)s(ort)f Fr(S)1574 2820 y Fm(f)1649 2805 y Ft(is)g(con)m(tained)i(in)e(the)i(cub)s(e)e(of) i(side)e Fr(q)f Fl(2)d Fn(Z)3135 2819 y Fo(+)1483 2994 y Fr(Q)g Ft(=)g([0)p Fr(;)15 b(q)25 b Fl(\000)20 b Ft(1])2013 2956 y Fm(d)2074 2994 y Fl(\\)g Fn(Z)2220 2956 y Fm(d)2271 2994 y Fr(:)118 3171 y Ft(W)-8 b(e)32 b(sho)m(w)e(that)h(the)f(limit) 1009 3369 y(\003)1072 3383 y Fm(Q)1132 3369 y Ft(\()p Fr(f)10 b Ft(\))51 b(=)89 b(lim)1429 3430 y Fm(N)7 b Fj(!1)1712 3308 y Ft(1)p 1673 3348 124 4 v 1673 3432 a Fr(N)1756 3405 y Fm(d)1842 3369 y Ft(log)17 b Fr(P)2046 3332 y Fm(")2082 3296 y Fk(\000)2139 3369 y Ft(exp)2324 3283 y Fk(X)2293 3480 y Fm(x)p Fj(2)p Fo(\003)2429 3491 y Fu(N)2502 3369 y Fr(f)29 b Fl(\016)21 b Fr(\022)2685 3383 y Fm(x)2729 3296 y Fk(\001)3456 3369 y Ft(\(B.9\))118 3646 y(exists.)51 b(T)-8 b(o)34 b(this)f(end)g(w)m(e)h(ma)m(y)h(assume) e Fr(f)41 b Fq(>)31 b Ft(0,)k(otherwise)e(replace)h Fr(f)43 b Ft(with)33 b Fr(f)e Ft(+)23 b Fl(k)p Fr(f)10 b Fl(k)3190 3660 y Fj(1)3264 3646 y Ft(.)52 b(By)34 b(shift)118 3754 y(in)m(v)-5 b(ariance)30 b(of)g Fr(P)722 3721 y Fm(")759 3754 y Ft(,)g(w)m(e)h(ma)m(y)g(replace)g(\003)1517 3768 y Fm(N)1614 3754 y Ft(in)e(\(4.45\))k(b)m(y)d(the)h(cub)s(e)1432 3938 y Fr(B)1501 3952 y Fm(N)1593 3938 y Ft(=)25 b([0)p Fr(;)15 b(N)31 b Fl(\000)20 b Ft(1])2064 3901 y Fm(d)2126 3938 y Fl(\\)f Fn(Z)2271 3901 y Fm(d)2323 3938 y Fr(:)118 4118 y Ft(W)-8 b(e)41 b(also)f(let)g Fr(B)686 4132 y Fm(N)753 4118 y Ft(\()p Fr(x)p Ft(\))g(denote)h(the)f(translate)g Fr(B)1842 4132 y Fm(N)1935 4118 y Ft(+)27 b Fr(x)p Ft(,)42 b Fr(x)f Fl(2)f Fn(Z)2411 4085 y Fm(d)2447 4118 y Ft(.)69 b(Let)41 b Fr(M)51 b Fl(2)40 b Fn(Z)3019 4132 y Fo(+)3074 4118 y Ft(,)i Fr(M)52 b(<)40 b(N)50 b Ft(and)118 4226 y(divide)32 b Fr(B)461 4240 y Fm(N)563 4226 y Ft(in)m(to)i(translates)h (of)f(the)h(cub)s(e)f Fr(B)1727 4240 y Fm(M)1806 4226 y Ft(.)53 b(If)34 b Fr(m)d Ft(=)h([)p Fr(N)r(=)-5 b(M)10 b Ft(],)37 b(the)e(in)m(teger)g(part,)g(then)f Fr(B)3491 4240 y Fm(N)3591 4226 y Ft(=)118 4336 y Fr(B)187 4350 y Fm(mM)355 4336 y Fl([)26 b Fr(R)511 4350 y Fm(N)s(;M)669 4336 y Ft(,)43 b(with)38 b Fl(j)p Fr(R)1047 4350 y Fm(N)s(;M)1205 4336 y Fl(j)k Fq(6)f Fr(c)1423 4350 y Fm(M)1502 4336 y Fr(N)1585 4303 y Fm(d)p Fj(\000)p Fo(1)1716 4336 y Ft(,)h(for)e(some)h Fr(M)10 b Ft(-dep)s(enden)m(t)39 b(constan)m(t)i Fr(c)3162 4350 y Fm(M)3242 4336 y Ft(.)69 b(W)-8 b(e)41 b(no)m(w)118 4444 y(enlarge)29 b(the)g(cub)s(es)f Fr(B)903 4458 y Fm(M)1011 4444 y Ft(con)m(tained)h(in)f Fr(B)1595 4458 y Fm(mM)1765 4444 y Ft(b)m(y)h(c)m(hanging)g(their)f (side)g(from)g Fr(M)39 b Ft(to)30 b Fr(M)d Ft(+)17 b Fr(q)j Ft(+)d Fr(l)r Ft(,)30 b(for)118 4552 y(some)h Fr(l)c(>)e Ft(0.)41 b(W)-8 b(e)32 b(c)m(ho)s(ose)f Fr(m)1132 4519 y Fm(d)1203 4552 y Ft(sites)f Fr(x)1458 4566 y Fm(i)1511 4552 y Fl(2)25 b Ft(\()p Fr(M)31 b Ft(+)20 b Fr(q)j Ft(+)c Fr(l)r Ft(\))p Fn(Z)2125 4519 y Fm(d)2162 4552 y Ft(,)30 b(in)f(suc)m(h)h(a)h(w)m(a)m(y)h(that,)1384 4834 y Fr(B)1453 4848 y Fm(mM)1635 4834 y Fl(\032)1754 4720 y Fm(m)1816 4696 y Fu(d)1753 4747 y Fk(\\)1746 4943 y Fm(i)p Fo(=1)1891 4834 y Fr(B)1960 4849 y Fm(M)7 b Fo(+)p Fm(q)r Fo(+)p Fm(l)2204 4834 y Ft(\()p Fr(x)2291 4848 y Fm(i)2320 4834 y Ft(\))15 b Fr(:)118 5092 y Ft(W)-8 b(e)32 b(ma)m(y)f(estimate)815 5282 y Fk(X)782 5479 y Fm(x)p Fj(2)p Fm(B)922 5490 y Fu(N)995 5368 y Fr(f)f Fl(\016)20 b Fr(\022)1178 5382 y Fm(x)1262 5368 y Fq(6)1405 5254 y Fm(m)1467 5231 y Fu(d)1388 5282 y Fk(X)1397 5477 y Fm(i)p Fo(=1)1705 5282 y Fk(X)1535 5483 y Fm(x)p Fj(2)p Fm(B)1675 5495 y Fu(M)5 b Fg(+)p Fu(q)r Fg(+)p Fu(l)1886 5483 y Fo(\()p Fm(x)1953 5493 y Fu(i)1979 5483 y Fo(\))2022 5368 y Fr(f)29 b Fl(\016)21 b Fr(\022)2205 5382 y Fm(x)2284 5368 y Ft(+)35 b Fr(c)2429 5382 y Fm(M)2523 5368 y Fr(N)2606 5331 y Fm(d)p Fj(\000)p Fo(1)2737 5368 y Fl(k)p Fr(f)10 b Fl(k)2882 5382 y Fj(1)2972 5368 y Fr(:)p eop %%Page: 34 34 34 33 bop 118 45 a Fo(34)1103 b(P)-6 b(.)31 b(CAPUTO)f(AND)g(J.-D.)f (DEUSCHEL)118 366 y Ft(Moreo)m(v)m(er,)47 b(there)42 b(is)e(a)i(\014nite)f(constan)m(t)i Fr(c)1651 381 y Fm(q)r(;l)1731 366 y Ft(,)h(dep)s(ending)39 b(on)j Fr(q)i Ft(and)d Fr(l)r Ft(,)k(suc)m(h)c(that)h(for)f(an)m(y)h Fr(i)j Ft(=)118 482 y(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(m)445 449 y Fm(d)516 482 y Ft(and)30 b(for)g(an)m(y)h Fr(M)k Fl(2)25 b Fn(Z)1278 496 y Fo(+)1137 637 y Fl(j)15 b Fr(B)1246 652 y Fm(M)7 b Fo(+)p Fm(q)r Fo(+)p Fm(l)1491 637 y Ft(\()p Fr(x)1578 651 y Fm(i)1607 637 y Ft(\))20 b Fl(n)h Fr(B)1797 651 y Fm(M)1876 637 y Ft(\()p Fr(x)1963 651 y Fm(i)1991 637 y Ft(\))15 b Fl(j)52 b Fq(6)e Fr(c)2278 652 y Fm(q)r(;l)2373 637 y Fr(M)2471 600 y Fm(d)p Fj(\000)p Fo(1)2617 637 y Fr(;)118 788 y Ft(and)30 b(therefore)421 934 y Fk(X)388 1131 y Fm(x)p Fj(2)p Fm(B)528 1142 y Fu(N)601 1021 y Fr(f)g Fl(\016)20 b Fr(\022)784 1035 y Fm(x)868 1021 y Fq(6)50 b Ft(\()p Fr(c)1063 1036 y Fm(q)r(;l)1159 1021 y Fr(m)1239 983 y Fm(d)1294 1021 y Fr(M)1392 983 y Fm(d)p Fj(\000)p Fo(1)1558 1021 y Ft(+)35 b Fr(c)1703 1035 y Fm(M)1798 1021 y Fr(N)1881 983 y Fm(d)p Fj(\000)p Fo(1)2011 1021 y Ft(\))15 b Fl(k)p Fr(f)10 b Fl(k)2206 1035 y Fj(1)2317 1021 y Ft(+)2440 907 y Fm(m)2502 884 y Fu(d)2423 934 y Fk(X)2432 1130 y Fm(i)p Fo(=1)2668 934 y Fk(X)2570 1136 y Fm(x)p Fj(2)p Fm(B)2710 1147 y Fu(M)2777 1136 y Fo(\()p Fm(x)2844 1146 y Fu(i)2871 1136 y Fo(\))2913 1021 y Fr(f)30 b Fl(\016)20 b Fr(\022)3096 1035 y Fm(x)3410 1021 y Ft(\(B.10\))118 1276 y(By)31 b(construction,)f(the)h(supp)s (orts)d Fr(S)1392 1290 y Fm(i)1450 1276 y Ft(of)j(the)f(functions)1500 1441 y Fr(f)1545 1455 y Fm(i)1613 1441 y Fl(\021)1808 1354 y Fk(X)1709 1556 y Fm(x)p Fj(2)p Fm(B)1849 1567 y Fu(M)1917 1556 y Fo(\()p Fm(x)1984 1566 y Fu(i)2010 1556 y Fo(\))2053 1441 y Fr(f)f Fl(\016)21 b Fr(\022)2236 1455 y Fm(x)118 1701 y Ft(are)31 b(con)m(tained)f(in)f Fr(B)856 1715 y Fm(M)7 b Fo(+)p Fm(q)1023 1701 y Ft(\()p Fr(x)1110 1715 y Fm(i)1139 1701 y Ft(\),)31 b(so)f(that)h Fr(d)p Ft(\()p Fr(S)1676 1715 y Fm(i)1704 1701 y Fr(;)15 b(S)1800 1715 y Fm(j)1837 1701 y Ft(\))25 b Fr(>)g(l)r Ft(,)31 b(for)e(all)g Fr(i)d Fl(6)p Ft(=)f Fr(j)36 b Ft(and,)29 b(setting)i Fr(L)25 b Ft(=)g Fr(M)k Ft(+)20 b Fr(q)s Ft(,)30 b(w)m(e)118 1809 y(see)36 b(that)g(Lemma)f(B.2)h (applies.)53 b(If)35 b Fr(l)i Ft(is)d(large)i(enough)f(w)m(e)g(can)h (then)f(use)f(the)i(estimate)g(\(B.7\))h(to)118 1917 y(obtain)1023 2149 y Fr(P)1094 2112 y Fm(")1131 2048 y Fk(\020)1209 2035 y Fm(m)1271 2012 y Fu(d)1200 2063 y Fk(Y)1201 2258 y Fm(i)p Fo(=1)1332 2149 y Ft(exp)15 b Fr(f)1531 2163 y Fm(i)1559 2048 y Fk(\021)1663 2149 y Fq(6)1808 2035 y Fm(m)1870 2012 y Fu(d)1800 2063 y Fk(Y)1801 2258 y Fm(i)p Fo(=1)1931 2048 y Fk(h)1974 2149 y Fr(P)2045 2112 y Fm(")2082 2149 y Ft(\(exp)g Fr(\032)p Ft(\()p Fr(l)r Ft(\))p Fr(f)2462 2163 y Fm(i)2490 2149 y Ft(\))2525 2048 y Fk(i)2569 2071 y Fo(1)p Fm(=\032)p Fo(\()p Fm(l)q Fo(\))3410 2149 y Ft(\(B.11\))118 2398 y(where)30 b Fr(\032)p Ft(\()p Fr(l)r Ft(\))c Fl(!)f Ft(1,)31 b(as)g Fr(l)c Fl(!)e(1)p Ft(.)41 b(Also,)30 b(notice)h(that)f(for)h(eac)m(h)g Fr(i)26 b Ft(=)f(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(m)2727 2365 y Fm(d)2767 2398 y Ft(,)642 2531 y(1)p 592 2571 147 4 v 592 2655 a Fr(\032)p Ft(\()p Fr(l)r Ft(\))779 2592 y(log)31 b Fr(P)997 2555 y Fm(")1034 2519 y Fk(\000)1091 2592 y Ft(exp)15 b Fr(\032)p Ft(\()p Fr(l)r Ft(\))p Fr(f)1436 2606 y Fm(i)1464 2519 y Fk(\001)1546 2592 y Fq(6)56 b Ft(log)31 b Fr(P)1891 2555 y Fm(")1928 2519 y Fk(\000)1985 2592 y Ft(exp)15 b Fr(f)2184 2606 y Fm(i)2212 2519 y Fk(\001)2289 2592 y Ft(+)35 b(\()p Fr(\032)p Ft(\()p Fr(l)r Ft(\))21 b Fl(\000)f Ft(1\))15 b Fr(M)2881 2555 y Fm(d)2937 2592 y Fl(k)p Fr(f)10 b Fl(k)3082 2606 y Fj(1)3172 2592 y Fr(:)118 2792 y Ft(Let)31 b(us)f(de\014ne)1190 2948 y(\000)1247 2962 y Fm(N)1315 2948 y Ft(\()p Fr(f)10 b Ft(\))40 b(=)g(log)17 b Fr(P)1795 2910 y Fm(")1831 2847 y Fk(\020)1901 2948 y Ft(exp)2088 2862 y Fk(X)2055 3059 y Fm(x)p Fj(2)p Fm(B)2195 3070 y Fu(N)2268 2948 y Fr(f)29 b Fl(\016)21 b Fr(\022)2451 2962 y Fm(x)2494 2847 y Fk(\021)2564 2948 y Fr(:)118 3198 y Ft(Using)30 b(shift)f(in)m(v)-5 b(ariance)29 b(of)i Fr(P)1183 3165 y Fm(")1220 3198 y Ft(,)f(from)g(\(B.11\))j(and)c (\(B.10\))k(w)m(e)e(ha)m(v)m(e)368 3356 y(\000)425 3370 y Fm(N)492 3356 y Ft(\()p Fr(f)10 b Ft(\))41 b Fq(6)f Fr(m)849 3319 y Fm(d)889 3356 y Ft(\000)946 3370 y Fm(M)1025 3356 y Ft(\()p Fr(f)10 b Ft(\))35 b(+)1291 3283 y Fk(\000)1333 3356 y Fr(m)1413 3319 y Fm(d)1469 3356 y Fr(M)1567 3319 y Fm(d)1622 3356 y Ft(\()p Fr(\032)p Ft(\()p Fr(l)r Ft(\))21 b Fl(\000)f Ft(1\))37 b(+)e Fr(c)2177 3371 y Fm(q)r(;l)2272 3356 y Fr(m)2352 3319 y Fm(d)2407 3356 y Fr(M)2505 3319 y Fm(d)p Fj(\000)p Fo(1)2671 3356 y Ft(+)g Fr(c)2816 3370 y Fm(M)2911 3356 y Fr(N)2994 3319 y Fm(d)p Fj(\000)p Fo(1)3124 3283 y Fk(\001)3166 3356 y Fl(k)p Fr(f)10 b Fl(k)3311 3370 y Fj(1)3386 3356 y Fr(:)118 3507 y Ft(F)-8 b(or)31 b(\014xed)f Fr(M)40 b Ft(and)30 b Fr(l)r Ft(,)h(this)e(implies) 832 3706 y(lim)15 b(sup)869 3785 y Fm(N)7 b Fj(!1)1135 3645 y Ft(\000)1192 3659 y Fm(N)1259 3645 y Ft(\()p Fr(f)j Ft(\))p 1135 3685 250 4 v 1198 3769 a Fr(N)1281 3742 y Fm(d)1445 3706 y Fq(6)1576 3645 y Ft(\000)1633 3659 y Fm(M)1712 3645 y Ft(\()p Fr(f)g Ft(\))p 1576 3685 261 4 v 1637 3769 a Fr(M)1735 3742 y Fm(d)1887 3706 y Ft(+)1998 3633 y Fk(\002)2036 3706 y Ft(\()p Fr(\032)p Ft(\()p Fr(l)r Ft(\))21 b Fl(\000)f Ft(1\))36 b(+)2561 3643 y Fr(c)2600 3658 y Fm(q)r(;l)p 2561 3685 119 4 v 2571 3769 a Fr(M)2690 3633 y Fk(\003)2728 3706 y Fl(k)p Fr(f)10 b Fl(k)2873 3720 y Fj(1)118 3908 y Ft(By)31 b(letting)f(\014rst)f Fr(M)36 b Fl(!)25 b(1)p Ft(,)30 b(and)g(then)g Fr(l)d Fl(!)e(1)p Ft(,)31 b(w)m(e)g(ha)m(v)m(e)1229 4107 y(lim)15 b(sup)1266 4185 y Fm(N)7 b Fj(!1)1532 4046 y Ft(\000)1589 4060 y Fm(N)1656 4046 y Ft(\()p Fr(f)j Ft(\))p 1532 4086 250 4 v 1595 4170 a Fr(N)1678 4143 y Fm(d)1841 4107 y Fq(6)66 b Ft(lim)15 b(inf)1996 4168 y Fm(M)7 b Fj(!1)2255 4046 y Ft(\000)2312 4060 y Fm(M)2391 4046 y Ft(\()p Fr(f)j Ft(\))p 2255 4086 261 4 v 2316 4170 a Fr(M)2414 4143 y Fm(d)2526 4107 y Fr(:)118 4314 y Ft(This)29 b(pro)m(v)m(es)i(the)f (existence)h(of)g(the)f(limit)e(in)i(\(4.45\).)1625 4513 y Fs(References)157 4658 y Fx([1])39 b(E.)33 b(Bolthausen,)i(J.)d(D.)h (Deusc)n(hel,)h(Critical)g(large)f(deviations)g(for)g(Gaussian)g (\014elds)f(in)g(the)g(phase)h(transition)276 4750 y(regime,)26 b Fp(A)n(nn.)h(Pr)l(ob)l(ab.)i Fd(21)p Fx(,1876-1920,)g(1993)157 4841 y([2])39 b(E.)20 b(Bolthausen,)i(J.)e(D.)g(Deusc)n(hel,)h(O.)e (Zeitouni,)j(En)n(tropic)e(repulsion)g(of)g(the)f(lattice)i(free)f (\014eld,)h Fp(Commun.)g(Math.)276 4932 y(Phys.)26 b Fd(170)p Fx(,)g(417-443,)i(1995)157 5024 y([3])39 b(W.)33 b(Bryc,)h(A.)f(Dem)n(b)r(o,)g(On)f(large)i(deviations)f(of)g(empirical) g(measures)f(for)i(stationary)f(Gaussian)h(pro)r(cesses,)276 5115 y Fp(Sto)l(chastic)c(Pr)l(o)l(c)l(esses)g(Appl.)25 b Fd(58)p Fx(,)h(23-34,)h(1995)157 5206 y([4])39 b(D.)30 b(Brydges,)j(J.)e(F)-6 b(r\177)-38 b(ohlic)n(h,)32 b(T.)f(Sp)r(encer,)g (The)g(random)f(w)n(alk)h(represen)n(tation)f(of)i(classical)h(spin)d (systems)g(and)276 5298 y(correlation)d(inequalities,)g Fp(Commun.)g(Math.)h(Phys.)e Fd(83)p Fx(,)g(123-150,)i(1982)157 5389 y([5])39 b(P)-6 b(.)24 b(Caputo,)h(J.D.)g(Deusc)n(hel,)f(Critical) i(large)g(deviations)e(for)h(massless)g(Gaussian)g(\014elds)f(with)h (long)f(range)h(in)n(ter-)276 5480 y(action,)i Fp(in)g(pr)l(ep)l(ar)l (ation)p Fx(.)p eop %%Page: 35 35 35 34 bop 440 45 a Fo(LAR)n(GE)31 b(DEVIA)-6 b(TIONS)31 b(AND)f(V)-8 b(ARIA)i(TIONAL)31 b(PRINCIPLE)g(F)n(OR)f(HARMONIC)g(CR)-6 b(YST)g(ALS)252 b(35)157 366 y Fx([6])39 b(T.)25 b(Ch)n(y)n(onobu,)e (S.)h(Kusuok)l(a,)g(The)g(large)h(deviations)g(principle)f(for)h(h)n (yp)r(ermixing)d(pro)r(cesses,)k Fp(Pr)l(ob)l(ab.)h(Th.)e(R)l(el.)276 458 y(Fields)h Fd(78)p Fx(,)g(627-649,)i(1988)157 549 y([7])39 b(F.)c(Comets,)i(Grandes)e(deviations)f(p)r(our)h(des)f(c)n (hamps)f(de)i(Gibbs)f(sur)g Fv(Z)2530 517 y Fu(d)2560 549 y Fx(,)j Fp(C.)e(R.)g(A)l(c)l(ad.)h(Sci.)f(Paris)h(I)d Fd(303)p Fx(,)276 640 y(511-513,)28 b(1986.)157 732 y([8])39 b(J.T.)28 b(Co)n(x,)f(A.)f(Grev)n(en,)h(T.)g(Shiga,)h(Finite)e(and)g (in\014nite)h(systems)e(of)j(in)n(teracting)f(di\013usions,)g Fp(Pr)l(ob)l(ab.)i(Th.)f(R)l(el.)276 823 y(Fields)e Fd(102)p Fx(,)g(165-197,)i(1995.)157 914 y([9])39 b(A.)25 b(Dem)n(b)r(o,)g(O.)h (Zeitouni,)g(Large)h(deviations,)f(Jones)h(and)e(Burtlett)h (publishers,)g(Boston)h(1993)118 1006 y([10])40 b(J.D.)21 b(Deusc)n(hel,)h(G.)f(Giacomin,)i(D.)e(Io\013e,)h(Large)f(deviations)g (and)g(concen)n(tration)g(prop)r(erties)g(for)h Fc(r)p Fb(')f Fx(in)n(terfaces,)276 1097 y Fp(pr)l(eprint)29 b Fx(1998)118 1188 y([11])40 b(J.D.)26 b(Deusc)n(hel,)g(D.W.)g(Stro)r (o)r(c)n(k,)g(Large)h(deviations,)f(Pure)g(and)f(Applied)g(Math.)i Fd(137)p Fx(,)f(Academic)f(Press)i(1989)118 1280 y([12])40 b(R.L.)30 b(Dobrushin,)g(Gaussian)h(random)e(\014elds:)42 b(Gibbsian)30 b(p)r(oin)n(t)g(of)g(view,)i Fp(in)p Fx(:)42 b(Multicomp)r(onen)n(t)29 b(random)g(sys-)276 1371 y(tems,)c(R.L.)h (Dobrushin)f(and)g(Y)-6 b(a.G.)26 b(Sinai)g(eds.,)h(Marcel)g(Dekk)n (er,)e(New)g(Y)-6 b(ork,)26 b(1980)118 1462 y([13])40 b(M.D.)31 b(Donsk)n(er,)h(S.R.S.)e(V)-6 b(aradhan,)31 b(Asymptotic)e(ev)l(aluation)i(of)h(certain)f(Mark)n(o)n(v)f(pro)r (cess)i(exp)r(ectations)f(for)276 1554 y(large)c(time)e(IV,)g Fp(Commun.)i(Pur)l(e)h(Appl.)f(Math.)f Fd(28)p Fx(,)g(1975,)i(1-47.)118 1645 y([14])40 b(M.D.)23 b(Donsk)n(er,)g(S.R.S.)g(V)-6 b(aradhan,)23 b(Large)g(deviations)g(for)h(stationary)f(Gaussian)h(pro) r(cesses,)h Fp(Commun.)f(Math.)276 1736 y(Phys.)i Fd(97)p Fx(,)g(187-210,)i(1985)118 1828 y([15])40 b(R.M.)26 b(Dudley)-6 b(,)25 b(Real)h(analysis)h(and)e(probabilit)n(y)-6 b(,)25 b(W)-6 b(adsw)n(orth)26 b(and)g(Bro)r(oks/Cole,)i(Mon)n(terey)-6 b(,)26 b(1989)118 1919 y([16])40 b(A.C.D.)26 b(v)l(an)f(En)n(ter,)g(R.) h(F)-6 b(ernandez,)25 b(A.D.)g(Sok)l(al,)h(Regularit)n(y)f(prop)r (erties)i(and)e(pathologies)i(of)f(p)r(osition-space)276 2010 y(renormalization-group)j(transformations:)40 b(scop)r(e)29 b(and)f(limitations)h(of)g(Gibbsian)f(theory)-6 b(,)28 b Fp(J.)i(Stat.)g(Phys.)f Fd(72)p Fx(,)276 2102 y(879-1167,)f(1993.)118 2193 y([17])40 b(H.)27 b(F\177)-38 b(ollmer,)29 b(On)d(en)n(trop)n(y)g (and)h(information)h(gain)g(in)g(random)e(\014elds,)i Fp(Z.)h(Wahrsch.)g(verw.)h(Gebiete)p Fx(,)f Fd(26)p Fx(,)g(207-)276 2284 y(217,)e(1973)118 2376 y([18])40 b(H.)30 b(F\177)-38 b(ollmer,)33 b(S.)d(Orey)-6 b(,)31 b(Large)g(deviations)g(for)g(the)g (empirical)f(\014eld)g(of)i(a)e(Gibbs)h(measure,)h Fp(A)n(nn.)f(Pr)l (ob)l(ab.)h Fd(16)p Fx(,)276 2467 y(1988,)27 b(961-977)118 2558 y([19])40 b(J.)26 b(F)-6 b(r\177)-38 b(ohlic)n(h,)25 b(C.)g(P\014ster,)h(On)e(the)h(absence)g(of)g(sp)r(on)n(taneous)h (symmetry)c(breaking)j(and)f(of)i(crystalline)g(ordering)276 2650 y(in)g(t)n(w)n(o-dimensional)f(systems,)h Fp(Commun.)h(Math.)g (Phys.)f Fd(81)p Fx(,)h(277-298,)g(1981.)118 2741 y([20])40 b(H.O.)26 b(Georgii,)h(Gibbs)f(measures)g(and)f(phase)h(transitions,)h (de)e(Gruyter)h(Studies)f(in)g(Math.)i(9,)f(1988.)118 2834 y([21])40 b(H.O.)28 b(Georgii,)i(Large)f(deviations)g(and)e(maxim) n(um)e(en)n(trop)n(y)i(principle)h(for)h(in)n(teracting)f(random)f (\014elds)h(on)g Fv(Z)3610 2802 y Fu(d)3640 2834 y Fx(,)276 2926 y Fp(A)n(nn.)f(Pr)l(ob)l(ab.)i Fd(21)p Fx(,)d(1845-1875,)j(1993) 118 3017 y([22])40 b(H.)30 b(K)r(\177)-41 b(unsc)n(h,)31 b(Thermo)r(dynamics)e(and)g(statistical)j(analysis)f(of)g(Gaussian)g (random)e(\014elds,)i Fp(Z.)g(Wahrsch.)h(verw.)276 3108 y(Gebiete)27 b Fd(58)p Fx(,)f(407-421,)i(1981)118 3200 y([23])40 b(H.)33 b(K)r(\177)-41 b(unsc)n(h,)35 b(Almost)d(sure)h(en)n (trop)n(y)f(and)h(v)l(ariational)h(principle)g(for)f(random)f(\014elds) h(with)h(un)n(b)r(ounded)d(state)276 3291 y(space,)c Fp(Z.)g(Wahrsch.)h(verw.)g(Gebiete)f(58)p Fx(,)f(69-85,)h(1981)118 3382 y([24])40 b(O.E.)c(Lanford)g(I)r(I)r(I,)e(En)n(trop)n(y)h(and)g (equilibrium)f(states)i(in)f(classical)j(statistical)f(mec)n(hanics,)h Fp(in)p Fx(:)53 b(Statistical)276 3474 y(mec)n(hanics)25 b(and)h(mathematical)f(problems,)g(Lecture)h(Notes)g(in)f(Ph)n(ysics)h Fd(20)p Fx(,)h(1-113,)g(Springer)e(Berlin)i(1973.)118 3565 y([25])40 b(G.F.)27 b(La)n(wler,)g(In)n(tersections)f(of)g(random) f(w)n(alks,)i(Birkh\177)-38 b(auser)26 b(Boston,)h(1991.)118 3656 y([26])40 b(J.L.)26 b(Leb)r(o)n(witz,)g(E.)f(Presutti,)h (Statistical)g(mec)n(hanics)e(of)h(un)n(b)r(ounded)e(spins)i(systems,)f Fp(Commun.)i(Math.)h(Phys.)276 3747 y Fd(50)p Fx(,)f(195-218,)i(1976) 118 3839 y([27])40 b(E.W.)26 b(Mon)n(troll,)h(B.J.)g(W)-6 b(est,)25 b(On)g(an)g(enric)n(hed)g(collection)j(of)e(sto)r(c)n(hastic) g(pro)r(cesses,)i Fp(in)d Fx(Fluctuation)g(phenom-)276 3930 y(ena,)i(Studies)f(in)g(statistical)i(mec)n(hanics)e(v)n(ol.)h(VI) r(I,)e(eds.)i(E.W.)g(Mon)n(troll)h(and)e(J.L.)h(Leb)r(o)n(witz,)h (North-Holland,)276 4021 y(1979)118 4113 y([28])40 b(E.)26 b(Nelson,)h(The)e(free)i(Mark)n(o)n(v)e(\014eld,)h(J.)h(F)-6 b(unct.)25 b(Anal.)g Fd(12)p Fx(,)i(211-227,)h(1973)118 4204 y([29])40 b(S.)26 b(Olla,)g(Large)h(deviations)f(for)h(Gibbs)e (random)g(\014elds,)h Fp(Pr)l(ob)l(ab.)j(Th.)e(R)l(el.)g(Fields)e Fd(77)p Fx(,)i(1988,)g(343-357)118 4295 y([30])40 b(C.)26 b(Preston,)h(Random)d(\014elds,)i(Lecture)g(Notes)g(in)f(Mathematics)h Fd(534)p Fx(,)h(Springer)f(1976)118 4387 y([31])40 b(Y.)24 b(A.)g(Rozano)n(v,)g(On)g(Gaussian)h(\014elds)f(with)g(giv)n(en)g (conditional)h(distributions.)g Fp(Th.)g(Pr)l(ob.)i(Appl.)c Fd(12)p 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