Content-Type: multipart/mixed; boundary="-------------0202080527404" This is a multi-part message in MIME format. ---------------0202080527404 Content-Type: text/plain; name="02-61.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-61.keywords" Ising model, Glauber dynamics, phase separation, spectral gap ---------------0202080527404 Content-Type: application/postscript; name="thierry.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="thierry.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: thierry_final.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o thierry.ps thierry_final.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.02.06:1219 %%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 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(of)g(phase)f(transition)118 4662 y(drastically)36 b(mo)s(di\014es)g (the)i(b)s(eha)m(vior)f(of)h(the)g(dynamics)f(and)g(new)h(ph)m(ysical)e (features)i(slo)m(w)g(do)m(wn)118 4770 y(the)f(relaxation;)i(among)e (those,)i(the)e(n)m(ucleation)f(and)f(the)i(in)m(terface)g(motions.)59 b(Metastabilit)m(y)36 b(is)p 118 4841 499 4 v 218 4932 a FB(1991)29 b Ft(Mathematics)g(Subje)l(ct)f(Classi\014c)l(ation.)39 b FB(Primary)26 b(54C40,)h(14E20;)h(Secondary)e(46E25,)h(20C20.)218 5024 y(During)j(the)f(\014rst)h(part)g(of)h(this)g(w)n(ork)f(w)n(e)g (ha)n(v)n(e)g(b)r(ene\014t)f(from)h(instructiv)n(e)g(con)n(v)n (ersations)h(with)g(D.)f(Io\013e)g(whom)118 5115 y(w)n(e)h(w)n(armly)e (thank.)47 b(The)30 b(second)h(part)f(of)h(this)f(w)n(ork)g(w)n(as)h (done)f(while)h(b)r(oth)f(authors)g(w)n(ere)h(visiting)f(the)g (Institut)118 5206 y(H.)i(P)n(oincar)n(\023)-36 b(e)33 b(during)f(the)f(sp)r(ecial)i(semester)f(dev)n(oted)f(to)h(\\Hydro)r (dynamic)f(Limits".)53 b(W)-6 b(e)31 b(w)n(ould)i(lik)n(e)f(to)g(w)n (armly)118 5298 y(thank)21 b(the)g(organizers,)k(S.)d(Olla)g(and)g(F.)g (Golse,)i(for)f(their)e(hospitalit)n(y)i(there)e(and)g(for)i(the)e(v)n (ery)g(stim)n(ulating)h(scien)n(ti\014c)118 5389 y(atmosphere.)35 b(T.)26 b(B.)g(also)i(ac)n(kno)n(wledges)f(Univ)n(ersities)f(of)g(Roma) f(I)r(I)h(and)f(Roma)g(I)r(I)r(I)g(for)i(their)f(in)n(vitation)g(when)f (this)118 5480 y(w)n(ork)h(started.)35 b(Finally)-6 b(,)26 b(w)n(e)g(w)n(ould)g(lik)n(e)g(to)g(thank)e(Y.)i(V)-6 b(elenik)25 b(and)g(N.)h(Y)-6 b(oshida)25 b(for)i(useful)f(commen)n (ts.)1872 5580 y Fp(1)p eop %%Page: 2 2 2 1 bop 118 45 a Fp(2)1106 b(T.)30 b(BODINEA)n(U)h(AND)f(F.)g(MAR)-6 b(TINELLI)118 366 y Fr(c)m(haracteristic)31 b(of)e(these)h(slo)m(w)g (phenomena)f(since)g(the)h(system)g(is)e(trapp)s(ed)h(for)g(a)h(v)m (ery)g(long)g(p)s(erio)s(d)118 474 y(of)42 b(time)f(in)f(a)h(lo)s(cal)g (equilibrium.)69 b(In)40 b(this)g(case,)46 b(the)41 b(relaxation)g(mec) m(hanism)g(is)f(so)i(slo)m(w)f(that)118 582 y(the)31 b(time)g(of)h(n)m(ucleation)e(can)i(b)s(e)e(expressed)h(in)f(terms)h (of)g(equilibrium)c(quan)m(tities.)42 b(In)31 b(particular,)118 690 y(it)36 b(w)m(as)i(pro)m(v)m(en)f(b)m(y)f(Martinelli)f(\(see)j(eg.) 60 b([Ma)q(])37 b(and)g(references)g(therin\))e(that)j(for)e(free)h(b)s (oundary)118 798 y(conditions)d(the)h(asymptotic)h(of)f(the)g(sp)s (ectral)g(gap)g(with)f(resp)s(ect)h(to)h(the)g(size)f(of)g(the)g (domains)f(is)118 906 y(related)c(to)g(the)g(surface)g(tension)f(and)g (the)h(main)f(mec)m(hanism)g(driving)e(the)j(system)g(to)g(equilibrium) 118 1014 y(is)h(n)m(ucleation)h(of)h(one)f(phase)g(inside)e(the)i (other.)47 b(A)32 b(complete)h(picture)e(of)i(the)f(n)m(ucleation)g (pro)s(cess)118 1122 y(in)20 b Fo(Z)280 1089 y Fp(2)335 1122 y Fr(in)g(the)h(framew)m(ork)g(of)g(metastabilit)m(y)f(w)m(as)h (obtained)f(b)m(y)h(Sc)m(honmann)f(and)g(Shlosman)f(in)g([SS2].)218 1338 y(In)28 b(this)g(pap)s(er,)g(w)m(e)i(are)f(in)m(terested)g(in)f(a) h(di\013eren)m(t)f(regime)h(in)f(whic)m(h)g(the)h(relaxation)f(to)i (equilib-)118 1446 y(rium)k(is)g(driv)m(en)g(b)m(y)i(the)f(slo)m(w)g (motion)h(of)f(the)h(in)m(terfaces.)56 b(This)34 b(is)h(the)g(case)i (of)f(the)f(Ising)g(mo)s(del)118 1554 y(in)40 b(a)i(large)g(b)s(o)m(x)g (with)e(plus)g(b)s(oundary)f(conditions.)73 b(When)42 b(a)g(droplet)e(of)i(the)g(min)m(us)e(phase)h(is)118 1662 y(surrounded)e(b)m(y)i(the)g(plus)f(phase,)j(it)e(tends)g(to)h (shrink)d(according)i(to)h(its)f(curv)-5 b(ature)40 b(under)g(the)118 1770 y(action)c(of)f(the)g(non-conserv)-5 b(ativ)m(e)36 b(dynamics)e(on)h(the)h(spins)d(close)i(to)h(the)f(in)m(terface.)56 b(This)34 b(subtle)118 1877 y(phenomenon)41 b(has)g(b)s(een)g(studied)e (rigorously)h(only)h(in)f(rare)i(instances:)62 b(b)m(y)42 b(Sp)s(ohn)d([Sp)o(])j(in)e(the)118 1985 y(case)34 b(of)f(Ising)f(mo)s (del)g(at)i(zero)f(temp)s(erature)g(\(see)h(also)f(Rezakhanlou,)h(Sp)s (ohn)c([RS]\),)35 b(b)m(y)d(Cha)m(y)m(es,)118 2093 y(Sc)m(honmann,)c (Swindle)e([CSS)o(])j(for)f(a)h(v)-5 b(arian)m(t)29 b(of)g(this)f(mo)s (del)f(and)h(b)m(y)h(De)g(Masi,)g(Orlandi,)e(Presutti,)118 2201 y(T)-8 b(riolo)32 b([DOPT1)q(,)h(DOPT2])g(for)g(the)g(Kac-Ising)g (mo)s(del.)47 b(Notice)34 b(also)f(that)h(the)f(motion)f(b)m(y)h(mean) 118 2309 y(curv)-5 b(ature)34 b(pla)m(ys)g(a)g(k)m(ey)h(role)f(in)f (the)h(coarsening)g(phenomenon,)h(as)f(it)g(has)g(b)s(een)f(sho)m(wn)h (recen)m(tly)118 2417 y(b)m(y)40 b(F)-8 b(on)m(tes,)45 b(Sc)m(honmann,)d(Sidora)m(vicius)37 b([FSS].)70 b(F)-8 b(or)41 b(p)s(ositiv)m(e)e(temp)s(eratures,)j(a)f(mathematical)118 2525 y(deriv)-5 b(ation)29 b(of)i(similar)c(results)j(seems)g(to)h(b)s (e)f(more)h(c)m(hallenging.)218 2741 y(A)26 b(w)m(a)m(y)i(to)f(capture) f(some)h(insigh)m(ts)e(in)m(to)h(the)h(slo)m(w)f(relaxation)g(driv)m (en)f(b)m(y)h(in)m(terface)h(motion)f(is)g(to)118 2849 y(estimate)33 b(sp)s(ectral)e(quan)m(tities)g(related)h(to)h(the)f (generator)h(of)f(the)g(Glaub)s(er)e(dynamics.)44 b(W)-8 b(e)34 b(pro)m(v)m(e)118 2957 y(that)i(for)g(an)m(y)g(dimension)d Fn(d)69 b Fm(>)f Fr(2,)38 b(in)c(the)i(phase)f(transition)g(regime)g (and)g(with)g(plus)e(b)s(oundary)118 3065 y(conditions,)26 b(the)h(logarithmic-Sob)s(olev)e(constan)m(t)j(for)f(a)g(domain)f(of)h (linear)e(size)i Fn(N)37 b Fr(div)m(erge)27 b(at)g(least)118 3173 y(lik)m(e)g Fn(N)365 3140 y Fp(2)432 3173 y Fr(\(up)h(to)g(some)h (logarithmic)d(corrections\).)41 b(This)26 b(can)i(b)s(e)f(considered)g (as)h(a)g(\014rst)f(c)m(haracteri-)118 3281 y(zation)h(of)g(the)g(slo)m (w)f(do)m(wn)g(of)h(the)g(dynamics)e(and)h(is)g(in)f(agreemen)m(t)k (with)c(the)i(heuristics)d(predicted)118 3388 y(b)m(y)f(the)g(motion)g (b)m(y)g(mean)g(curv)-5 b(ature.)39 b(In)23 b(the)h(same)h(setting)f (but)f Fn(d)j Fr(=)f(2,)h(w)m(e)f(pro)m(v)m(e)f(that)h(the)f(in)m(v)m (erse)118 3496 y(of)37 b(the)h(sp)s(ectral)e(gap)i(gro)m(ws)f(at)h (least)f(lik)m(e)g Fn(N)47 b Fr(\(up)36 b(to)i(logarithmic)e (corrections\).)62 b(In)36 b(dimension)118 3604 y Fn(d)51 b Fm(>)f Fr(3)31 b(our)f(argumen)m(t)h(fails)d(to)k(pro)s(duce)d(a)i (result)e(on)h(the)h(div)m(ergence)f(of)h(the)f(sp)s(ectral)g(gap.)218 3820 y(Let)38 b(us)e(stress)h(that)h(w)m(e)g(ha)m(v)m(e)h(not)e(b)s (een)g(able)g(to)h(deriv)m(e)f(matc)m(hing)g(upp)s(er)e(b)s(ounds;)k (the)f(b)s(est)118 3930 y(existing)24 b(b)s(ounds)f(ha)m(v)m(e)k(b)s (een)d(pro)m(v)m(ed)i(only)e(in)g Fn(d)i Fr(=)f(2)h(and)e(are)i(of)f (the)h(form)f(exp)2916 3857 y Fl(\000)2958 3853 y Fk(p)p 3034 3853 83 4 v 77 x Fn(N)10 b Fr(\(log)17 b Fn(N)10 b Fr(\))3403 3897 y Fp(2)3443 3857 y Fl(\001)3509 3930 y Fr(\(see)118 4039 y([YW)q(]\).)41 b(Ho)m(w)m(ev)m(er,)32 b(an)e(exact)h(computation)f(for)g(a)g(to)m(y)h(mo)s(del)e(based)g(on)h (mean)g(curv)-5 b(ature)30 b(motion)118 4147 y(seems)k(to)g(con\014rm)f (that)h(the)f(p)s(olynomial)e(asymptotics)j(w)m(e)f(obtain)g(are)h (correct)h(\(see)f(Section)f(7\).)118 4255 y(The)41 b(pro)s(of)f(b)s (oils)f(do)m(wn)i(to)g(b)s(ound)e(the)j(v)-5 b(ariational)40 b(form)m(ula)g(for)h(the)g(P)m(oincar)m(\023)-43 b(e)43 b(and)d(the)h(Log-)118 4363 y(Sob)s(olev)32 b(inequalities)f(b)m(y)i(c) m(ho)s(osing)f(an)h(appropriate)f(test)i(function.)48 b(This)31 b(reduces)i(the)g(problem)118 4470 y(to)e(a)g(computation)f (under)f(the)h(equilibrium)c(Gibbs)j(measure.)40 b(The)30 b(main)f(di\016cult)m(y)g(is)h(to)h(reco)m(v)m(er)118 4578 y(p)s(olynomial)h(b)s(ounds)f(b)m(y)j(using)f(only)g(the)h(exp)s (onen)m(tial)f(estimates)i(pro)m(vided)e(b)m(y)h(the)g(equilibrium)118 4686 y(theory)k(of)f(phase)g(segregation)i(\(see)f([BIV)q(])g(and)f (references)g(therein\).)61 b(This)36 b(is)h(ac)m(hiev)m(ed)h(b)m(y)f (the)118 4794 y(c)m(hoice)31 b(of)g(a)g(subtle)e(test)i(function)e (whic)m(h)g(w)m(as)i(suggested)g(some)g(y)m(ears)g(ago)h(b)m(y)e(H.T.)h (Y)-8 b(au.)218 5010 y(The)32 b(second)g(part)h(of)f(the)h(pap)s(er)e (\(section)i(6\))h(applies)c(the)j(result)e(on)i(the)g(lo)m(w)m(er)f(b) s(ound)f(on)h(the)118 5118 y(in)m(v)m(erse)c(of)g(the)g(sp)s(ectral)f (gap)h(to)h(in)m(v)m(estigate)g(the)f(relaxation)f(in)f(the)i (in\014nite)e(domain)h Fo(Z)3266 5085 y Fp(2)3301 5118 y Fr(.)39 b(Thanks)118 5226 y(to)33 b(an)f(heuristic)f(argumen)m(t)h (based)g(on)h(the)f(motion)g(b)m(y)g(mean)h(curv)-5 b(ature,)32 b(Fisher)g(and)f(Huse)h([HF)q(])118 5334 y(conjectured)f(that)g(the)g (equilibrium)26 b(time)k(auto-correlation)h(of)g(the)g(spin)d(at)k(the) e(origin)g(deca)m(ys)h(as)118 5442 y(a)37 b(stretc)m(hed)g(exp)s(onen)m (tial)f(exp\()p Fk(\000)1337 5368 y(p)p 1413 5368 33 4 v 74 x Fn(t)o Fr(\))h(in)f Fn(d)f Fr(=)h(2.)59 b(W)-8 b(e)38 b(pro)m(vide)e(a)h(\014rst)e(step)i(to)m(w)m(ards)g(a)g (rigorous)p eop %%Page: 3 3 3 2 bop 569 45 a Fp(SOME)30 b(NEW)h(RESUL)-6 b(TS)31 b(ON)g(THE)f(KINETIC)h(ISING)h(MODEL)e(IN)h(A)f(PURE)g(PHASE)416 b(3)118 366 y Fr(pro)s(of)25 b(b)m(y)h(sho)m(wing)f(that)i(a)f (dynamical)f(quan)m(tit)m(y)h(strictly)f(related)h(to)h(the)f (auto-correlation)h(cannot)118 474 y(relax)j(faster)h(than)f(exp\()p Fk(\000)1051 401 y(p)p 1127 401 33 4 v 73 x Fn(t)p Fr(\).)1057 785 y(2.)51 b Fq(The)35 b(model)e(and)g(the)h(Main)g(Resul)-6 b(ts)218 947 y Fr(In)31 b(this)f(section)i(w)m(e)g(de\014ne)f(the)h(mo) s(del)e(and)h(\014x)g(some)h(useful)e(the)i(notation,)g(recall)f(some)h (basic)118 1054 y(facts)c(ab)s(out)e(the)h(Ising)f(mo)s(del)g(b)s(elo)m (w)g(the)h(critical)e(p)s(oin)m(t)h(and)g(\014nally)f(state)j(our)e(t)m (w)m(o)j(main)c(results.)118 1305 y(2.1.)54 b FC(The)38 b(standard)g(Ising)h(mo)s(del.)44 b Fr(Let)34 b(\003)g(b)s(e)e(a)i (generic)g(\014nite)e(subset)h(of)g Fo(Z)3046 1272 y Fj(d)3082 1305 y Fr(,)i(with)d Fn(d)60 b Fm(>)h Fr(2.)118 1413 y(Eac)m(h)30 b(site)f Fn(i)h Fr(in)d(\003)j(indexes)e(a)h(spin)f Fn(\033)1405 1427 y Fj(i)1462 1413 y Fr(whic)m(h)g(tak)m(es)j(v)-5 b(alues)28 b Fk(\006)p Fr(1.)41 b(The)28 b(spin)g(con\014gurations)g Fk(f)p Fn(\033)3463 1427 y Fj(i)3492 1413 y Fk(g)3537 1427 y Fj(i)p Fi(2)p Fp(\003)118 1521 y Fr(ha)m(v)m(e)k(a)e (statistical)h(w)m(eigh)m(t)f(determined)f(b)m(y)i(the)f(Hamiltonian) 1098 1749 y Fn(H)1185 1712 y Fp(\026)-39 b Fj(\033)1228 1749 y Fr(\()p Fn(\033)s Fr(\))26 b(=)f Fk(\000)1556 1688 y Fr(1)p 1556 1728 46 4 v 1556 1811 a(2)1677 1663 y Fl(X)1667 1848 y Fy(i;j)s Fh(2)p Fv(\003)1636 1905 y Fh(j)p Fy(i)p Fh(\000)p Fy(j)s Fh(j)p Fv(=1)1875 1749 y Fn(\033)1927 1763 y Fj(i)1955 1749 y Fn(\033)2007 1763 y Fj(j)2064 1749 y Fk(\000)2233 1663 y Fl(X)2165 1850 y Fy(i)p Fh(2)p Fv(\003)p Fy(;j)s Fh(2)p Fv(\003)2400 1836 y Fy(c)2192 1906 y Fh(j)p Fy(i)p Fh(\000)p Fy(j)s Fh(j)p Fv(=1)2457 1749 y Fn(\033)2509 1763 y Fj(i)2542 1749 y Fr(\026)-50 b Fn(\033)2589 1763 y Fj(j)2641 1749 y Fn(;)118 2087 y Fr(where)35 b(\026)-50 b Fn(\033)28 b Fr(=)d Fk(f)5 b Fr(\026)-50 b Fn(\033)654 2101 y Fj(i)683 2087 y Fk(g)728 2101 y Fj(i)p Fi(2)p Fp(\003)848 2082 y Fy(c)915 2087 y Fr(are)31 b(b)s(oundary)d(conditions)h(outside)h (\003.)218 2195 y(The)g(Gibbs)e(measure)j(asso)s(ciated)f(to)i(the)e (spin)f(system)h(with)f(b)s(oundary)g(conditions)k(\026)-50 b Fn(\033)34 b Fr(is)880 2423 y Fk(8)p Fn(\033)28 b Fr(=)d Fk(f)p Fn(\033)1204 2437 y Fj(x)1248 2423 y Fk(g)1293 2437 y Fj(x)p Fi(2)p Fp(\003)1434 2423 y Fn(;)197 b(\026)1715 2385 y Fp(\026)-39 b Fj(\033)1711 2445 y Fp(\003)1764 2423 y Fr(\()p Fn(\033)s Fr(\))26 b(=)2087 2361 y(1)p 2021 2402 178 4 v 2021 2485 a Fn(Z)2093 2454 y Fp(\026)-38 b Fj(\033)2083 2514 y(\014)s(;)p Fp(\003)2224 2423 y Fr(exp)2378 2349 y Fl(\000)2419 2423 y Fk(\000)p Fn(\014)5 b(H)2633 2385 y Fp(\026)-39 b Fj(\033)2676 2423 y Fr(\()p Fn(\033)s Fr(\))2801 2349 y Fl(\001)2859 2423 y Fn(;)118 2696 y Fr(where)33 b Fn(\014)38 b Fr(is)32 b(the)h(in)m(v)m(erse)g(of)g (the)g(temp)s(erature)g(\()p Fn(\014)i Fr(=)2055 2661 y Fp(1)p 2047 2676 51 4 v 2047 2728 a Fj(T)2108 2696 y Fr(\))f(and)e Fn(Z)2428 2663 y Fp(\026)-38 b Fj(\033)2418 2724 y(\014)s(;)p Fp(\003)2567 2696 y Fr(is)32 b(the)h(partition)f (function.)47 b(If)118 2813 y(the)33 b(b)s(oundary)e(conditions)g(are)i (uniformly)d(equal)i(to)i(1)f(\(resp.)48 b Fk(\000)p Fr(1\),)34 b(the)f(Gibbs)e(measure)i(will)d(b)s(e)118 2923 y(denoted)h(b)m(y)f Fn(\026)643 2885 y Fp(+)643 2951 y(\003)732 2923 y Fr(\(resp.)40 b Fn(\026)1050 2885 y Fi(\000)1050 2951 y Fp(\003)1109 2923 y Fr(\).)218 3031 y(The)28 b(phase)g(transition)f(regime)i(o)s(ccurs)f(at)i(lo)m(w)e (temp)s(erature)h(and)f(is)g(c)m(haracterized)h(b)m(y)g(sp)s(on)m(ta-) 118 3139 y(neous)h(magnetization)h(in)e(the)i(thermo)s(dynamic)d (limit.)39 b(There)30 b(is)f(a)i(critical)e(v)-5 b(alue)30 b Fn(\014)3148 3153 y Fj(c)3214 3139 y Fr(suc)m(h)g(that)803 3329 y Fk(8)p Fn(\014)g(>)25 b(\014)1082 3343 y Fj(c)1117 3329 y Fn(;)234 b Fr(lim)1340 3397 y Fp(\003)p Fi(!)p Fg(Z)1509 3379 y Fy(d)1555 3329 y Fn(\026)1610 3291 y Fp(+)1610 3357 y(\003)1669 3329 y Fr(\()p Fn(\033)1756 3343 y Fp(0)1795 3329 y Fr(\))26 b(=)f Fk(\000)52 b Fr(lim)2038 3397 y Fp(\003)p Fi(!)p Fg(Z)2207 3379 y Fy(d)2253 3329 y Fn(\026)2308 3291 y Fi(\000)2308 3357 y Fp(\003)2367 3329 y Fr(\()p Fn(\033)2454 3343 y Fp(0)2493 3329 y Fr(\))26 b(=)f Fn(m)2730 3291 y Fi(\003)2794 3329 y Fn(>)g Fr(0)15 b Fn(:)500 b Fr(\(2.1\))118 3575 y(F)-8 b(urthermore,)33 b(in)e(the)i(thermo)s(dynamic)d(limit)g(the)j(measures)f Fn(\026)2418 3537 y Fp(+)2418 3604 y(\003)2509 3575 y Fr(and)g Fn(\026)2743 3537 y Fi(\000)2743 3604 y Fp(\003)2834 3575 y Fr(con)m(v)m(erge)i(\(w)m(eakly\))g(to)118 3703 y(t)m(w)m(o)28 b(distinct)c(Gibbs)h(measures)h Fn(\026)1311 3670 y Fp(+)1396 3703 y Fr(and)f Fn(\026)1623 3670 y Fi(\000)1708 3703 y Fr(whic)m(h)g(are)h(measures)g(on)g(the)h(space)f Fk(f\006)p Fr(1)p Fk(g)3216 3670 y Fg(Z)3268 3647 y Fy(d)3302 3703 y Fr(.)39 b(Eac)m(h)27 b(of)118 3821 y(these)k(measures)g (represen)m(ts)f(a)h(pure)f(state.)43 b(In)30 b(dimension)e Fn(d)52 b Fm(>)f Fr(3,)31 b(w)m(e)g(also)g(denote)g(b)m(y)3329 3797 y(^)3316 3821 y Fn(\014)3367 3835 y Fj(c)3453 3821 y Fm(>)51 b Fn(\014)3626 3835 y Fj(c)118 3929 y Fr(the)30 b(\\slab)e(critical)g(p)s(oin)m(t")h(\(see)h([A)m(CCFR)q(])f(and)g([Pi) o(]\))h(whic)m(h)e(is)g(conjectured)i(to)g(coincide)e(with)g Fn(\014)3601 3943 y Fj(c)3636 3929 y Fr(.)118 4047 y(F)-8 b(or)33 b(con)m(v)m(enience)f(w)m(e)g(set)1073 4023 y(^)1060 4047 y Fn(\014)1111 4061 y Fj(c)1173 4047 y Fr(=)27 b Fn(\014)1322 4061 y Fj(c)1389 4047 y Fr(in)j(dimension)f(2.)45 b(Our)31 b(pro)s(ofs)g(rely)f(on)i(results)e(of)i(equilibrium)118 4161 y(phase)f(co)s(existence)h(for)f(the)g(Ising)f(mo)s(del)f(whic)m (h)h(are)i(restricted)e(to)i(v)-5 b(alues)31 b Fn(\014)g(>)3031 4137 y Fr(^)3018 4161 y Fn(\014)3069 4175 y Fj(c)3135 4161 y Fr(\(for)g(tec)m(hnical)118 4269 y(reasons\).)218 4484 y(The)f(next)g(step)h(is)f(to)h(quan)m(tify)f(the)g(co)s (existence)i(of)f(the)f(t)m(w)m(o)i(pure)e(states)h(de\014ned)f(ab)s(o) m(v)m(e.)42 b(Due)118 4592 y(to)c(the)f(lattice)h(structure,)h(the)e (surface)g(tension)g(is)f(an-isotropic.)61 b(Let)37 b(\003)g(=)f Fk(f\000)p Fn(N)5 b(;)15 b(:)g(:)g(:)i(;)e(N)10 b Fk(g)3455 4559 y Fj(d)3496 4592 y Fr(,)39 b(let)116 4702 y Fn(~)-43 b(n)38 b Fr(b)s(e)f(a)i(v)m(ector)h(in)c Fo(S)877 4669 y Fj(d)p Fi(\000)p Fp(1)1040 4702 y Fr(suc)m(h)h(that)g Fn(~)-43 b(n)25 b Fk(\001)d Fn(~)-41 b(e)1630 4716 y Fp(1)1708 4702 y Fn(>)38 b Fr(0)g(and)g(let)43 b(\026)-50 b Fn(\033)41 b Fr(b)s(e)c(the)i(follo)m(wing)d(mixed)h(b)s(oundary)118 4810 y(conditions)1007 5060 y Fk(8)p Fn(i)25 b Fk(2)g Fr(\003)1263 5022 y Fj(c)1298 5060 y Fn(;)202 b Fr(\026)-50 b Fn(\033)1572 5074 y Fj(i)1626 5060 y Fr(=)1722 4904 y Fl(\()1795 4998 y Fr(+1)p Fn(;)273 b Fr(if)88 b Fn(~)-43 b(n)20 b Fk(\001)g Fn(i)51 b Fm(>)f Fr(0)p Fn(;)1795 5128 y Fk(\000)p Fr(1)p Fn(;)273 b Fr(if)88 b Fn(~)-43 b(n)20 b Fk(\001)g Fn(i)26 b(<)f Fr(0)p Fn(:)118 5342 y Fr(The)37 b(partition)g(function)f(with)h(mixed)f(b)s(oundary)g (conditions)g(is)h(denoted)h(b)m(y)f Fn(Z)3065 5304 y Fi(\006)3058 5372 y Fj(\014)s(;N)3188 5342 y Fr(\()n Fn(~)-43 b(n)p Fr(\))38 b(and)f(the)118 5471 y(one)31 b(with)e(b)s(oundary)f(conditions)h(uniformly)e(equal)j(to)h(+1)g(b)m (y)f Fn(Z)2443 5433 y Fp(+)2436 5501 y Fj(\014)s(;N)2566 5471 y Fr(.)p eop %%Page: 4 4 4 3 bop 118 45 a Fp(4)1106 b(T.)30 b(BODINEA)n(U)h(AND)f(F.)g(MAR)-6 b(TINELLI)118 366 y FC(De\014nition)35 b(2.1.)46 b Ff(The)32 b(surfac)-5 b(e)33 b(tension)f(in)g(the)g(dir)-5 b(e)g(ction)31 b Fn(~)-42 b(n)25 b Fk(2)g Fo(S)2484 333 y Fj(d)p Fi(\000)p Fp(1)2609 366 y Ff(,)32 b(with)e Fn(~)-43 b(n)18 b Fk(\001)d Fn(~)-41 b(e)3024 380 y Fp(1)3089 366 y Fn(>)25 b Fr(0)p Ff(,)32 b(is)g(de\014ne)-5 b(d)118 474 y(by)1185 696 y Fn(\034)10 b Fr(\()n Fn(~)-43 b(n)p Fr(\))26 b(=)64 b(lim)1482 757 y Fj(N)7 b Fi(!1)1716 696 y Fk(\000)1797 635 y Fr(\()n Fn(~)-43 b(n;)12 b(~)-42 b(e)1969 649 y Fp(1)2009 635 y Fr(\))p 1797 676 248 4 v 1814 759 a Fn(N)1897 733 y Fj(d)p Fi(\000)p Fp(1)2084 696 y Fr(log)2227 619 y Fn(Z)2296 580 y Fi(\006)2289 648 y Fj(\014)s(;N)2419 619 y Fr(\()n Fn(~)f(n)p Fr(\))p 2227 676 318 4 v 2290 765 a Fn(Z)2359 727 y Fp(+)2352 794 y Fj(\014)s(;N)2569 696 y Fn(:)881 b Fr(\(2.2\))218 939 y(W)-8 b(e)32 b(refer)f(to)h (Messager,)g(Miracle-Sol)m(\023)-43 b(e)32 b(and)e(Ruiz)g([MMR)q(])i (for)e(a)i(deriv)-5 b(ation)30 b(of)h(the)g(thermo)s(dy-)118 1047 y(namic)i(limit)f(\(2.2\))r(.)51 b(Asso)s(ciated)34 b(in)f(a)h(natural)f(w)m(a)m(y)i(to)g(the)f(surface)f(tension)h(is)f (the)h(W)-8 b(ul\013)33 b(shap)s(e)118 1155 y(whic)m(h)f(describ)s(es)g (the)h(optimal)f(shap)s(e)h(of)g(a)h(droplet)e(of)i(the)f(min)m(us)f (phase)h(immersed)f(in)g(the)h(plus)118 1263 y(phase.)118 1426 y FC(De\014nition)i(2.2.)46 b Ff(The)33 b(Wul\013)f(shap)-5 b(e)35 b(is)d(the)h(c)-5 b(onvex)33 b(set)g(in)g Fo(R)2352 1393 y Fj(d)2431 1426 y Ff(given)f(by)1076 1596 y FC(W)26 b Fr(=)1377 1510 y Fl(\\)1305 1715 y Fj(~)-34 b(n)p Fi(2)p Fg(S)1439 1696 y Fy(d)p Fh(\000)p Fv(1)1563 1495 y Fl(n)1624 1596 y Fn(x)25 b Fk(2)g Fo(R)1846 1559 y Fj(d)1893 1596 y Fr(;)201 b Fn(x)20 b Fk(\001)e Fn(~)-42 b(n)24 b Fm(6)h Fn(\034)10 b Fr(\()n Fn(~)-43 b(n)p Fr(\))2587 1495 y Fl(o)2679 1596 y Fn(:)771 b Fr(\(2.3\))218 1883 y(The)23 b(W)-8 b(ul\013)23 b(shap)s(e)g(with)f(v)m(olume)i(1)g(is)f(denoted)g (b)m(y)2002 1860 y Fl(c)2002 1883 y Fo(W)2087 1850 y Fj(d)2133 1883 y Fr(.)39 b(Finally)21 b(in)i(what)g(follo)m(ws)g(w)m(e) h(will)d(c)m(ho)s(ose)118 2001 y(for)26 b(simplicit)m(y)e(the)i (\014nite)g(set)h(\003)f(as)h(the)f(domain)g Fo(W)1937 2015 y Fj(N)2035 2001 y Fr(=)f Fn(N)2214 1977 y Fl(c)2214 2001 y Fo(W)2299 1968 y Fj(d)2357 2001 y Fk(\\)12 b Fo(Z)2495 1968 y Fj(d)2531 2001 y Fr(,)28 b(instead)d(of)i(a)g(cub)s(e)e(of)i (side)e Fn(N)10 b Fr(.)118 2109 y(The)26 b(corresp)s(onding)f(Gibbs)g (measure)i(on)f Fo(W)1700 2123 y Fj(N)1800 2109 y Fr(with)f(+)i(b)s (oundary)d(conditions)i(will)e(b)s(e)i(denoted)g(b)m(y)118 2217 y Fn(\026)173 2179 y Fp(+)173 2245 y Fj(N)240 2217 y Fr(.)118 2491 y(2.2.)54 b FC(The)35 b(Glaub)s(er)g(dynamics.)45 b Fr(The)30 b(sto)s(c)m(hastic)h(dynamics)f(w)m(e)g(w)m(an)m(t)i(to)f (study)f(is)f(de\014ned)g(b)m(y)118 2599 y(the)i(Mark)m(o)m(v)h (generator)f(giv)m(en)g(b)m(y)1264 2760 y(\()p Fk(L)1362 2722 y Fp(+)1362 2789 y Fj(N)1429 2760 y Fn(f)10 b Fr(\)\()p Fn(\033)s Fr(\))26 b(=)1805 2674 y Fl(X)1766 2871 y Fj(x)p Fi(2)p Fg(W)1913 2882 y Fy(N)1992 2760 y Fn(c)2031 2723 y Fp(+)2031 2783 y Fj(x)2090 2760 y Fr(\()p Fn(\033)s Fr(\))p Fk(r)2291 2774 y Fj(x)2336 2760 y Fn(f)10 b Fr(\()p Fn(\033)s Fr(\))118 3008 y(where)20 b(the)g(v)-5 b(alues)20 b(of)g Fn(\033)k Fr(outside)19 b Fo(W)1335 3022 y Fj(N)1429 3008 y Fr(are)h(k)m(ept)h(\014xed)f(iden)m(tical)f(to)i(+1)f(and)g Fk(r)2808 3022 y Fj(x)2852 3008 y Fn(f)10 b Fr(\()p Fn(\033)s Fr(\))25 b(=)g([)p Fn(f)10 b Fr(\()p Fn(\033)3323 2975 y Fj(x)3367 3008 y Fr(\))21 b Fk(\000)e Fn(f)10 b Fr(\()p Fn(\033)s Fr(\)])q(.)118 3116 y(On)30 b(the)g(\015ip)f(rates)i Fn(c)845 3130 y Fj(x)889 3116 y Fr(\()p Fn(\033)s Fr(\))g(w)m(e)g (assume)191 3244 y(\(i\))42 b Fn(k)379 3211 y Fi(\000)p Fp(1)499 3244 y Fk(\024)25 b Fn(c)634 3211 y Fp(+)634 3266 y Fj(x)693 3244 y Fr(\()p Fn(\033)s Fr(\))h Fk(\024)f Fn(k)34 b Fr(for)c(some)g Fn(k)k Fr(and)c(an)m(y)g Fn(x;)15 b(\033)166 3353 y Fr(\(ii\))41 b(rev)m(ersibilit)m(y)28 b(w.r.t.)41 b(the)30 b(Gibbs)f(measure)h Fn(\026)1907 3315 y Fp(+)1907 3382 y Fj(N)141 3461 y Fr(\(iii\))40 b(\014nite)29 b(range)118 3624 y FC(Remark)34 b(2.1.)47 b Ff(It)29 b(is)g(p)-5 b(ossible)30 b(to)g(che)-5 b(ck)29 b(\(se)-5 b(e)29 b(e.g.)40 b Fr([Li])29 b Ff(or)h Fr([Ma)q(])p Ff(\))f(that)h(it)f(is)g(p)-5 b(ossible)31 b(to)e(extend)h(the)118 3734 y(ab)-5 b(ove)32 b(de\014nition)g(of)f(the)g(gener)-5 b(ator)33 b Fk(L)1488 3696 y Fp(+)1488 3763 y Fj(N)1585 3734 y Ff(dir)-5 b(e)g(ctly)33 b(to)e(the)g(whole)i(lattic)-5 b(e)31 b Fo(Z)2739 3701 y Fj(d)2806 3734 y Ff(and)h(get)f(a)g(wel)5 b(l)31 b(de\014ne)-5 b(d)118 3862 y(Markov)28 b(pr)-5 b(o)g(c)g(ess)30 b(on)f Fr(\012)24 b(:=)i Fk(f)p Fr(0)p Fn(;)15 b Fr(1)p Fk(g)1303 3829 y Fg(Z)1355 3806 y Fy(d)1389 3862 y Ff(.)41 b(We)27 b(wil)5 b(l)28 b(r)-5 b(efer)28 b(to)h(the)f(latter)h(as)f(the)g(in\014nite)g(volume)g(Glaub)-5 b(er)118 3970 y(dynamics.)118 4133 y Fr(The)30 b(Diric)m(hlet)f(form)h (asso)s(ciated)h(to)g Fk(L)1502 4095 y Fp(+)1502 4162 y Fj(N)1599 4133 y Fr(tak)m(es)h(the)f(form)1197 4306 y Fk(E)1253 4268 y Fp(+)1245 4334 y Fj(N)1312 4306 y Fr(\()p Fn(f)5 b(;)15 b(f)10 b Fr(\))25 b(=)1688 4219 y Fl(X)1648 4417 y Fj(x)p Fi(2)p Fg(W)1795 4428 y Fy(N)1874 4306 y Fn(\026)1929 4268 y Fp(+)1929 4334 y Fj(N)1996 4232 y Fl(\000)2053 4306 y Fn(c)2092 4320 y Fj(x)2136 4306 y Fr(\()p Fn(\033)s Fr(\))p Fk(jr)2362 4320 y Fj(x)2407 4306 y Fn(f)10 b Fk(j)2487 4268 y Fp(2)2541 4232 y Fl(\001)118 4553 y Fr(and,)37 b(thanks)e(to)h(assumption)e(\(i\))h(on)g(the)h (\015ip)e(rates)i(it)f(is)f(uniformly)f(b)s(ounded)g(from)i(ab)s(o)m(v) m(e)i(and)118 4661 y(from)30 b(b)s(elo)m(w)g(b)m(y)1226 4785 y Fn(\026)1281 4747 y Fp(+)1281 4814 y Fj(N)1348 4711 y Fl(\000)1445 4699 y(X)1405 4896 y Fj(x)p Fi(2)p Fg(W)1552 4907 y Fy(N)1631 4785 y Fk(jr)1732 4799 y Fj(x)1776 4785 y Fn(f)10 b Fk(j)1856 4747 y Fp(2)1910 4711 y Fl(\001)1977 4785 y Fr(:=)25 b Fn(\026)2153 4747 y Fp(+)2153 4814 y Fj(N)2220 4711 y Fl(\000)2277 4785 y Fk(jr)p Fn(f)10 b Fk(j)2458 4747 y Fp(2)2512 4711 y Fl(\001)118 5009 y Fr(Tw)m(o)26 b(k)m(ey)g(quan)m(tities)f(measure)g(the)h(time)f(scale) h(on)f(whic)m(h)f(relaxation)h(to)h(equilibrium)21 b(o)s(ccurs.)39 b(The)118 5117 y(\014rst)24 b(one,)i(denoted)e(b)m(y)h Fn(S)1001 5131 y Fj(N)1068 5117 y Fr(,)g(is)f(the)g(in)m(v)m(erse)h(of) f(the)h(sp)s(ectral)e(gap)i(of)g(the)f(generator,)j(while)c(the)h (other)118 5224 y(one)35 b(the)g(logarithmic)f(Sob)s(olev)f(constan)m (t)j Fn(L)1710 5238 y Fj(N)1777 5224 y Fr(.)54 b(They)35 b(are)g(b)s(oth)f(c)m(haracterized)i(b)m(y)e(a)h(v)-5 b(ariational)118 5332 y(principle)27 b(in)i(that)i(they)g(are)f(the)h (optimal)e(constan)m(ts)j(in)d(the)h(P)m(oincar)m(\023)-43 b(e)32 b(inequalit)m(y)1071 5480 y Fn(\026)1126 5442 y Fp(+)1126 5509 y Fj(N)1193 5480 y Fr(\()p Fn(f)5 b(;)15 b(f)10 b Fr(\))25 b Fk(\024)g Fn(c)15 b Fk(E)1639 5442 y Fp(+)1631 5509 y Fj(N)1699 5480 y Fr(\()p Fn(f)5 b(;)15 b(f)10 b Fr(\))p Fn(;)197 b Fk(8)15 b Fn(f)34 b Fk(2)25 b Fn(L)2429 5443 y Fp(2)2468 5480 y Fr(\()p Fn(d\026)2605 5442 y Fp(+)2605 5509 y Fj(N)2673 5480 y Fr(\))p eop %%Page: 5 5 5 4 bop 569 45 a Fp(SOME)30 b(NEW)h(RESUL)-6 b(TS)31 b(ON)g(THE)f(KINETIC)h(ISING)h(MODEL)e(IN)h(A)f(PURE)g(PHASE)416 b(5)118 366 y Fr(and)30 b(in)f(the)i(logarithmic)e(Sob)s(olev)g (inequalit)m(y)593 512 y Fn(\026)648 474 y Fp(+)648 541 y Fj(N)715 512 y Fr(\()p Fn(f)805 475 y Fp(2)859 512 y Fr(log)16 b Fn(f)1046 475 y Fp(2)1085 512 y Fr(\))26 b Fk(\024)f Fn(c)15 b Fk(E)1352 474 y Fp(+)1344 541 y Fj(N)1411 512 y Fr(\()p Fn(f)5 b(;)15 b(f)10 b Fr(\))15 b Fn(;)197 b Fk(8)15 b Fn(f)35 b Fk(2)24 b Fn(L)2156 475 y Fp(2)2196 512 y Fr(\()p Fn(d\026)2333 474 y Fp(+)2333 541 y Fj(N)2400 512 y Fr(\))92 b(with)29 b Fn(\026)2789 474 y Fp(+)2789 541 y Fj(N)2856 512 y Fr(\()p Fn(f)2946 475 y Fp(2)2985 512 y Fr(\))d(=)e(1)118 658 y(resp)s(ectiv)m(ely)-8 b(.)73 b(As)42 b(it)e(is)h(w)m(ell)f(kno)m(wn)h(the)g(quan)m(tit)m(y)h Fn(S)2116 672 y Fj(N)2224 658 y Fr(measures)f(the)h(relaxation)e(time)h (in)f(an)118 766 y Fn(L)180 733 y Fp(2)219 766 y Fr(\()p Fn(d\026)356 728 y Fp(+)356 794 y Fj(N)424 766 y Fr(\))30 b(sense)g(while)d Fn(L)1020 780 y Fj(N)1117 766 y Fr(measures)i(the)h (relaxation)g(time)f(in)f(an)i Fn(L)2588 733 y Fi(1)2692 766 y Fr(sense)g(\(w)m(orst)g(case)h(for)e(the)118 900 y(initial)i(condition\).)49 b(More)35 b(precisely)-8 b(,)34 b(if)e Fn(P)1653 853 y Fp(\(+)p Fj(;N)7 b Fp(\))1640 925 y Fj(t)1883 900 y Fr(denotes)34 b(the)g(Mark)m(o)m(v)h(semigroup)e (generated)h(b)m(y)118 1011 y Fk(L)181 973 y Fp(+)181 1040 y Fj(N)278 1011 y Fr(and)c Fn(f)40 b Fr(is)29 b(an)h(arbitrary)g (function)f(with)g Fn(\026)1763 973 y Fp(+)1763 1040 y Fj(N)1830 1011 y Fr(\()p Fn(f)10 b Fr(\))25 b(=)g(0)31 b(then)1123 1203 y Fn(\026)1178 1165 y Fp(+)1178 1232 y Fj(N)1245 1130 y Fl(\000)1302 1203 y Fr([)p Fn(P)1398 1155 y Fp(\(+)p Fj(;N)7 b Fp(\))1385 1227 y Fj(t)1594 1203 y Fn(f)j Fr(])1674 1166 y Fp(2)1729 1130 y Fl(\001)1796 1203 y Fk(\024)25 b Fn(\026)1947 1165 y Fp(+)1947 1232 y Fj(N)2014 1203 y Fr(\()p Fn(f)2104 1166 y Fp(2)2143 1203 y Fr(\))15 b(exp\()p Fk(\000)2493 1142 y Fn(t)p 2448 1182 123 4 v 2448 1266 a(S)2504 1280 y Fj(N)2581 1203 y Fr(\))g Fn(:)218 1389 y Fr(In)29 b(man)m(y)h(cases)h(e.g.)41 b(at)31 b(high)e(temp)s(erature)g(the)i(t)m(w)m(o)g(quan)m(tities)e (are)i(of)f(the)g(same)g(order)g(but)f(it)118 1497 y(ma)m(y)i(v)m(ery)h (w)m(ell)d(happ)s(en)g(that)j(they)e(are)i(quite)e(di\013eren)m(t.)41 b(W)-8 b(e)32 b(will)c(argue)j(later)g(on)f(that)i(the)f(Ising)118 1605 y(mo)s(del)e(b)s(elo)m(w)h(the)g(critical)g(temp)s(erature)g(is)f (actually)h(one)h(of)f(these)h(cases.)118 1767 y(2.3.)54 b FC(Main)35 b(Results.)46 b Fr(W)-8 b(e)31 b(are)g(\014nally)d(in)i(a) g(p)s(osition)f(to)i(state)h(our)d(main)h(results.)118 1927 y FC(Theorem)k(2.1.)46 b Ff(Assume)40 b Fn(d)f Fr(=)f(2)j Ff(and)g Fn(\014)i(>)c(\014)1837 1941 y Fj(c)1872 1927 y Ff(.)64 b(Ther)-5 b(e)41 b(exists)f(a)g(c)-5 b(onstant)42 b Fn(\024)e Ff(dep)-5 b(ending)42 b(on)e Fn(\014)118 2035 y Ff(such)33 b(that)1441 2211 y Fr(lim)1402 2271 y Fj(N)7 b Fi(!1)1631 2149 y Fr(\(log)17 b Fn(N)10 b Fr(\))1917 2116 y Fj(\024)p 1631 2190 332 4 v 1755 2273 a Fn(N)1972 2211 y(S)2028 2225 y Fj(N)2120 2211 y Fr(=)25 b(+)p Fk(1)1097 b Fr(\(2.4\))118 2405 y FC(Remark)34 b(2.2.)47 b Ff(As)37 b(we)h(alr)-5 b(e)g(ady)40 b(p)-5 b(ointe)g(d)40 b(out)f(in)e(the)i(intr)-5 b(o)g(duction,)41 b(in)d(dimension)h(gr)-5 b(e)g(ater)39 b(than)118 2513 y(two)26 b(our)g(choic)-5 b(e)26 b(of)f(the)h(test)f(function)g(to)h(b) -5 b(e)25 b(inserte)-5 b(d)26 b(in)f(the)h(Poinc)-5 b(ar)n(\023)-44 b(e)26 b(ine)-5 b(quality)26 b(do)-5 b(es)26 b(not)g(pr)-5 b(ovide)118 2620 y(any)33 b(non)h(trivial)f(information.)218 2781 y Fr(The)d(next)g(result)f(concerns)i(the)f(large)h Fn(N)40 b Fr(b)s(eha)m(vior)30 b(of)g(the)h(logarithmic)e(Sob)s(olev)g (constan)m(t.)118 2947 y FC(Theorem)34 b(2.2.)46 b Ff(Assume)37 b Fn(d)66 b Fm(>)f Fr(2)37 b Ff(and)h Fn(\014)g(>)1831 2923 y Fr(^)1818 2947 y Fn(\014)1869 2961 y Fj(c)1904 2947 y Ff(.)55 b(Ther)-5 b(e)37 b(exists)g(a)g(c)-5 b(onstant)39 b Fn(\024)e Ff(dep)-5 b(ending)38 b(on)f Fn(\014)118 3055 y Ff(and)d Fn(d)f Ff(such)f(that)1438 3230 y Fr(lim)1399 3291 y Fj(N)7 b Fi(!1)1628 3169 y Fr(\(log)17 b Fn(N)10 b Fr(\))1914 3136 y Fj(\024)p 1628 3209 V 1732 3293 a Fn(N)1815 3266 y Fp(2)1969 3230 y Fn(L)2031 3244 y Fj(N)2123 3230 y Fr(=)25 b(+)p Fk(1)1094 b Fr(\(2.5\))218 3462 y(Finally)29 b(w)m(e)j(in)m(v)m(estigate)h(in)d Fn(d)d Fr(=)g(2)32 b(the)g(relaxation)f(in)f(the)i(plus)d(phase)i(for)h(the)f (in\014nite)f(v)m(olume)118 3570 y(dynamics.)38 b(F)-8 b(or)26 b(this)e(purp)s(ose,)g(let)h(us)g(consider)f(an)g(arbitrary)g (coupling)g(of)h(the)g(Glaub)s(er)f(dynamics)118 3678 y(in)41 b(the)i(in\014nite)d(v)m(olume)i Fo(Z)1115 3645 y Fp(2)1150 3678 y Fr(.)77 b(The)42 b(t)m(w)m(o)i(pro)s(cesses)e(at)h (time)f Fn(t)h Fr(are)f(denoted)h(b)m(y)f(\()p Fn(\033)3205 3645 y Fj(\021)3248 3678 y Fr(\()p Fn(t)p Fr(\))p Fn(;)21 b Fr(~)-51 b Fn(\033)3446 3645 y Fj(!)3497 3678 y Fr(\()p Fn(t)p Fr(\)\),)118 3786 y(where)38 b(\()p Fn(\021)s(;)15 b(!)s Fr(\))39 b(are)g(the)g(initial)c(spin)h(con\014gurations.)64 b(The)38 b(join)m(t)g(exp)s(ectation)h(of)f(the)g(pro)s(cess)g(is)118 3903 y(denoted)29 b(b)m(y)593 3880 y(^)585 3903 y Fo(E)17 b Fr(.)40 b(The)28 b(initial)f(conditions)g(will)f(in)i(general)h(b)s (e)g(c)m(hosen)g(w.r.t.)40 b(the)30 b(pro)s(duct)d(measure)118 4011 y Fn(d)7 b Fr(^)-52 b Fn(\026)220 3978 y Fp(+)279 4011 y Fr(\()p Fn(\021)s(;)15 b(!)s Fr(\))27 b(=)e Fn(d\026)722 3978 y Fp(+)781 4011 y Fr(\()p Fn(\021)s Fr(\))p Fn(d\026)1001 3978 y Fp(+)1061 4011 y Fr(\()p Fn(!)s Fr(\),)31 b(where)f Fn(\026)1565 3978 y Fp(+)1654 4011 y Fr(is)g(the)g(Gibbs)f(measure)h (in)f(the)i(+)f(pure)f(phase.)118 4172 y FC(Theorem)34 b(2.3.)46 b Ff(Ther)-5 b(e)37 b(exist)f(p)-5 b(ositive)37 b(c)-5 b(onstants)38 b Fn(C)2049 4186 y Fp(1)2088 4172 y Fn(;)15 b(C)2193 4186 y Fp(2)2269 4172 y Ff(and)37 b Fn(\024)f Ff(indep)-5 b(endent)38 b(of)e(the)h(choic)-5 b(e)36 b(of)118 4280 y(the)d(c)-5 b(oupling)34 b(such)e(that)378 4467 y Fk(8)p Fn(t)25 b(>)g Fr(0)p Fn(;)854 4343 y Fl(Z)976 4467 y Fn(d)7 b Fr(^)-52 b Fn(\026)1078 4430 y Fp(+)1137 4467 y Fr(\()p Fn(\021)s(;)15 b(!)s Fr(\))1379 4444 y(^)1370 4467 y Fo(E)1432 4393 y Fl(\000)1473 4467 y Fn(\033)1528 4423 y Fj(\021)1525 4493 y Fp(0)1571 4467 y Fr(\()p Fn(t)p Fr(\))25 b Fk(6)p Fr(=)30 b(~)-50 b Fn(\033)1850 4430 y Fj(!)1847 4490 y Fp(0)1901 4467 y Fr(\()p Fn(t)p Fr(\))2004 4393 y Fl(\001)2097 4467 y Fm(>)50 b Fn(C)2283 4481 y Fp(1)2337 4467 y Fr(exp)2491 4393 y Fl(\000)2553 4467 y Fk(\000)20 b Fn(C)2709 4481 y Fp(2)2749 4389 y Fk(p)p 2824 4389 33 4 v 2824 4467 a Fn(t)15 b Fr(\(log)i Fn(t)p Fr(\))3108 4430 y Fj(\024)3153 4393 y Fl(\001)3210 4467 y Fn(:)240 b Fr(\(2.6\))118 4669 y FC(Remark)34 b(2.3.)47 b Ff(A)n(lthough)37 b(we)g(b)-5 b(elieve)37 b(that)h(the)f(quantity)h (c)-5 b(onsider)g(e)g(d)39 b(in)e(the)g(the)-5 b(or)g(em)39 b(is)e(a)g(go)-5 b(o)g(d)118 4777 y(me)g(asur)g(e)33 b(of)f(the)g(time)f(auto{c)-5 b(orr)g(elation)36 b(in)c(the)f(plus)i (phase)f(of)g(the)g(spin)g(at)g(the)g(origin,)g(the)g(latter)118 4885 y(is)h(unfortunately)h(only)f(b)-5 b(ounde)g(d)34 b(fr)-5 b(om)34 b Fe(ab)s(o)m(v)m(e)g Ff(by)f(the)g(LHS)f(of)52 b Fr(\(2.6\))r Ff(.)41 b(We)33 b(have)g(in)f(fact)284 5086 y Fn(\026)339 5048 y Fp(+)413 4985 y Fl(\020)467 5012 y(\000)509 5086 y Fn(P)567 5100 y Fj(t)597 5086 y Fr(\()p Fn(\033)684 5100 y Fp(0)724 5086 y Fr(\))20 b Fk(\000)g Fn(m)950 5048 y Fi(\003)990 5012 y Fl(\001)1031 5035 y Fp(2)1071 4985 y Fl(\021)1150 5086 y Fr(=)25 b Fn(\026)1301 5048 y Fp(+)1375 4958 y Fl(\022)1442 5012 y(\000)1484 5086 y Fn(P)1542 5100 y Fj(t)1572 5086 y Fr(\()p Fn(\033)1659 5100 y Fp(0)1698 5086 y Fr(\))c Fk(\000)1845 4962 y Fl(Z)1966 5086 y Fn(d\026)2068 5048 y Fp(+)2148 5063 y Fr(~)2127 5086 y Fn(P)2185 5100 y Fj(t)2215 5086 y Fr(\()p Fn(\033)2302 5100 y Fp(0)2342 5086 y Fr(\))2377 5012 y Fl(\001)2419 5035 y Fp(2)2458 4958 y Fl(\023)470 5367 y Fr(=)566 5243 y Fl(Z)687 5367 y Fn(d\026)789 5329 y Fp(+)848 5367 y Fr(\()p Fn(\021)s Fr(\))981 5211 y Fl( )1055 5238 y(\022)1121 5243 y(Z)1243 5367 y Fn(d\026)1345 5329 y Fp(+)1404 5367 y Fr(\()p Fn(!)s Fr(\))1557 5343 y(^)1549 5367 y Fo(E)1610 5293 y Fl(\000)1652 5367 y Fn(\033)1707 5322 y Fj(\021)1704 5393 y Fp(0)1749 5367 y Fr(\()p Fn(t)p Fr(\))g Fk(\000)j Fr(~)-49 b Fn(\033)2019 5329 y Fj(!)2016 5389 y Fp(0)2069 5367 y Fr(\()p Fn(t)p Fr(\))2172 5293 y Fl(\001)2214 5238 y(\023)2281 5261 y Fp(2)2321 5211 y Fl(!)2433 5367 y Fm(6)25 b Fr(4)2589 5243 y Fl(Z)2711 5367 y Fn(d)7 b Fr(^)-52 b Fn(\026)2813 5329 y Fp(+)2872 5367 y Fr(\()p Fn(\021)s(;)15 b(!)s Fr(\))3098 5343 y(^)3090 5367 y Fo(E)3152 5293 y Fl(\000)3194 5367 y Fn(\033)3249 5322 y Fj(\021)3246 5393 y Fp(0)3291 5367 y Fr(\()p Fn(t)p Fr(\))26 b Fk(6)p Fr(=)j(~)-49 b Fn(\033)3571 5329 y Fj(!)3568 5389 y Fp(0)3621 5367 y Fr(\()p Fn(t)p Fr(\))3724 5293 y Fl(\001)3782 5367 y Fn(:)p eop %%Page: 6 6 6 5 bop 118 45 a Fp(6)1106 b(T.)30 b(BODINEA)n(U)h(AND)f(F.)g(MAR)-6 b(TINELLI)118 366 y FC(Remark)34 b(2.4.)47 b Ff(A)e(r)-5 b(elate)g(d)49 b(r)-5 b(esult)47 b(at)g Fn(\014)55 b Fr(=)50 b(+)p Fk(1)c Ff(was)i(pr)-5 b(ove)g(d)48 b(r)-5 b(e)g(c)g(ently)48 b(in)e Fr([FSS])g Ff(for)h(the)g(zer)-5 b(o)118 474 y(temp)g(er)g(atur)g(e)35 b(dynamics)f(\(se)-5 b(e)33 b(the)-5 b(or)g(em)35 b(1.2)e(ther)-5 b(e\).)1407 666 y Fr(3.)86 b Fq(Lar)n(ge)34 b(Devia)-6 b(tions)118 828 y Fr(In)35 b(this)g(section)h(w)m(e)g(recall)f(some)i(results)d(on) i(the)g(large)g(deviations)f(for)g(the)h(Gibbs)f(measure)g Fn(\026)3594 789 y Fp(+)3594 856 y Fj(N)118 952 y Fr(when)k Fn(\014)45 b(>)586 928 y Fr(^)573 952 y Fn(\014)624 966 y Fj(c)659 952 y Fr(.)68 b(Our)38 b(pro)s(ofs)h(rely)g(on)g(a)h(w)m (eak)g(description)e(of)i(phase)f(segregation)i(in)d(terms)h(of)118 1060 y Fo(L)173 1027 y Fp(1)218 1060 y Fr({norm.)g(In)26 b(dimension)e(2,)j(more)g(precise)f(results)f(can)h(b)s(e)g(found)f(in) g(Io\013e,)j(Sc)m(honmann)d([ISc])i(and)118 1168 y(in)f(P\014ster,)i(V) -8 b(elenik)27 b([PV])h(\(for)g Fo(L)1286 1135 y Fp(1)1332 1168 y Fr(-concen)m(tration)h(statemen)m(ts\).)41 b(The)27 b(reader)h(is)e(referred)h(to)i([BIV])118 1275 y(for)h(a)h(surv)m(ey)f (on)g(phase)g(co)s(existence)i(and)d(a)i(complete)g(list)e(of)i (references.)118 1491 y(W)-8 b(e)31 b(consider)e(our)g(microscopic)g (Ising)g(mo)s(del)f(em)m(b)s(edded)h(in)2302 1468 y Fl(c)2302 1491 y Fo(W)2387 1458 y Fj(d)2434 1491 y Fr(.)40 b(Let)2661 1468 y Fl(c)2661 1491 y Fo(W)2746 1505 y Fj(N)2845 1491 y Fr(=)2965 1456 y Fp(1)p 2951 1471 64 4 v 2951 1523 a Fj(N)3024 1491 y Fo(Z)3089 1458 y Fj(d)3144 1491 y Fk(\\)3223 1468 y Fl(c)3223 1491 y Fo(W)3308 1458 y Fj(d)3384 1491 y Fr(and)29 b(let)118 1623 y Fn(K)k Fr(b)s(e)25 b(a)i(mesoscopic)f(scale)g(\(ev)m(en)m(tually)h(dep)s(ending)c(on)j Fn(N)10 b Fr(\).)40 b(The)25 b(domain)2797 1599 y Fl(c)2797 1623 y Fo(W)2882 1590 y Fj(d)2955 1623 y Fr(is)g(partitioned)f(in)m(to) 118 1745 y(b)s(o)m(xes)374 1722 y Fl(b)369 1745 y Fo(B)423 1759 y Fj(N)s(;K)576 1745 y Fr(,)31 b(eac)m(h)g(of)g(them)f(con)m (taining)g Fn(K)1698 1712 y Fj(d)1769 1745 y Fr(sites)g(of)2075 1722 y Fl(c)2075 1745 y Fo(W)2160 1759 y Fj(N)2233 1745 y Fr(:)660 1973 y Fn(j)g Fk(2)25 b Fo(Z)878 1935 y Fj(d)914 1973 y Fn(;)46 b(x)1037 1987 y Fj(j)1099 1973 y Fr(=)25 b Fn(j)1247 1911 y(K)p 1247 1952 84 4 v 1247 2035 a(N)1366 1973 y Fk(2)1452 1949 y Fl(c)1452 1973 y Fo(W)1537 1987 y Fj(N)1610 1973 y Fn(;)1838 1949 y Fl(b)1833 1973 y Fo(B)1887 1987 y Fj(N)s(;K)2040 1973 y Fr(\()p Fn(x)2127 1987 y Fj(j)2164 1973 y Fr(\))h(=)f Fn(x)2373 1987 y Fj(j)2429 1973 y Fr(+)2520 1845 y Fl(\025)2568 1973 y Fk(\000)2671 1911 y Fn(K)p 2649 1952 129 4 v 2649 2035 a Fr(2)p Fn(N)2787 1973 y(;)2860 1911 y(K)p 2838 1952 V 2838 2035 a Fr(2)p Fn(N)2976 1845 y Fl(\025)3024 1867 y Fj(d)3095 1973 y Fn(:)118 2200 y Fr(Let)d Fo(B)327 2214 y Fj(K)401 2200 y Fr(\()p Fn(N)10 b(x)571 2214 y Fj(j)608 2200 y Fr(\))22 b(b)s(e)e(the)i(microscopic)f(coun)m(terpart)h (of)1992 2176 y Fl(b)1987 2200 y Fo(B)2042 2214 y Fj(N)s(;K)2195 2200 y Fr(\()p Fn(x)2282 2214 y Fj(j)2319 2200 y Fr(\),)i(i.e.)37 b(the)22 b(sites)f(of)2993 2176 y Fl(c)2993 2200 y Fo(W)3078 2214 y Fj(N)3172 2200 y Fr(in)3274 2176 y Fl(b)3269 2200 y Fo(B)3324 2214 y Fj(N)s(;K)3477 2200 y Fr(\()p Fn(x)3564 2214 y Fj(j)3601 2200 y Fr(\).)118 2327 y(These)27 b(b)s(o)m(xes)g(are) g(cen)m(tered)h(on)f(the)g(sites)f(of)1706 2303 y Fl(c)1706 2327 y Fo(W)1791 2341 y Fj(N)s(;K)1970 2327 y Fr(=)2075 2291 y Fj(K)p 2075 2306 65 4 v 2076 2358 a(N)2150 2327 y Fo(Z)2215 2294 y Fj(d)2264 2327 y Fk(\\)2338 2303 y Fl(c)2338 2327 y Fo(W)2423 2294 y Fj(d)2469 2327 y Fr(.)39 b(As)27 b(the)g(domain)f(is)g(not)h(regular)118 2458 y(some)f(b)s(o)m(xes)g(ma)m(y)g(not)f(\014t)h(inside)1300 2434 y Fl(c)1300 2458 y Fo(W)1385 2472 y Fj(N)1458 2458 y Fr(,)h(therefore)e(at)i(the)e(b)s(oundary)f(w)m(e)i(consider)e(a)i (relaxed)f(notion)118 2566 y(of)31 b(b)s(o)m(xes.)118 2674 y(Finally)-8 b(,)22 b(the)f(lo)s(cal)g(magnetization)h(is)e (de\014ned)g(as)h(a)h(piece-wise)f(constan)m(t)h(function)e(on)h(the)h (partition)118 2789 y Fk(f)168 2765 y Fl(b)163 2789 y Fo(B)218 2803 y Fj(N)s(;K)371 2789 y Fr(\()p Fn(x)458 2803 y Fj(j)495 2789 y Fr(\))p Fk(g)p Fr(:)885 2985 y Fk(8)p Fn(y)28 b Fk(2)1100 2961 y Fl(b)1095 2985 y Fo(B)1149 2999 y Fj(N)s(;K)1302 2985 y Fr(\()p Fn(x)1389 2999 y Fj(j)1426 2985 y Fr(\))p Fn(;)198 b Fk(M)1793 2999 y Fj(N)s(;K)1940 2985 y Fr(\()p Fn(y)s Fr(\))26 b(=)2257 2923 y(1)p 2190 2964 180 4 v 2190 3047 a Fk(j)p Fo(B)2270 3061 y Fj(K)2344 3047 y Fk(j)2511 2898 y Fl(X)2394 3100 y Fj(i)p Fi(2)p Fg(B)2504 3111 y Fy(K)2568 3100 y Fp(\()p Fj(N)7 b(x)2698 3110 y Fy(j)2731 3100 y Fp(\))2774 2985 y Fn(\033)2826 2999 y Fj(i)2869 2985 y Fn(:)581 b Fr(\(3.7\))118 3246 y(The)28 b(lo)s(cal)f(order)h(parameter)h Fk(M)1293 3260 y Fj(N)s(;K)1440 3246 y Fr(\()p Fn(y)s Fr(\))f(c)m(haracterizes)i (the)f(lo)s(cal)e(equilibrium)c(of)29 b(the)f(mesoscopic)118 3354 y(b)s(o)m(x)35 b(con)m(taining)f Fn(y)s Fr(.)54 b(The)34 b(k)m(ey)i(result)d(concerning)h(the)h(lo)s(cal)g(order)f (parameters)h(is)f(a)h(trivial)e(con-)118 3462 y(sequence)j(of)f(the)g (results)f(obtained)h(b)m(y)g(Pisztora)g([Pi])g(and)f(it)h(is)f(based)h (on)g(the)h(follo)m(wing)d(coarse)118 3582 y(grained)23 b(description.)36 b(T)-8 b(o)24 b(eac)m(h)g(b)s(o)m(x)1430 3558 y Fl(b)1425 3582 y Fo(B)1479 3596 y Fj(N)s(;K)1632 3582 y Fr(\()p Fn(x)1719 3596 y Fj(j)1756 3582 y Fr(\))g(w)m(e)g(asso)s (ciate)g(a)g(mesoscopic)f(phase)g(lab)s(el)f Fn(u)3355 3596 y Fj(N)s(;K)3502 3582 y Fr(\()p Fn(x)3589 3596 y Fj(j)3626 3582 y Fr(\))118 3690 y(taking)30 b(v)-5 b(alues)30 b(in)f Fk(f\000)p Fr(1)p Fn(;)15 b Fr(0)p Fn(;)g Fr(1)p Fk(g)597 3838 y Fn(u)649 3852 y Fj(N)s(;K)796 3838 y Fr(\()p Fn(x)883 3852 y Fj(j)920 3838 y Fr(\))25 b(=)g(1)1121 3866 y Fi(fjM)1260 3877 y Fy(N)r(;K)1388 3866 y Fp(\()p Fj(x)1455 3876 y Fy(j)1488 3866 y Fp(\))p Fi(\000)p Fj(m)1632 3847 y Fh(\003)1669 3866 y Fi(j)19 b Fd(6)1793 3839 y Fv(1)p 1793 3851 31 3 v 1793 3892 a(4)1833 3866 y Fj(m)1895 3847 y Fh(\003)1931 3866 y Fi(g)1991 3838 y Fk(\000)h Fr(1)2127 3866 y Fi(fjM)2266 3877 y Fy(N)r(;K)2394 3866 y Fp(\()p Fj(x)2461 3876 y Fy(j)2494 3866 y Fp(\)+)p Fj(m)2638 3847 y Fh(\003)2675 3866 y Fi(j)f Fd(6)2798 3839 y Fv(1)p 2798 3851 V 2798 3892 a(4)2839 3866 y Fj(m)2901 3847 y Fh(\003)2937 3866 y Fi(g)2992 3838 y Fn(:)118 4019 y Fr(The)32 b(distribution)c(of)33 b(the)f(v)-5 b(ariables)31 b Fk(f)p Fn(u)1544 4033 y Fj(N)s(;K)1692 4019 y Fr(\()p Fn(x)1779 4033 y Fj(j)1816 4019 y Fr(\))p Fk(g)i Fr(under)d Fn(\026)2243 3981 y Fp(+)2243 4048 y Fj(N)2342 4019 y Fr(is)i(dominated)f(b)m(y)h(Bernoulli)e(P)m(erco-) 118 4127 y(lation.)118 4399 y FC(Theorem)k(3.1)h Fr(\([Pi]\))p FC(.)46 b Ff(F)-7 b(or)44 b(any)f Fn(\014)49 b(>)1604 4375 y Fr(^)1591 4399 y Fn(\014)1642 4413 y Fj(c)1720 4399 y Ff(ther)-5 b(e)44 b(exists)f Fn(c)2255 4414 y Fj(\014)2346 4399 y Fn(>)g Fr(0)g Ff(and)g Fn(\015)49 b Fk(2)15 b Fr(]0)p Fn(;)g Fr(1[)44 b Ff(such)f(that)h(the)118 4507 y(fol)5 b(lowing)34 b(holds)g(uniformly)g(in)e Fn(N)10 b Ff(:)354 4670 y Fk(8f)p Fn(x)502 4684 y Fp(1)541 4670 y Fn(;)15 b(:)g(:)g(:)i(;)e(x)795 4685 y Fj(`)828 4670 y Fk(g)26 b(2)985 4647 y Fl(c)985 4670 y Fo(W)1070 4684 y Fj(N)s(;K)1223 4670 y Fn(;)108 b(\026)1411 4632 y Fp(+)1411 4699 y Fj(N)1493 4670 y Fr(\()q Fn(u)1581 4684 y Fj(N)s(;K)1728 4670 y Fr(\()p Fn(x)1815 4684 y Fp(1)1854 4670 y Fr(\))26 b(=)f(0)p Fn(;)15 b(:)g(:)g(:)i(;)e(u)2310 4684 y Fj(N)s(;K)2458 4670 y Fr(\()p Fn(x)2545 4685 y Fj(`)2578 4670 y Fr(\))26 b(=)e(0\))42 b Fm(6)2952 4597 y Fl(\000)2993 4670 y Fn(\032)3040 4684 y Fj(K)3109 4597 y Fl(\001)3150 4619 y Fj(`)3198 4670 y Fn(;)252 b Fr(\(3.8\))118 4824 y Ff(with)34 b Fn(\032)363 4838 y Fj(K)456 4824 y Fr(=)25 b(exp\()p Fk(\000)p Fn(c)836 4839 y Fj(\014)884 4824 y Fn(K)968 4791 y Fj(\015)1012 4824 y Fr(\))p Ff(.)118 5043 y FC(Remark)34 b(3.1.)47 b Ff(F)-7 b(or)28 b(the)g(next)g(r)-5 b(esults)28 b(\(Pr)-5 b(op)g(ositions)31 b(3.1,)e(3.2,)g(3.3\))f(to)g(hold)h(true)f (the)g(mesosc)-5 b(opic)118 5151 y(sc)g(ale)32 b Fn(K)39 b Ff(has)32 b(to)g(b)-5 b(e)32 b(chosen)g(just)g(lar)-5 b(ge)32 b(enough)g(\(dep)-5 b(ending)33 b(on)f Fn(\014)37 b Ff(and)32 b(on)g(some)h(extr)-5 b(a)32 b(p)-5 b(ar)g(ameter)118 5259 y Fn(\016)s Ff(\).)63 b(However)40 b(in)f(the)h(next)f(se)-5 b(ctions)41 b(it)e(wil)5 b(l)40 b(b)-5 b(e)39 b(essential)h(to)g(r)-5 b(elate)41 b Fn(K)46 b Ff(with)40 b(the)g(b)-5 b(asic)39 b(sc)-5 b(ale)40 b Fn(N)118 5372 y Ff(via)33 b(the)g(sc)-5 b(aling)34 b(r)-5 b(elation)34 b Fn(K)f Fk(\031)25 b Fr(\(log)17 b Fn(N)10 b Fr(\))1541 5339 y Fp(1)p Fj(=\015)1689 5372 y Ff(and)33 b(ther)-5 b(efor)g(e)35 b(we)e(wil)5 b(l)33 b(adopt)i(this)e(choic)-5 b(e)34 b(right)f(away)118 5480 y(and)i(denote)g(the)g(c)-5 b(orr)g(esp)g(onding)37 b(mesosc)-5 b(opic)36 b(phase)g(lab)-5 b(els)35 b(simply)g(by)f Fn(u)2742 5494 y Fj(N)2810 5480 y Ff(.)46 b(Mor)-5 b(e)g(over,)36 b(sinc)-5 b(e)34 b(the)p eop %%Page: 7 7 7 6 bop 569 45 a Fp(SOME)30 b(NEW)h(RESUL)-6 b(TS)31 b(ON)g(THE)f(KINETIC)h(ISING)h(MODEL)e(IN)h(A)f(PURE)g(PHASE)416 b(7)118 366 y Ff(blo)-5 b(cks)32 b(with)g(lab)-5 b(el)31 b Fr(0)h Ff(wil)5 b(l)31 b(play)h(an)g(imp)-5 b(ortant)33 b(r)-5 b(ole)33 b(in)d(the)i(pr)-5 b(o)g(of)33 b(of)e(the)h(main)f(r)-5 b(esults,)33 b(they)e(wil)5 b(l)31 b(b)-5 b(e)118 474 y(r)g(eferr)g(e)g(d)34 b(to)g(as)f(the)g(b)-5 b(ad)33 b(blo)-5 b(cks.)218 647 y Fr(In)33 b(order)g(to)h(state)h(the)e(other)h (results)e(on)i(the)g(large)f(deviations)g(of)g Fn(\026)2720 609 y Fp(+)2720 676 y Fj(N)2821 647 y Fr(w)m(e)h(need)f(to)h(in)m(tro)s (duce)118 776 y(some)d(more)f(notation.)41 b(F)-8 b(or)32 b(an)m(y)e Fn(\016)f(>)c Fr(0,)31 b(the)g Fn(\016)s Fr(-neigh)m(b)s (orho)s(o)s(d)e(of)i Fn(v)d Fk(2)d Fo(L)2692 743 y Fp(1)2737 776 y Fr(\()2772 752 y Fl(c)2772 776 y Fo(W)2857 743 y Fj(d)2904 776 y Fr(\))30 b(is)g(de\014ned)f(b)m(y)1053 956 y Fk(V)7 b Fr(\()p Fn(v)s(;)15 b(\016)s Fr(\))28 b(=)d Fk(f)p Fn(v)1532 918 y Fi(0)1581 956 y Fk(2)g Fo(L)1722 918 y Fp(1)1767 956 y Fr(\()1802 932 y Fl(c)1802 956 y Fo(W)1888 918 y Fj(d)1934 956 y Fr(\))31 b Fk(j)91 b(k)p Fn(v)2208 918 y Fi(0)2252 956 y Fk(\000)20 b Fn(v)s Fk(k)2435 970 y Fp(1)2500 956 y Fn(<)25 b(\016)s Fk(g)15 b Fn(:)218 1149 y Fr(Let)34 b Fk(O)j Fr(b)s(e)c(an)h(op)s(en)f(set)i (con)m(taining)1564 1126 y Fl(c)1564 1149 y Fo(W)1649 1116 y Fj(d)1695 1149 y Fr(.)52 b(The)33 b(set)i(of)f(functions)e(of)i (b)s(ounded)e(v)-5 b(ariation)33 b(in)g Fk(O)118 1267 y Fr(taking)25 b(v)-5 b(alues)24 b(in)g Fk(f\000)p Fr(1)p Fn(;)15 b Fr(1)p Fk(g)27 b Fr(and)d(uniformly)f(equal)h(to)i(1)f (outside)2377 1243 y Fl(c)2377 1267 y Fo(W)2462 1234 y Fj(d)2533 1267 y Fr(is)f(denoted)h(b)m(y)g(BV)r(\()3247 1243 y Fl(c)3247 1267 y Fo(W)3332 1234 y Fj(d)3379 1267 y Fn(;)15 b Fk(f\006)p Fr(1)p Fk(g)p Fr(\))118 1390 y(\(see)38 b([EG)q(])f(for)g(a)g(review\).)61 b(F)-8 b(or)38 b(a)f(giv)m(en)g Fn(a)g(>)f Fr(0,)k(the)d(set)g(of)h(functions)d(in)h(BV)r(\()3033 1366 y Fl(c)3033 1390 y Fo(W)3118 1357 y Fj(d)3165 1390 y Fn(;)15 b Fk(f\006)p Fr(1)p Fk(g)p Fr(\))39 b(with)118 1497 y(p)s(erimeter)30 b(smaller)g(than)h Fn(a)g Fr(is)f(denoted)h(b)m (y)g Fk(C)1749 1511 y Fj(a)1791 1497 y Fr(.)43 b(Finally)29 b(w)m(e)i(de\014ne)g(the)g(W)-8 b(ul\013)31 b(functional)e Fk(W)3487 1512 y Fj(\014)3565 1497 y Fr(on)118 1625 y(BV)r(\()287 1602 y Fl(c)287 1625 y Fo(W)372 1592 y Fj(d)419 1625 y Fn(;)15 b Fk(f\006)p Fr(1)p Fk(g)p Fr(\))36 b(as)f(follo)m(ws.)54 b(F)-8 b(or)35 b(an)m(y)h Fn(v)g Fk(2)c Fr(BV)r(\()1886 1602 y Fl(c)1886 1625 y Fo(W)1971 1592 y Fj(d)2018 1625 y Fn(;)15 b Fk(f\006)p Fr(1)p Fk(g)p Fr(\))q(,)37 b(there)e(exists)f(a) h(generalized)g(notion)118 1733 y(of)j(the)g(b)s(oundary)e(of)j(the)f (set)g Fk(f)p Fn(v)k Fr(=)c Fk(\000)p Fr(1)p Fk(g)g Fr(called)g (reduced)f(b)s(oundary)f(and)h(denoted)h(b)m(y)g Fn(@)3425 1700 y Fi(\003)3465 1733 y Fn(v)s Fr(.)64 b(If)118 1841 y Fk(f)p Fn(v)45 b Fr(=)40 b Fk(\000)p Fr(1)p Fk(g)g Fr(is)f(a)h(regular)f(set,)k(then)c Fn(@)1516 1808 y Fi(\003)1556 1841 y Fn(v)k Fr(coincides)38 b(with)h(the)g(usual)g(b)s (oundary)e Fn(@)5 b(v)s Fr(.)69 b(Then)39 b(one)118 1949 y(de\014nes)1368 2138 y Fk(W)1458 2153 y Fj(\014)1505 2138 y Fr(\()p Fn(v)s Fr(\))27 b(:=)1770 2014 y Fl(Z)1820 2220 y Fj(@)1861 2202 y Fh(\003)1897 2220 y Fj(v)1953 2138 y Fn(\034)10 b Fr(\()20 b Fn(~)-65 b(n)2093 2152 y Fj(x)2137 2138 y Fr(\))15 b Fn(d)p Fk(H)2311 2152 y Fj(x)2371 2138 y Fn(;)118 2349 y Fr(where)33 b Fk(H)461 2363 y Fj(x)537 2349 y Fr(is)f(the)h Fn(d)22 b Fk(\000)g Fr(1)33 b(Hausdor\013)g(measure.)48 b(The)32 b(W)-8 b(ul\013)33 b(functional)e Fk(W)2801 2364 y Fj(\014)2881 2349 y Fr(can)i(b)s(e)g (extended)f(on)118 2477 y Fo(L)173 2444 y Fp(1)218 2477 y Fr(\()253 2453 y Fl(c)253 2477 y Fo(W)338 2444 y Fj(d)385 2477 y Fr(\))f(b)m(y)f(setting)679 2703 y Fk(W)769 2718 y 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Fn(;)15 b Fk(f\006)p Fr(1)p Fk(g)p Fr(\))r Fn(;)2131 3073 y Fl(\014)2131 3127 y(\014)2192 3026 y(Z)2256 3215 y Fc(b)2243 3233 y Fg(W)2303 3214 y Fy(d)2364 3150 y Fn(m)2444 3113 y Fi(\003)2499 3150 y Fn(v)2543 3164 y Fj(r)2596 3150 y Fn(dr)2687 3073 y Fl(\014)2687 3127 y(\014)2742 3150 y Fm(6)25 b Fn(m)2918 3022 y Fl(\033)3002 3150 y Fn(:)402 b Fr(\(3.10\))118 3370 y(If)38 b(w)m(e)i(denote)f(b)m(y)f Fk(D)865 3384 y Fj(m)971 3370 y Fr(the)h(set)g(of)g(minimizers)c(of)46 b(\(3.10\))41 b(it)e(has)f(b)s(een)g(pro)m(v)m(en)h(b)m(y)g([T)-8 b(a])39 b(that)h(in)118 3480 y Fo(R)178 3447 y Fj(d)255 3480 y Fr(the)32 b(minimizer)c(is)i(unique)g(up)g(to)i(translations)e (and)h(giv)m(en)g(b)m(y)g(suitable)f(dilation)f(of)i(the)h(W)-8 b(ul\013)118 3598 y(shap)s(e)30 b(\(2.3\))r(.)40 b(In)30 b(particular)f(the)h(in)m(terfacial)g(energy)g(of)2135 3574 y Fl(c)2135 3598 y Fo(W)2220 3565 y Fj(d)2296 3598 y Fr(is)g(giv)m(en)g(b)m(y)1234 3810 y Fn(\034)1284 3772 y Fi(\003)1349 3810 y Fr(=)24 b Fk(W)1534 3825 y Fj(\014)1581 3810 y Fr(\()1616 3786 y Fl(c)1616 3810 y Fo(W)1702 3772 y Fj(d)1748 3810 y Fr(\))i(=)1905 3686 y Fl(Z)1955 3892 y Fj(@)2010 3875 y Fc(b)1996 3892 y Fg(W)2057 3873 y Fy(d)2118 3810 y Fn(\034)10 b Fr(\()n Fn(~)-43 b(n)2258 3824 y Fj(x)2302 3810 y Fr(\))p Fn(d)p Fk(H)2461 3824 y Fj(x)2520 3810 y Fn(:)884 b Fr(\(3.11\))118 4021 y(All)29 b(that)i(b)s(eing)e(said)g(the)i(results)e(w)m(e)i(are)g(going)f(to)h (use)f(can)h(b)s(e)f(summarized)f(as)h(follo)m(ws.)118 4193 y FC(Prop)s(osition)36 b(3.1.)46 b Ff(Ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fn(C)7 b Fr(\()p Fn(\014)e Fr(\))26 b Fn(>)f Fr(0)33 b Ff(such)f(that)i(for)g(any)f Fn(\016)c(>)c Fr(0)571 4402 y Fk(8)p Fn(a)g(>)g Fr(0)p Fn(;)201 b Fr(lim)15 b(sup)1099 4480 y Fj(N)7 b Fi(!1)1475 4340 y Fr(1)p 1391 4381 214 4 v 1391 4464 a Fn(N)1474 4438 y Fj(d)p Fi(\000)p Fp(1)1630 4402 y Fr(log)16 b Fn(\026)1817 4363 y Fp(+)1817 4430 y Fj(N)1899 4402 y Fr(\()q Fn(u)1987 4416 y Fj(N)2079 4402 y Fk(62)25 b(V)7 b Fr(\()p Fk(C)2311 4416 y Fj(a)2353 4402 y Fn(;)15 b(\016)s Fr(\)\))43 b Fm(6)i Fk(\000)20 b Fn(C)7 b Fr(\()p Fn(\014)e Fr(\))15 b Fn(a;)118 4646 y Ff(wher)-5 b(e)34 b Fk(V)7 b Fr(\()p Fk(C)521 4660 y Fj(a)563 4646 y Fn(;)15 b(\016)s Fr(\))34 b Ff(is)f(the)g Fn(\016)s Ff(-neighb)-5 b(orho)g(o)g(d)36 b(of)d Fk(C)1735 4660 y Fj(a)1809 4646 y Ff(in)f Fo(L)1975 4613 y Fp(1)2021 4646 y Fr(\()2056 4622 y Fl(c)2056 4646 y Fo(W)2141 4613 y Fj(d)2187 4646 y Fr(\))p Ff(.)118 4819 y Fr(This)g(prop)s(osition)e(tells)j(us)f(that)i(only)f(the)g (con\014gurations)f(close)i(to)g(the)f(compact)i(set)f Fk(C)3329 4833 y Fj(a)3403 4819 y Fr(ha)m(v)m(e)h(a)118 4927 y(con)m(tribution)29 b(whic)m(h)g(is)h(of)g(a)h(surface)f(order.) 118 5035 y(The)g(precise)g(asymptotic)g(related)h(to)g(surface)f (tension)g(are)118 5208 y FC(Prop)s(osition)36 b(3.2.)46 b Ff(Uniformly)34 b(over)f Fn(\016)c(>)c Fr(0)895 5411 y(lim)15 b(inf)919 5471 y Fj(N)7 b Fi(!1)1282 5349 y Fr(1)p 1198 5390 V 1198 5473 a Fn(N)1281 5447 y Fj(d)p Fi(\000)p Fp(1)1437 5411 y Fr(log)16 b Fn(\026)1624 5372 y Fp(+)1624 5439 y Fj(N)1691 5337 y Fl(\000)1733 5411 y Fk(k)p Fn(u)1830 5425 y Fj(N)1918 5411 y Fr(+)k(1)p Fk(k)2099 5425 y Fp(1)2164 5411 y Fm(6)25 b Fn(\016)2303 5337 y Fl(\001)2396 5411 y Fm(>)50 b Fk(\000)p Fn(\034)2638 5373 y Fi(\003)2693 5411 y Fn(:)p eop %%Page: 8 8 8 7 bop 118 45 a Fp(8)1106 b(T.)30 b(BODINEA)n(U)h(AND)f(F.)g(MAR)-6 b(TINELLI)118 375 y FC(Prop)s(osition)36 b(3.3.)46 b Ff(F)-7 b(or)34 b(al)5 b(l)33 b Fn(v)j Ff(in)d Fr(BV)q(\()1551 352 y Fl(c)1551 375 y Fo(W)1637 342 y Fj(d)1683 375 y Fn(;)15 b Fk(f\006)p Fr(1)p Fk(g)p Fr(\))34 b Ff(such)f(that)h Fk(W)2481 390 y Fj(\014)2528 375 y Fr(\()p Fn(v)s Fr(\))g Ff(is)e(\014nite)h(and)g(for)h Fn(\016)29 b(>)c Fr(0)670 576 y(lim)15 b(sup)707 654 y Fj(N)7 b Fi(!1)1082 514 y Fr(1)p 998 555 214 4 v 998 638 a Fn(N)1081 612 y Fj(d)p Fi(\000)p Fp(1)1237 576 y Fr(log)17 b Fn(\026)1425 537 y Fp(+)1425 604 y Fj(N)1492 502 y Fl(\000)1533 576 y Fk(k)p Fn(u)1630 590 y Fj(N)1718 576 y Fk(\000)j Fn(v)s Fk(k)1901 590 y Fp(1)1967 576 y Fm(6)25 b Fn(\016)2106 502 y Fl(\001)2173 576 y Fm(6)45 b Fk(\000)20 b(W)2470 591 y Fj(\014)2517 576 y Fr(\()p Fn(v)s Fr(\))i(+)d Fn(")p Fr(\()p Fn(\016)s Fr(\))c Fn(;)118 784 y Ff(wher)-5 b(e)34 b Fn(")p Fr(\()p Fn(\016)s Fr(\))g Ff(vanishes)g(as)f Fn(\016)j Ff(go)-5 b(es)33 b(to)h(0.)1402 986 y Fr(4.)51 b Fq(The)34 b(test)f(function)218 1148 y Fr(In)f(this)f(section)i(w)m (e)h(de\014ne)e(the)h(test)g(function)f(that,)i(if)e(plugged)f(in)m(to) i(the)g(P)m(oincar)m(\023)-43 b(e)34 b(and)e(loga-)118 1256 y(rithmic)c(Sob)s(olev)g(inequalities,)f(will)g(pro)m(v)m(e)j (theorems)g(\(2.1\))h(and)e(\(2.2\).)42 b(As)29 b(w)m(e)h(men)m(tioned) f(in)f(the)118 1364 y(in)m(tro)s(duction,)h(the)h(form)g(of)h(function) e(describ)s(ed)f(b)s(elo)m(w)i(w)m(as)g(suggested)i(b)m(y)e(H.T.)g(Y)-8 b(au.)218 1472 y(Fix)42 b Fn(\025)k Fk(2)15 b Fr(])603 1436 y Fp(1)p 603 1451 36 4 v 603 1503 a(2)648 1472 y Fn(\034)698 1439 y Fi(\003)738 1472 y Fn(;)g(\034)828 1439 y Fi(\003)868 1472 y Fr([,)46 b(where)c Fn(\034)1289 1439 y Fi(\003)1372 1472 y Fr(is)f(de\014ned)h(in)f(\(3.11\))s(.)78 b(Let)43 b Fn(g)j Fr(b)s(e)d(a)g(smo)s(oth)g(non)f(increasing)118 1580 y(function)29 b(suc)m(h)h(that)1167 1792 y Fn(g)s Fr(\()p Fn(s)p Fr(\))c(=)1448 1636 y Fl(\()1521 1730 y Fr(1)p Fn(;)274 b Fr(if)90 b Fn(s)25 b Fm(6)2182 1694 y Fi(\000)p Fj(m)2299 1671 y Fh(\003)p 2182 1709 154 4 v 2241 1761 a Fp(2)2361 1730 y Fn(;)1521 1860 y Fr(0)p Fn(;)274 b Fr(if)90 b Fn(s)50 b Fm(>)2233 1824 y Fi(\000)p Fj(m)2350 1800 y Fh(\003)p 2233 1839 V 2292 1891 a Fp(4)2411 1860 y Fn(:)118 2063 y Fr(The)25 b(mesoscopic)h(scale)f Fn(K)32 b Fr(is)25 b(c)m(hosen)g(equal)g(to)h(\()p Fn(b)15 b Fr(log)i Fn(N)10 b Fr(\))2148 1990 y Fv(1)p 2146 2002 36 3 v 2146 2043 a Fy(\015)2196 2063 y Fr(,)26 b(where)f Fn(\015)30 b Fr(w)m(as)c(in)m(tro)s(duced)e(in)g(\(3.8\))j(and)118 2171 y Fn(b)i Fr(is)f(a)h(constan)m(t)h(whic)m(h)e(will)e(b)s(e)i (\014xed)g(later.)41 b(The)28 b(test)i(function)d Fn(f)38 b Fr(has)29 b(the)g(follo)m(wing)e(expression:)671 2428 y Fn(f)10 b Fr(\()p Fn(\033)s Fr(\))26 b(=)f(exp)1127 2245 y Fl(0)1127 2408 y(@)1216 2366 y Fn(\025K)1353 2333 y Fj(d)p 1216 2407 178 4 v 1263 2490 a Fn(N)1419 2341 y Fl(X)1468 2537 y Fj(j)1565 2428 y Fn(g)s Fr(\()p Fk(M)1755 2442 y Fj(N)s(;K)1903 2428 y Fr(\()p Fn(x)1990 2442 y Fj(j)2027 2428 y Fr(\)\))2097 2245 y Fl(1)2097 2408 y(A)2208 2428 y Fn(;)197 b Fk(8)p Fn(\033)28 b Fk(2)d(f\000)p Fr(1)p Fn(;)15 b Fr(1)p Fk(g)2938 2390 y Fg(W)3000 2401 y Fy(N)3083 2428 y Fn(:)321 b Fr(\(4.12\))118 2700 y(The)43 b(factor)h Fn(K)675 2667 y Fj(d)759 2700 y Fr(stands)f(for)g(the)h(v)m (olume)f(of)g(the)h(b)s(o)m(xes)g Fo(B)2308 2714 y Fj(K)2426 2700 y Fr(whic)m(h)e(equals)h(to)h Fn(K)3196 2667 y Fj(d)3280 2700 y Fr(\(with)e(the)118 2808 y(exception)32 b(of)f(some)h(b)s(o)m (xes)f(along)g(the)h(b)s(oundary\).)42 b(Notice)32 b(that)g Fn(f)40 b Fr(is)30 b(a)i(non)f(increasing)f(function)118 2916 y(of)h(the)f(spins.)118 3024 y(There)g(are)h(three)f(main)g (features)g(of)h Fn(f)39 b Fr(that)31 b(mak)m(e)h(it)e(quite)f (e\013ectiv)m(e.)43 b(These)30 b(are:)202 3154 y FC(i\))42 b Fr(The)28 b(v)-5 b(ariance)28 b(of)h Fn(f)38 b Fr(almost)29 b(coincides)e(with)h Fn(\026)1973 3116 y Fp(+)1973 3183 y Fj(N)2040 3154 y Fr(\()p Fn(f)2130 3121 y Fp(2)2169 3154 y Fr(\))h(or,)g(put)f(it)g(in)f(another)i(w)m(a)m(y)-8 b(,)31 b Fn(\026)3314 3116 y Fp(+)3314 3183 y Fj(N)3381 3154 y Fr(\()p Fn(f)3471 3121 y Fp(2)3510 3154 y Fr(\))26 b Fk(\035)314 3269 y Fn(\026)369 3231 y Fp(+)369 3298 y Fj(N)436 3269 y Fr(\()p Fn(f)10 b Fr(\))561 3236 y Fp(2)600 3269 y Fr(;)173 3386 y FC(ii\))42 b Fr(The)29 b(en)m(trop)m(y)i(of)g Fn(f)990 3353 y Fp(2)1059 3386 y Fr(w.r.t.)41 b(to)31 b Fn(\026)1478 3347 y Fp(+)1478 3414 y Fj(N)1575 3386 y Fr(is)f(of)g(order)g Fn(N)2091 3353 y Fj(d)p Fi(\000)p Fp(1)2222 3386 y Fr(;)144 3541 y FC(iii\))42 b Fr(Let)31 b(us)e(denote)j(b)m(y)e Fn(\026)1068 3497 y Fp(+)p Fj(;f)1068 3569 y(N)1218 3541 y Fr(the)h(w)m(eigh)m(ted)g (measure)2118 3490 y Fj(d\026)2196 3457 y Fv(+)p Fy(;f)2196 3513 y(N)p 2118 3520 185 4 v 2142 3585 a Fj(d\026)2220 3555 y Fv(+)2220 3608 y Fy(N)2338 3541 y Fr(=)2505 3505 y Fp(1)p 2444 3520 158 4 v 2444 3591 a Fj(Z)2497 3558 y Fv(+)p Fy(;f)2492 3614 y(N)2627 3541 y Fn(f)2682 3508 y Fp(2)2751 3541 y Fr(where)f Fn(Z)3083 3497 y Fp(+)p Fj(;f)3076 3569 y(N)3228 3541 y Fr(:=)c Fn(\026)3405 3503 y Fp(+)3405 3569 y Fj(N)3472 3541 y Fr(\()p Fn(f)3562 3508 y Fp(2)3601 3541 y Fr(\).)314 3713 y(Then)36 b(under)g Fn(\026)878 3668 y Fp(+)p Fj(;f)878 3741 y(N)1036 3713 y Fr(the)i(t)m(ypical)f(n)m(um)m(b)s(er)f(of)i(non)f(zero)i(terms)e(in) g Fk(jr)p Fn(f)10 b Fk(j)2893 3680 y Fp(2)2969 3713 y Fr(is)37 b(of)h(the)g(order)f(of)314 3829 y Fn(N)397 3796 y Fj(d)p Fi(\000)p Fp(1)527 3829 y Fr(.)118 3959 y(It)e(is)f(clear)h(that)g(once)h(these)f(prop)s(erties)e(are)i (established)e(then)i(the)g(pro)s(of)f(of)h(theorems)g(2.1)h(and)118 4067 y(2.2)c(should)c(follo)m(w)i(quite)f(easily)-8 b(.)118 4175 y(In)m(tuitiv)m(ely)41 b(the)h(pro)s(of)g(of)g FC(i)p Fr(\))p Fn(;)15 b FC(ii)q Fr(\))43 b(and)e FC(iii)p Fr(\))i(is)e(based) h(on)g(the)h(follo)m(wing)d(simple)g(heuristic.)75 b(The)118 4282 y(function)39 b Fn(f)50 b Fr(assigns)40 b(an)g(exp)s(onen)m(tial)g (w)m(eigh)m(t)h(to)g(the)g(con\014gurations)f(with)f(a)i(large)g(n)m (um)m(b)s(er)e(of)118 4390 y(mesoscopic)32 b(b)s(o)m(xes)f(with)f(lab)s (el)g Fn(u)1317 4404 y Fj(N)1411 4390 y Fr(=)c Fk(\000)p Fr(1)32 b(b)s(ecause)f(of)g(the)h(c)m(hoice)g(of)f(the)h(function)e Fn(g)s Fr(.)44 b(According)118 4498 y(to)24 b(the)f(large)g(deviation)f (theory)-8 b(,)25 b(among)e(the)g(con\014gurations)g(fa)m(v)m(ored)g(b) m(y)g Fn(f)10 b Fr(,)24 b(those)f(with)f(the)h(largest)118 4606 y Fn(\026)173 4568 y Fp(+)173 4635 y Fj(N)267 4606 y Fr(w)m(eigh)m(t)k(form)f(a)h(W)-8 b(ul\013)26 b(droplet)g(of)g(a)h (certain)g(size.)39 b(Therefore,)27 b(to)h(compute)f Fn(\026)3024 4568 y Fp(+)3024 4635 y Fj(N)3091 4606 y Fr(\()p Fn(f)10 b Fr(\))26 b(or)h Fn(\026)3405 4568 y Fp(+)3405 4635 y Fj(N)3472 4606 y Fr(\()p Fn(f)3562 4573 y Fp(2)3601 4606 y Fr(\),)118 4714 y(w)m(e)d(will)e(need)h(to)i (compare,)h(for)d(a)i(giv)m(en)e(W)-8 b(ul\013)24 b(droplet,)g(the)g (gain)g(giv)m(en)g(b)m(y)f(the)i(exp)s(onen)m(tial)d(factor)118 4822 y(in)30 b Fn(f)41 b Fr(or)32 b Fn(f)479 4789 y Fp(2)549 4822 y Fr(with)f(the)h Fn(\026)971 4784 y Fp(+)971 4851 y Fj(N)1069 4822 y Fr(probabilit)m(y)d(of)j(creating)g(the)g(droplet)f (itself.)43 b(It)32 b(turns)e(out,)j(due)e(to)h(the)118 4930 y(precise)d(c)m(hoice)h(of)g(the)f(parameter)h Fn(\025)p Fr(,)g(that)g(the)f(balance)h(for)f Fn(f)38 b Fr(is)29 b(negativ)m(e)h(and)f(no)g(W)-8 b(ul\013)29 b(droplet)118 5038 y(will)e(app)s(ear,)j(while)e(the)i(balance)g(is)f(p)s(ositiv)m(e) g(for)h Fn(f)1957 5005 y Fp(2)2026 5038 y Fr(and)f(the)h(t)m(ypical)g (spin)e(con\014gurations)h(under)118 5154 y Fn(\026)173 5110 y Fp(+)p Fj(;f)173 5183 y(N)327 5154 y Fr(will)j(consist)j(of)f(a) h(W)-8 b(ul\013)35 b(droplet)e(of)i(the)g Ff(minus)h(phase)g Fr(of)f(v)m(olume)g Fk(\031)d Fn(N)2970 5121 y Fj(d)3010 5154 y Fr(.)53 b(That)35 b(accoun)m(ts)118 5262 y(for)d FC(i)p Fr(\))h(and)f FC(ii)p Fr(\).)47 b(Giv)m(en)32 b(the)h(ab)s(o)m(v)m(e)h(picture,)e(it)g(is)f(also)i(clear)f(that)h FC(iii)p Fr(\))g(holds)e(simply)f(b)s(ecause)i(the)118 5370 y(non)c(zero)h(terms)g(in)e Fk(jr)p Fn(f)10 b Fk(j)1020 5337 y Fp(2)1088 5370 y Fr(come)29 b(only)f(from)g(the)h(bad)f(b)s(o)m (xes,)h(again)g(b)s(ecause)f(of)h(the)g(c)m(hoice)g(of)g(the)118 5480 y(function)e Fn(g)s Fr(.)40 b(The)28 b(b)s(oundary)e(of)i(the)g(W) -8 b(ul\013)28 b(droplet)f(pro)s(duces)g Fn(O)s Fr(\()p Fn(N)2554 5447 y Fj(d)p Fi(\000)p Fp(1)2685 5480 y Fr(\))h(of)g(suc)m (h)g(b)s(o)m(xes)g(while)e(the)p eop %%Page: 9 9 9 8 bop 569 45 a Fp(SOME)30 b(NEW)h(RESUL)-6 b(TS)31 b(ON)g(THE)f(KINETIC)h(ISING)h(MODEL)e(IN)h(A)f(PURE)g(PHASE)416 b(9)118 366 y Fr(inside)30 b(of)j(the)g(droplet)e(t)m(ypically)g(do)s (es)i(not)f(con)m(tain)h Ff(any)g Fr(bad)f(b)s(o)m(x)g(b)s(ecause)h(of) g(the)f(c)m(hoice)i(of)e(the)118 474 y(mesoscopic)h(scale)g Fn(K)7 b Fr(.)48 b(W)-8 b(ere)35 b Fn(K)k Fr(b)s(e)32 b(large)h(but)g(indep)s(enden)m(t)d(of)j Fn(N)43 b Fr(then)33 b(w)m(e)g(w)m(ould)f(alw)m(a)m(ys)h(ha)m(v)m(e)118 582 y(a)e(densit)m(y)f(of)g(bad)g(b)s(o)m(xes)g(and)g(the)h(whole)e (construction)h(w)m(ould)f(break)i(do)m(wn.)118 775 y(4.1.)54 b FC(The)34 b(V)-9 b(ariance)36 b(of)f Fn(f)10 b FC(.)45 b Fr(W)-8 b(e)31 b(are)g(\014rst)f(going)g(to)h(c)m(hec)m(k)h(that)1015 996 y(lim)976 1057 y Fj(N)7 b Fi(!1)1205 932 y Fn(\026)1260 894 y Fp(+)1260 961 y Fj(N)1327 932 y Fr(\()p Fn(f)e(;)15 b(f)10 b Fr(\))p 1205 975 337 4 v 1230 1065 a Fn(\026)1285 1027 y Fp(+)1285 1093 y Fj(N)1352 1065 y Fr(\()p Fn(f)1442 1038 y Fp(2)1482 1065 y Fr(\))1577 996 y(=)64 b(lim)1673 1057 y Fj(N)7 b Fi(!1)1902 932 y Fn(\026)1957 894 y Fp(+)1957 961 y Fj(N)2025 932 y Fr(\()p Fn(f)2115 899 y Fp(2)2154 932 y Fr(\))20 b Fk(\000)g Fn(\026)2355 894 y Fp(+)2355 961 y Fj(N)2422 932 y Fr(\()p Fn(f)10 b Fr(\))2547 899 y Fp(2)p 1902 975 685 4 v 2101 1065 a Fn(\026)2156 1027 y Fp(+)2156 1093 y Fj(N)2223 1065 y Fr(\()p Fn(f)2313 1038 y Fp(2)2352 1065 y Fr(\))2622 996 y(=)25 b(1)15 b Fn(:)626 b Fr(\(4.13\))118 1246 y(The)30 b(function)f(uniformly)e (equal)j(to)h Fk(\000)p Fr(1)g(in)1679 1223 y Fl(c)1679 1246 y Fo(W)1764 1213 y Fj(d)1841 1246 y Fr(is)e(denoted)h(b)m(y)h Fk(\000)p Fr(1)-25 b(I.)40 b(Let)31 b Fn(")26 b(>)f Fr(0,)31 b(then)246 1457 y Fn(\026)301 1419 y Fp(+)301 1486 y Fj(N)368 1457 y Fr(\()p Fn(f)458 1420 y Fp(2)497 1457 y Fr(\))51 b Fm(>)f Fn(\026)759 1419 y Fp(+)759 1486 y Fj(N)842 1384 y Fl(\000)883 1457 y Fn(f)938 1420 y Fp(2)992 1457 y Fr(1)1037 1476 y Fi(f)p Fj(u)1113 1487 y Fy(N)1171 1476 y Fi(2V)6 b Fp(\()p Fi(\000)p Fp(1)-25 b(I)q Fj(;")p Fp(\))p Fi(g)1505 1384 y Fl(\001)1597 1457 y Fm(>)50 b Fr(exp)1872 1329 y Fl(\022)1949 1396 y Fr(2)p Fn(\025)p 1949 1436 99 4 v 1957 1520 a(N)2058 1457 y(N)2141 1420 y Fj(d)2181 1457 y Fr(\(1)21 b Fk(\000)f Fn(")p Fr(\))2450 1329 y Fl(\023)2533 1457 y Fn(\026)2588 1419 y Fp(+)2588 1486 y Fj(N)2670 1457 y Fr(\()p Fn(u)2757 1471 y Fj(N)2850 1457 y Fk(2)25 b(V)7 b Fr(\()p Fk(\000)p Fr(1)-25 b(I)p Fn(;)15 b(")p Fr(\)\))32 b Fn(;)118 1687 y Fr(where)24 b(w)m(e)i(used)e(the)h(fact)h(that)f(if)f Fn(u)1352 1701 y Fj(N)1419 1687 y Fr(\()p Fn(x)p Fr(\))i(=)f Fk(\000)p Fr(1)g(then)f Fk(M)2114 1701 y Fj(N)s(;K)2262 1687 y Fr(\()p Fn(x)p Fr(\))i Fn(<)f Fk(\000)2587 1652 y Fj(m)2649 1628 y Fh(\003)p 2586 1667 V 2618 1719 a Fp(2)2695 1687 y Fr(.)38 b(Prop)s(osition)23 b(3.2)j(implies)118 1795 y(that)31 b(for)f Fn(N)40 b Fr(large)31 b(enough)852 1979 y Fn(\026)907 1941 y Fp(+)907 2008 y Fj(N)974 1979 y Fr(\()p Fn(f)1064 1942 y Fp(2)1103 1979 y Fr(\))51 b Fm(>)f Fr(exp)1464 1878 y Fl(\020)1518 1979 y Fn(N)1601 1942 y Fj(d)p Fi(\000)p Fp(1)1732 1905 y Fl(\000)1773 1979 y Fr(2)p Fn(\025)p Fr(\(1)22 b Fk(\000)e Fn(")p Fr(\))h Fk(\000)f Fn(\034)2303 1942 y Fi(\003)2362 1979 y Fk(\000)g Fn(o)p Fr(\()p Fn(")p Fr(\))2609 1905 y Fl(\001)2652 1878 y(\021)2737 1979 y Fn(;)667 b Fr(\(4.14\))118 2168 y(where)30 b Fn(o)p Fr(\()p Fn(")p Fr(\))h(v)-5 b(anishes)30 b(as)g Fn(")h Fr(go)s(es)g(to)g(0.)118 2277 y(Next)g(w)m(e)g(examine)f Fn(\026)885 2239 y Fp(+)885 2306 y Fj(N)952 2277 y Fr(\()p Fn(f)10 b Fr(\))31 b(and)e(pro)m(v)m(e)j(that)1318 2487 y(lim)1279 2548 y Fj(N)7 b Fi(!1)1593 2426 y Fr(1)p 1509 2466 214 4 v 1509 2550 a Fn(N)1592 2523 y Fj(d)p Fi(\000)p Fp(1)1747 2487 y Fr(log)17 b Fn(\026)1935 2449 y Fp(+)1935 2516 y Fj(N)2002 2487 y Fr(\()p Fn(f)10 b Fr(\))25 b(=)g(0)15 b Fn(:)1096 b Fr(\(4.15\))118 2697 y(The)43 b(deriv)-5 b(ation)42 b(of)i(an)g(upp)s(er)d(b)s(ound)h(for)h Fn(\026)1789 2658 y Fp(+)1789 2725 y Fj(N)1856 2697 y Fr(\()p Fn(f)10 b Fr(\))44 b(requires)e(some)i(tec)m(hnicalities.)79 b(First)43 b(\014x)g(a)118 2815 y(constan)m(t)35 b Fn(a)30 b(>)732 2780 y Fj(\025)p 677 2795 153 4 v 677 2848 a(C)5 b Fp(\()p Fj(\014)s Fp(\))873 2815 y Fr(where)32 b Fn(C)7 b Fr(\()p Fn(\014)e Fr(\))34 b(app)s(ears)e(in)g(Prop)s(osition)g(3.1.) 50 b(Then)32 b(Prop)s(osition)g(3.1)i(implies)118 2936 y(that)d(for)f(an)m(y)h Fn(\016)e(>)c Fr(0)678 3135 y(lim)15 b(sup)715 3213 y Fj(N)7 b Fi(!1)1091 3074 y Fr(1)p 1007 3114 214 4 v 1007 3198 a Fn(N)1090 3171 y Fj(d)p Fi(\000)p Fp(1)1245 3135 y Fr(log)17 b Fn(\026)1433 3097 y Fp(+)1433 3164 y Fj(N)1515 3062 y Fl(\000)1557 3135 y Fn(f)24 b Fr(1)1671 3154 y Fi(f)p Fj(u)1747 3165 y Fy(N)1806 3154 y Fi(62V)6 b Fp(\()p Fi(C)1967 3162 y Fy(a)2004 3154 y Fj(;\016)r Fp(\))p Fi(g)2125 3062 y Fl(\001)2207 3135 y Fm(6)25 b Fn(\025)20 b Fk(\000)g Fn(C)7 b Fr(\()p Fn(\014)e Fr(\))15 b Fn(a)26 b(<)f Fr(0)15 b Fn(:)494 b Fr(\(4.16\))118 3360 y(Then)24 b(\014x)g Fn(")h(>)g Fr(0)g(and)f(recall)g(that)h Fk(C)1353 3374 y Fj(a)1419 3360 y Fr(is)f(compact)i(for)e(the)h Fo(L)2201 3327 y Fp(1)2271 3360 y Fr(top)s(ology)-8 b(.)40 b(According)24 b(to)h(Prop)s(osition)118 3468 y(3.3,)30 b(for)e Fn(\016)k Fr(small)27 b(enough)h(the)g(set)h Fk(V)7 b Fr(\()p Fk(C)1484 3482 y Fj(a)1526 3468 y Fn(;)15 b(\016)s Fr(\))30 b(can)f(b)s(e)e(co)m(v)m(ered)j(b)m(y)e(a)h(\014nite) e(union)f Fk([)3023 3435 y Fj(`)3023 3493 y(i)p Fp(=1)3141 3468 y Fk(V)7 b Fr(\()p Fn(v)3283 3482 y Fj(i)3312 3468 y Fn(;)15 b(")3394 3482 y Fj(i)3423 3468 y Fr(\))29 b(suc)m(h)118 3576 y(that)i(for)f(an)m(y)h Fn(i)26 b Fm(6)e Fn(`)31 b Fr(and)e Fn(N)41 b Fr(large)30 b(enough)981 3713 y(1)p 897 3754 V 897 3837 a Fn(N)980 3811 y Fj(d)p Fi(\000)p Fp(1)1135 3775 y Fr(log)17 b Fn(\026)1323 3736 y Fp(+)1323 3803 y Fj(N)1390 3701 y Fl(\000)1432 3775 y Fn(u)1484 3789 y Fj(N)1576 3775 y Fk(2)25 b(V)7 b Fr(\()p Fn(v)1804 3789 y Fj(i)1833 3775 y Fn(;)15 b(")1915 3789 y Fj(i)1944 3775 y Fr(\))1979 3701 y Fl(\001)2046 3775 y Fm(6)45 b Fk(\000)20 b(W)2343 3790 y Fj(\014)2390 3775 y Fr(\()p Fn(v)2469 3789 y Fj(i)2498 3775 y Fr(\))g(+)g Fn(")15 b(;)703 b Fr(\(4.17\))118 3967 y(where)30 b Fn(")423 3981 y Fj(i)477 3967 y Fk(2)25 b Fr(\(0)p Fn(;)15 b(")p Fr(\).)42 b(Noticing)30 b(that)665 4220 y Fn(\026)720 4181 y Fp(+)720 4248 y Fj(N)787 4220 y Fr(\()p Fn(f)10 b Fr(\))25 b Fm(6)1100 4106 y Fj(`)1048 4133 y Fl(X)1057 4329 y Fj(i)p Fp(=1)1195 4220 y Fn(\026)1250 4181 y Fp(+)1250 4248 y Fj(N)1332 4146 y Fl(\000)1374 4220 y Fn(f)f Fr(1)1488 4238 y Fi(f)p Fj(u)1564 4249 y Fy(N)1622 4238 y Fi(2V)6 b Fp(\()p Fj(v)1780 4248 y Fy(i)1807 4238 y Fj(;")1860 4248 y Fy(i)1885 4238 y Fp(\))p Fi(g)1952 4146 y Fl(\001)2014 4220 y Fr(+)20 b Fn(\026)2160 4181 y Fp(+)2160 4248 y Fj(N)2242 4146 y Fl(\000)2284 4220 y Fn(f)k Fr(1)2398 4238 y Fi(f)p Fj(u)2474 4249 y Fy(N)2532 4238 y Fi(62V)6 b Fp(\()p Fi(C)2693 4246 y Fy(a)2731 4238 y Fj(;\016)r Fp(\))p Fi(g)2851 4146 y Fl(\001)2923 4220 y Fn(;)481 b Fr(\(4.18\))118 4467 y(and)30 b(com)m(bining)g(\(4.16\))j(with)c (\(4.17\))r(,)i(w)m(e)f(get)193 4720 y Fn(\026)248 4682 y Fp(+)248 4749 y Fj(N)315 4720 y Fr(\()p Fn(f)10 b Fr(\))26 b Fm(6)628 4607 y Fj(`)577 4634 y Fl(X)585 4829 y Fj(i)p Fp(=1)723 4720 y Fr(exp)877 4619 y Fl(\020)932 4720 y Fn(N)1015 4683 y Fj(d)p Fi(\000)p Fp(1)1145 4647 y Fl(\000)1187 4720 y Fn(\025)p Fk(j)p Fn(v)1309 4734 y Fj(i)1337 4720 y Fk(j)21 b(\000)f(W)1564 4735 y Fj(\014)1611 4720 y Fr(\()p Fn(v)1690 4734 y Fj(i)1718 4720 y Fr(\))h(+)f Fn(")p Fr(\(1)h(+)f Fn(\025)p Fr(\))2187 4647 y Fl(\001)2229 4619 y(\021)2304 4720 y Fr(+)f(exp)2548 4619 y Fl(\020)2603 4720 y Fn(N)2686 4683 y Fj(d)p Fi(\000)p Fp(1)2816 4647 y Fl(\000)2858 4720 y Fn(\025)h Fk(\000)g Fn(C)7 b Fr(\()p Fn(\014)e Fr(\))p Fn(a)3268 4647 y Fl(\001)3310 4619 y(\021)3395 4720 y Fn(;)118 4968 y Fr(where)24 b Fk(j)p Fn(v)444 4982 y Fj(i)472 4968 y Fk(j)g Fr(denotes)h(the)f(v)m(olume)g (of)g(the)g(set)h Fk(f)p Fn(v)1773 4982 y Fj(i)1827 4968 y Fr(=)g Fk(\000)p Fr(1)p Fk(g)p Fr(.)39 b(T)-8 b(o)25 b(c)m(hec)m(k)g(that)g(the)f(spin)e(con\014guration)i(in)118 5078 y Fk(f)p Fn(u)215 5092 y Fj(N)308 5078 y Fk(2)h(V)7 b Fr(\()p Fn(v)536 5092 y Fj(i)565 5078 y Fn(;)15 b(")647 5092 y Fj(i)676 5078 y Fr(\))p Fk(g)30 b Fr(ha)m(v)m(e)g(a)g(n)m(um)m (b)s(er)e(of)i(blo)s(c)m(ks)e(of)i(the)f(order)g(of)h Fn(N)2457 5045 y Fj(d)2497 5078 y Fk(j)p Fn(v)2566 5092 y Fj(i)2595 5078 y Fk(j)p Fr(,)g(it)f(is)f(enough)h(to)h(regularize)118 5186 y Fn(v)162 5200 y Fj(i)221 5186 y Fr(b)m(y)g(a)h(p)s(olyhedral)c (set)k(\(see)h(Thm)d(2.5.1)j(in)d([BIV)q(]\).)118 5309 y(By)i(the)f(v)m(ery)h(de\014nition)d(of)j(the)f(W)-8 b(ul\013)30 b(v)-5 b(ariational)29 b(problem,)h(for)g(an)m(y)g Fn(v)f Fk(2)c Fr(BV)r(\()2996 5285 y Fl(c)2996 5309 y Fo(W)3081 5276 y Fj(d)3127 5309 y Fn(;)15 b Fk(f\006)p Fr(1)p Fk(g)p Fr(\))1171 5480 y Fk(W)1261 5495 y Fj(\014)1308 5480 y Fr(\()p Fn(v)s Fr(\))52 b Fm(>)e Fn(\034)1648 5443 y Fi(\003)1688 5480 y Fk(j)p Fn(v)s Fk(j)1785 5443 y Fp(\()p Fj(d)p Fi(\000)p Fp(1\))p Fj(=d)2093 5480 y Fm(>)g Fn(\034)2264 5443 y Fi(\003)2304 5480 y Fk(j)p Fn(v)s Fk(j)15 b Fn(;)988 b Fr(\(4.19\))p eop %%Page: 10 10 10 9 bop 118 45 a Fp(10)1071 b(T.)30 b(BODINEA)n(U)h(AND)f(F.)g(MAR)-6 b(TINELLI)118 375 y Fr(where)30 b(w)m(e)h(ha)m(v)m(e)g(used)f(the)h (fact)g(that)g Fk(j)p Fn(v)s Fk(j)26 b Fm(6)f Fk(j)1709 352 y Fl(c)1709 375 y Fo(W)1794 342 y Fj(d)1840 375 y Fk(j)h Fr(=)f(1.)41 b(As)30 b Fn(\025)c(<)f(\034)2457 342 y Fi(\003)2496 375 y Fr(,)1098 558 y Fn(\026)1153 520 y Fp(+)1153 587 y Fj(N)1220 558 y Fr(\()p Fn(f)10 b Fr(\))25 b Fm(6)g Fn(`)15 b Fr(exp)1673 457 y Fl(\020)1728 558 y Fn(N)1811 521 y Fj(d)p Fi(\000)p Fp(1)1941 558 y Fn(")p Fr(\(1)22 b(+)e Fn(\025)p Fr(\))2264 457 y Fl(\021)2339 558 y Fr(+)f(1)c Fn(:)915 b Fr(\(4.20\))118 741 y(Since)29 b Fn(")i Fr(w)m(as)g(arbitrary)-8 b(,)30 b(this)f(implies)f(\(4.15\))r (.)118 849 y(Com)m(bining)g(\(4.14\))r(,)h(\(4.15\))i(and)e(the)f(fact) i(that)f(2)p Fn(\025)d(>)f(\034)2100 816 y Fi(\003)2140 849 y Fr(,)k(w)m(e)g(\014nally)e(deriv)m(e)h(\(4.13\))j(b)m(y)d(c)m(ho) s(osing)h Fn(")118 957 y Fr(small)g(enough.)118 1138 y(4.2.)54 b FC(The)34 b(En)m(trop)m(y)h(of)g Fn(f)1104 1105 y Fp(2)1143 1138 y FC(.)46 b Fr(W)-8 b(e)31 b(will)d(pro)m(v)m(e)j (that)1150 1302 y Fn(C)22 b(N)1320 1264 y Fj(d)p Fi(\000)p Fp(1)1476 1302 y Fm(6)j Fn(\026)1627 1257 y Fp(+)p Fj(;f)1627 1330 y(N)1746 1302 y Fr(\(log)17 b Fn(f)1969 1264 y Fp(2)2008 1302 y Fr(\))k Fk(\000)f Fr(log)c Fn(\026)2342 1263 y Fp(+)2342 1330 y Fj(N)2409 1302 y Fr(\()p Fn(f)2499 1264 y Fp(2)2538 1302 y Fr(\))f Fn(;)816 b Fr(\(4.21\))118 1457 y(for)30 b(a)h(suitable)e(constan)m(t)j Fn(C)7 b Fr(.)118 1565 y(Using)30 b(the)g(previous)f(strategy)-8 b(,)33 b(w)m(e)d(c)m(hec)m(k)i(that)f(for)f Fn(")c(>)f Fr(0)31 b(and)f(for)g Fn(N)40 b Fr(large)30 b(enough)621 1807 y Fn(\026)676 1769 y Fp(+)676 1836 y Fj(N)743 1807 y Fr(\()p Fn(f)833 1770 y Fp(2)872 1807 y Fr(\))c Fm(6)1095 1694 y Fj(`)1044 1721 y Fl(X)1052 1916 y Fj(i)p Fp(=1)1190 1807 y Fr(exp)1344 1706 y Fl(\020)1398 1807 y Fn(N)1481 1770 y Fj(d)p Fi(\000)p Fp(1)1612 1734 y Fl(\000)1654 1807 y Fr(2)p Fn(\025)p Fk(j)p Fn(v)1821 1821 y Fj(i)1850 1807 y Fk(j)20 b(\000)g(W)2076 1822 y Fj(\014)2123 1807 y Fr(\()p Fn(v)2202 1821 y Fj(i)2231 1807 y Fr(\))g(+)g Fn(")p Fr(\(1)h(+)f Fn(\025)p Fr(\))2699 1734 y Fl(\001)2741 1706 y(\021)2816 1807 y Fr(+)g(1)15 b Fn(:)118 2049 y Fr(Inequalit)m(y)31 b(\(4.19\))h(implies)681 2296 y Fn(\026)736 2258 y Fp(+)736 2325 y Fj(N)803 2296 y Fr(\()p Fn(f)893 2259 y Fp(2)932 2296 y Fr(\))108 b Fm(6)1306 2183 y Fj(`)1254 2210 y Fl(X)1263 2405 y Fj(i)p Fp(=1)1401 2296 y Fr(exp)1555 2196 y Fl(\020)1609 2296 y Fn(N)1692 2259 y Fj(d)p Fi(\000)p Fp(1)1823 2223 y Fl(\000)1864 2296 y Fr(\(2)p Fn(\025)21 b Fk(\000)f Fn(\034)2159 2259 y Fi(\003)2199 2296 y Fr(\))p Fk(j)p Fn(v)2303 2310 y Fj(i)2332 2296 y Fk(j)g Fr(+)g Fn(")p Fr(\(1)h(+)f Fn(\025)p Fr(\))2790 2223 y Fl(\001)2832 2196 y(\021)2907 2296 y Fr(+)g(1)15 b Fn(;)1075 2554 y Fm(6)108 b Fn(`)15 b Fr(exp)1461 2453 y Fl(\020)1516 2554 y Fn(N)1599 2517 y Fj(d)p Fi(\000)p Fp(1)1729 2481 y Fl(\000)1771 2554 y Fr(2)p Fn(\025)21 b Fk(\000)f Fn(\034)2031 2517 y Fi(\003)2090 2554 y Fr(+)g Fn(")p Fr(\(1)i(+)d Fn(\025)p Fr(\))2503 2481 y Fl(\001)2546 2453 y(\021)2620 2554 y Fr(+)h(1)15 b Fn(:)633 b Fr(\(4.22\))118 2732 y(It)31 b(remains)e(to)i(c)m(hec)m(k)h(that)f(for)f Fn(")c(>)f Fr(0)30 b(and)g Fn(N)40 b Fr(large)31 b(enough)440 2946 y Fn(\026)495 2902 y Fp(+)p Fj(;f)495 2975 y(N)615 2873 y Fl(\000)672 2946 y Fr(log)16 b Fn(f)859 2909 y Fp(2)898 2873 y Fl(\001)965 2946 y Fr(=)1071 2885 y(2)p Fn(\025K)1253 2852 y Fj(d)p 1071 2925 223 4 v 1141 3009 a Fn(N)1304 2946 y(\026)1359 2902 y Fp(+)p Fj(;f)1359 2975 y(N)1478 2845 y Fl(\020)1607 2860 y(X)1532 3077 y Fj(x)p Fi(2)1633 3059 y Fc(b)1619 3077 y Fg(W)1680 3088 y Fy(N)r(;K)1829 2946 y Fn(g)s Fr(\()p Fk(M)2019 2960 y Fj(N)s(;K)2167 2946 y Fr(\()p Fn(x)p Fr(\)\))2324 2845 y Fl(\021)2430 2946 y Fm(>)50 b Fr(\(1)21 b Fk(\000)f Fn(")p Fr(\)2)p Fn(\025N)3001 2909 y Fj(d)p Fi(\000)p Fp(1)3148 2946 y Fn(:)256 b Fr(\(4.23\))118 3225 y(This)29 b(is)g(a)i(consequence)g (of)g(the)f(follo)m(wing)f(estimate.)41 b(F)-8 b(or)32 b(an)m(y)e Fn(")c(>)f Fr(0)31 b(and)e Fn(N)41 b Fr(large)30 b(enough)969 3381 y(\(1)21 b Fk(\000)f Fn(o)p Fr(\()p Fn(")p Fr(\)\))p Fn(\026)1407 3342 y Fp(+)1407 3409 y Fj(N)1476 3381 y Fr(\()p Fn(f)1566 3343 y Fp(2)1605 3381 y Fr(\))26 b Fm(6)f Fn(\026)1817 3342 y Fp(+)1817 3409 y Fj(N)1899 3307 y Fl(\000)1940 3381 y Fn(f)1995 3343 y Fp(2)2034 3381 y Fr(1)2079 3399 y Fi(f)p Fj(u)2155 3410 y Fy(N)2213 3399 y Fi(2V)6 b Fp(\()p Fi(\000)p Fp(1)-25 b(I)q Fj(;")p Fp(\))p Fi(g)2547 3307 y Fl(\001)2619 3381 y Fn(:)785 b Fr(\(4.24\))118 3551 y(Let)31 b Fk(F)k Fr(=)477 3477 y Fl(\000)518 3551 y Fk(V)7 b Fr(\()p Fk(\000)p Fr(1)-25 b(I)q Fn(;)15 b(")p Fr(\))858 3477 y Fl(\001)901 3500 y Fj(c)935 3551 y Fr(.)41 b(First)30 b(notice)g(that)477 3707 y(sup)475 3785 y Fj(v)r Fi(2F)631 3707 y Fk(f\000W)7 b Fr(\()p Fn(v)s Fr(\))21 b(+)f(2)p Fn(\025)p Fk(j)p Fn(v)s Fk(jg)42 b Fm(6)h Fr(sup)1467 3785 y Fj(v)r Fi(2F)1623 3707 y Fk(fj)p Fn(v)s Fk(j)p Fr(\()p Fk(\000)p Fn(\034)1921 3669 y Fi(\003)1982 3707 y Fr(+)20 b(2)p Fn(\025)p Fr(\))p Fk(g)41 b Fm(6)25 b Fr(\()p Fk(\000)p Fn(\034)2544 3669 y Fi(\003)2604 3707 y Fr(+)20 b(2)p Fn(\025)p Fr(\)\(1)i Fk(\000)e Fn(")p Fr(\))15 b Fn(:)118 3922 y Fr(W)-8 b(e)26 b(pro)s(ceed)e(as)g(b)s(efore)g(and)g(co)m(v)m(er)i(the)e(set)h Fk(F)18 b(\\)8 b(V)f Fr(\()p Fk(C)1955 3936 y Fj(a)1996 3922 y Fn(;)15 b(\016)s Fr(\))26 b(with)d(a)i(\014nite)e(n)m(um)m(b)s (er)g(of)i(neigh)m(b)s(orho)s(o)s(ds.)118 4030 y(This)k(implies)e(that) k(for)f(an)m(y)h Fn(\016)e(>)c Fr(0)31 b(and)f Fn(N)40 b Fr(large)30 b(enough)580 4187 y Fn(\026)635 4149 y Fp(+)635 4216 y Fj(N)717 4114 y Fl(\000)758 4187 y Fn(f)813 4150 y Fp(2)852 4187 y Fr(1)897 4206 y Fi(f)p Fj(u)973 4217 y Fy(N)1031 4206 y Fi(2F)7 b(g)1175 4114 y Fl(\001)1257 4187 y Fm(6)25 b Fn(`)15 b Fr(exp)1560 4114 y Fl(\000)1602 4187 y Fr(\()p Fk(\000)p Fn(\034)1758 4150 y Fi(\003)1817 4187 y Fr(+)20 b(2)p Fn(\025)h Fr(+)f Fn(o)p Fr(\()p Fn(\016)s Fr(\)\)\(1)j Fk(\000)d Fn(")p Fr(\))p Fn(N)2664 4150 y Fj(d)p Fi(\000)p Fp(1)2795 4114 y Fl(\001)2857 4187 y Fr(+)g(1)15 b Fn(:)118 4342 y Fr(On)30 b(the)g(other)h(hand,)836 4520 y Fn(\026)891 4482 y Fp(+)891 4549 y Fj(N)958 4419 y Fl(\020)1013 4520 y Fn(f)1068 4483 y Fp(2)1107 4520 y Fr(1)1152 4539 y Fi(f)p Fj(u)1228 4550 y Fy(N)1286 4539 y Fi(2V)6 b Fp(\()p Fi(\000)p Fp(1)-25 b(I)p Fj(;")p Fp(\))p Fi(g)1619 4419 y Fl(\021)1724 4520 y Fm(>)50 b Fr(exp)1999 4447 y Fl(\000)2041 4520 y Fr(\()p Fk(\000)p Fn(\034)2197 4483 y Fi(\003)2257 4520 y Fr(+)20 b(2)p Fn(\025)p Fr(\))p Fn(N)2564 4483 y Fj(d)p Fi(\000)p Fp(1)2695 4447 y Fl(\001)2752 4520 y Fn(:)118 4703 y Fr(Th)m(us,)30 b(for)g Fn(N)40 b Fr(large)31 b(enough,)f(w)m(e)h(deriv)m(e)f(\(4.24\)) r(.)41 b(This)28 b(implies)g(that)377 4886 y Fn(K)461 4849 y Fj(d)501 4886 y Fn(\026)556 4848 y Fp(+)556 4915 y Fj(N)623 4785 y Fl(\020)677 4886 y Fn(f)732 4849 y Fp(2)786 4886 y Fr(\()896 4800 y Fl(X)821 5017 y Fj(x)p Fi(2)922 4999 y Fc(b)908 5017 y Fg(W)969 5028 y Fy(N)r(;K)1118 4886 y Fn(g)s Fr(\()p Fk(M)1308 4900 y Fj(N)s(;K)1456 4886 y Fr(\()p Fn(x)p Fr(\)\)\))1648 4785 y Fl(\021)1754 4886 y Fm(>)50 b Fn(\026)1930 4848 y Fp(+)1930 4915 y Fj(N)2013 4813 y Fl(\000)2054 4886 y Fn(f)2109 4849 y Fp(2)2163 4886 y Fr(\(1)2243 4905 y Fi(f)p Fj(u)2319 4916 y Fy(N)2378 4905 y Fi(2V)6 b Fp(\()p Fi(\000)p Fp(1)-25 b(I)p Fj(;")p Fp(\))p Fi(g)2711 4886 y Fr(\))2746 4813 y Fl(\001)2803 4886 y Fr(\(1)21 b Fk(\000)f Fn(")p Fr(\))p Fn(N)3155 4849 y Fj(d)3211 4886 y Fn(:)118 5170 y Fr(The)30 b(inequalit)m(y)g(\(4.24\))j(leads)c(to)484 5353 y Fn(K)568 5316 y Fj(d)608 5353 y Fn(\026)663 5315 y Fp(+)663 5382 y Fj(N)730 5252 y Fl(\020)784 5353 y Fn(f)839 5316 y Fp(2)893 5353 y Fr(\()1003 5267 y Fl(X)928 5484 y Fj(x)p Fi(2)1029 5466 y Fc(b)1015 5484 y Fg(W)1076 5495 y Fy(N)r(;K)1225 5353 y Fn(g)s Fr(\()p Fk(M)1415 5367 y Fj(N)s(;K)1563 5353 y Fr(\()p Fn(x)p Fr(\)\)\))1755 5252 y Fl(\021)1861 5353 y Fm(>)50 b Fn(\026)2037 5315 y Fp(+)2037 5382 y Fj(N)2120 5280 y Fl(\000)2161 5353 y Fn(f)2216 5316 y Fp(2)2255 5280 y Fl(\001)2312 5353 y Fr(\(1)21 b Fk(\000)f Fn(")p Fr(\)\(1)h Fk(\000)f Fn(o)p Fr(\()p Fn(")p Fr(\)\))p Fn(N)3047 5316 y Fj(d)3104 5353 y Fn(:)p eop %%Page: 11 11 11 10 bop 569 45 a Fp(SOME)30 b(NEW)h(RESUL)-6 b(TS)31 b(ON)g(THE)f(KINETIC)h(ISING)h(MODEL)e(IN)h(A)f(PURE)g(PHASE)381 b(11)118 366 y Fr(Therefore)30 b(\(4.23\))i(is)d(complete.)41 b(Com)m(bining)28 b(\(4.22\))k(and)e(\(4.23\))r(,)g(w)m(e)g(obtain)f (for)h Fn(N)40 b Fr(large)30 b(enough)182 526 y Fn(\026)237 481 y Fp(+)p Fj(;f)237 554 y(N)357 526 y Fr(\(log)17 b Fn(f)580 488 y Fp(2)619 526 y Fr(\))j Fk(\000)g Fr(log)d Fn(\026)953 487 y Fp(+)953 554 y Fj(N)1020 526 y Fr(\()p Fn(f)1110 488 y Fp(2)1149 526 y Fr(\))108 b Fm(>)1471 452 y Fl(\000)1513 526 y Fr(\(1)21 b Fk(\000)f Fn(o)p Fr(\()p Fn(")p Fr(\)\)2)p Fn(\025)i Fk(\000)e Fr(\()p Fk(\000)p Fn(\034)2263 488 y Fi(\003)2323 526 y Fr(+)g(2)p Fn(\025)g Fr(+)g Fn(")2665 488 y Fi(0)2689 526 y Fr(\(1)h(+)f Fn(\025)p Fr(\)\))3004 452 y Fl(\001)3046 526 y Fn(N)3129 488 y Fj(d)p Fi(\000)p Fp(1)3280 526 y Fk(\000)g Fr(log)d Fn(`)e(;)1292 676 y Fm(>)1471 603 y Fl(\000)1513 676 y Fn(\034)1563 639 y Fi(\003)1623 676 y Fk(\000)20 b Fn(o)p Fr(\()p Fn(")p Fr(\)2)p Fn(\025)i Fk(\000)d Fn(")2122 639 y Fi(0)2146 676 y Fr(\(1)i(+)f Fn(\025)p Fr(\))2426 603 y Fl(\001)2483 676 y Fn(N)2566 639 y Fj(d)p Fi(\000)p Fp(1)2717 676 y Fk(\000)g Fr(log)d Fn(`)e(:)118 822 y Fr(F)-8 b(or)31 b(an)m(y)f Fn(\014)h(>)25 b(\014)681 836 y Fj(c)746 822 y Fr(\(the)31 b(true)e(critical)g(p)s(oin)m(t\))h (it)f(is)g(kno)m(wn)h(that)h Fn(\034)2421 789 y Fi(\003)2485 822 y Fn(>)25 b Fr(0.)41 b(Th)m(us,)30 b(b)m(y)g(c)m(ho)s(osing)g Fn(")g Fr(and)118 930 y Fn(")160 897 y Fi(0)214 930 y Fr(small)f(enough,)h(w)m(e)h(deriv)m(e)f(\(4.21\))j(for)d Fn(N)40 b Fr(large)31 b(enough.)118 1092 y(4.3.)54 b FC(The)37 b(Diric)m(hlet)h(F)-9 b(orm)37 b(of)g Fn(f)10 b FC(.)45 b Fr(The)32 b(Diric)m(hlet)g(form)g(asso)s(ciated)h(to)g Fn(f)42 b Fr(can)33 b(b)s(e)f(b)s(ounded)e(as)118 1200 y(follo)m(ws.)40 b(There)30 b(is)f Fn(C)871 1214 y Fp(1)936 1200 y Fn(>)c Fr(0)31 b(suc)m(h)f(that)h(for)f Fn(N)40 b Fr(large)31 b(enough)435 1371 y Fn(\026)490 1333 y Fp(+)490 1400 y Fj(N)557 1298 y Fl(\000)599 1371 y Fk(jr)p Fn(f)10 b Fk(j)780 1334 y Fp(2)819 1298 y Fl(\001)886 1371 y Fm(6)25 b Fn(C)1047 1385 y Fp(1)1086 1371 y Fn(\025)1139 1334 y Fp(2)1179 1371 y Fn(N)1262 1334 y Fj(d)p Fi(\000)p Fp(3)1392 1371 y Fn(K)1476 1334 y Fj(d)1542 1371 y Fn(\026)1597 1333 y Fp(+)1597 1400 y Fj(N)1664 1298 y Fl(\000)1705 1371 y Fn(f)1760 1334 y Fp(2)1799 1298 y Fl(\001)1866 1371 y Fr(=)g Fn(C)2027 1385 y Fp(1)2066 1371 y Fn(\025)2119 1334 y Fp(2)2159 1371 y Fn(N)2242 1334 y Fj(d)p Fi(\000)p Fp(3)2372 1371 y Fr(\()p Fn(b)15 b Fr(log)i Fn(N)10 b Fr(\))2724 1299 y Fy(d)p 2723 1311 36 3 v 2723 1352 a(\015)2773 1371 y Fn(\026)2828 1333 y Fp(+)2828 1400 y Fj(N)2895 1298 y Fl(\000)2936 1371 y Fn(f)2991 1334 y Fp(2)3030 1298 y Fl(\001)3087 1371 y Fn(:)317 b Fr(\(4.25\))118 1517 y(By)31 b(T)-8 b(a)m(ylor)30 b(expansion)320 1676 y Fk(jr)p Fn(f)10 b Fk(j)501 1639 y Fp(2)565 1676 y Fr(=)693 1590 y Fl(X)661 1787 y Fj(i)p Fi(2)p Fg(W)793 1798 y Fy(N)872 1676 y Fk(jr)973 1690 y Fj(i)1001 1676 y Fn(f)g Fk(j)1081 1639 y Fp(2)1228 1676 y Fr(=)1482 1590 y Fl(X)1407 1806 y Fj(x)p Fi(2)1508 1789 y Fc(b)1494 1806 y Fg(W)1554 1817 y Fy(N)s(;K)1772 1590 y Fl(X)1703 1791 y Fj(i)p Fi(2)p Fg(B)1813 1802 y Fy(K)1878 1791 y Fp(\()p Fj(x)p Fp(\))1987 1676 y Fk(jr)2088 1690 y Fj(i)2116 1676 y Fn(f)g Fk(j)2196 1639 y Fp(2)1228 2009 y Fm(6)108 b Fn(f)1462 1971 y Fp(2)1516 1908 y Fl(\020)1586 2009 y Fr(2)p Fn(K)1715 1971 y Fj(d)1780 1947 y Fn(\025)1833 1915 y Fp(2)p 1765 1988 123 4 v 1765 2071 a Fn(N)1848 2045 y Fp(2)1897 2009 y Fk(k)p Fn(g)1988 1971 y Fi(0)2013 2009 y Fk(k)2058 1971 y Fp(2)2058 2031 y Fi(1)2148 1908 y Fl(\021)2293 1923 y(X)2218 2139 y Fj(x)p Fi(2)2318 2122 y Fc(b)2305 2139 y Fg(W)2365 2150 y Fy(N)r(;K)2514 2009 y Fr(1)2559 2042 y Fi(f\000)2659 2015 y Fy(m)2713 2001 y Fh(\003)p 2659 2027 91 3 v 2689 2068 a Fv(2)2779 2042 y Fd(6)20 b Fi(M)2938 2053 y Fy(N)r(;K)3066 2042 y Fp(\()p Fj(x)p Fp(\))g Fd(6)g Fi(\000)3320 2015 y Fy(m)3374 2001 y Fh(\003)p 3319 2027 V 3349 2068 a Fv(4)3420 2042 y Fi(g)1228 2342 y Fm(6)108 b Fn(f)1462 2304 y Fp(2)1516 2241 y Fl(\020)1586 2342 y Fr(2)p Fn(K)1715 2304 y Fj(d)1780 2280 y Fn(\025)1833 2247 y Fp(2)p 1765 2321 123 4 v 1765 2404 a Fn(N)1848 2378 y Fp(2)1897 2342 y Fk(k)p Fn(g)1988 2304 y Fi(0)2013 2342 y Fk(k)2058 2304 y Fp(2)2058 2364 y Fi(1)2148 2241 y Fl(\021)2202 2342 y Fk(Q)2276 2356 y Fj(N)2359 2342 y Fn(:)118 2533 y Fr(where)20 b Fk(Q)445 2547 y Fj(N)533 2533 y Fr(denotes)g(the)h(n)m(um)m(b)s(er)e(of)h(blo)s(c)m(ks)g(in)f Fo(W)1859 2547 y Fj(N)1952 2533 y Fr(with)g(a)m(v)m(eraged)j (magnetization)f(in)e([)p Fk(\000)3298 2497 y Fj(m)3360 2473 y Fh(\003)p 3298 2512 99 4 v 3330 2564 a Fp(2)3407 2533 y Fn(;)c Fk(\000)3528 2497 y Fj(m)3590 2473 y Fh(\003)p 3528 2512 V 3559 2564 a Fp(4)3636 2533 y Fr(].)118 2663 y(Using)30 b(the)g(notation)h Fn(\026)949 2618 y Fp(+)p Fj(;f)949 2691 y(N)1099 2663 y Fr(in)m(tro)s(duced)e(in)g FC(ii)p Fr(\))h(ab)s(o)m(v)m(e)i(w)m(e)f(can)f(write)964 2878 y Fn(\026)1019 2840 y Fp(+)1019 2907 y Fj(N)1086 2804 y Fl(\000)1128 2878 y Fk(jr)p Fn(f)10 b Fk(j)1309 2840 y Fp(2)1348 2804 y Fl(\001)1415 2878 y Fm(6)25 b Fn(c)1550 2892 y Fp(1)1599 2816 y Fn(\025)1652 2783 y Fp(2)1692 2816 y Fn(K)1776 2783 y Fj(d)p 1599 2857 217 4 v 1646 2940 a Fn(N)1729 2914 y Fp(2)1826 2878 y Fk(k)p Fn(g)1917 2840 y Fi(0)1941 2878 y Fk(k)1986 2840 y Fp(2)1986 2900 y Fi(1)2077 2878 y Fn(\026)2132 2840 y Fp(+)2132 2907 y Fj(N)2199 2878 y Fr(\()p Fn(f)2289 2840 y Fp(2)2328 2878 y Fr(\))g Fn(\026)2443 2834 y Fp(+)p Fj(;f)2443 2907 y(N)2563 2878 y Fr(\()p Fk(Q)2672 2892 y Fj(N)2740 2878 y Fr(\))15 b Fn(:)118 3055 y Fr(The)30 b(estimate)h(\(4.25\))i (will)27 b(follo)m(w)j(from)g(the)h(fact)g(that)g(for)f Fn(N)40 b Fr(large)31 b(enough)1480 3214 y Fn(\026)1535 3169 y Fp(+)p Fj(;f)1535 3242 y(N)1654 3140 y Fl(\000)1696 3214 y Fk(Q)1770 3228 y Fj(N)1837 3140 y Fl(\001)1904 3214 y Fm(6)25 b Fr(2)p Fn(N)2128 3176 y Fj(d)p Fi(\000)p Fp(1)2275 3214 y Fn(:)1129 b Fr(\(4.26\))118 3359 y(This)29 b(b)s(oils)f(do)m(wn)i(to)h(c)m(hec)m(k)h(that)996 3510 y Fn(\026)1051 3466 y Fp(+)p Fj(;f)1051 3539 y(N)1186 3409 y Fl(\020)1240 3510 y Fk(Q)1314 3524 y Fj(N)1397 3510 y Fr(1)1442 3532 y Fi(fQ)1535 3543 y Fy(N)1593 3532 y Fj(>N)1711 3513 y Fy(d)p Fh(\000)p Fv(1)1826 3532 y Fi(g)1865 3409 y Fl(\021)1960 3510 y Fm(6)25 b Fn(N)2139 3473 y Fj(d)2194 3510 y Fr(exp\()p Fk(\000)p Fn(cN)2561 3473 y Fj(d)p Fi(\000)p Fp(1)2692 3510 y Fr(\))15 b Fn(;)662 b Fr(\(4.27\))118 3694 y(where)30 b Fn(c)h Fr(is)e(a)i(p)s(ositiv)m(e)e (constan)m(t.)42 b(As)31 b Fn(\026)1546 3655 y Fp(+)1546 3722 y Fj(N)1613 3694 y Fr(\()p Fn(f)1703 3661 y Fp(2)1742 3694 y Fr(\))50 b Fm(>)h Fr(1,)31 b(w)m(e)g(see)g(that)631 3909 y Fn(\026)686 3865 y Fp(+)p Fj(;f)686 3938 y(N)821 3808 y Fl(\020)875 3909 y Fk(Q)949 3923 y Fj(N)1032 3909 y Fr(1)1077 3931 y Fi(fQ)1170 3942 y Fy(N)1228 3931 y Fj(>N)1346 3912 y Fy(d)p Fh(\000)p Fv(1)1460 3931 y Fi(g)1500 3808 y Fl(\021)1594 3909 y Fm(6)40 b Fr(exp\(2)p Fn(\025N)2060 3871 y Fj(d)p Fi(\000)p Fp(1)2192 3909 y Fr(\))p Fn(\026)2282 3871 y Fp(+)2282 3938 y Fj(N)2364 3808 y Fl(\020)2418 3909 y Fk(Q)2492 3923 y Fj(N)2585 3909 y Fn(>)25 b(N)2764 3871 y Fj(d)p Fi(\000)p Fp(1)2895 3808 y Fl(\021)2975 3847 y Fn(N)3058 3814 y Fj(d)p 2974 3888 125 4 v 2974 3971 a Fn(K)3058 3945 y Fj(d)3123 3909 y Fn(:)281 b Fr(\(4.28\))118 4081 y(Remem)m(b)s(er)31 b(that)i(the)f(o)s(ccurrence)f(of)h(bad)f(blo) s(c)m(ks)g(is)g(dominated)g(b)m(y)h(Bernoulli)d(p)s(ercolation)i(with) 118 4188 y(parameter)38 b Fn(\032)607 4202 y Fj(K)712 4188 y Fr(=)e Fn(N)902 4155 y Fi(\000)p Fj(c)988 4167 y Fy(\014)1029 4155 y Fj(b)1063 4188 y Fr(.)61 b(Therefore,)39 b(for)e Fn(b)g Fr(large)g(enough,)i(it)d(is)g(then)h(quite)g(simple)e (to)j(c)m(hec)m(k)118 4296 y(that)31 b(there)g(is)e Fn(c)d(>)f Fr(0)30 b(suc)m(h)g(that)h(for)g Fn(N)40 b Fr(large)30 b(enough,)1132 4465 y Fn(\026)1187 4426 y Fp(+)1187 4493 y Fj(N)1269 4364 y Fl(\020)1324 4465 y Fk(Q)1398 4479 y Fj(N)1490 4465 y Fn(>)25 b(N)1669 4427 y Fj(d)p Fi(\000)p Fp(1)1800 4364 y Fl(\021)1895 4465 y Fm(6)40 b Fr(exp)2160 4391 y Fl(\000)2222 4465 y Fk(\000)19 b Fn(cN)2434 4427 y Fj(d)p Fi(\000)p Fp(1)2565 4391 y Fl(\001)2622 4465 y Fn(:)118 4638 y Fr(Com)m(bining)28 b(the)j(previous)e(b)s(ound)f (with)h(\(4.28\))r(,)i(w)m(e)g(deriv)m(e)f(\(4.27\))r(.)118 4800 y FC(Remark)k(4.1.)47 b Ff(It)32 b(would)i(b)-5 b(e)33 b(p)-5 b(ossible)34 b(to)f(derive)f(sharp)-5 b(er)35 b(estimates)f(for)43 b Fr(\(4.26\))r Ff(.)f(One)32 b(exp)-5 b(e)g(cts)1472 5009 y Fn(\026)1527 4965 y Fp(+)p Fj(;f)1527 5038 y(N)1647 4936 y Fl(\000)1689 5009 y Fk(Q)1763 5023 y Fj(N)1830 4936 y Fl(\001)1897 5009 y Fm(6)25 b Fn(c)2042 4948 y(N)2125 4915 y Fj(d)p Fi(\000)p Fp(1)p 2042 4988 215 4 v 2042 5072 a Fn(K)2126 5045 y Fj(d)p Fi(\000)p Fp(1)2282 5009 y Fn(:)118 5181 y Ff(Nevertheless)41 b(this)g(would)g (not)h(b)-5 b(e)40 b(enough)h(to)g(derive)f(an)h(asymptotic)i(for)e (the)f(sp)-5 b(e)g(ctr)g(al)43 b(gap)e(and)118 5289 y(the)29 b(L)-5 b(o)g(g-Sob)g(olev)30 b(c)-5 b(onstant)30 b(without)g(a)f(lo)-5 b(garithmic)31 b(c)-5 b(orr)g(e)g(ction)31 b(:)39 b(on)29 b(\014nite)g(mesosc)-5 b(opic)30 b(sc)-5 b(ales,)30 b(we)118 5397 y(c)-5 b(annot)34 b(c)-5 b(ontr)g(ol)35 b(the)e(test)g(function.)p eop %%Page: 12 12 12 11 bop 118 45 a Fp(12)1071 b(T.)30 b(BODINEA)n(U)h(AND)f(F.)g(MAR)-6 b(TINELLI)1092 366 y Fr(5.)85 b Fq(Pr)n(oof)34 b(of)g(Theorems)f(2.1)i (and)e(2.2)118 528 y Fr(W)-8 b(e)32 b(are)e(in)g(p)s(osition)e(to)j (pro)m(v)m(e)g(the)g(\014rst)e(t)m(w)m(o)j(main)d(results.)118 690 y(5.1.)54 b FC(Pro)s(of)36 b(of)f(Theorem)f(2.1.)46 b Fr(By)30 b(de\014nition)1424 896 y Fn(S)1480 910 y Fj(N)1572 896 y Fk(\025)1678 832 y Fn(\026)1733 794 y Fp(+)1733 861 y Fj(N)1800 832 y Fr(\()p Fn(\036;)15 b(\036)p Fr(\))p 1678 876 342 4 v 1711 959 a Fk(E)8 b Fr(\()p Fn(\036;)15 b(\036)p Fr(\))2029 896 y Fn(;)197 b Fk(8)p Fn(\036)118 1091 y Fr(When)30 b Fn(\036)h Fr(is)e(equal)h(to)h(our)e (test)j(function)d Fn(f)39 b Fr(the)30 b(ab)s(o)m(v)m(e)i(ratio)e(can)h (b)s(e)e(b)s(ounded)g(from)g(b)s(elo)m(w)h(using)118 1199 y(\(4.25\))j(and)d(\(4.13\))i(b)m(y)1755 1347 y Fn(N)1838 1314 y Fp(3)p Fi(\000)p Fj(d)p 1609 1387 507 4 v 1609 1474 a Fn(C)1674 1488 y Fp(2)1713 1474 y Fr(\(log)17 b Fn(N)10 b Fr(\))1999 1448 y Fj(d=\015)2140 1408 y Fn(;)1264 b Fr(\(5.29\))118 1626 y(where)30 b Fn(C)446 1640 y Fp(2)511 1626 y Fr(=)25 b Fn(C)672 1640 y Fp(1)711 1626 y Fn(\024)763 1593 y Fj(d=\015)879 1626 y Fn(\025)932 1593 y Fp(2)972 1626 y Fr(.)p 3595 1626 4 62 v 3599 1568 55 4 v 3599 1626 V 3653 1626 4 62 v 118 1842 a(Clearly)k(in)g(dimension)f Fn(d)51 b Fm(>)f Fr(3)31 b(the)f(test)i(function)d(do)s(es)h(not)g(pro) m(vide)g(an)m(y)g(information.)118 2004 y(5.2.)54 b FC(Pro)s(of)36 b(of)f(Theorem)f(2.2.)46 b Fr(Fix)30 b Fn(d)50 b Fm(>)h Fr(2.)41 b(By)30 b(de\014nition)1295 2210 y Fn(L)1357 2224 y Fj(N)1474 2210 y Fm(>)1605 2146 y Fn(\026)1660 2108 y Fp(+)1660 2174 y Fj(N)1727 2146 y Fr(\()p Fn(\036)1816 2113 y Fp(2)1871 2146 y Fr(log)17 b Fn(\036)2058 2113 y Fp(2)2098 2146 y Fr(\))p 1605 2189 528 4 v 1702 2278 a Fk(E)1758 2240 y Fp(+)1750 2307 y Fj(N)1817 2278 y Fr(\()p Fn(\036;)e(\036)p Fr(\))2143 2210 y Fn(;)197 b Fk(8)15 b Fn(\036)118 2416 y Fr(When)30 b Fn(\036)h Fr(is)e(equal)h(to)h(our)e(test)j(function)d Fn(f)39 b Fr(the)30 b(ab)s(o)m(v)m(e)i(ratio)e(can)h(b)s(e)e(b)s(ounded)g(from) g(b)s(elo)m(w)h(using)118 2524 y(\(4.25\))j(and)d(\(4.21\))i(b)m(y)1821 2669 y Fn(N)1904 2636 y Fp(2)p 1629 2709 507 4 v 1629 2796 a Fn(C)1694 2810 y Fp(3)1733 2796 y Fr(\(log)17 b Fn(N)10 b Fr(\))2019 2770 y Fj(d=\015)3429 2730 y Fr(\(5.30\))118 2946 y(where)30 b Fn(C)446 2960 y Fp(3)511 2946 y Fr(=)742 2910 y Fj(C)p 617 2925 305 4 v 617 2986 a(C)667 2995 y Fv(1)701 2986 y Fj(\024)742 2966 y Fy(d=\015)846 2986 y Fj(\025)887 2967 y Fv(2)932 2946 y Fr(.)p 3595 2946 4 62 v 3599 2888 55 4 v 3599 2946 V 3653 2946 4 62 v 542 3302 a(6.)50 b Fq(Slo)n(w)34 b(do)n(wn)f(of)h(the)g(Gla)n(uber)f(d) n(ynamics)g(in)h(tw)n(o)g(dimensions)118 3464 y Fr(In)f(this)g(section) h(w)m(e)g(will)d(deriv)m(e)j(some)g(consequences)g(from)g(the)g(t)m(w)m (o)h(dimensional)c(upp)s(er)g(b)s(ound)118 3572 y(on)d(the)g(in)m(v)m (erse)g(sp)s(ectral)g(gap)g(for)g Fn(\014)i(>)25 b(\014)1569 3586 y Fj(c)1632 3572 y Fr(on)j(the)h(sp)s(eed)e(of)h(relaxation)g(of)g (the)g(Glaub)s(er)f(dynamics)118 3680 y(to)g(its)e(equilibrium.)34 b(In)25 b(particular)f(w)m(e)i(will)d(pro)m(v)m(e)j(theorem)g(2.3.)41 b(The)25 b(notation)h(will)d(b)s(e)i(that)h(\014xed)118 3787 y(in)j(section)i(2)f(and)g(section)h(3.)118 3949 y(6.1.)54 b FC(A)40 b(\014rst)g(\014nite)g(v)m(olume)g(b)s(ound.)46 b Fr(The)34 b(\014rst)g(simple)f(consequence)j(of)f(Theorem)g(2.1,)i (is)d(a)118 4057 y(b)s(ound)28 b(on)j(the)f(dynamical)f(ev)m(olution)h (of)h(the)f(test)h(function)e(\(4.12\))k(itself.)118 4215 y FC(Prop)s(osition)j(6.1.)46 b Ff(F)-7 b(or)34 b(any)f Fn(N)43 b Ff(lar)-5 b(ge)33 b(enough,)221 4410 y Fk(8)p Fn(t)25 b(>)g Fr(0)p Fn(;)201 b(\026)752 4372 y Fp(+)752 4439 y Fj(N)835 4309 y Fl(\020)889 4337 y(\000)931 4410 y Fn(P)1002 4362 y Fp(\(+)p Fj(;N)7 b Fp(\))989 4434 y Fj(t)1198 4410 y Fn(f)1253 4337 y Fl(\001)1294 4359 y Fp(2)1334 4309 y Fl(\021)1438 4410 y Fm(>)51 b Fn(\026)1615 4372 y Fp(+)1615 4439 y Fj(N)1682 4410 y Fr(\()p Fn(f)1772 4373 y Fp(2)1811 4410 y Fr(\))15 b(exp)2015 4282 y Fl(\022)2082 4410 y Fk(\000)p Fr(2)p Fn(t)2241 4349 y Fr(\(log)i Fn(N)10 b Fr(\))2527 4316 y Fj(\024)p 2241 4389 332 4 v 2365 4473 a Fn(N)2582 4282 y Fl(\023)2664 4337 y(\000)2706 4410 y Fr(1)20 b Fk(\000)g Fr(exp\()p Fk(\000)p Fn(c)3146 4425 y Fj(\025)3192 4410 y Fn(N)10 b Fr(\))3310 4337 y Fl(\001)3367 4410 y Fn(;)37 b Fr(\(6.31\))118 4606 y Ff(wher)-5 b(e)34 b Fn(c)414 4621 y Fj(\025)492 4606 y Ff(is)f(a)g(p)-5 b(ositive)33 b(c)-5 b(onstant)35 b(dep)-5 b(ending)33 b(on)g Fn(\025)p Ff(.)218 4763 y Fr(This)j(result)h(pro)m(vides)g(a)h(\014rst)g(\(admittedly)f(w)m (eak\))i(clue)f(on)g(the)g(relaxation)f(time)h(of)g(the)h(dy-)118 4872 y(namics.)g(Let)30 b(us)e(assume)g(that)i(the)f(Mark)m(o)m(v)h (pro)s(cess)e(generated)i(b)m(y)f Fn(L)2640 4834 y Fp(+)2640 4901 y Fj(N)2736 4872 y Fr(is)f(attractiv)m(e)i(\(see)g([Li)o(])f(or) 118 4980 y([Ma)q(]\).)41 b(This)29 b(is)g(the)h(case)h(if)f(for)f (example)h(the)h(\015ip)d(rates)j(w)m(ere)f(those)h(of)f(the)h(Metrop)s (olis)e(or)h(of)g(the)118 5090 y(Heat)24 b(Bath)g(dynamics.)37 b(Let)23 b Fk(B)1202 5052 y Fi(\000)1199 5118 y Fj(N)1266 5090 y Fr(\()p Fn(\033)s Fr(\))g(b)s(e)g(the)g(n)m(um)m(b)s(er)e(blo)s (c)m(ks)h Fo(B)2326 5104 y Fj(K)2423 5090 y Fr(for)h(whic)m(h)e(the)i (spin)e(con\014guration)118 5209 y Fn(\033)34 b Fr(in)29 b Fk(f\006)p Fr(1)p Fk(g)516 5176 y Fg(W)577 5187 y Fy(N)675 5209 y Fr(has)h(a)m(v)m(eraged)j(magnetization)e(smaller)e(than)h Fk(\000)2412 5173 y Fj(m)2474 5150 y Fh(\003)p 2412 5188 99 4 v 2443 5240 a Fp(4)2520 5209 y Fr(.)41 b(W)-8 b(e)31 b(set)1210 5427 y(\011)1281 5441 y Fj(N)1348 5427 y Fr(\()p Fn(\033)s Fr(\))26 b(=)f(exp)1749 5299 y Fl(\022)1826 5366 y Fn(\025)15 b(K)1978 5333 y Fj(d)p 1826 5406 193 4 v 1881 5490 a Fn(N)2028 5427 y Fk(B)2091 5389 y Fi(\000)2088 5456 y Fj(N)2155 5427 y Fr(\()p Fn(\033)s Fr(\))2280 5299 y Fl(\023)2378 5427 y Fn(;)p eop %%Page: 13 13 13 12 bop 569 45 a Fp(SOME)30 b(NEW)h(RESUL)-6 b(TS)31 b(ON)g(THE)f(KINETIC)h(ISING)h(MODEL)e(IN)h(A)f(PURE)g(PHASE)381 b(13)118 366 y Fr(where)30 b Fn(K)i Fr(=)25 b(\()p Fn(b)15 b Fr(log)i Fn(N)10 b Fr(\))926 333 y Fp(1)p Fj(=\015)1072 366 y Fr(and)29 b Fn(b;)15 b(\015)36 b Fr(are)31 b(as)g(in)e(the)h (previous)f(section.)118 476 y(Since)i Fk(B)420 438 y Fi(\000)417 504 y Fj(N)515 476 y Fr(is)f(a)i(non)f(increasing)f (function)g(of)i(the)g(spin)d(v)-5 b(ariables,)31 b(the)g(monotonicit)m (y)h(inequalities)118 584 y(for)e(attractiv)m(e)j(pro)s(cesses)d(imply) 341 795 y Fn(P)412 747 y Fp(\(+)p Fj(;N)7 b Fp(\))399 819 y Fj(t)609 722 y Fl(\000)651 795 y Fr(\011)722 809 y Fj(N)788 722 y Fl(\001)830 795 y Fr(\()p Fk(\000)p Fr(\))51 b Fm(>)1143 694 y Fl(\020)1198 795 y Fn(\026)1253 757 y Fp(+)1253 824 y Fj(N)1335 694 y Fl(\020)1389 722 y(\000)1431 795 y Fn(P)1502 747 y Fp(\(+)p Fj(;N)7 b Fp(\))1489 819 y Fj(t)1698 795 y Fr(\011)1769 809 y Fj(N)1836 722 y Fl(\001)1878 744 y Fp(2)1917 694 y Fl(\021)q(\021)2026 717 y Fp(1)p Fj(=)p Fp(2)2186 795 y Fm(>)2308 694 y Fl(\020)2362 795 y Fn(\026)2417 757 y Fp(+)2417 824 y Fj(N)2499 694 y Fl(\020)2553 722 y(\000)2595 795 y Fn(P)2666 747 y Fp(\(+)p Fj(;N)g Fp(\))2653 819 y Fj(t)2863 795 y Fn(f)2918 722 y Fl(\001)2959 744 y Fp(2)2998 694 y Fl(\021\021)3107 717 y Fp(1)p Fj(=)p Fp(2)3247 795 y Fn(;)118 988 y Fr(where)33 b(the)g(sym)m(b)s(ol)e(\()p Fk(\000)p Fr(\))j(denotes)f(the)g (con\014guration)g(in)e Fo(W)2268 1002 y Fj(N)2374 988 y Fr(for)i(whic)m(h)f(all)f(the)j(spins)c(are)k(equal)118 1096 y(to)d Fk(\000)p Fr(1.)41 b(Inequalit)m(y)31 b(\(6.31\))h(implies) c(that)j(there)f(is)g Fn(")25 b(>)g Fr(0)31 b(suc)m(h)f(that)h(for)f (all)g Fn(N)40 b Fr(large)30 b(enough)673 1316 y Fn(P)744 1268 y Fp(\(+)p Fj(;N)7 b Fp(\))731 1340 y Fj(t)941 1243 y Fl(\000)983 1316 y Fr(\011)1054 1330 y Fj(N)1121 1243 y Fl(\001)1162 1316 y Fr(\()p Fk(\000)p Fr(\))51 b Fm(>)f Fr(exp)1630 1188 y Fl(\022)1696 1316 y Fr(\(2)p Fn(\025)21 b Fk(\000)f Fn(\034)1991 1279 y Fi(\003)2051 1316 y Fk(\000)g Fn(")p Fr(\))2229 1255 y Fn(N)p 2229 1295 83 4 v 2248 1379 a Fr(2)2343 1316 y Fk(\000)g Fn(t)2477 1255 y Fr(\(log)c Fn(N)10 b Fr(\))2762 1222 y Fj(\024)p 2477 1295 332 4 v 2601 1379 a Fn(N)2818 1188 y Fl(\023)2915 1316 y Fn(:)118 1537 y Fr(On)30 b(the)g(other)h(hand,)e(as)i(in)e(the)i(deriv)-5 b(ation)29 b(of)37 b(\(4.15\))s(,)30 b(one)h(can)f(c)m(hec)m(k)i(that) 1336 1745 y(lim)1297 1806 y Fj(N)7 b Fi(!1)1545 1684 y Fr(1)p 1526 1724 83 4 v 1526 1807 a Fn(N)1634 1745 y Fr(log)17 b Fn(\026)1822 1707 y Fp(+)1822 1774 y Fj(N)1889 1671 y Fl(\000)1930 1745 y Fr(\011)2001 1759 y Fj(N)2068 1671 y Fl(\001)2135 1745 y Fr(=)25 b(0)15 b Fn(:)118 1978 y Fr(Therefore)43 b(for)f(an)m(y)h(time)f(smaller)f(than)1755 1942 y Fj(N)1818 1919 y Fv(2)p 1658 1957 292 4 v 1658 2012 a Fp(\(log)13 b Fj(N)7 b Fp(\))1878 1993 y Fv(2)p Fy(\024)2002 1978 y Fr(the)43 b(quan)m(tit)m(y)g Fn(P)2618 1930 y Fp(\(+)p Fj(;N)7 b Fp(\))2605 2002 y Fj(t)2814 1904 y Fl(\000)2856 1978 y Fr(\011)2927 1992 y Fj(N)2994 1978 y Fr(\()p Fn(\033)s Fr(\))3119 1904 y Fl(\001)3162 1978 y Fr(\()p Fk(\000)p Fr(\))43 b(is)e(m)m(uc)m(h)118 2100 y(larger)28 b(than)g(the)g(equilibrium)23 b(exp)s(ectation)29 b(of)f(\011)1878 2114 y Fj(N)1973 2100 y Fr(:)40 b(in)27 b(the)h(ab)s(o)m(v)m(e)h(sp)s(ecial)e(sense)h(the)g(system)h(has)118 2208 y(not)i(y)m(et)g(relaxed.)118 2381 y FC(Remark)j(6.1.)47 b Ff(It)24 b(is)g(imp)-5 b(ortant)27 b(to)d(observe)g(that)i(in)e(the)g (ab)-5 b(ove)25 b(r)-5 b(e)g(asoning)26 b(we)e(have)g(never)g(use)-5 b(d)25 b(the)118 2489 y(information)36 b(that)f(the)f(lo)-5 b(garithmic)36 b(Sob)-5 b(olev)35 b(c)-5 b(onstant)36 b(is)d(lar)-5 b(ger)35 b(than)g Fk(\031)27 b Fn(N)2865 2456 y Fp(2)2905 2489 y Ff(.)45 b(Unfortunately)35 b(we)118 2613 y(have)30 b(not)f(b)-5 b(e)g(en)29 b(able)h(to)f(establish)h (anything)h(like)d(pr)-5 b(op)g(osition)33 b(6.1)c(for)h(the)f(entr)-5 b(opy)31 b(of)3186 2539 y Fl(\000)3243 2613 y Fn(P)3314 2565 y Fp(\(+)p Fj(;N)7 b Fp(\))3301 2637 y Fj(t)3511 2613 y Fn(f)3580 2539 y Fl(\001)3622 2562 y Fp(2)118 2747 y Ff(with)34 b(the)f(exp)-5 b(onent)34 b Fn(t)889 2702 y Fp(\(log)13 b Fj(N)7 b Fp(\))1109 2679 y Fy(\024)p 888 2726 262 4 v 987 2779 a Fj(N)1192 2747 y Ff(r)-5 b(eplac)g(e)g(d)35 b(by)e Fn(t)1699 2702 y Fp(\(log)13 b Fj(N)7 b Fp(\))1919 2679 y Fy(\024)p 1698 2726 V 1780 2781 a Fj(N)1843 2762 y Fv(2)1970 2747 y Ff(.)118 2933 y(Pr)-5 b(o)g(of.)43 b Fr(W)-8 b(e)32 b(set)e Fn(\036)c Fr(=)f Fn(f)k Fk(\000)20 b Fn(\026)1093 2895 y Fp(+)1093 2961 y Fj(N)1160 2933 y Fr(\()p Fn(f)10 b Fr(\).)41 b(The)30 b(sp)s(ectral)f(decomp)s(osition)g(of)i Fk(L)2645 2895 y Fp(+)2645 2961 y Fj(N)2742 2933 y Fr(implies)938 3153 y Fn(\026)993 3115 y Fp(+)993 3182 y Fj(N)1075 3052 y Fl(\020)1129 3080 y(\000)1171 3153 y Fn(P)1242 3105 y Fp(\(+)p Fj(;N)7 b Fp(\))1229 3177 y Fj(t)1438 3153 y Fn(\036)1492 3080 y Fl(\001)1534 3102 y Fp(2)1574 3052 y Fl(\021)1653 3153 y Fr(=)1749 3029 y Fl(Z)1840 3056 y Fi(1)1800 3236 y Fp(0)1945 3153 y Fn(d\027)2037 3168 y Fj(\036)2083 3153 y Fr(\()p Fn(#)p Fr(\))15 b(exp)q(\()p Fk(\000)p Fr(2)p Fn(t#)p Fr(\))g Fn(;)118 3364 y Fr(where)30 b Fn(\027)426 3379 y Fj(\036)502 3364 y Fr(denotes)h(the)g(sp)s(ectral) e(measure)i(asso)s(ciated)f(to)i Fn(\036)p Fr(.)40 b(By)31 b(Jensen)f(inequalit)m(y)-8 b(,)562 3601 y Fn(\026)617 3563 y Fp(+)617 3629 y Fj(N)699 3500 y Fl(\020)753 3527 y(\000)795 3601 y Fn(P)866 3553 y Fp(\(+)p Fj(;N)7 b Fp(\))853 3625 y Fj(t)1063 3601 y Fn(\036)1117 3527 y Fl(\001)1159 3550 y Fp(2)1198 3500 y Fl(\021)1303 3601 y Fm(>)1424 3473 y Fl(\022)1491 3477 y(Z)1582 3503 y Fi(1)1541 3683 y Fp(0)1687 3601 y Fn(d\027)1779 3616 y Fj(\036)1825 3601 y Fr(\()p Fn(#)p Fr(\))1949 3473 y Fl(\023)2057 3601 y Fr(exp)2211 3473 y Fl(\022)2278 3601 y Fk(\000)p Fr(2)p Fn(t)2437 3460 y Fl(R)2497 3486 y Fi(1)2480 3565 y Fp(0)2587 3533 y Fn(#)15 b(d\027)2748 3548 y Fj(\036)2795 3533 y Fr(\()p Fn(#)p Fr(\))p 2437 3580 483 4 v 2464 3593 a Fl(R)2524 3619 y Fi(1)2507 3698 y Fp(0)2629 3666 y Fn(d\027)2721 3681 y Fj(\036)2768 3666 y Fr(\()p Fn(#)p Fr(\))2929 3473 y Fl(\023)3026 3601 y Fn(:)118 3827 y Fr(By)31 b(de\014nition)d(of)i(the)h(sp)s (ectral)f(measure)681 4036 y Fn(\026)736 3998 y Fp(+)736 4065 y Fj(N)803 3963 y Fl(\000)845 4036 y Fn(\036)899 3999 y Fp(2)938 3963 y Fl(\001)1005 4036 y Fr(=)1101 3913 y Fl(Z)1192 3939 y Fi(1)1152 4119 y Fp(0)1297 4036 y Fn(d\027)1389 4051 y Fj(\036)1435 4036 y Fr(\()p Fn(#)p Fr(\))92 b(and)e Fk(E)1944 3998 y Fp(+)1936 4065 y Fj(N)2003 4036 y Fr(\()p Fn(f)5 b(;)15 b(f)10 b Fr(\))40 b(=)2370 3913 y Fl(Z)2460 3939 y Fi(1)2420 4119 y Fp(0)2550 4036 y Fn(#)15 b(d\027)2711 4051 y Fj(\036)2758 4036 y Fr(\()p Fn(#)p Fr(\))25 b Fn(:)118 4253 y Fr(Therefore)30 b(the)h(b)s(ound)d (on)i(the)h(sp)s(ectral)f(gap)g(\(see)i(\(5.29\))r(\))f(implies)c(that) k(for)f Fn(N)41 b Fr(large)30 b(enough,)784 4473 y Fn(\026)839 4435 y Fp(+)839 4502 y Fj(N)922 4372 y Fl(\020)976 4399 y(\000)1017 4473 y Fn(P)1088 4425 y Fp(\(+)p Fj(;N)7 b Fp(\))1075 4497 y Fj(t)1285 4473 y Fn(\036)1339 4399 y Fl(\001)1381 4422 y Fp(2)1420 4372 y Fl(\021)1525 4473 y Fm(>)50 b Fn(\026)1701 4435 y Fp(+)1701 4502 y Fj(N)1768 4399 y Fl(\000)1810 4473 y Fn(\036)1864 4435 y Fp(2)1904 4399 y Fl(\001)1986 4473 y Fr(exp)2140 4345 y Fl(\022)2207 4473 y Fk(\000)p Fr(2)p Fn(t)2366 4411 y Fr(\(log)17 b Fn(N)10 b Fr(\))2652 4378 y Fj(\024)p 2366 4452 332 4 v 2490 4535 a Fn(N)2707 4345 y Fl(\023)2804 4473 y Fn(:)600 b Fr(\(6.32\))118 4693 y(According)30 b(to)h(\(4.15\))s(,)f (there)h(is)e(a)i(constan)m(t)h Fn(c)1755 4708 y Fj(\025)1825 4693 y Fn(>)25 b Fr(0)31 b(suc)m(h)f(that)1208 4862 y Fn(\026)1263 4824 y Fp(+)1263 4891 y Fj(N)1330 4862 y Fr(\()p Fn(f)10 b Fr(\))1455 4824 y Fp(2)1520 4862 y Fm(6)25 b Fn(\026)1671 4824 y Fp(+)1671 4891 y Fj(N)1738 4862 y Fr(\()p Fn(f)1828 4824 y Fp(2)1867 4862 y Fr(\))15 b(exp\()p Fk(\000)p Fn(c)2201 4877 y Fj(\025)2247 4862 y Fn(N)10 b Fr(\))15 b Fn(:)118 5027 y Fr(The)30 b(former)g(inequalit)m (y)f(com)m(bined)g(with)g(\(6.32\))k(leads)d(to)440 5247 y Fn(\026)495 5209 y Fp(+)495 5276 y Fj(N)577 5147 y Fl(\020)632 5174 y(\000)673 5247 y Fn(P)744 5200 y Fp(\(+)p Fj(;N)7 b Fp(\))731 5272 y Fj(t)941 5247 y Fn(f)996 5174 y Fl(\001)1037 5196 y Fp(2)1076 5147 y Fl(\021)1181 5247 y Fm(>)50 b Fn(\026)1357 5209 y Fp(+)1357 5276 y Fj(N)1424 5174 y Fl(\000)1466 5247 y Fn(f)1521 5210 y Fp(2)1560 5174 y Fl(\001)1642 5247 y Fr(exp)1796 5119 y Fl(\022)1863 5247 y Fk(\000)p Fr(2)p Fn(t)2022 5186 y Fr(\(log)17 b Fn(N)10 b Fr(\))2308 5153 y Fj(\024)p 2022 5227 V 2146 5310 a Fn(N)2363 5119 y Fl(\023)2445 5174 y(\000)2487 5247 y Fr(1)20 b Fk(\000)g Fr(exp\()p Fk(\000)p Fn(c)2927 5262 y Fj(\025)2973 5247 y Fn(N)10 b Fr(\))3091 5174 y Fl(\001)3148 5247 y Fn(:)256 b Fr(\(6.33\))118 5463 y(This)29 b(concludes)g(the)i(pro)s(of.)p 3595 5463 4 62 v 3599 5405 55 4 v 3599 5463 V 3653 5463 4 62 v eop %%Page: 14 14 14 13 bop 118 45 a Fp(14)1071 b(T.)30 b(BODINEA)n(U)h(AND)f(F.)g(MAR)-6 b(TINELLI)118 366 y Fr(6.2.)54 b FC(Pro)s(of)36 b(of)f(Theorem)f(2.3.) 118 527 y Ff(Pr)-5 b(o)g(of.)43 b Fr(The)31 b(\014rst)g(step)h(is)e(to) j(reform)m(ulate)e(the)h(LHS)f(of)39 b(\(2.6\))33 b(in)d(terms)i(of)g (mesoscopic)f(v)-5 b(ariables.)118 634 y(F)d(or)35 b(an)m(y)g(site)f Fn(x)e Fk(2)f Fn(K)22 b Fo(Z)972 601 y Fp(2)1041 634 y Fr(w)m(e)35 b(de\014ne)e Fn(\020)1492 590 y Fj(\021)1485 646 y(x)1534 634 y Fr(\()p Fn(t)p Fr(\))h(to)h(b)s(e)f(the)h(indicator) e(function)g(of)h(the)g(ev)m(en)m(t)i(that)f(the)118 748 y(magnetization)26 b(in)d(the)i(b)s(o)m(x)g Fo(B)1179 762 y Fj(K)1253 748 y Fr(\()p Fn(x)p Fr(\))h(for)f(the)g(pro)s(cess)f Fn(\033)2052 715 y Fj(\021)2095 748 y Fr(\()p Fn(t)p Fr(\))h(in)f Fo(B)2378 762 y Fj(K)2453 748 y Fr(\()p Fn(x)p Fr(\))h(is)f(smaller)g(than)g Fk(\000)3282 712 y Fj(m)3344 689 y Fh(\003)p 3282 727 99 4 v 3314 779 a Fp(4)3390 748 y Fr(.)39 b(Then)118 856 y(w)m(e)31 b(ha)m(v)m(e)535 1008 y(^)-53 b Fn(\026)582 971 y Fp(+)656 907 y Fl(\020)718 985 y Fr(^)711 1008 y Fo(E)771 935 y Fl(\000)813 1008 y Fn(\020)860 964 y Fj(\021)853 1034 y Fp(0)901 1008 y Fr(\()p Fn(t)p Fr(\))26 b Fk(6)p Fr(=)1134 984 y(~)1126 1008 y Fn(\020)1173 971 y Fj(!)1166 1031 y Fp(0)1223 1008 y Fr(\()p Fn(t)p Fr(\))1326 935 y Fl(\001)1368 907 y(\021)1531 1008 y Fm(6)115 b Fr(^)-52 b Fn(\026)1765 971 y Fp(+)1839 907 y Fl(\020)1901 985 y Fr(^)1893 1008 y Fo(E)1954 935 y Fl(\000)2010 1008 y Fk(9)15 b Fn(i)26 b Fk(2)e Fo(B)2273 1022 y Fj(K)2347 1008 y Fr(\(0\))p Fn(;)108 b(\033)2650 964 y Fj(\021)2647 1035 y(i)2692 1008 y Fr(\()p Fn(t)p Fr(\))26 b Fk(6)p Fr(=)k(~)-50 b Fn(\033)2972 971 y Fj(!)2969 1031 y(i)3022 1008 y Fr(\()p Fn(t)p Fr(\))3140 935 y Fl(\001)3183 907 y(\021)1531 1207 y Fm(6)108 b Fn(K)1794 1170 y Fp(2)1840 1207 y Fr(^)-52 b Fn(\026)1888 1170 y Fp(+)1962 1106 y Fl(\020)2024 1184 y Fr(^)2016 1207 y Fo(E)2077 1133 y Fl(\000)2134 1207 y Fn(\033)2189 1163 y Fj(\021)2186 1233 y Fp(0)2231 1207 y Fr(\()p Fn(t)p Fr(\))25 b Fk(6)p Fr(=)30 b(~)-50 b Fn(\033)2510 1170 y Fj(!)2507 1230 y Fp(0)2561 1207 y Fr(\()p Fn(t)p Fr(\))2679 1133 y Fl(\001)2721 1106 y(\021)2806 1207 y Fn(;)118 1377 y Fr(where)30 b(w)m(e)h(used)e(the)i(in)m(v)-5 b(ariance)30 b(b)m(y)g(spatial)f(translation)h(in)f(the)h(last)g (inequalit)m(y)-8 b(.)118 1486 y(Let)35 b Fn(N)45 b Fr(b)s(e)33 b(a)i(large)g(in)m(teger,)h(c)m(ho)s(ose)g(as)e(usual)g(the)g (mesoscopic)h(scale)g Fn(K)k Fr(=)32 b(\()p Fn(b)15 b Fr(log)i Fn(N)10 b Fr(\))3230 1453 y Fp(1)p Fj(=\015)3379 1486 y Fr(and)34 b(let)118 1598 y Fn(L)25 b Fr(=)312 1562 y Fj(N)p 311 1577 65 4 v 311 1629 a(K)385 1598 y Fr(.)41 b(By)31 b(rep)s(eating)e(the)i(previous)e(computation)h(on)g(a) h(coarse)h(grained)d(lev)m(el,)h(w)m(e)h(get)125 1782 y(^)-52 b Fn(\026)173 1744 y Fp(+)232 1681 y Fl(\020)294 1758 y Fr(^)286 1782 y Fo(E)347 1708 y Fl(\000)531 1696 y(X)404 1898 y Fj(i)p Fi(2)p Fg(W)535 1909 y Fy(N)599 1898 y Fi(\\)p Fj(K)5 b Fg(Z)760 1879 y Fv(2)804 1782 y Fn(\020)851 1738 y Fj(\021)844 1809 y(i)893 1782 y Fr(\()p Fn(t)p Fr(\))25 b Fk(6)p Fr(=)1244 1696 y Fl(X)1117 1898 y Fj(i)p Fi(2)p Fg(W)1249 1909 y Fy(N)1312 1898 y Fi(\\)p Fj(K)5 b Fg(Z)1474 1879 y Fv(2)1526 1758 y Fr(~)1518 1782 y Fn(\020)1565 1744 y Fj(!)1558 1804 y(i)1615 1782 y Fr(\()p Fn(t)p Fr(\))1718 1708 y Fl(\001)1760 1681 y(\021)1922 1782 y Fm(6)116 b Fr(^)-53 b Fn(\026)2156 1744 y Fp(+)2230 1681 y Fl(\020)2292 1758 y Fr(^)2285 1782 y Fo(E)2345 1708 y Fl(\000)2387 1782 y Fk(9)15 b Fn(i)25 b Fk(2)g Fo(W)2680 1796 y Fj(N)2773 1782 y Fk(\\)20 b Fn(K)7 b Fo(Z)3003 1744 y Fp(2)3038 1782 y Fn(;)106 b(\020)3216 1738 y Fj(\021)3209 1809 y(i)3258 1782 y Fr(\()p Fn(t)p Fr(\))25 b Fk(6)p Fr(=)3490 1758 y(~)3482 1782 y Fn(\020)3529 1744 y Fj(!)3522 1804 y(i)3579 1782 y Fr(\()p Fn(t)p Fr(\))3682 1708 y Fl(\001)3724 1681 y(\021)1922 2058 y Fm(6)108 b Fn(L)2163 2021 y Fp(2)2210 2058 y Fr(^)-52 b Fn(\026)2258 2021 y Fp(+)2332 1957 y Fl(\020)2394 2035 y Fr(^)2386 2058 y Fo(E)2447 1984 y Fl(\000)2488 2058 y Fn(\020)2535 2014 y Fj(\021)2528 2084 y Fp(0)2577 2058 y Fr(\()p Fn(t)p Fr(\))25 b Fk(6)p Fr(=)2809 2034 y(~)2801 2058 y Fn(\020)2848 2021 y Fj(!)2841 2081 y Fp(0)2898 2058 y Fr(\()p Fn(t)p Fr(\))3001 1984 y Fl(\001)3043 1957 y(\021)3128 2058 y Fn(:)118 2227 y Fr(Let)31 b(no)m(w)1282 2375 y Fk(B)1345 2336 y Fi(\000)1342 2403 y Fj(N)1409 2375 y Fr(\()p Fn(\033)1499 2330 y Fj(\021)1496 2399 y(t)1542 2375 y Fr(\))25 b(=)1825 2288 y Fl(X)1698 2491 y Fj(i)p Fi(2)p Fg(W)1830 2502 y Fy(N)1894 2491 y Fi(\\)p Fj(K)5 b Fg(Z)2055 2472 y Fv(2)2099 2375 y Fn(\020)2146 2330 y Fj(\021)2139 2402 y(i)2187 2375 y Fr(\()p Fn(t)p Fr(\))15 b Fn(:)118 2622 y Fr(The)30 b(previous)f (results)g(imply)618 2808 y(^)-52 b Fn(\026)666 2769 y Fp(+)666 2836 y Fj(N)748 2707 y Fl(\020)810 2784 y Fr(^)803 2808 y Fo(E)863 2734 y Fl(\000)905 2808 y Fn(\033)960 2763 y Fj(\021)957 2834 y Fp(0)1002 2808 y Fr(\()p Fn(t)p Fr(\))26 b Fk(6)p Fr(=)k(~)-50 b Fn(\033)1282 2770 y Fj(!)1279 2830 y Fp(0)1332 2808 y Fr(\()p Fn(t)p Fr(\))1435 2734 y Fl(\001)1478 2707 y(\021)1582 2808 y Fm(>)1752 2746 y Fr(1)p 1714 2787 123 4 v 1714 2870 a Fn(N)1797 2844 y Fp(2)1853 2808 y Fr(^)e Fn(\026)1901 2770 y Fp(+)1975 2707 y Fl(\020)2037 2784 y Fr(^)2029 2808 y Fo(E)2090 2734 y Fl(\000)2131 2808 y Fk(B)2194 2769 y Fi(\000)2191 2836 y Fj(N)2258 2808 y Fr(\()p Fn(\033)2348 2763 y Fj(\021)2345 2832 y(t)2391 2808 y Fr(\))26 b Fk(6)p Fr(=)f Fk(B)2611 2769 y Fi(\000)2608 2836 y Fj(N)2674 2808 y Fr(\()5 b(~)-50 b Fn(\033)2764 2770 y Fj(!)2761 2830 y(t)2815 2808 y Fr(\))2850 2734 y Fl(\001)2892 2707 y(\021)2977 2808 y Fn(:)427 b Fr(\(6.34\))118 2981 y(In)29 b(the)g(second)h(step,)f(w)m (e)h(are)g(going)f(to)h(decouple)f(the)h(estimates)g(of)f(the)h(join)m (t)f(pro)s(cess.)40 b(The)28 b(main)118 3089 y(ph)m(ysical)f(idea)h(w)m (as)g(already)g(con)m(tained)h(in)e(the)h(Fisher,)g(Huse)g(pap)s(er)f ([HF)q(])h(and)g(it)g(go)s(es)h(as)f(follo)m(ws.)118 3197 y(W)-8 b(e)38 b(force)f(one)g(large)f(droplet)g(of)g(the)h(min)m (us)e(phase)h(of)g(radius)f Fk(\031)g Fn(N)10 b Fr(,)38 b(around)d(the)i(origin)e(in)g(e.g.)118 3305 y(the)d(initial)d (distribution)f(of)k Fn(\033)1197 3260 y Fj(\021)1194 3329 y(t)1239 3305 y Fr(,)g(b)m(y)f(pa)m(ying)h(a)g(price)e Fk(\031)d Fr(exp\()p Fk(\000)p Fn(\034)2411 3272 y Fi(\003)2466 3305 y Fn(N)10 b Fr(\).)45 b(This)29 b(droplet)i(should)f(relax)118 3418 y(only)c(in)f(a)i(time)f(scale)h(prop)s(ortional)e(to)i(its)f (initial)e(area)j(and)f(therefore,)i(if)e Fn(N)35 b Fr(=)25 b Fn(A)3036 3344 y Fk(p)p 3112 3344 33 4 v 74 x Fn(t)h Fr(with)g Fn(A)g Fr(large)118 3526 y(enough,)k(the)g(distribution)c(of) j Fk(B)1272 3488 y Fi(\000)1269 3555 y Fj(N)1336 3526 y Fr(\()p Fn(\033)1426 3482 y Fj(\021)1423 3550 y(t)1469 3526 y Fr(\))h(at)g(time)g Fn(t)f Fr(giv)m(en)h(the)f(ab)s(o)m(v)m(e)i (initial)c(unlik)m(ely)g(ev)m(en)m(t)k(should)118 3641 y(b)s(e)d(quite)f(di\013eren)m(t)h(from)g(that)h(of)f Fk(B)1396 3603 y Fi(\000)1393 3670 y Fj(N)1460 3641 y Fr(\()p Fn(\033)1550 3608 y Fj(!)1547 3664 y(t)1601 3641 y Fr(\).)41 b(Apparen)m(tly)27 b(in)g(order)h(to)h(carry)f(rigorously)f (the)h(ab)s(o)m(v)m(e)118 3749 y(program)35 b(one)g(needs)g(a)g(m)m(uc) m(h)g(more)g(precise)g(con)m(trol)g(on)g(the)g(life)f(time)g(of)h(a)h (droplet)e(than)g(what)118 3857 y(w)m(e)c(ha)m(v)m(e)g(b)s(een)f(able)f (to)i(obtain.)40 b(Actually)29 b(that)g(is)g(not)g(true)g(and)g(all)f (what)h(w)m(e)h(need)f(is)f(something)118 3965 y(not)j(more)f(precise)g (than)g(prop)s(osition)e(6.1)k(\(see)f(Lemma)f(6.1)i(b)s(elo)m(w\).)118 4073 y(F)-8 b(rom)38 b(a)g(tec)m(hnical)g(p)s(oin)m(t)e(of)i(view)f(it) 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y Fi(\000)2469 5361 y Fj(N)2536 5333 y Fr(\()5 b(~)-50 b Fn(\033)2626 5295 y Fj(!)2623 5355 y(t)2677 5333 y Fr(\))26 b Fm(6)e Fn(\013L)2953 5295 y Fp(2)2993 5259 y Fl(\001)1486 5480 y Fm(>)108 b Fo(E)1720 5407 y Fl(\000)1768 5480 y Fk(B)1831 5442 y Fi(\000)1828 5509 y Fj(N)1894 5480 y Fr(\()p Fn(\033)1984 5436 y Fj(\021)1981 5504 y(t)2027 5480 y Fr(\))26 b Fn(>)f(\013L)2304 5443 y Fp(2)2343 5407 y Fl(\001)2405 5480 y Fk(\000)20 b Fo(E)2551 5407 y Fl(\000)2598 5480 y Fk(B)2661 5442 y Fi(\000)2658 5509 y Fj(N)2725 5480 y Fr(\()5 b(~)-50 b Fn(\033)2815 5443 y Fj(!)2812 5503 y(t)2866 5480 y Fr(\))26 b Fn(>)f(\013L)3143 5443 y Fp(2)3183 5407 y Fl(\001)3240 5480 y Fn(;)p eop %%Page: 15 15 15 14 bop 569 45 a Fp(SOME)30 b(NEW)h(RESUL)-6 b(TS)31 b(ON)g(THE)f(KINETIC)h(ISING)h(MODEL)e(IN)h(A)f(PURE)g(PHASE)381 b(15)118 370 y Fr(where)24 b Fo(E)40 b Fr(refers)24 b(to)g(the)h (marginal)d(of)1431 347 y(^)1424 370 y Fo(E)16 b Fr(,)26 b(i.e.)38 b(to)25 b(the)f(usual)e(Glaub)s(er)h(dynamics.)37 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1057 y Fp(+)247 994 y Fl(\020)309 1071 y Fr(^)301 1095 y Fo(E)362 1021 y Fl(\000)404 1095 y Fk(B)467 1056 y Fi(\000)464 1123 y Fj(N)531 1095 y Fr(\()p Fn(\033)621 1050 y Fj(\021)618 1119 y(t)663 1095 y Fr(\))26 b Fk(6)p Fr(=)f Fk(B)883 1056 y Fi(\000)880 1123 y Fj(N)947 1095 y Fr(\()5 b(~)-50 b Fn(\033)1037 1057 y Fj(!)1034 1117 y(t)1088 1095 y Fr(\))1123 1021 y Fl(\001)1165 994 y(\021)1269 1095 y Fm(>)51 b Fr(exp\()p Fk(\000)p Fr(2)p Fn(\034)1731 1057 y Fi(\003)1770 1095 y Fn(N)10 b Fr(\))1903 994 y Fl(\020)1958 1095 y Fn(\026)2013 1057 y Fp(+)p Fj(;f)2129 1034 y Fv(2)2183 1021 y Fl(\000)2224 1095 y Fo(E)2279 1021 y Fl(\000)2327 1095 y Fk(B)2390 1056 y Fi(\000)2387 1123 y Fj(N)2454 1095 y Fr(\()p Fn(\033)2544 1050 y Fj(\021)2541 1119 y(t)2586 1095 y Fr(\))26 b Fn(>)f(\013L)2863 1057 y Fp(2)2902 1021 y Fl(\001\001)3006 1095 y Fk(\000)20 b Fr(exp\()p Fk(\000)p Fn(C)3407 1109 y Fj(\013)3456 1095 y Fn(N)10 b Fr(\))3574 994 y Fl(\021)3659 1095 y Fn(:)118 1278 y Fr(It)41 b(is)f(at)i(this)e(stage)i(that)g(w)m(e)f(are)h(going)f(to)g (use)g(the)g(information)f(on)g(the)i(sp)s(ectral)e(gap.)73 b(The)118 1385 y(necessary)34 b(dynamical)e(estimate)j(is)d(pro)m (vided)h(b)m(y)g(the)h(follo)m(wing)e(Lemma)i(whic)m(h)e(will)f(b)s(e)i (deriv)m(ed)118 1493 y(later.)118 1664 y FC(Lemma)g(6.1.)46 b Ff(We)36 b(\014x)g Fn(\013)h Ff(such)g(that)g(the)g(p)-5 b(ar)g(ameter)39 b Fr(\(2)p Fn(\025)p Fr(\()p Fn(\013)25 b Fk(\000)d Fr(1\))i(+)f Fn(\034)2668 1631 y Fi(\003)2707 1664 y Fr(\))37 b Ff(is)f(ne)-5 b(gative.)53 b(Then,)38 b(for)118 1772 y Fn(N)43 b Ff(lar)-5 b(ge)33 b(enough,)g(the)g(fol)5 b(lowing)34 b(ine)-5 b(quality)33 b(holds)138 1980 y Fk(8)p Fn(t)25 b(>)g Fr(0)p Fn(;)615 1856 y Fl(Z)736 1980 y Fn(d\026)838 1942 y Fp(+)p Fj(;f)954 1919 y Fv(2)992 1980 y Fr(\()p Fn(\021)s Fr(\))i Fo(E)1191 1906 y Fl(\000)1239 1980 y Fk(B)1302 1941 y Fi(\000)1299 2008 y Fj(N)1366 1980 y Fr(\()p Fn(\033)1456 1935 y Fj(\021)1453 2004 y(t)1498 1980 y Fr(\))f Fn(>)f(\013L)1775 1942 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Fj(\021)959 1622 y(t)1004 1598 y Fr(\))1039 1524 y Fl(\001)1096 1598 y Fo(E)1151 1524 y Fl(\000)1199 1598 y Fr(1)1244 1628 y Fi(B)1292 1598 y Fh(\000)1290 1651 y Fy(N)1348 1628 y Fp(\()p Fj(\033)1417 1596 y Fy(\021)1415 1648 y(t)1456 1628 y Fp(\))20 b Fd(>)g Fj(\013L)1671 1609 y Fv(2)1710 1524 y Fl(\001)1751 1497 y(\021)1831 1598 y Fr(=)25 b Fn(\026)1982 1560 y Fp(+)2056 1497 y Fl(\020)2110 1598 y Fn(f)2165 1560 y Fp(2)2204 1598 y Fr(\()p Fn(\021)s Fr(\))15 b Fo(E)2393 1524 y Fl(\000)2440 1598 y Fr(1)2485 1628 y Fi(B)2533 1598 y Fh(\000)2531 1651 y Fy(N)2590 1628 y Fp(\()p Fj(\033)2659 1596 y Fy(\021)2657 1649 y Fv(2)p Fy(t)2716 1628 y Fp(\))20 b Fd(>)f Fj(\013L)2930 1609 y Fv(2)2969 1524 y Fl(\001)3011 1497 y(\021)3096 1598 y Fn(:)118 1778 y Fr(This)29 b(concludes)g(the)i(Lemma.)p 3595 1778 4 62 v 3599 1720 55 4 v 3599 1778 V 3653 1778 4 62 v 118 1996 a FC(Prop)s(osition)36 b(6.2.)46 b Ff(In)33 b(dimension)h Fn(d)25 b Fr(=)g(2)p Ff(,)33 b(for)g(any)h Fn(\014)c(>)25 b(\014)2269 2010 y Fj(c)2337 1996 y Ff(then)859 2196 y Fk(8)p Fn(N)5 b(;)201 b(\026)1269 2158 y Fp(+)1328 2196 y Fr(\()p Fn(f)1418 2158 y Fp(2)1457 2196 y Fr(\))20 b Fk(\000)g Fn(\026)1658 2158 y Fp(+)1717 2196 y Fr(\()p Fn(f)10 b Fr(\))1842 2158 y Fp(2)1932 2196 y Fm(>)50 b Fn(C)2259 2134 y(N)p 2135 2175 332 4 v 2135 2258 a Fr(\(log)16 b Fn(N)10 b Fr(\))2420 2232 y Fj(\024)2476 2196 y Fn(\026)2531 2158 y Fp(+)2589 2196 y Fr(\()p Fk(jr)p Fn(f)g Fk(j)2805 2158 y Fp(2)2845 2196 y Fr(\))15 b Fn(;)118 2407 y Ff(wher)-5 b(e)34 b Fn(f)42 b Ff(is)32 b(the)h(test)g(function)g (intr)-5 b(o)g(duc)g(e)g(d)35 b(in)40 b Fr(\(4.12\))r Ff(.)118 2576 y(Pr)-5 b(o)g(of.)43 b Fr(The)g(upp)s(er)e(b)s(ound)h (\(4.25\))k(on)d(the)h(Diric)m(hlet)e(form)h(is)g(unc)m(hanged)g(under) f Fn(\026)3263 2543 y Fp(+)3365 2576 y Fr(since)h(it)118 2684 y(b)s(oils)31 b(do)m(wn)h(to)i(estimating)e(the)h(n)m(um)m(b)s(er) f(of)h(bad)f(blo)s(c)m(ks)h(in)e(the)i(region)g Fo(W)2824 2698 y Fj(N)2930 2684 y Fr(b)m(y)f(using)g(Bernoulli)118 2792 y(p)s(ercolation.)61 b(The)37 b(lo)m(w)m(er)h(b)s(ound)d(\(4.14\)) 40 b(holds)c(also)h(for)g Fn(\026)2278 2759 y Fp(+)2337 2792 y Fr(\()p Fn(f)2427 2759 y Fp(2)2466 2792 y Fr(\))h(b)s(ecause)f (it)g(in)m(v)m(olv)m(es)h(only)e(the)118 2900 y(computation)30 b(of)h(an)f(ev)m(en)m(t)i(lo)s(calized)d(in)g Fo(W)1680 2914 y Fj(N)1754 2900 y Fr(.)218 3008 y(Th)m(us)g(it)h(remains)f(only)h (to)h(c)m(hec)m(k)h(that)1299 3203 y(lim)15 b(sup)1336 3282 y Fj(N)7 b Fi(!1)1647 3142 y Fr(1)p 1628 3182 83 4 v 1628 3266 a Fn(N)1736 3203 y Fr(log)16 b Fn(\026)1923 3166 y Fp(+)1982 3203 y Fr(\()p Fn(f)10 b Fr(\))26 b Fm(6)e Fr(0)15 b Fn(:)1116 b Fr(\(6.38\))218 3437 y(Let)413 3414 y Fl(b)394 3437 y Fn(D)51 b Fr(=)c([)p Fk(\000)p Fn(R)q(;)15 b(R)q Fr(])939 3404 y Fp(2)1026 3437 y Fk(\032)48 b Fo(R)1205 3404 y Fp(2)1250 3437 y Fr(,)f(where)d Fn(R)g Fr(will)e(b)s(e)h(c)m(hosen)h(large)g(enough;)51 b(in)42 b(particular)h(suc)m(h)118 3560 y(that)322 3537 y Fl(c)322 3560 y Fo(W)419 3527 y Fp(2)502 3560 y Fk(\032)37 b Fr([)p Fk(\000)p Fn(R)q(=)p Fr(2)p Fn(;)15 b(R)q(=)p Fr(2])1091 3527 y Fp(2)1133 3560 y Fr(.)62 b(Let)38 b Fk(C)1438 3574 y Fj(N)1543 3560 y Fr(b)s(e)f(the)h(set)g(of)g(spin)e (con\014gurations)h(whic)m(h)f(con)m(tain)i(a)h Fk(\003)p Fr({)118 3668 y(connected)32 b(circuit)f(of)g(+)g(spins)f(inside)f([)p Fk(\000)p Fn(N)5 b(;)15 b(N)10 b Fr(])1845 3635 y Fp(2)1906 3668 y Fk(n)21 b Fr([)p Fk(\000)p Fn(N)r(=)p Fr(2)p Fn(;)15 b(N)r(=)p Fr(2])2463 3635 y Fp(2)2537 3668 y Fr(separating)31 b(\([)p Fk(\000)p Fn(N)5 b(;)15 b(N)10 b Fr(])3335 3635 y Fp(2)3375 3668 y Fr(\))3410 3635 y Fj(c)3477 3668 y Fr(from)118 3776 y([)p Fk(\000)p Fn(N)r(=)p Fr(2)p Fn(;)15 b(N)r(=)p Fr(2])609 3743 y Fp(2)652 3776 y Fr(.)40 b(As)31 b Fn(\014)f(>)25 b(\014)1080 3790 y Fj(c)1146 3776 y Fr(and)k Fn(d)d Fr(=)f(2,)31 b(there)f(is)g Fn(c)1955 3791 y Fj(\014)2028 3776 y Fn(>)25 b Fr(0)30 b(suc)m(h)g(that)1344 3939 y Fn(\026)1399 3902 y Fp(+)1457 3939 y Fr(\()p Fk(C)1545 3902 y Fj(c)1540 3962 y(N)1608 3939 y Fr(\))c Fm(6)40 b Fr(exp\()p Fk(\000)p Fn(c)2064 3954 y Fj(\014)2111 3939 y Fn(N)10 b Fr(\))15 b Fn(:)118 4097 y Fr(By)31 b(c)m(ho)s(osing)f Fn(R)h Fr(suc)m(h)f(that)h Fn(R)q(c)1241 4112 y Fj(\014)1314 4097 y Fn(>)24 b Fr(2)p Fn(\025)p Fr(,)32 b(w)m(e)e(get)284 4260 y Fn(\026)339 4223 y Fp(+)398 4260 y Fr(\()p Fn(f)10 b Fr(\))25 b Fm(6)g Fn(\026)699 4223 y Fp(+)758 4260 y Fr(\()p Fn(f)f Fr(1)907 4274 y Fi(C)944 4285 y Fy(N)5 b(R)1053 4260 y Fr(\))21 b(+)f Fn(\026)1255 4223 y Fp(+)1313 4260 y Fr(\()p Fk(C)1401 4223 y Fj(c)1396 4283 y(N)7 b(R)1517 4260 y Fr(\))31 b(exp\()p Fn(\025N)10 b Fr(\))26 b Fm(6)f Fn(\026)2105 4223 y Fp(+)2164 4260 y Fr(\()p Fn(f)f Fr(1)2313 4274 y Fi(C)2350 4285 y Fy(N)5 b(R)2459 4260 y Fr(\))21 b(+)f(exp\(\()p Fn(\025)h Fk(\000)e Fn(c)3018 4275 y Fj(\014)3066 4260 y Fn(R)q Fr(\))p Fn(N)10 b Fr(\))15 b Fn(:)118 4425 y Fr(Conditionning)34 b(with)h(resp)s(ect)i(to)h(the)f(+)f(circuit)g (whic)m(h)f(is)h(the)h(closest)h(to)f(\([)p Fk(\000)p Fn(R)q(N)5 b(;)15 b(R)q(N)10 b Fr(])3367 4392 y Fj(d)3408 4425 y Fr(\))3443 4392 y Fj(c)3515 4425 y Fr(and)118 4533 y(then)30 b(using)f(the)i(fact)g(that)g Fn(f)39 b Fr(is)30 b(non-increasing,)f(w)m(e)i(obtain)f(b)m(y)g(FK)m(G)1444 4691 y Fn(\026)1499 4653 y Fp(+)1558 4691 y Fr(\()p Fn(f)10 b Fr(1)1693 4705 y Fi(C)1730 4716 y Fy(N)5 b(R)1838 4691 y Fr(\))26 b Fm(6)f Fn(\026)2050 4653 y Fp(+)2050 4719 y Fj(N)7 b(R)2170 4691 y Fr(\()p Fn(f)j Fr(\))15 b Fn(:)118 4856 y Fr(where)28 b Fn(\026)434 4818 y Fp(+)434 4885 y Fj(N)7 b(R)583 4856 y Fr(denotes)29 b(no)m(w)f(the)h(Gibbs)e(measure) h(in)f Fk(f\000)p Fn(N)10 b(R)q(;)15 b(:)g(:)g(:)i(;)e(N)10 b(R)q Fk(g)2639 4823 y Fp(2)2707 4856 y Fr(with)27 b(+)i(b)s(oundary)d (condi-)118 4964 y(tions.)218 5072 y(A)m(t)i(this)e(p)s(oin)m(t,)h(w)m (e)h(pro)s(ceed)f(as)g(in)f(subsection)g(4.1.)41 b(The)26 b(only)h(di\013erence)f(is)h(that)g(the)h(estimates)118 5189 y(are)j(in)e Fo(L)431 5156 y Fp(1)476 5189 y Fr(\()530 5166 y Fl(b)511 5189 y Fn(D)s Fr(\))i(instead)e(of)i Fo(L)1127 5156 y Fp(1)1172 5189 y Fr(\()1207 5166 y Fl(c)1207 5189 y Fo(W)1292 5156 y Fp(2)1338 5189 y Fr(\).)41 b(This)28 b(implies)375 5390 y(lim)15 b(sup)413 5468 y Fj(N)7 b Fi(!1)723 5329 y Fr(1)p 704 5369 V 704 5453 a Fn(N)812 5390 y Fr(log)17 b Fn(\026)1000 5352 y Fp(+)1000 5419 y Fj(N)7 b(R)1120 5390 y Fr(\()p Fn(f)j Fr(\))26 b Fm(6)212 b Fr(sup)1382 5488 y Fj(v)r Fi(2)p Fp(BV\()1610 5471 y Fc(b)1595 5488 y Fj(D)s(;)p Fi(f\006)p Fp(1)p Fi(g)p Fp(\))1879 5289 y Fl(n)1959 5390 y Fk(\000)20 b(W)2140 5405 y Fj(\014)2187 5390 y Fr(\()p Fn(v)s Fr(\))h(+)f Fn(\025)2469 5313 y Fl(\014)2469 5368 y(\014)2500 5390 y Fk(f)p Fn(v)29 b Fr(=)c Fk(\000)p Fr(1)p Fk(g)c(\\)2976 5367 y Fl(c)2976 5390 y Fo(W)3061 5353 y Fp(2)3107 5313 y Fl(\014)3107 5368 y(\014)3137 5289 y(o)3213 5390 y Fn(:)p eop %%Page: 17 17 17 16 bop 569 45 a Fp(SOME)30 b(NEW)h(RESUL)-6 b(TS)31 b(ON)g(THE)f(KINETIC)h(ISING)h(MODEL)e(IN)h(A)f(PURE)g(PHASE)381 b(17)118 366 y Fr(T)-8 b(o)33 b(deriv)m(e)f(\(6.38\))s(,)h(it)f (remains)f(to)j(c)m(hec)m(k)g(that)f(the)g(RHS)f(is)f(negativ)m(e.)49 b(Either)31 b Fk(j)p Fn(v)s Fk(j)59 b Fm(>)e Fr(1)33 b(in)f(whic)m(h)118 474 y(case,)g(w)m(e)f(get)366 643 y Fk(\000W)527 658 y Fj(\014)573 643 y Fr(\()p Fn(v)s Fr(\))22 b(+)d Fn(\025)855 565 y Fl(\014)855 620 y(\014)886 643 y Fk(f)p Fn(v)29 b Fr(=)c Fk(\000)p Fr(1)p Fk(g)c(\\)1362 619 y Fl(c)1362 643 y Fo(W)1447 605 y Fp(2)1493 565 y Fl(\014)1493 620 y(\014)1548 643 y Fm(6)45 b Fk(\000)20 b Fn(\034)1805 605 y Fi(\003)1845 643 y Fk(j)p Fn(v)s Fk(j)1942 605 y Fp(\()p Fj(d)p Fi(\000)p Fp(1\))p Fj(=d)2220 643 y Fr(+)g Fn(\025)2364 565 y Fl(\014)2364 620 y(\014)2394 619 y(c)2394 643 y Fo(W)2479 605 y Fp(2)2525 565 y Fl(\014)2525 620 y(\014)2580 643 y Fm(6)45 b Fk(\000)20 b Fn(\034)2837 605 y Fi(\003)2897 643 y Fr(+)g Fn(\025)25 b(<)g Fr(0)15 b Fn(;)118 802 y Fr(or)30 b Fk(j)p Fn(v)s Fk(j)d Fn(<)e Fr(1)30 b(and)g(\(4.19\))j(applies)784 975 y Fk(\000W)945 990 y Fj(\014)991 975 y Fr(\()p Fn(v)s Fr(\))22 b(+)e Fn(\025)1274 898 y Fl(\014)1274 952 y(\014)1304 975 y Fk(f)p Fn(v)29 b Fr(=)c Fk(\000)p Fr(1)p Fk(g)c(\\)1780 952 y Fl(c)1780 975 y Fo(W)1865 938 y Fp(2)1911 898 y Fl(\014)1911 952 y(\014)1966 975 y Fm(6)46 b Fk(\000)19 b Fn(\034)2223 938 y Fi(\003)2263 975 y Fk(j)p Fn(v)s Fk(j)i Fr(+)f Fn(\025)p Fk(j)p Fn(v)s Fk(j)26 b Fn(<)f Fr(0)15 b Fn(:)p 3595 1134 4 62 v 3599 1076 55 4 v 3599 1134 V 3653 1134 4 62 v 118 1303 a FC(Remark)34 b(6.2.)47 b Ff(As)36 b(a)i(c)-5 b(onse)g(quenc)g(e)37 b(of)h(the)f(pr)-5 b(o)g(of)39 b(of)f(Pr)-5 b(op)g(osition)40 b(6.2,)e(we)g(se)-5 b(e)37 b(that)h(the)f(inverse)118 1411 y(of)e(the)h(sp)-5 b(e)g(ctr)g(al)38 b(gap)d(asso)-5 b(ciate)g(d)38 b(to)e(the)f(Glaub)-5 b(er)36 b(dynamics)h(in)e(the)g(cub)-5 b(e)35 b Fr([)p Fk(\000)p Fn(N)5 b(;)15 b(N)10 b Fr(])3110 1378 y Fp(2)3185 1411 y Ff(gr)-5 b(ows)37 b(faster)118 1523 y(than)434 1487 y Fj(N)p 335 1502 262 4 v 335 1556 a Fp(\(log)13 b Fj(N)7 b Fp(\))555 1537 y Fy(\024)606 1523 y Ff(.)182 1755 y Fr(7.)51 b Fq(A)34 b(one)f(dimensional)g(bir)-6 b(th)34 b(and)g(dea)-6 b(th)33 b(pr)n(ocess)f(f)n(or)i(the)f(dr)n (oplet)g(ev)n(olution)218 1917 y Fr(In)24 b(this)h(section)g(w)m(e)h (discuss)d(a)j(simple)d(one)j(dimensional)c(to)m(y)27 b(mo)s(del)d(whic)m(h)g(mimics)g(the)h(random)118 2025 y(ev)m(olution)34 b(of)h(the)g(v)m(olume)f(of)h(a)g(droplet)e(of)i(the) g(min)m(us)e(phase)h(in)f(a)i(large)g(cub)s(e)e(of)i(side)f Fn(N)44 b Fr(in)33 b Fo(Z)3625 1992 y Fj(d)118 2133 y Fr(under)40 b(a)h(Glaub)s(er)f(dynamics)g(with)f(plus)h(b)s(oundary)f (condition.)71 b(The)41 b(mo)s(del)f(go)s(es)i(as)f(follo)m(ws.)118 2241 y(Let)36 b Fn(\013)f Fr(:=)520 2205 y Fj(d)p Fi(\000)p Fp(1)p 520 2220 127 4 v 565 2272 a Fj(d)692 2241 y Fr(and)g(consider)g (a)h(birth)e(and)h(death)h(pro)s(cess)g(on)f(the)h(in)m(tegers)h(\003)d (:=)g Fk(f)p Fr(0)p Fn(;)15 b(:)g(:)g(:)j(;)d(N)3550 2208 y Fj(d)3591 2241 y Fk(g)p Fr(,)118 2349 y(rev)m(ersible)29 b(with)g(resp)s(ect)i(to)g(the)f(measure)1485 2546 y Fn(\026)p Fr(\()p Fn(x)p Fr(\))c(:=)1831 2484 y(1)p 1819 2525 69 4 v 1819 2608 a Fn(Z)1913 2546 y Fr(exp\()p Fk(\000)p Fn(x)2210 2508 y Fj(\013)2259 2546 y Fr(\))118 2735 y(and)31 b(with)f(birth)f(rate)j Fn(b)p Fr(\()p Fn(x)p Fr(\))c(=)e(\()p Fn(x)c Fk(_)e Fr(1\))1476 2702 y Fj(\013)1526 2735 y Fn(;)42 b(x)27 b(<)g(N)10 b Fr(.)43 b(By)32 b(rev)m(ersibilit)m(y)d (the)i(death)h(rate)g Fn(d)p Fr(\()p Fn(x)p Fr(\))g(is)e(giv)m(en)118 2843 y(b)m(y)969 2964 y Fn(d)p Fr(\()p Fn(x)21 b Fr(+)f(1\))26 b(:=)f(\()p Fn(x)20 b Fk(_)g Fr(1\))1710 2926 y Fj(\013)1776 2964 y Fr(exp)o(\(\()p Fn(x)h Fr(+)f(1\))2228 2926 y Fj(\013)2299 2964 y Fk(\000)g Fn(x)2442 2926 y Fj(\013)2491 2964 y Fr(\))p Fn(;)41 b(x)25 b(>)g Fr(0)118 3104 y(One)e(easily)g(c)m (hec)m(ks)i(that)f(the)g(drift)e(giv)m(en)i(b)m(y)f Fn(b)p Fr(\()p Fn(x)p Fr(\))7 b Fk(\000)g Fn(d)p Fr(\()p Fn(x)p Fr(\))25 b(is)d(negativ)m(e)j(and)e(prop)s(ortional)f(to)i Fn(\013x)3486 3071 y Fp(2)p Fj(\013)p Fi(\000)p Fp(1)118 3211 y Fr(for)j(large)f Fn(x)p Fr(.)40 b(The)26 b(connection)h(with)e (the)i(ev)m(olution)f(of)h(a)g(large)g(droplet)f(of)h(the)g(min)m(us)e (phase)h(under)118 3319 y(the)k(Glaub)s(er)f(dynamics)g(with)g(plus)f (b)s(oundary)g(condition)h(in)g(a)i(large)f(cub)s(e)f(of)i(side)e Fn(N)40 b Fr(in)29 b Fo(Z)3423 3286 y Fj(d)3489 3319 y Fr(is)g(as)118 3427 y(follo)m(ws.)51 b(The)33 b(v)-5 b(ariable)33 b Fn(x)h Fr(represen)m(ts)f(the)i(v)m(olume)e(of)h(the)g (droplet)f(at)i(time)f Fn(t)f Fr(whic)m(h)g(is)g(assumed)118 3535 y(to)28 b(form)f(a)h(compact)h(set)f(without)f(holes.)39 b(The)27 b(rate)h Fn(b)p Fr(\()p Fn(x)p Fr(\))h(should)c(then)i(b)s(e)g (in)m(terpreted)g(as)h(the)g(rate)118 3643 y(with)c(whic)m(h)g(a)i (plus)d(spin)g(just)i(outside)f(the)i(b)s(oundary)d(of)j(the)f(droplet) f(\015ips)g(to)i(min)m(us)d(one)j(and)e(gets)118 3751 y(attac)m(hed)29 b(to)f(the)g(droplet)e(while)g(the)h(rate)h Fn(d)p Fr(\()p Fn(x)p Fr(\))h(represen)m(ts)e(just)g(the)g(opp)s(osite) g(pro)s(cess)g(in)f(whic)m(h)g(a)118 3859 y(min)m(us)i(spin)g(at)j(the) f(b)s(oundary)e(\015ips)g(to)j(plus)d(one)i(and)g(gets)h(detac)m(hed)g (from)e(the)h(droplet.)40 b(Clearly)118 3967 y(b)s(oth)34 b(these)h(rates)g(should)e(b)s(e)h(prop)s(ortional)e(to)j(the)g(size)f (of)h(the)g(b)s(oundary)d(whic)m(h,)j(for)f(roundish)118 4075 y(shap)s(e,)e(is)f(of)h(order)f(of)h Fn(x)995 4042 y Fj(\013)1045 4075 y Fr(.)45 b(Finally)30 b(the)i(drift)f(comes)h (from)g(the)g(rev)m(ersibilit)m(y)d(condition)i(together)118 4183 y(with)i(the)i(fact)h(that)f(the)f(equilibrium)c(distribution)h (of)k(the)f(droplet)g(v)m(olume)g(should)f(b)s(eha)m(v)m(e)i(lik)m(e) 118 4291 y(the)g(measure)f Fn(\026)p Fr(\()p Fn(x)p Fr(\))h(ab)s(o)m(v) m(e)h(according)f(to)g(the)g(results)f(of)g(section)h(3.)54 b(Quite)34 b(nicely)f(the)i(drift)e(one)118 4399 y(gets)26 b(out)e(of)h(these)g(natural)e(h)m(yp)s(otheses)h(is)g(of)g(the)h(same) g(order)f(of)g(that)h(prescrib)s(ed)d(b)m(y)i(an)h(ev)m(olution)118 4507 y(b)m(y)30 b(mean)h(curv)-5 b(ature)1597 4609 y Fn(d)p 1581 4649 81 4 v 1581 4732 a(dt)1671 4670 y(x)1723 4633 y Fp(1)p Fj(=d)1859 4670 y Fr(=)25 b Fk(\000)2095 4609 y Fr(1)p 2036 4649 163 4 v 2036 4736 a Fn(x)2088 4710 y Fp(1)p Fj(=d)118 4840 y Fr(Our)j(goal)h(no)m(w)g(is)f(to)i (compute)f(the)h(precise)e(asymptotic)h(as)g Fn(N)36 b Fk(!)25 b(1)k Fr(of)g(the)g(in)m(v)m(erse)g(sp)s(ectral)f(gap)118 4947 y Fn(S)5 b Fr(\()p Fn(N)g(;)15 b(d)p Fr(\))39 b(and)d(logarithmic) g(Sob)s(olev)g(constan)m(t)j Fn(L)p Fr(\()p Fn(N)5 b(;)15 b(d)p Fr(\))38 b(of)f(the)h(ab)s(o)m(v)m(e)g(pro)s(cess)f(in)f(order)h (to)g(test)118 5055 y(the)31 b(accuracy)g(of)g(the)f(b)s(ounds)e(pro)m (v)m(ed)j(in)e(section)i(2.3.)118 5225 y FC(Theorem)j(7.1.)46 b Ff(F)-7 b(or)34 b(any)f Fn(d)26 b Fk(\025)f Fr(2)33 b Ff(ther)-5 b(e)33 b(exists)g(a)g(p)-5 b(ositive)34 b(c)-5 b(onstant)34 b Fn(k)28 b Fr(=)d Fn(k)s Fr(\()p Fn(d)p Fr(\))34 b Ff(such)f(that)214 5372 y Fb(\(i\))881 5337 y Fj(N)944 5313 y Fv(2)p 881 5352 98 4 v 910 5404 a Fj(k)1014 5372 y Fm(6)25 b Fn(L)p Fr(\()p Fn(N)5 b(;)15 b(d)p Fr(\))26 b Fm(6)f Fn(k)36 b(N)1695 5339 y Fp(2)1734 5372 y Fn(;)201 b Fk(8)p Fn(d)51 b Fm(>)f Fr(2)p eop %%Page: 18 18 18 17 bop 118 45 a Fp(18)1071 b(T.)30 b(BODINEA)n(U)h(AND)f(F.)g(MAR)-6 b(TINELLI)166 366 y Fb(\(ii\))881 331 y Fj(N)p 881 346 64 4 v 893 398 a(k)979 366 y Fm(6)25 b Fn(S)5 b Fr(\()p Fn(N)g(;)15 b Fr(2\))27 b Fm(6)e Fn(k)s(N)118 582 y Fb(\(iii\))749 b Fn(S)5 b Fr(\()p Fn(N)g(;)15 b(d)p Fr(\))26 b Fm(6)f Fn(k)s(;)202 b Fk(8)p Fn(d)50 b Fm(>)g Fr(3)118 772 y Ff(Pr)-5 b(o)g(of.)43 b Fr(W)-8 b(e)30 b(apply)d(the)h(metho)s(d)g(of)h (Hardy)f(inequalities)e(en)m(visaged)j(in)e([Mi]\))i(in)e(order)h(to)h (compute)118 880 y(sharp)36 b(upp)s(er)e(and)i(lo)m(w)m(er)h(b)s(ounds) e(on)h(the)h(quan)m(tities)f(of)h(in)m(terest.)59 b(W)-8 b(e)38 b(b)s(egin)d(with)h(the)g(in)m(v)m(erse)118 988 y(sp)s(ectral)30 b(gap)g(and)g(de\014ne)1110 1253 y Fn(B)1179 1267 y Fp(+)1238 1253 y Fr(\()p Fn(i)p Fr(\))c(:=)f(sup)1495 1329 y Fj(x>i)1623 1152 y Fl(\020)1776 1139 y Fj(x)1730 1166 y Fl(X)1693 1362 y Fj(y)r Fp(=)p Fj(i)p Fp(+1)2067 1191 y Fr(1)p 1924 1232 331 4 v 1924 1315 a Fn(\026)p Fr(\()p Fn(y)s Fr(\))p Fn(b)p Fr(\()p Fn(y)s Fr(\))2280 1152 y Fl(\021)2350 1166 y(X)2349 1362 y Fj(y)r Fi(\025)p Fj(x)2496 1253 y Fn(\026)p Fr(\()p Fn(y)s Fr(\))1110 1586 y Fn(B)1179 1600 y Fi(\000)1238 1586 y Fr(\()p Fn(i)p Fr(\))h(:=)f(sup)1495 1663 y Fj(x)g Fr(3,)36 b(while)c Fn(B)921 3705 y Fi(\000)980 3691 y Fr(\()p Fn(i)p Fr(\))h Fk(\031)e Fn(i)1247 3658 y Fp(1)p Fi(\000)p Fp(2)p Fj(\013)1437 3691 y Fr(exp\()p Fn(i)1642 3658 y Fj(\013)1692 3691 y Fr(\))k(for)f(an)m(y)g Fn(d)p Fr(.)53 b(Notice)35 b(that)f Fn(B)2763 3705 y Fp(+)2822 3691 y Fr(\()p Fn(i)p Fr(\))f(=)e Fk(1)j Fr(if)f Fn(i)f(<)f Fr(0)k(and)118 3801 y Fn(B)187 3815 y Fi(\000)246 3801 y Fr(\()p Fn(i)p Fr(\))26 b(=)f Fk(1)31 b Fr(if)e Fn(i)c(>)g(N)909 3768 y Fj(d)950 3801 y Fr(.)218 3909 y(W)-8 b(e)31 b(no)m(w)g(turn)e (to)i(the)g(analysis)e(of)h(the)h(logarithmic)e(Sob)s(olev)g(constan)m (t.)42 b(W)-8 b(e)32 b(de\014ne)769 4174 y Fn(A)837 4188 y Fp(+)897 4174 y Fr(\()p Fn(i)p Fr(\))26 b(:=)f(sup)1154 4250 y Fj(x>i)1282 4073 y Fl(\020)1435 4060 y Fj(x)1389 4087 y Fl(X)1351 4283 y Fj(y)r Fp(=)p Fj(i)p Fp(+1)1726 4112 y Fr(1)p 1583 4153 331 4 v 1583 4236 a Fn(\026)p Fr(\()p Fn(y)s Fr(\))p Fn(b)p Fr(\()p Fn(y)s Fr(\))1939 4073 y Fl(\021)2008 4174 y Fr(log)2126 4073 y Fl(\020)2377 4112 y Fr(1)p 2190 4153 421 4 v 2190 4168 a Fl(P)2286 4263 y Fj(y)r Fi(\025)p Fj(x)2437 4236 y Fn(\026)p Fr(\()p Fn(y)s Fr(\))2620 4073 y Fl(\021)2690 4087 y(X)2690 4283 y Fj(y)r Fi(\025)p Fj(x)2837 4174 y Fn(\026)p Fr(\()p Fn(y)s Fr(\))769 4507 y Fn(A)837 4521 y Fi(\000)897 4507 y Fr(\()p Fn(i)p Fr(\))h(:=)f(sup)1154 4584 y Fj(x