Content-Type: multipart/mixed; boundary="-------------0401070054274" This is a multi-part message in MIME format. ---------------0401070054274 Content-Type: text/plain; name="04-4.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-4.comments" bodineau@math.jussieu.fr rhs@math.ucla.edu shlosman@cpt.univ-mrs.fr ---------------0401070054274 Content-Type: text/plain; name="04-4.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-4.keywords" crystals, SOS model, facets, interfaces ---------------0401070054274 Content-Type: application/postscript; name="bss.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bss.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: bss.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips bss -o bss.ps %DVIPSParameters: dpi=600, compressed 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b(Usually)-8 b(,)31 b(to)h(pro)m(v)m(e)h(a)e(result)h (of)g(suc)m(h)h(kind,)f(one)g(has)h(to)e(obtain)g(the)324 3678 y(lo)m(w)m(er)37 b(estimate)f(on)h(the)h(probabilit)m(y)d(of)i (\\nice")f(b)s(eha)m(vior)h(of)g(the)h(random)e(surface)324 3798 y(w)m(e)45 b(are)f(in)m(terested)h(in,)h(together)e(with)g(the)g (upp)s(er)h(estimate)e(on)h(the)h(probabilit)m(y)324 3919 y(of)i(its)h(\\ugly")e(b)s(eha)m(vior.)90 b(Here)48 b(the)h(latter)d(is)i(easy)-8 b(,)52 b(while)47 b(the)i(former)e(is)g (v)m(ery)324 4039 y(hard,)29 b(since)g(this)e(is)h(the)h(question)f(ab) s(out)g(the)g(t)m(ypical)g(b)s(eha)m(vior)g(of)f(the)i(collection)d(of) 324 4159 y(con)m(tours)35 b(whic)m(h)h(are)f(strongly)f(in)m (teracting,)g(see)i([FPS].)51 b(W)-8 b(e)35 b(manage)f(to)g(establish) 324 4280 y(our)j(result)h(b)m(y)h(ha)m(ving)e(only)h(the)g(upp)s(er)g (estimate.)59 b(This)38 b(is)f(b)s(oth)g(the)i(strong)e(and)324 4400 y(the)e(w)m(eak)h(p)s(oin)m(t)d(of)h(our)h(approac)m(h;)h(w)m(e)g (pro)m(v)m(e)f(our)g(theorem,)g(but)g(w)m(e)g(do)g(not)f(ha)m(v)m(e)324 4521 y(the)f(tec)m(hnique)h(to)e(obtain)f(the)i(complete)f(con)m(trol)g (o)m(v)m(er)i(our)e(mo)s(del.)1918 5251 y(5)p eop %%Page: 6 6 6 5 bop 324 548 a Fu(2)161 b(Statemen)l(t)51 b(of)j(the)f(Main)i (Result)324 767 y Fv(Let)34 b Fs(')c Fv(=)g Ft(f)o Fs(')813 782 y Fr(s)878 767 y Ft(2)e Fl(Z)p Ft(g)j Fv(b)s(e)j(an)g(in)m(teger)g (v)-5 b(alued)33 b(random)g(\014eld,)h(de\014ned)i(for)d Fs(s)d Ft(2)g Fl(Z)3352 731 y Fq(2)3389 767 y Fs(:)k Fv(Its)324 887 y(distribution)c(is)j(de\014ned)h(b)m(y)f(the)g (\\Solid-on-Solid")28 b(Hamiltonian)1418 1115 y Fs(H)c Fv(\()p Fs(')p Fv(\))j(=)1843 1020 y Fp(X)1803 1242 y Fr(s;t)p Fo(2)p Fn(Z)1978 1218 y Fg(2)2007 1242 y Fq(:)1794 1313 y Fo(j)o Fr(s)p Fo(\000)p Fr(t)p Fo(j)p Fq(=1)2053 1115 y Ft(j)o Fs(')2144 1130 y Fr(s)2203 1115 y Ft(\000)c Fs(')2367 1130 y Fr(t)2396 1115 y Ft(j)17 b Fs(:)324 1522 y Fv(Namely)-8 b(,)38 b(let)f(\003)f Ft(\032)g Fl(Z)1139 1486 y Fq(2)1214 1522 y Fv(b)s(e)i(a)f(\014nite)g(b)s(o)m(x,)j Ft(j)o Fv(\003)p Ft(j)c Fs(<)g Ft(1)p Fs(;)h Fv(the)h(con\014guration)f (\(b)s(oundary)324 1643 y(condition\))32 b Fs( )854 1658 y Fo(\001)911 1643 y Fv(b)s(e)h(giv)m(en)h(outside)f(\003)p Fs(;)g Fv(and)g(the)h(parameter)f Fs(\014)h(>)29 b Fv(0)k(\(in)m(v)m (erse)h(temp)s(er-)324 1763 y(ature\))e(is)h(\014xed.)44 b(Then)34 b(w)m(e)f(de\014ne)h(the)f(distribution)e Fs(Q)2445 1778 y Fr(\014)s(;)p Fq(\003)p Fr(; )2661 1763 y Fv(on)i(the)g (con\014gurations)324 1883 y(\012)394 1898 y Fq(\003)475 1883 y Fv(=)28 b Ft(f)o Fs(')692 1898 y Fr(s)757 1883 y Fv(:)f Fs(s)h Ft(2)g Fv(\003)p Ft(g)k Fv(b)m(y)894 2126 y Fs(Q)971 2141 y Fr(\014)s(;)p Fq(\003)p Fr(; )1172 2126 y Fv(\()p Fs(')p Fv(\))27 b(=)h(exp)1608 2016 y Fp(n)1675 2126 y Ft(\000)p Fs(\014)6 b(H)1894 2141 y Fq(\003)1963 2016 y Fp(\020)2023 2126 y Fs(')2114 2012 y Fp(\014)2114 2072 y(\014)2114 2131 y(\014)2175 2126 y Fs( )2242 2016 y Fp(\021)q(o)2417 2126 y Fs(=)33 b(Z)23 b Fv(\()p Fs(\014)6 b(;)17 b Fv(\003)p Fs(;)g( )t Fv(\))e Fs(:)446 b Fv(\(1\))324 2363 y(Here)1004 2578 y Fs(H)1085 2593 y Fq(\003)1154 2467 y Fp(\020)1214 2578 y Fs(')1305 2463 y Fp(\014)1305 2523 y(\014)1305 2583 y(\014)1366 2578 y Fs( )1433 2467 y Fp(\021)1520 2578 y Fv(=)1711 2483 y Fp(X)1624 2705 y Fr(s)p Fo(2)p Fq(\003)p Fr(;t)p Fo(2)p Fn(Z)1895 2682 y Fg(2)1923 2705 y Fq(:)1662 2777 y Fo(j)p Fr(s)p Fo(\000)p Fr(t)p Fo(j)o Fq(=1)1960 2578 y Ft(j)o Fv(\()p Fs(')22 b Ft(_)h Fs( )t Fv(\))2305 2607 y Fr(s)2364 2578 y Ft(\000)f Fv(\()p Fs(')g Ft(_)h Fs( )t Fv(\))2781 2607 y Fr(t)2810 2578 y Ft(j)17 b Fs(;)324 2981 y Fv(\()p Fs(')22 b Ft(_)g Fs( )t Fv(\))641 3010 y Fr(t)698 2981 y Fv(equals)28 b(to)f Fs(')1167 2996 y Fr(t)1224 2981 y Fv(for)g Fs(t)h Ft(2)g Fv(\003)g(and)f(to)g Fs( )1982 2996 y Fr(t)2040 2981 y Fv(for)g Fs(t)39 b(=)-60 b Ft(2)28 b Fv(\003)p Fs(;)f Fv(and)h(the)g(partition)d(function)324 3102 y Fs(Z)e Fv(\()p Fs(\014)6 b(;)17 b Fv(\003)p Fs(;)g( )t Fv(\))31 b(is)h(a)h(normalizing)c(factor,)j(making)f(\(1\))h(a)h (probabilit)m(y)d(distribution.)470 3222 y(Our)f(mo)s(del)f(of)h(the)g (crystal)g(will)e(b)s(e)j(the)f(distribution)f(obtained)g(from)g Fs(Q)3245 3237 y Fr(\014)s(;)p Fq(\003)p Fr(; )3459 3222 y Fv(b)m(y)324 3343 y(a)k(suitable)g(conditioning.)41 b(Namely)-8 b(,)31 b(w)m(e)j(will)c(consider)j(the)g(case)h(when)469 3566 y Ft(\017)49 b Fv(\003)27 b(=)h(\003)835 3581 y Fr(N)929 3566 y Fv(=)g Ft(f)p Fs(s)f Fv(:)h(1)f Ft(\024)i Fs(s)1439 3581 y Fr(i)1494 3566 y Ft(\024)g Fs(N)5 b(;)17 b(i)27 b Fv(=)h(1)p Fs(;)17 b Fv(2)p Ft(g)31 b Fv(is)i(a)f(square)i Fs(N)e Ft(\002)23 b Fs(N)5 b(;)469 3768 y Ft(\017)49 b Fs( )631 3783 y Fo(\001)683 3768 y Ft(\021)28 b Fv(0)p Fs(;)469 3970 y Ft(\017)49 b Fv(the)33 b(v)m(olume)f(constrain)m(t)1570 4185 y Fl(V)1636 4200 y Fr(N)1709 4185 y Fv(\()p Fs(')p Fv(\))c(=)1980 4091 y Fp(X)1988 4302 y Fr(s)p Fo(2)p Fq(\003)2141 4185 y Fs(')2205 4200 y Fr(s)2269 4185 y Ft(\025)g Fs(\025N)2519 4144 y Fq(3)3437 4185 y Fv(\(2\))568 4489 y(is)k(imp)s(osed,)f(with)i Fs(\025)27 b(>)h Fv(0)k(\014xed.)470 4734 y(W)-8 b(e)29 b(denote)h(the)f(conditional)d(distribution)g Fs(Q)2213 4749 y Fr(\014)s(;)p Fq(\003)2325 4760 y Fm(N)2383 4749 y Fr(; )r Fq(=0)2562 4623 y Fp(\020)2621 4734 y Fs(')2713 4619 y Fp(\014)2713 4679 y(\014)2713 4739 y(\014)2774 4734 y Fl(V)2840 4749 y Fr(N)2913 4734 y Fv(\()p Fs(')p Fv(\))i Ft(\025)g Fs(\025N)3331 4698 y Fq(3)3371 4623 y Fp(\021)3459 4734 y Fv(b)m(y)324 4882 y Fs(P)387 4897 y Fr(\014)s(;N)533 4882 y Fv(\()p Fs(')p Fv(\).)42 b(W)-8 b(e)28 b(do)h(not)f(k)m(eep)h Fs(\025)f Fv(in)g(this)g(notation,)f (since)i(it)e(will)f(b)s(e)i(\014xed)i(throughout)324 5002 y(the)k(pap)s(er.)46 b(W)-8 b(e)34 b(will)d(use)j(the)g(notation)e Fs(Q)1967 5017 y Fr(\014)s(;N)2131 5002 y Fv(for)g(the)i(unconditional) e(distribution)1918 5251 y(6)p eop %%Page: 7 7 7 6 bop 324 548 a Fs(Q)401 563 y Fr(\014)s(;)p Fq(\003)513 574 y Fm(N)570 563 y Fr(; )r Fq(=0)749 548 y Fv(\()p Ft(\001)p Fv(\))16 b Fs(:)32 b Fv(W)-8 b(e)31 b(de\014ne)h(the)g (crystal)f Fc(C)17 b Fv(\()p Fs(')p Fv(\))30 b(to)h(b)s(e)g(the)h(b)s (o)s(dy)f(b)s(elo)m(w)g(the)g(graph)g(of)324 668 y Fs(')c Fv(:)996 854 y Fc(C)17 b Fv(\()p Fs(')p Fv(\))27 b(=)1346 774 y Fp(\010)1404 854 y Fv(\()p Fs(s;)17 b(h)p Fv(\))27 b Ft(2)h Fl(Z)1816 813 y Fq(3)1881 854 y Fv(:)g Fs(s)f Ft(2)h Fv(\003)p Fs(;)17 b Fv(0)27 b Ft(\024)h Fs(h)g Ft(\024)g Fs(')17 b Fv(\()p Fs(s)p Fv(\))2788 774 y Fp(\011)2862 854 y Fs(:)470 1041 y Fv(T)-8 b(o)35 b(form)m(ulate)e(our)h(results)i (ab)s(out)e(the)h(facets)g(w)m(e)h(ha)m(v)m(e)g(to)e(in)m(tro)s(duce)h (the)g(lev)m(el)324 1161 y(sets.)44 b(So)31 b(for)g(ev)m(ery)i Fs(')e Fv(and)g(ev)m(ery)i(in)m(teger)e Fs(l)f(>)e Fv(0)j(w)m(e)h (denote)g(b)m(y)g Fs(D)19 b Fv(\()p Fs(';)e(l)r Fv(\))31 b(the)h(subset)324 1281 y(of)e(all)e(sites)j Fs(s)f Fv(in)g(\003)p Fs(;)g Fv(where)i Fs(')17 b Fv(\()o Fs(s)p Fv(\))28 b Ft(\025)g Fs(l)r(:)i Fv(W)-8 b(e)31 b(iden)m(tify)f Fs(D)19 b Fv(\()p Fs(';)e(l)r Fv(\))30 b(with)h(the)f(union)g(of)g(the)324 1402 y(closed)40 b(unit)g(squares)h(cen)m(tered)h(at)e(the)h(corresp)s (onding)f(p)s(oin)m(ts)g Fs(s:)g Fv(The)h(connected)324 1522 y(comp)s(onen)m(ts)d(of)e(the)i(top)s(ological)c(b)s(oundary)j(of) g Fs(D)19 b Fv(\()p Fs(';)e(l)r Fv(\))37 b(will)e(b)s(e)j(called)e(con) m(tours.)324 1642 y(The)f(set)f(of)f(all)f(con)m(tours)j(will)c(b)s(e)j (denoted)h(b)m(y)g(\001)17 b(\()p Fs(';)g(l)r Fv(\))f Fs(:)34 b Fv(The)h(sets)g Fs(D)19 b Fv(\()p Fs(';)e(l)r Fv(\))33 b(can)h(b)s(e)324 1763 y(disconnected;)d(w)m(e)d(denote)h(b)m (y)g Fs(D)1583 1778 y Fr(i)1627 1763 y Fv(\()p Fs(';)17 b(l)r Fv(\))28 b(the)g(collections)e(of)i(connected)h(comp)s(onen)m(ts) 324 1883 y(of)44 b Fs(D)20 b Fv(\()p Fs(';)d(l)r Fv(\))f Fs(;)45 b(i)k Fv(=)g(1)p Fs(;)17 b Fv(2)p Fs(;)g(:::)44 b Fv(whic)m(h)h(are)g(m)m(utually)f(external.)80 b(They)47 b(will)c(b)s(e)i(called)324 2004 y Fh(sections)p Fv(.)k(By)35 b Fs(@)5 b(D)1075 2019 y Fr(i)1120 2004 y Fv(\()p Fs(';)17 b(l)r Fv(\))35 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Fv(Our)g(initial)c(h)m(yp)s(othesis) 324 3838 y(w)m(as)32 b(that)f(the)h(lev)m(el)f(set)h Fh(F)1329 3853 y Fq(1)1385 3838 y Fv(\()p Fs(')p Fv(\))f({)g(the)h (\\First)e(F)-8 b(acet")31 b({)g(is)g(the)h(facet)f(sough)m(t,)i(in)d (the)324 3959 y(sense)e(that)f Ft(j)p Fh(F)872 3974 y Fq(1)928 3959 y Fv(\()p Fs(')p Fv(\))p Ft(j)g(\025)h Fv(\(1)22 b Ft(\000)h Fs(a)17 b Fv(\()p Fs(\014)6 b Fv(\))o(\))17 b Fs(N)1784 3923 y Fq(2)1824 3959 y Fs(:)27 b Fv(Ho)m(w)m(ev)m(er)i(at) d(presen)m(t)j(w)m(e)f(cannot)f(pro)m(v)m(e)h(nor)324 4079 y(dispro)m(v)m(e)37 b(this)f(statemen)m(t,)i(and)e(w)m(e)i(think)e (that)g(it)f(is)h(not)g(v)-5 b(alid.)53 b(In)37 b(particular)e(w)m(e) 324 4200 y(cannot)h(rule)g(out)g(the)h(case)g(of)f Ft(j)o Fh(F)1617 4215 y Fq(1)1674 4200 y Fv(\()p Fs(')p Fv(\))p Ft(j)d(\030)i Fs(N)2075 4163 y Fq(2)2115 4200 y Fs(=)p Fv(2)p Fs(;)h Fv(sa)m(y)-8 b(.)55 b(Still,)35 b(w)m(e)i(can)f(sho)m(w)i (that)e(a)324 4320 y(sharply)c(lo)s(calized)f(jump)h(of)g(the)h (function)f Ft(j)p Fs(D)19 b Fv(\()p Fs(';)e(l)r Fv(\))p Ft(j)32 b Fv(happ)s(ens)h(for)g(t)m(ypical)e Fs(')p Fv(-s:)324 4509 y Fh(Theorem)37 b(3)49 b Ff(Supp)-5 b(ose)43 b(the)g(temp)-5 b(er)g(atur)g(e)44 b(is)f(low)g(enough.)71 b(Then)43 b(for)g(the)h(typic)-5 b(al)324 4630 y(crystal)35 b Fe(the)41 b(\\Se)-6 b(c)g(ond)41 b(F)-9 b(ac)j(et")42 b(is)g(lar)-6 b(ge)p Ff(:)815 4816 y Fs(P)878 4831 y Fr(\014)s(;N)1025 4735 y Fp(\010)1083 4816 y Fs(')27 b Fv(:)h Ft(j)p Fh(F)1327 4831 y Fq(2)1383 4816 y Fv(\()p Fs(')p Fv(\))p Ft(j)f(\025)h Fv(\(1)22 b Ft(\000)h Fs(a)17 b Fv(\()p Fs(\014)6 b Fv(\))o(\))17 b Fs(N)2239 4775 y Fq(2)2279 4735 y Fp(\011)2365 4816 y Ft(!)27 b Fv(1)34 b Ff(as)h Fs(N)j Ft(!)27 b(1)p Fs(;)324 5002 y Ff(with)34 b(some)h Fs(a)17 b Fv(\()o Fs(\014)6 b Fv(\))28 b Ft(!)f Fv(0)34 b Ff(as)h Fs(\014)e Ft(!)28 b(1)p Fs(:)1918 5251 y Fv(7)p eop %%Page: 8 8 8 7 bop 470 548 a Fv(This)32 b(result)f(means)g(that)g(the)h(crystal)f Fc(C)17 b Fv(\()p Fs(')p Fv(\))31 b(indeed)g(has)h(a)f(horizon)m(tal)f (facet,)h(in)324 668 y(the)37 b(follo)m(wing)d(sense:)54 b(the)37 b(lev)m(el)g(of)f(heigh)m(t)h Fs(L)17 b Fv(\()p Fs(')p Fv(\))24 b Ft(\000)i Fv(1)37 b(is)f(almost)f(\014lled)h(with)g (sites,)324 789 y(whereas)k(at)e(the)h(lev)m(els)g(ab)s(o)m(v)m(e)h Fs(L)17 b Fv(\()p Fs(')p Fv(\))38 b(only)g(few)h(sites)g(b)s(elong)f (to)g(the)h(crystal.)62 b(W)-8 b(e)324 909 y(do)37 b(not)g(kno)m(w)h (what)f(happ)s(ens)h(at)f(the)g(lev)m(el)g Fs(L)17 b Fv(\()p Fs(')p Fv(\))f Fs(;)37 b Fv(i.e.)57 b(ho)m(w)37 b(big)f(the)i(First)e(F)-8 b(acet)324 1029 y(really)31 b(is.)470 1150 y(It)36 b(is)g(kno)m(wn)i(that)e(the)g(SOS-mo)s(del)e (undergo)s(es)k(the)e(roughening)g(transition)e(in)324 1270 y(temp)s(erature,)53 b(see)d([F)-8 b(rSp].)94 b(A)m(t)49 b(lo)m(w)g(temp)s(eratures)h(the)f(\(unconstrained\))h(SOS-)324 1391 y(mo)s(del)42 b(\(without)h(condition)f(\(2\)\))h(with)g(zero)h(b) s(oundary)g(conditions)f(is)g(lo)s(calized,)324 1511 y(while)h(at)h(high)f(temp)s(eratures)i(it)e(div)m(erges)i (logarithmically)40 b(with)45 b Fs(N)10 b Fv(.)82 b(It)45 b(is)g(rea-)324 1631 y(sonable)39 b(to)g(conjecture)h(that)g(the)f (roughening)g(temp)s(erature)g Fs(T)2816 1595 y Fr(S)t(O)r(S)2802 1656 y(r)3008 1631 y Fv(is)g(critical)e(for)324 1752 y(our)f(problem)e(as)i(w)m(ell.)53 b(In)37 b(particular,)e(it)g(will)e (mean)j(that)f(for)h(ev)m(ery)i(temp)s(erature)324 1872 y Fs(\014)385 1836 y Fo(\000)p Fq(1)506 1872 y Fs(>)28 b(T)681 1836 y Fr(S)t(O)r(S)667 1897 y(r)834 1872 y Fs(;)k Fv(ev)m(ery)j Fs(")27 b(>)g Fv(0)33 b(and)f(for)g(ev)m(ery)j(pair)c Fs(m;)17 b(n)33 b Fv(of)f(in)m(tegers)751 2149 y Fs(P)814 2164 y Fr(\014)s(;N)961 2009 y Fp(\032)1035 2149 y Fs(')c Fv(:)1192 2082 y Ft(j)o Fs(D)20 b Fv(\()p Fs(';)d(L)g Fv(\()o Fs(')p Fv(\))22 b(+)g Fs(m)p Fv(\))p Ft(j)p 1192 2126 768 4 v 1205 2218 a(j)o Fs(D)d Fv(\()p Fs(';)e(L)g Fv(\()p Fs(')p Fv(\))22 b Ft(\000)h Fs(n)p Fv(\))p Ft(j)1997 2149 y Fs(<)k Fv(1)22 b Ft(\000)h Fs(")2317 2009 y Fp(\033)2419 2149 y Ft(!)k Fv(0)32 b(as)h Fs(N)38 b Ft(!)28 b(1)p Fs(;)319 b Fv(\(4\))324 2421 y(for)26 b(an)m(y)h(v)-5 b(alue)26 b(of)g(the)h(parameter)f Fs(a)h(>)h Fv(0)p Fs(;)e Fv(used)i(in)e(the)g(de\014nition)g(of)g(the)h(lev)m(el)f(heigh) m(t)324 2541 y Fs(L)17 b Fv(\()p Fs(')p Fv(\))f Fs(:)28 b Fv(As)h(our)f(theorem)g(sho)m(ws,)i(the)f(b)s(eha)m(vior)e(opp)s (osite)h(to)g(\(4\))f(tak)m(es)j(place)d(at)h(lo)m(w)324 2662 y(temp)s(eratures,)f(and)e(w)m(e)h(conjecture)g(that)f(it)f(is)h (the)g(case)h(for)e(all)f(temp)s(eratures)j(b)s(elo)m(w)324 2782 y Fs(T)395 2746 y Fr(S)t(O)r(S)381 2807 y(r)547 2782 y Fs(:)470 2902 y Fv(T)-8 b(o)38 b(form)m(ulate)e(the)j(No)e (Hairs)h(theorem)f(w)m(e)i(in)m(tro)s(duce)f(the)g(b)s(oundary)h Fs(@)5 b Fh(F)3376 2917 y Fr(i)3422 2902 y Fv(\()p Fs(')p Fv(\))324 3023 y(of)30 b(the)g Fs(i)p Fv(-th)g(facet)h(to)e(b)s(e)i (just)f(the)h(b)s(oundary)g Fs(@)5 b(D)20 b Fv(\()p Fs(';)d(L)g Fv(\()o Fs(')p Fv(\))22 b Ft(\000)h Fs(i)f Fv(+)g(1\).)43 b(The)31 b(theorem)324 3143 y(states)47 b(that)f(inside)g Fs(@)5 b Fh(F)1257 3158 y Fq(2)1314 3143 y Fv(\()p Fs(')p Fv(\))46 b(the)h(surface)g Fs(')f Fv(is)g(almost)e(\015at,)50 b(up)c(to)g(logarithmic)324 3264 y(excitations.)324 3492 y Fh(Theorem)37 b(4)49 b Ff(Ther)-5 b(e)37 b(exists)g Fs(\014)1511 3507 y Fq(0)1588 3492 y Ff(such)h(that)g(for)f(any)h Fs(\014)g(>)33 b(\014)2620 3507 y Fq(0)2660 3492 y Ff(,)38 b(one)f(c)-5 b(an)37 b(\014nd)g Fs(C)j(>)33 b Fv(0)324 3612 y Ff(for)h(which)g(the)h(fol)5 b(lowing)34 b(holds)379 3832 y Fs(P)442 3847 y Fr(\014)s(;N)588 3832 y Ft(f)p Fs(')27 b Fv(:)63 b Ft(9)p Fs(s)28 b Ft(2)g Fv(In)m(t)34 b(\()p Fs(@)5 b Fh(F)1364 3847 y Fq(2)1421 3832 y Fv(\()p Fs(')p Fv(\)\))16 b Fs(;)117 b Ft(j)o Fs(')1850 3847 y Fr(s)1909 3832 y Ft(\000)23 b Fs(L)p Fv(\()p Fs(')p Fv(\))p Ft(j)k Fs(>)h(C)c Fv(ln)15 b Fs(N)10 b Ft(g)28 b(!)g Fv(0)104 b Ff(as)35 b Fs(N)j Ft(!)27 b(1)p Fs(:)324 4165 y Fu(3)161 b(Zero)53 b(temp)t(erature)e(Ising)j(crystal)324 4384 y Fv(In)47 b(this)f(section)g(w)m(e)i(discuss)f(the)g(relations)e 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b(w)m(e)h(are)f(lo)s(oking)f(in)m(to)g(the)i(follo) m(wing)c(problem:)42 b(let)32 b Fs(K)39 b Fv(b)s(e)324 909 y(an)c(in)m(teger,)g(and)g(w)m(e)h(consider)f(the)g(family)2018 884 y(~)2005 909 y Ft(V)2066 924 y Fr(K)2169 909 y Fv(of)f(all)f (subsets)k Fs(V)53 b Ft(\032)32 b Fl(Z)3048 873 y Fq(3)3120 909 y Fv(con)m(taining)324 1029 y(precisely)g Fs(K)40 b Fv(sites.)j(F)-8 b(or)32 b(ev)m(ery)i Fs(V)53 b Fv(w)m(e)34 b(de\014ne)f(the)g(v)-5 b(alue)32 b Ft(j)o Fs(@)5 b(V)23 b Ft(j)31 b Fv(to)h(b)s(e)h(the)f(n)m(um)m(b)s(er)h(of)324 1150 y(plaquettes)e(\(of)e(the)h(dual)g(lattice\))e(in)h(the)i(b)s (oundary)f(of)g Fs(V)21 b Fv(;)30 b(in)f(other)i(w)m(ords,)g Ft(j)p Fs(@)5 b(V)22 b Ft(j)30 b Fv(is)324 1270 y(the)36 b(area)h(of)e(the)i(surface)g Fs(@)5 b(V)g(:)38 b Fv(W)-8 b(e)36 b(de\014ne)i Ft(V)2030 1285 y Fr(K)2132 1270 y Ft(\032)2257 1245 y Fv(~)2244 1270 y Ft(V)2305 1285 y Fr(K)2409 1270 y Fv(to)e(b)s(e)h(the)f(subset)i(consisting)324 1391 y(of)32 b Ff(minimal)44 b Fs(V)21 b Fv(-s:)1287 1611 y Fs(V)50 b Ft(2)28 b(V)1549 1626 y Fr(K)1645 1611 y Ft(,)f(j)p Fs(@)5 b(V)22 b Ft(j)28 b Fv(=)59 b(min)2094 1689 y Fr(W)10 b Fo(2)2227 1672 y Fq(~)2218 1689 y Fo(V)2262 1700 y Fm(K)2336 1611 y Ft(j)p Fs(@)5 b(W)14 b Ft(j)j Fs(:)324 1887 y Fv(In)38 b(the)g(follo)m(wing)d(w)m(e)k(will)c(not)j (distinguish)e(the)i(elemen)m(ts)g(of)f Ft(V)2811 1902 y Fr(K)2917 1887 y Fv(whic)m(h)i(di\013er)e(b)m(y)324 2007 y(translation)i(only)-8 b(,)43 b(th)m(us)f Ft(V)1354 2022 y Fr(K)1464 2007 y Fv(b)s(ecomes)f(a)g(\014nite)g(set,)j(so)e(w)m (e)g(can)f(endo)m(w)i(it)d(with)h(a)324 2128 y(uniform)31 b(probabilit)m(y)f(distribution.)42 b(W)-8 b(e)33 b(w)m(an)m(t)h(to)e (tak)m(e)i Fs(K)39 b Fv(to)33 b(in\014nit)m(y)f(and)g(to)h(lo)s(ok)324 2248 y(on)41 b(the)g(t)m(ypical)g(prop)s(erties)g(of)f(the)i(crystal)f (shap)s(es.)70 b(Ho)m(w)m(ev)m(er)43 b(what)f(w)m(e)g(will)c(see)324 2368 y(dep)s(ends)j(on)f(the)g(v)-5 b(alues)40 b(of)f Fs(K)r(:)g Fv(In)h(case)h Fs(K)47 b Fv(=)40 b Fs(M)2286 2332 y 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3640 y Fq(1)2281 3625 y Ft(\000)22 b Fs(D)s Fv(\))g(+)g Fs(p)17 b Fv(\()p Fs(V)2783 3640 y Fq(2)2844 3625 y Fv(+)22 b Fs(D)s Fv(\))16 b Fs(:)324 3840 y Fh(Pro)s(of.)43 b Fv(W)-8 b(e)33 b(will)d(sho)m(w)k(that)e(the)h(di\013erence)1081 4048 y Fs(p)17 b Fv(\()o Fs(V)1241 4063 y Fq(1)1303 4048 y Ft(\000)22 b Fs(D)s Fv(\))g(+)g Fs(p)17 b Fv(\()p Fs(V)1805 4063 y Fq(2)1866 4048 y Fv(+)22 b Fs(D)s Fv(\))g Ft(\000)g Fs(p)17 b Fv(\()p Fs(V)2368 4063 y Fq(1)2407 4048 y Fv(\))22 b Ft(\000)h Fs(p)17 b Fv(\()o Fs(V)2727 4063 y Fq(2)2767 4048 y Fv(\))324 4256 y(is)29 b(of)g(the)i(order)e(of)h Fs(p)17 b Fv(\()o Fs(V)1213 4271 y Fq(1)1253 4256 y Fv(\))f(+)g Fs(p)h Fv(\()p Fs(V)1560 4271 y Fq(2)1599 4256 y Fv(\))30 b(and)g(negativ)m(e.)42 b(Since)30 b(the)g(function)g Fs(p)17 b Fv(\()o Fs(x)p Fv(\))30 b(equals)324 4377 y(appro)m(ximately) h(to)h(4)1134 4305 y Ft(p)p 1217 4305 56 4 v 72 x Fs(x)h Fv({)f(more)g(precisely)-8 b(,)1419 4585 y(4)1468 4509 y Ft(p)p 1551 4509 V 76 x Fs(x)28 b Ft(\024)g Fs(p)17 b Fv(\()o Fs(x)p Fv(\))28 b Fs(<)g Fv(4)2116 4509 y Ft(p)p 2199 4509 V 76 x Fs(x)22 b Fv(+)g(4)p Fs(;)938 b Fv(\(13\))324 4794 y({)32 b(it)g(is)g(enough)h(to)f(sho)m(w)i(that)e(the)h (di\013erence)1164 4910 y Fp(p)p 1264 4910 302 4 v 92 x Fs(V)1321 5017 y Fq(1)1382 5002 y Ft(\000)23 b Fs(D)i Fv(+)1686 4910 y Fp(p)p 1786 4910 301 4 v 92 x Fs(V)1843 5017 y Fq(2)1904 5002 y Fv(+)d Fs(D)j Ft(\000)2207 4910 y Fp(p)p 2307 4910 97 4 v 92 x Fs(V)2364 5017 y Fq(1)2426 5002 y Ft(\000)2525 4910 y Fp(p)p 2625 4910 V 92 x Fs(V)2682 5017 y Fq(2)1894 5251 y Fv(13)p eop %%Page: 14 14 14 13 bop 324 548 a Fv(is)32 b(of)g(the)h(order)g(of)1067 471 y Ft(p)p 1150 471 97 4 v 77 x Fs(V)1207 563 y Fq(1)1268 548 y Fv(+)1366 471 y Ft(p)p 1449 471 V 77 x Fs(V)1506 563 y Fq(2)1578 548 y Fv(and)g(negativ)m(e.)44 b(Let)32 b(us)h(rewrite)g(the)g(di\013erence)g(as)956 758 y Fp(p)p 1056 758 V 92 x Fs(V)1113 865 y Fq(1)1169 679 y Fp( )1248 699 y(r)p 1347 699 287 4 v 1347 850 a Fv(1)22 b Ft(\000)1534 782 y Fs(D)p 1528 827 97 4 v 1528 918 a(V)1585 933 y Fq(1)1656 850 y Ft(\000)h Fv(1)1805 679 y Fp(!)1906 850 y Fv(+)2004 758 y Fp(p)p 2103 758 V 2103 850 a Fs(V)2160 865 y Fq(2)2216 679 y Fp( )2295 699 y(r)p 2395 699 286 4 v 151 x Fv(1)f(+)2580 782 y Fs(D)p 2574 827 97 4 v 2574 918 a(V)2631 933 y Fq(2)2702 850 y Ft(\000)h Fv(1)2851 679 y Fp(!)324 1169 y Fv(and)32 b(use)i(the)f(T)-8 b(a)m(ylor)32 b(expansion)h(of)f(the)h(function)2270 1090 y Ft(p)p 2353 1090 225 4 v 79 x Fv(1)22 b(+)g Fs(x)q(:)32 b Fv(W)-8 b(e)33 b(get)830 1379 y Fp(p)p 929 1379 97 4 v 929 1470 a Fs(V)986 1485 y Fq(1)1042 1300 y Fp( )1121 1319 y(r)p 1221 1319 287 4 v 151 x Fv(1)22 b Ft(\000)1408 1403 y Fs(D)p 1401 1447 97 4 v 1401 1539 a(V)1458 1554 y Fq(1)1530 1470 y Ft(\000)g Fv(1)1678 1300 y Fp(!)1779 1470 y Fv(+)1877 1379 y Fp(p)p 1977 1379 V 91 x Fs(V)2034 1485 y Fq(2)2090 1300 y Fp( )2169 1319 y(r)p 2268 1319 286 4 v 2268 1470 a Fv(1)g(+)2453 1403 y Fs(D)p 2447 1447 97 4 v 2447 1539 a(V)2504 1554 y Fq(2)2576 1470 y Ft(\000)g Fv(1)2724 1300 y Fp(!)857 1774 y Fv(=)971 1707 y(1)p 971 1751 49 4 v 971 1843 a(2)1046 1634 y Fp(\022)1120 1774 y Ft(\000)1255 1707 y Fs(D)p 1207 1751 180 4 v 1207 1771 a Ft(p)p 1290 1771 97 4 v 77 x Fs(V)1347 1863 y Fq(1)1419 1774 y Fv(+)1574 1707 y Fs(D)p 1527 1751 180 4 v 1527 1771 a Ft(p)p 1610 1771 97 4 v 77 x Fs(V)1667 1863 y Fq(2)1716 1634 y Fp(\023)1811 1774 y Ft(\000)1921 1707 y Fv(1)p 1921 1751 49 4 v 1921 1843 a(8)1996 1634 y Fp(\022)2156 1707 y Fs(D)2240 1671 y Fq(2)p 2080 1751 276 4 v 2080 1848 a Fs(V)2137 1863 y Fq(1)2176 1771 y Ft(p)p 2259 1771 97 4 v 77 x Fs(V)2316 1863 y Fq(1)2387 1774 y Fv(+)2571 1707 y Fs(D)2655 1671 y Fq(2)p 2495 1751 276 4 v 2495 1848 a Fs(V)2552 1863 y Fq(2)2592 1771 y Ft(p)p 2675 1771 97 4 v 77 x Fs(V)2732 1863 y Fq(2)2781 1634 y Fp(\023)2876 1774 y Fv(+)g Fs(:::)333 b Fv(\(14\))324 2046 y(No)m(w,)35 b(since)g Fs(V)873 2061 y Fq(1)943 2046 y Fs(<)30 b(V)1106 2061 y Fq(2)1145 2046 y Fs(;)35 b Fv(the)f(con)m(ten)m(ts)i(of)e(all)e(the)j(o)s(dd)f (brac)m(k)m(ets)i(are)e(negativ)m(e,)h(while)324 2167 y(the)j(ev)m(en)i(co)s(e\016cien)m(ts)f(are)e(also)g(negativ)m(e,)j(so) e(the)g(di\013erence)h(is)e(negativ)m(e)h(as)g(w)m(ell.)324 2287 y(Finally)-8 b(,)35 b(since)i(all)e(the)i(v)-5 b(alues)37 b Fs(V)1587 2302 y Fq(1)1626 2287 y Fs(;)17 b(V)1727 2302 y Fq(2)1803 2287 y Fv(and)37 b Fs(D)i Fv(are)e(of)f(the)h(same)g (order,)h(the)f(second)324 2407 y(term)625 2368 y Fr(D)685 2345 y Fg(2)p 567 2384 211 4 v 567 2446 a Fr(V)608 2455 y Fg(1)642 2394 y Fo(p)p 701 2394 76 3 v 52 x Fr(V)742 2455 y Fg(1)809 2407 y Fv(+)975 2368 y Fr(D)1035 2345 y Fg(2)p 917 2384 211 4 v 917 2446 a Fr(V)958 2455 y Fg(2)993 2394 y Fo(p)p 1052 2394 76 3 v 52 x Fr(V)1093 2455 y Fg(2)1170 2407 y Fv(is)32 b(of)g(the)h(order)g(of)1913 2330 y Ft(p)p 1996 2330 97 4 v 77 x Fs(V)2053 2422 y Fq(1)2114 2407 y Fv(+)2212 2330 y Ft(p)p 2295 2330 V 77 x Fs(V)2352 2422 y Fq(2)2392 2407 y Fs(;)f Fv(and)h(the)g(pro)s(of)f (follo)m(ws.)p 3454 2407 49 49 v 324 2696 a FA(4.3)135 b(Estimates)47 b(on)e(the)g(large)h(con)l(tours)324 2881 y Fv(In)33 b(this)f(section)h(w)m(e)g(will)d(pro)m(v)m(e)k(the)f (estimate)f(\(9\))16 b Fs(:)470 3001 y Fv(F)-8 b(or)38 b(a)h(giv)m(en)g(in)m(teger)f Fs(`)g Ft(\025)h Fv(2)g(and)g(a)f (compatible)f(collection)g(of)h(large)g(con)m(tours)324 3122 y(\()p Fs(\015)5 b(;)17 b Fv(\000\))30 b(=)g(\()p Ft(f)p Fs(\015)836 3137 y Fr(i)864 3122 y Ft(g)16 b Fs(;)h Ft(f)p Fv(\000)1085 3137 y Fr(j)1121 3122 y Ft(g)p Fv(\))34 b(w)m(e)i(denote)f(b)m(y)g Fs(')30 b Ft(\030)i Fv(\()p Fs(\015)5 b(;)17 b Fv(\000)p Fs(;)g(`)p Fv(\))33 b(the)i(heigh)m(t)f (con\014gurations)g Fs(')324 3242 y Fv(whic)m(h)f(satisfy:)469 3470 y Ft(\017)49 b Fv(the)33 b(v)m(olume)f(constrain)m(t)65 b Fl(V)1629 3485 y Fr(N)1702 3470 y Fv(\()p Fs(')p Fv(\))27 b(=)1973 3396 y Fp(P)2078 3499 y Fr(s)p Fo(2)p Fq(\003)2227 3470 y Fs(')2291 3485 y Fr(s)2356 3470 y Ft(\025)h Fs(\025N)2606 3434 y Fq(3)2646 3470 y Fs(;)469 3674 y Ft(\017)49 b Fs(L)p Fv(\()p Fs(')p Fv(\))27 b(=)h Fs(`;)469 3877 y Ft(\017)49 b Fv(the)36 b(only)f(exterior)h(large)f(con)m(tours)i(on)f (the)g(lev)m(el)g(sets)h Fh(F)2759 3892 y Fq(1)2835 3877 y Fv(and)f Fh(F)3098 3892 y Fq(2)3173 3877 y Fv(are)g(giv)m(en)568 3998 y(b)m(y)d(\()p Fs(\015)5 b(;)17 b Fv(\000\).)470 4226 y(Then)34 b(w)m(e)f(ha)m(v)m(e)394 4634 y Fs(P)457 4649 y Fr(\014)s(;N)587 4634 y Fv(\()p Ft(L)p Fv(\))27 b Ft(\024)h Fs(P)927 4649 y Fr(\014)s(;N)1057 4634 y Fv(\()p Ft(H)1180 4593 y Fr(c)1215 4634 y Fv(\))22 b(+)1745 4567 y(1)p 1383 4611 773 4 v 1383 4702 a Fs(Q)1460 4717 y Fr(\014)s(;N)1590 4702 y Fv(\()32 b Fl(V)1726 4717 y Fr(N)1827 4702 y Ft(\025)c Fs(\025N)2077 4674 y Fq(3)2117 4702 y Fv(\))2190 4509 y Fr(K)5 b(N)2182 4539 y Fp(X)2194 4752 y Fr(`)p Fq(=2)2463 4539 y Fp(X)2407 4755 y Fq(\()p Fr(\015)t(;)p Fq(\000\))p Fo(2L)2826 4539 y Fp(X)2743 4755 y Fr(')p Fo(\030)p Fq(\()p Fr(\015)t(;)p Fq(\000)p Fr(;`)p Fq(\))3101 4634 y Fs(Q)3178 4649 y Fr(\014)s(;N)3308 4634 y Fv(\()p Fs(')p Fv(\))17 b Fs(:)1894 5251 y Fv(14)p eop %%Page: 15 15 15 14 bop 470 548 a Fv(F)-8 b(or)32 b(a)g(giv)m(en)h(triplet)e(\()p Fs(\015)5 b(;)17 b Fv(\000)p Fs(;)g(`)p Fv(\),)32 b(w)m(e)h(de\014ne)h (the)f(erasing)f(map)694 842 y Fs(')c Fb( )42 b Fv(^)-64 b Fs(')27 b Fv(=)1108 672 y Fp( )1202 842 y Fv(^)-64 b Fs(')1251 857 y Fr(s)1315 842 y Fv(=)28 b Fs(')1483 857 y Fr(s)1542 842 y Ft(\000)1641 748 y Fp(X)1701 958 y Fr(i)1802 842 y Fv(1)1851 858 y Fo(f)p Fr(s)p Fo(2)p Fq(In)n(t)q(\()p Fr(\015)2119 868 y Fm(i)2146 858 y Fq(\))p Fo(g)2235 842 y Ft(\000)2334 748 y Fp(X)2390 958 y Fr(j)2495 842 y Fv(1)2544 858 y Fo(f)p Fr(s)p Fo(2)p Fq(In)n(t)q(\(\000)2820 868 y Fm(j)2853 858 y Fq(\))p Fo(g)2920 672 y Fp(!)2998 985 y Fr(s)p Fo(2)p Fq(\003)3164 842 y Fs(:)197 b Fv(\(15\))324 1159 y(It)32 b(maps)h(injectiv)m(ely)f(the)h(set)g Ft(f)p Fs(')27 b Ft(\030)i Fv(\()p Fs(\015)5 b(;)17 b Fv(\000)p Fs(;)g(`)p Fv(\))p Ft(g)31 b Fv(in)m(to)h(the)h(set)891 1283 y Fp(\()1004 1453 y Fl(V)1070 1468 y Fr(N)1143 1453 y Fv(\()p Fs(')p Fv(\))28 b Ft(\025)g Fs(\025N)1561 1412 y Fq(3)1623 1453 y Ft(\000)1723 1358 y Fp(X)1783 1568 y Fr(i)1883 1453 y Ft(j)p Fv(In)m(t\()p Fs(\015)2124 1468 y Fr(i)2152 1453 y Fv(\))p Ft(j)22 b(\000)2340 1358 y Fp(X)2395 1568 y Fr(j)2500 1453 y Ft(j)p Fv(In)m(t\(\000)2751 1468 y Fr(j)2788 1453 y Fv(\))p Ft(j)2854 1283 y Fp(\))2967 1453 y Fs(:)324 1756 y Fv(Eviden)m(tly)-8 b(,)861 1975 y Fs(Q)938 1990 y Fr(\014)s(;N)1068 1975 y Fv(\()p Fs(')p Fv(\))27 b(=)h(exp)1504 1894 y Fp(\000)1572 1975 y Ft(\000)23 b Fs(\014)1749 1880 y Fp(X)1809 2090 y Fr(i)1910 1975 y Ft(j)p Fs(\015)1989 1990 y Fr(i)2016 1975 y Ft(j)f(\000)g Fs(\014)2243 1880 y Fp(X)2298 2090 y Fr(j)2403 1975 y Ft(j)p Fv(\000)2492 1990 y Fr(j)2528 1975 y Ft(j)2556 1894 y Fp(\001)2634 1975 y Fs(Q)2711 1990 y Fr(\014)s(;N)2841 1975 y Fv(\()16 b(^)-65 b Fs(')p Fv(\))17 b Fs(:)324 2277 y Fv(Therefore)33 b(w)m(e)h(ha)m(v)m(e)g(the)f(\\P)m(eierls)f (estimate")324 2552 y Fs(P)387 2567 y Fr(\014)s(;N)517 2552 y Fv(\()p Ft(L)p Fv(\))27 b Ft(\024)h Fs(P)857 2567 y Fr(\014)s(;N)987 2552 y Fv(\()p Ft(H)1110 2511 y Fr(c)1144 2552 y Fv(\))22 b(+)1311 2428 y Fr(K)5 b(N)1302 2457 y Fp(X)1315 2670 y Fr(`)p Fq(=2)1583 2457 y Fp(X)1528 2673 y Fq(\()p Fr(\015)t(;)p Fq(\000\))p Fo(2L)1799 2552 y Fv(exp)1965 2471 y Fp(\000)2032 2552 y Ft(\000)23 b Fs(\014)2209 2457 y Fp(X)2269 2667 y Fr(i)2370 2552 y Ft(j)p Fs(\015)2449 2567 y Fr(i)2476 2552 y Ft(j)f(\000)h Fs(\014)2703 2457 y Fp(X)2759 2667 y Fr(j)2863 2552 y Ft(j)p Fv(\000)2952 2567 y Fr(j)2989 2552 y Ft(j)3017 2471 y Fp(\001)3388 2552 y Fv(\(16\))1269 2899 y Ft(\002)1378 2818 y Fs(Q)1455 2833 y Fr(\014)s(;N)1585 2737 y Fp(\000)1663 2818 y Fl(V)1729 2833 y Fr(N)1803 2818 y Fv(\()p Fs(')p Fv(\))k Ft(\025)h Fs(\025N)2220 2782 y Fq(3)2282 2818 y Ft(\000)2382 2743 y Fp(P)2487 2847 y Fr(i)2532 2818 y Ft(j)p Fv(In)m(t\()p Fs(\015)2773 2833 y Fr(i)2801 2818 y Fv(\))p Ft(j)22 b(\000)2988 2743 y Fp(P)3093 2847 y Fr(j)3146 2818 y Ft(j)p Fv(In)m(t)q(\(\000)3398 2833 y Fr(j)3434 2818 y Fv(\))p Ft(j)3500 2737 y Fp(\001)p 1378 2876 2168 4 v 1998 2972 a Fs(Q)2075 2987 y Fr(\014)s(;N)2205 2892 y Fp(\000)2283 2972 y Fl(V)2349 2987 y Fr(N)2422 2972 y Fv(\()p Fs(')p Fv(\))28 b Ft(\025)g Fs(\025N)2840 2944 y Fq(3)2880 2892 y Fp(\001)3572 2899 y Fs(:)470 3171 y Fv(The)23 b(imp)s(ortan)m(t)e(quan)m(tit)m(y)i(is)e(the)i(total) e(v)m(olume)g(of)h(the)h(in)m(teriors)e(of)h(the)h(con)m(tours,)324 3292 y(th)m(us)38 b(w)m(e)f(are)g(going)f(to)g(a)m(v)m(erage)i(on)f (all)d(the)k(p)s(ossible)e(con)m(tour)h(shap)s(es)h(in)e(order)h(to)324 3412 y(retain)h(only)g(the)h(information)c(on)k(the)g(v)m(olume.)60 b(Fix)38 b Fs(V)60 b Ft(2)38 b Fv([)2686 3373 y Fr(a)p 2686 3389 38 4 v 2687 3447 a Fq(2)2733 3412 y Fs(N)2821 3376 y Fq(2)2861 3412 y Fs(;)17 b Fv(\(1)26 b Ft(\000)h Fs(a)p Fv(\))p Fs(N)3299 3376 y Fq(2)3339 3412 y Fv(])38 b(and)324 3533 y(consider)33 b(the)g(collection)d(\()p Fs(\015)1394 3548 y Fr(i)1422 3533 y Fv(\))i(of)h(the)g(large)e(con)m (tours)i(suc)m(h)h(that)1585 3663 y Fp(X)1645 3873 y Fr(i)1745 3757 y Ft(j)p Fv(In)m(t)q(\()p Fs(\015)1987 3772 y Fr(i)2014 3757 y Fv(\))p Ft(j)28 b Fv(=)f Fs(V)5 b(:)1088 b Fv(\(17\))324 4047 y(The)48 b(optimal)c(shap)s(e)k(for)e(a)h (con)m(tour)h(of)e(v)m(olume)h Fs(V)69 b Fv(is)46 b(a)h(square)h(of)f (side)g(length)324 4167 y Fs(L)28 b Fv(=)f Fs(L)17 b Fv(\()p Fs(V)22 b Fv(\))32 b(with)g(\(p)s(ossibly\))g(an)h(additional)d (la)m(y)m(er)i(of)g Fs(r)20 b Fv(\()p Fs(V)h Fv(\))33 b(sites)f(suc)m(h)i(that)885 4374 y Fs(V)49 b Fv(=)27 b Fs(L)17 b Fv(\()p Fs(V)22 b Fv(\))1332 4326 y Fq(2)1393 4374 y Fv(+)g Fs(r)e Fv(\()p Fs(V)h Fv(\))c Fs(;)212 b(r)19 b Fv(\()p Fs(V)i Fv(\))28 b Ft(2)g(f)p Fv(0)p Fs(;)17 b(:)g(:)g(:)e(;)i Fv(2)p Fs(L)g Fv(\()p Fs(V)k Fv(\))p Ft(g)c Fs(:)324 4582 y Fv(In)39 b(this)f(case)h(the)g(follo)m (wing)d(isop)s(erimetric)g(inequalit)m(y)i(holds)g(uniformly)f(o)m(v)m (er)i(the)324 4702 y(collection)30 b(\()p Fs(\015)846 4717 y Fr(i)874 4702 y Fv(\))j(whic)m(h)g(satisfy)f(the)h(v)m(olume)f (constrain)m(t)h(\(17\))27 b(:)1556 4832 y Fp(X)1616 5042 y Fr(i)1716 4927 y Ft(j)p Fs(\015)1795 4942 y Fr(i)1823 4927 y Ft(j)g(\025)h Fv(4)p Fs(L)17 b Fv(\()p Fs(V)k Fv(\))34 b Fs(:)1894 5251 y Fv(15)p eop %%Page: 16 16 16 15 bop 470 548 a Fv(Let)35 b Fs(\014)708 512 y Fo(0)761 548 y Fv(=)c Fs(\014)e Ft(\000)24 b Fv(10.)48 b(Summing)32 b(o)m(v)m(er)k(all)c(the)j(collections)e(of)h(con)m(tours)h(suc)m(h)h (that)324 594 y Fp(P)429 697 y Fr(i)474 668 y Ft(j)p Fv(In)m(t\()p Fs(\015)715 683 y Fr(i)743 668 y Fv(\))p Ft(j)27 b Fv(=)h Fs(V)21 b Fv(,)33 b(w)m(e)g(get)523 876 y Fp(X)537 1092 y Fq(\()p Fr(\015)600 1102 y Fm(i)627 1092 y Fq(\))684 971 y Fv(exp)849 890 y Fp(\000)917 971 y Ft(\000)23 b Fs(\014)1094 876 y Fp(X)1154 1086 y Fr(i)1255 971 y Ft(j)p Fs(\015)1334 986 y Fr(i)1361 971 y Ft(j)1389 890 y Fp(\001)1462 971 y Ft(\024)28 b Fv(exp)q(\()p Ft(\000)p Fs(\014)1892 929 y Fo(0)1916 971 y Fv(4)p Fs(L)17 b Fv(\()o Fs(V)22 b Fv(\)\))2257 770 y Fp(0)2257 950 y(@)2343 876 y(X)2357 1092 y Fq(\()p Fr(\015)2420 1102 y Fm(i)2447 1092 y Fq(\))2504 876 y Fp(Y)2556 1086 y Fr(i)2648 971 y Fv(exp)2814 890 y Fp(\000)2881 971 y Ft(\000)h Fv(10)p Ft(j)p Fs(\015)3158 986 y Fr(i)3185 971 y Ft(j)3213 890 y Fp(\001)3258 770 y(1)3258 950 y(A)1462 1242 y Ft(\024)28 b Fv(exp)q(\()p Ft(\000)p Fv(4)p Fs(\014)1941 1201 y Fo(0)1964 1242 y Fs(L)17 b Fv(\()p Fs(V)22 b Fv(\)\))17 b Fs(;)324 1424 y Fv(where)36 b(the)f(\014nal)e(inequalit)m(y)h(is)g (obtained)g(b)m(y)h(taking)f(in)m(to)g(accoun)m(t)h(the)g(en)m(trop)m (y)h(of)324 1544 y(a)c(single)g(large)f(con)m(tour)583 1741 y Fp(X)596 1957 y Fq(\()p Fr(\015)659 1967 y Fm(i)686 1957 y Fq(\))743 1741 y Fp(Y)795 1951 y Fr(i)887 1836 y Fv(exp)1053 1755 y Fp(\000)1121 1836 y Ft(\000)22 b Fv(10)p Ft(j)p Fs(\015)1397 1851 y Fr(i)1424 1836 y Ft(j)1452 1755 y Fp(\001)1525 1836 y Ft(\024)1631 1665 y Fp( )1709 1836 y Fv(1)g(+)1953 1741 y Fp(X)1878 1953 y Fr(`)p Fo(\025)p Fr(K)16 b Fq(ln)c Fr(N)2188 1836 y Fv(3)2237 1795 y Fr(`)2287 1836 y Fv(exp)q(\()p 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3858 y Ft(2)36 b Fs(@)5 b(D)21 b Fv(\()p Fs(';)c(L)g Fv(\()o Fs(')p Fv(\))22 b Ft(\000)h Fs(i)p Fv(\))16 b Fs(:)38 b Fv(The)324 3978 y(v)-5 b(alue)32 b Fs(H)40 b Fv(will)30 b(b)s(e)j(called)e(the)i (length)f(of)g(the)h(hair.)470 4099 y(Clearly)-8 b(,)46 b(our)e(statemen)m(t)h(is)e(equiv)-5 b(alen)m(t)44 b(to)g(pro)m(ving)g (that)f(the)i(probabilit)m(y)d(of)324 4219 y(o)s(ccurrence)27 b(of)e(a)h(hair)f(with)g(length)h Fs(H)35 b(>)27 b(C)d Fv(ln)16 b Fs(N)36 b Fv(v)-5 b(anishes)27 b(in)e(the)h(limit)c(as)k Fs(N)39 b Ft(!)27 b(1)p Fs(:)324 4340 y Fv(W)-8 b(e)33 b(denote)g(suc)m(h)h(an)f(ev)m(en)m(t)h(b)m(y)g Fs(A)1628 4355 y Fr(C)1687 4340 y Fs(:)470 4460 y Fv(In)26 b(what)f(follo)m(ws)f (w)m(e)j(will)c(treat)i(only)g(the)h(up-hairs,)g(since)g(the)f(case)i (of)d(the)i(do)m(wn-)324 4580 y(hairs)38 b(is)h(simpler,)h(as)f(there)h (is)f(no)g(v)m(olume)f(constrain)m(t)i(\(see)g(the)f(argumen)m(t)g(of)g (the)324 4701 y(subsection)44 b(4.5\);)k(therefore,)e(in)c(the)h(rest)h 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Fs(h)1074 3540 y Fr(r)1112 3525 y Fv(\()p Fs(N)10 b Fv(\))28 b(=)1407 3385 y Fp(\032)1524 3463 y Fv(4)p Fs(;)730 b Fv(if)64 b Fs(r)30 b(<)e(R)2705 3427 y Fo(0)2704 3489 y Fr(N)2788 3463 y Fs(;)1524 3586 y(C)1594 3601 y Fq(2)1649 3586 y Fv(2)1698 3549 y Fr(r)r(=)p Fq(2)1807 3586 y Fs(N)1895 3549 y Fo(\000)p Fq(1)2006 3586 y Fv(ln)16 b Fs(N)5 b(;)116 b Fv(if)64 b Fs(r)30 b Ft(\025)e Fs(R)2706 3549 y Fo(0)2705 3611 y Fr(N)2789 3586 y Fs(;)324 3787 y Fv(where)34 b Fs(C)676 3802 y Fq(2)747 3787 y Fv(is)e(c)m(hosen)j (suc)m(h)f(that)1104 4046 y Fs(h)1160 4062 y Fr(R)1213 4039 y Fa(0)1213 4085 y Fm(N)1276 4046 y Fv(\()p Fs(N)10 b Fv(\))28 b(=)1581 3906 y Ft(p)p 1664 3906 52 4 v 72 x Fs(a)17 b(C)1802 3993 y Fq(2)1858 3978 y Fv(ln)f Fs(N)p 1581 4023 464 4 v 1600 4043 a Fp(q)p 1699 4043 327 4 v 1699 4151 a Fs(v)1746 4167 y Fr(R)1799 4144 y Fa(0)1799 4190 y Fm(N)1861 4151 y Fv(\()p Fs(N)10 b Fv(\))2082 4046 y(=)2224 3906 y Ft(p)p 2307 3906 52 4 v 72 x Fs(a)p 2195 4023 193 4 v 2195 4043 a Ft(p)p 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Fq(3)3416 548 y Fv(.)54 b(If)324 668 y(the)31 b(length)f(of)h(\000)g(exceeds)i Fs(C)24 b Fv(ln)15 b Fs(N)10 b Fv(,)32 b(then)g(w)m(e)g(de\014ne)g(the)f(v)-5 b(alue)30 b Fs(r)2777 683 y Fq(0)2848 668 y Fv(as)h(the)g(\014rst)g (index)324 789 y Fs(r)36 b Fv(for)c(whic)m(h)i(the)f(b)s(ound)h Fs(H)1384 804 y Fr(r)1421 789 y Fv(\(\000\))29 b Fs(<)f(h)1747 804 y Fr(r)1786 789 y Fv(\()p Fs(N)10 b Fv(\))33 b(is)g(violated.)44 b(Then)34 b(de\014ne)g Fs(`)f Fv(as)h(the)f(\014rst)324 909 y(index)g(suc)m(h)h(that)1535 1129 y Ft(j)p Fv(In)m(t)17 b(\()p Fs(\015)1793 1144 y Fr(`)1826 1129 y Fv(\))p Ft(j)27 b(\024)h Fs(v)2071 1144 y Fr(r)2103 1153 y Fg(0)2142 1129 y Fv(\()p Fs(N)10 b Fv(\))17 b Fs(:)324 1349 y Fv(T)-8 b(o)36 b(summarize,)f(to)g(an)m(y)i(\000)e(with)h(length)f(larger)g (than)g Fs(C)24 b Fv(ln)16 b Fs(N)46 b Fv(w)m(e)37 b(asso)s(ciate)e(a)h (pair)324 1469 y(\()p Fs(r)406 1484 y Fq(0)445 1469 y Fs(;)17 b(`)p Fv(\))25 b(and)h(the)f(subsequence)30 b(of)24 b(con)m(tours)j Ft(f)p Fs(\015)2072 1484 y Fr(`)2104 1469 y Fs(;)17 b(\015)2199 1484 y Fr(`)p Fq(+1)2322 1469 y Fs(;)g(:)g(:)g(:)e(;)i(\015)2591 1484 y Fr(`)p Fq(+)p Fr(h)2716 1492 y Fm(r)2745 1507 y Fg(0)2783 1484 y Fo(\000)p Fq(1)2877 1469 y Ft(g)28 b(\032)g Fv(\000)p Fs(;)d Fv(for)g(whic)m(h) 324 1601 y(\(29\))k(holds.)43 b(W)-8 b(e)31 b(denote)g(b)m(y)1431 1520 y Fp(\000)1477 1601 y Fs(r)1521 1616 y Fq(0)1560 1601 y Fs(;)17 b(`;)g Ft(f)p Fs(\015)1790 1616 y Fr(`)1822 1601 y Fs(;)g(:)g(:)g(:)f(;)h(\015)2092 1616 y Fr(`)p Fq(+)p Fr(h)2217 1624 y Fm(r)2246 1639 y Fg(0)2283 1616 y Fo(\000)p Fq(1)2377 1601 y Ft(g)2427 1520 y Fp(\001)2503 1601 y Fv(the)31 b(class)f(of)g(all)e(suc)m(h)k(hairs)324 1721 y(\000)p Fs(:)470 1841 y Fv(The)h(strategy)f(of)f(the)h(pro)s(of)f (will)f(b)s(e)i(to)f(apply)g(a)h(P)m(eierls)f(t)m(yp)s(e)i(estimate)e (\(under)324 1962 y(the)24 b(v)m(olume)f(constrain)m(t\))h(to)f(the)h (section)g(of)f(the)h(hair)f(made)g(of)g(the)h(exterior)g(con)m(tours) 324 2082 y Ft(f)p Fs(\015)425 2097 y Fr(`)457 2082 y Fs(;)17 b(:)g(:)g(:)f(;)h(\015)727 2097 y Fr(`)p Fq(+)p Fr(h)852 2105 y Fm(r)881 2120 y Fg(0)918 2097 y Fo(\000)p Fq(1)1013 2082 y Ft(g)p Fv(.)324 2423 y Fs(P)387 2438 y Fr(\014)s(;N)533 2423 y Ft(fA)663 2438 y Fr(C)722 2423 y Ft(g)27 b(\024)2529 b Fv(\(31\))351 2674 y Ft(\024)480 2547 y Fr(R)533 2558 y Fm(N)463 2579 y Fp(X)457 2789 y Fr(r)489 2798 y Fg(0)523 2789 y Fq(=1)630 2549 y Fr(C)680 2558 y Fg(3)726 2549 y Fq(ln)12 b Fr(N)673 2579 y Fp(X)685 2792 y Fr(`)p Fq(=1)904 2579 y Fp(X)910 2795 y Fo(f)p Fr(\015)981 2805 y Fm(i)1008 2795 y Fo(g)1065 2674 y Fs(P)1128 2689 y Fr(\014)s(;N)1274 2593 y Fp(\010)1332 2674 y Fs(')33 b Fv(has)g(a)f(hair)g(\000)g(in)g(the)h(class)2505 2593 y Fp(\000)2551 2674 y Fs(r)2595 2689 y Fq(0)2634 2674 y Fs(;)17 b(`;)g Ft(f)p Fs(\015)2864 2689 y Fr(`)2896 2674 y Fs(;)g(:)g(:)g(:)f(;)h(\015)3166 2689 y Fr(`)p Fq(+)p Fr(h)3291 2697 y Fm(r)3320 2712 y Fg(0)3357 2689 y Fo(\000)p Fq(1)3451 2674 y Ft(g)3501 2593 y Fp(\001\011)3638 2674 y Fs(;)324 3005 y Fv(where)31 b(the)g(sum)f(is)g(o)m(v)m(er)i(the) e(collections)f(of)h(con)m(tours)h Ft(f)p Fs(\015)2512 3020 y Fr(`)2545 3005 y Fs(;)17 b(:)g(:)g(:)f(;)h(\015)2815 3020 y Fr(`)p Fq(+)p Fr(h)2940 3028 y Fm(r)2969 3043 y Fg(0)3006 3020 y Fo(\000)p Fq(1)3100 3005 y Ft(g)p Fv(.)43 b(In)30 b(order)324 3125 y(to)42 b(estimate)g(\(31\))o(,)j(t)m (w)m(o)e(cases)h(ha)m(v)m(e)g(to)e(b)s(e)h(distinguished.)72 b(Either)43 b Fs(r)3068 3140 y Fq(0)3150 3125 y Fv(is)f(smaller)324 3245 y(than)36 b Fs(R)630 3209 y Fo(0)629 3270 y Fr(N)732 3245 y Fv(and)g(all)e(the)j(con)m(tours)g Fs(\015)1684 3260 y Fr(`)1716 3245 y Fs(;)17 b(:)g(:)g(:)f(;)h(\015)1986 3260 y Fr(`)p Fq(+)p Fr(h)2111 3268 y Fm(r)2140 3283 y Fg(0)2177 3260 y Fo(\000)p Fq(1)2307 3245 y Fv(are)36 b(large,)g(or)g Fs(r)2910 3260 y Fq(0)2985 3245 y Fv(is)g(larger)f (than)324 3366 y Fs(R)399 3330 y Fo(0)398 3391 y Fr(N)499 3366 y Fv(and)g(one)f(has)h(to)f(rely)g(on)g(more)f(delicate)h (estimates,)g(taking)g(in)m(to)f(accoun)m(t)i(the)324 3486 y(fact)d(that)h(these)g(con)m(tours)h(migh)m(t)d(b)s(e)i(small.) 324 3725 y(Case)g(1)g(:)43 b Fs(r)753 3740 y Fq(0)820 3725 y Ft(\024)28 b Fs(R)1000 3688 y Fo(0)999 3750 y Fr(N)1067 3725 y Fv(.)470 3965 y(In)33 b(this)f(case)i Fs(h)1045 3980 y Fr(r)1077 3989 y Fg(0)1143 3965 y Fv(=)28 b(4)k(and)h(w)m(e)g(ha)m(v)m(e)1446 4150 y Fq(3)1391 4180 y Fp(X)1406 4390 y Fr(i)p Fq(=0)1552 4275 y Ft(j)p Fv(In)m(t)17 b(\()p Fs(\015)1810 4290 y Fr(`)p Fq(+)p Fr(i)1921 4275 y Fv(\))p Ft(j)27 b(\024)i Fv(4)p Fs(v)2216 4290 y Fr(r)2248 4299 y Fg(0)2286 4275 y Fv(\()p Fs(N)10 b Fv(\))17 b Fs(;)1397 4660 y Fq(3)1343 4690 y Fp(X)1357 4900 y Fr(i)p Fq(=0)1503 4785 y Ft(j)p Fs(\015)1582 4800 y Fr(`)p Fq(+)p Fr(i)1693 4785 y Ft(j)28 b(\025)g Fv(4\(4)1990 4695 y Fp(p)p 2089 4695 373 4 v 2089 4785 a Fs(v)2136 4800 y Fr(r)2168 4809 y Fg(0)2202 4800 y Fq(+1)2297 4785 y Fv(\()p Fs(N)10 b Fv(\)\))17 b Fs(:)1894 5251 y Fv(23)p eop %%Page: 24 24 24 23 bop 470 548 a Fv(Applying)33 b(the)h(P)m(eierls)g(estimate)f(as)h (in)f(the)i(pro)s(of)d(of)i(Theorem)g(6)g(\(see)g(relation)324 668 y(\(15\))o(\),)f(w)m(e)g(get)g(for)f(ev)m(ery)i Fs(r)1351 683 y Fq(0)1418 668 y Fv(=)28 b([0)p Fs(;)17 b(R)1717 632 y Fo(0)1716 693 y Fr(N)1783 668 y Fv(])33 b(and)f(ev)m(ery)j(heigh) m(t)d Fs(`)c Ft(\024)g Fs(C)2829 683 y Fq(3)2885 668 y Fv(ln)16 b Fs(N)556 788 y Fp(X)561 1004 y Fo(f)p Fr(\015)632 1014 y Fm(i)659 1004 y Fo(g)717 882 y Fs(P)780 897 y Fr(\014)s(;N)926 882 y Ft(f)p Fv(\000)27 b Ft(2)h Fv(\()p Fs(r)1240 897 y Fq(0)1280 882 y Fs(;)17 b(`;)g Ft(f)p Fs(\015)1510 897 y Fr(`)1541 882 y Fs(;)g(:)g(:)g(:)f(;)h(\015)1811 897 y Fr(`)p Fq(+3)1934 882 y Ft(g)p Fv(\))o Ft(g)1317 b Fv(\(32\))762 1241 y Ft(\024)867 1146 y Fp(X)873 1362 y Fo(f)p Fr(\015)944 1372 y Fm(i)971 1362 y Fo(g)1028 1241 y Fv(exp)1194 1070 y Fp( )1272 1241 y Ft(\000)p Fs(\014)1481 1116 y Fq(3)1427 1146 y Fp(X)1442 1356 y Fr(i)p Fq(=0)1588 1241 y Ft(j)p Fs(\015)1667 1256 y Fr(`)p Fq(+)p Fr(i)1778 1241 y Ft(j)1806 1070 y Fp(!)1943 1173 y Fs(Q)2020 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Fa(0)1598 2157 y Fm(N)1660 2117 y Fv(\()p Fs(N)10 b Fv(\))57 b Ft(\025)g Fv(10)49 b(\(see)h(\(30\)\),)j Fs(h)2722 2132 y Fr(r)2754 2141 y Fg(0)2793 2117 y Fv(\()p Fs(N)10 b Fv(\))2957 2032 y Fp(p)p 3057 2032 373 4 v 85 x Fs(v)3104 2132 y Fr(r)3136 2141 y Fg(0)3171 2132 y Fq(+1)3265 2117 y Fv(\()p Fs(N)g Fv(\))57 b(=)324 2219 y Fp(p)p 423 2219 58 4 v 433 2256 a Fr(a)p 433 2273 38 4 v 434 2330 a Fq(2)481 2296 y Fs(C)551 2311 y Fq(2)607 2296 y Fv(ln)16 b Fs(N)53 b Fv(\(see)43 b(\(34\)\),)i(while)1612 2231 y Fq(ln)1671 2239 y Fv(\()1709 2231 y Fr(v)1743 2239 y Fm(r)1772 2254 y Fg(0)1811 2231 y Fq(\()p Fr(N)7 b Fq(\))1928 2239 y Fv(\))p 1600 2273 381 4 v 1600 2285 a Ft(p)p 1683 2285 297 4 v 62 x Fr(v)1717 2356 y Fm(r)1746 2371 y Fg(0)1781 2356 y(+1)1862 2347 y Fq(\()p Fr(N)g Fq(\))2035 2296 y Ft(\024)45 b Fv(1.)73 b(Th)m(us)44 b(summing)d(\(35\))h(o)m(v)m(er)i (the)324 2458 y(indices)32 b Fs(r)688 2473 y Fq(0)760 2458 y Fv(and)h Fs(`)p Fv(,)f(w)m(e)i(obtain)722 2640 y Fr(R)775 2651 y Fm(N)706 2673 y Fp(X)617 2889 y Fr(r)649 2898 y Fg(0)683 2889 y Fq(=)p Fr(R)791 2866 y Fa(0)791 2912 y Fm(N)849 2889 y Fq(+1)956 2642 y Fr(C)1006 2651 y Fg(3)1052 2642 y Fq(ln)12 b Fr(N)999 2673 y Fp(X)1011 2885 y Fr(`)p Fq(=0)1230 2673 y Fp(X)1235 2888 y Fo(f)p Fr(\015)1306 2898 y Fm(i)1333 2888 y Fo(g)1391 2767 y Fs(P)1454 2782 y Fr(\014)s(;N)1600 2687 y Fp(\010)1658 2767 y Fv(\000)28 b Ft(\030)1852 2687 y Fp(\000)1898 2767 y Fs(r)1942 2782 y Fq(0)1981 2767 y Fs(;)17 b(`;)g Ft(f)p Fs(\015)2211 2782 y Fr(`)2243 2767 y Fs(;)g(:)g(:)g(:)e(;)i (\015)2512 2782 y Fr(`)p Fq(+)p Fr(h)2637 2790 y Fm(r)2666 2805 y Fg(0)2704 2782 y Fo(\000)p Fq(1)2798 2767 y Ft(g)2848 2687 y Fp(\001\011)628 3138 y Ft(\024)839 3011 y Fr(R)892 3022 y Fm(N)822 3043 y Fp(X)733 3260 y Fr(r)765 3269 y Fg(0)799 3260 y Fq(=)p Fr(R)907 3237 y Fa(0)907 3283 y Fm(N)965 3260 y Fq(+1)1072 3013 y Fr(C)1122 3022 y Fg(3)1168 3013 y Fq(ln)12 b Fr(N)1115 3043 y Fp(X)1127 3255 y Fr(`)p Fq(=0)1319 3138 y Fs(N)1407 3097 y Fq(2)1463 3138 y Fv(exp)1629 2997 y Fp(\022)1702 2992 y(r)p 1802 2992 72 4 v 1812 3070 a Fs(a)p 1812 3115 52 4 v 1813 3206 a Fv(2)1873 3138 y Fs(C)1943 3153 y Fq(2)1999 3138 y Fv(ln)k Fs(N)2202 2997 y Fp(\022)2275 3138 y Fv(1)22 b Ft(\000)h Fv(4\()p Fs(\014)k Ft(\000)c Fv(10\))e(+)2992 3070 y(4)p Fs(\014)p 2980 3115 132 4 v 2980 3135 a Ft(p)p 3063 3135 49 4 v 82 x Fv(5)3122 2997 y Fp(\023\023)628 3410 y Ft(\024)28 b Fv(exp)17 b(\()p Ft(\000)p Fs(C)1083 3425 y Fr(\014)1147 3410 y Fv(ln)f Fs(N)10 b Fv(\))34 b Fs(;)324 3630 y Fv(where)g Fs(C)676 3645 y Fr(\014)755 3630 y Fv(is)e(p)s(ositiv)m(e)g(constan)m(t)i(for)e Fs(\014)38 b Fv(large)31 b(enough.)470 3751 y(Com)m(bining)g(this)h(with)g (\(33\),)g(w)m(e)i(conclude)f(the)g(pro)s(of.)470 3871 y Fh(Ac)m(kno)m(wledgmen)m(t.)92 b Ff(We)46 b(would)g(like)g(to)h (thank)f(P.)g(F)-7 b(err)i(ari,)48 b(D.)e(Io\013e,)i(M.)324 3991 y(Pr\177)-50 b(ahofer)34 b(and)f(H.)i(Sp)-5 b(ohn)33 b(for)i(very)f(inter)-5 b(esting)34 b(discussions)f(on)h(step)g (\015uctuations.)324 4137 y(R.S.)26 b(acknow)5 b(le)-5 b(dges)24 b(the)j(\014nancial)e(supp)-5 b(ort)26 b(of)g(the)h(NSF)f(gr) -5 b(ant)26 b(DMS-0300672.)40 b(S.S.)324 4257 y(acknow)5 b(le)-5 b(dges)33 b(the)h(\014nancial)g(supp)-5 b(ort)35 b(of)g(the)f(RFFI)g(gr)-5 b(ant)35 b(03-01-00444.)324 4590 y Fu(References)324 4809 y Fv([B])175 b(T.)37 b(Bo)s(dineau.)e Ff(The)j(Wul\013)g(c)-5 b(onstruction)38 b(in)g(thr)-5 b(e)g(e)38 b(and)g(mor)-5 b(e)37 b(dimensions)p Fv(,)622 4929 y(Comm.)31 b(Math.)i(Ph)m(ys.)i Fh(207)p Fv(,)d(197{229)f (\(1999\).)1894 5251 y(26)p eop %%Page: 27 27 27 26 bop 324 548 a Fv([BFL])50 b(J.)30 b(Bricmon)m(t,)f(J.-R.)g(F)-8 b(on)m(taine,)30 b(and)f(J.L.)h(Leb)s(o)m(witz:)42 b Ff(Surfac)-5 b(e)32 b(tension,)g(p)-5 b(er-)622 668 y(c)g(olation,)34 b(and)g(r)-5 b(oughening)p Fv(,)32 b(J.)h(Statist.)e(Ph)m(ys.)k(29,)d (no.)h(2,)f(193{203)f(\(1982\).)324 872 y([BFM])49 b(J.)44 b(Bricmon)m(t,)i(J.)f(F)-8 b(rohlic)m(h)43 b(and)h(A.)g(El)g(Mellouki:) 65 b Ff(R)-5 b(andom)45 b(surfac)-5 b(es)45 b(in)622 992 y(statistic)-5 b(al)41 b(me)-5 b(chanics:)54 b(r)-5 b(oughening,)42 b(r)-5 b(ounding,)41 b(wetting,)h(...)e Fv(,)h(J.)e(Statist.)622 1112 y(Ph)m(ys.)34 b Fh(42)p Fv(,)f(no.)g(5-6,)e(743{798)g(\(1986\).)324 1316 y([BI])140 b(T.)39 b(Bo)s(dineau)g(and)g(D.)g(Io\013e,)81 b Ff(Stability)41 b(of)g(interfac)-5 b(es)40 b(and)g(sto)-5 b(chastic)41 b(dy-)622 1436 y(namics)34 b(in)g(the)h(r)-5 b(e)g(gime)34 b(of)h(p)-5 b(artial)34 b(wetting)p Fv(,)f(preprin)m(t)f(\(2003\).)324 1640 y([BIV])67 b(T.)34 b(Bo)s(dineau,)g(D.)f(Io\013e)h(and)g(Y.)g(V)-8 b(elenik:)46 b Ff(R)n(igor)-5 b(ous)35 b(pr)-5 b(ob)g(abilistic)35 b(analysis)622 1760 y(of)f(e)-5 b(quilibrium)35 b(crystal)g(shap)-5 b(es,)34 b Fv(J.)e(Math.)h(Ph)m(ys.)i Fh(41,)d Fv(1033{1098)f (\(2000\).)324 1963 y([CP])108 b(R.)26 b(Cerf)g(and)g(A.)g(Pizstora.)f Ff(On)k(the)f(Wul\013)h(crystal)g(in)f(the)h(Ising)f(mo)-5 b(del)p Fv(,)26 b(Ann.)622 2084 y(Probab.)33 b Fh(28)p Fv(,)f(no.)h(3,)f(947{1017)f(\(2000\).)324 2287 y([CK])98 b(R.)34 b(Cerf)g(and)g(R.)g(Ken)m(y)m(on:)48 b Ff(The)35 b(low-temp)-5 b(er)g(atur)g(e)36 b(exp)-5 b(ansion)34 b(of)i(the)g(Wul\013)622 2408 y(crystal)27 b(in)f(the)g(3D)g(Ising)g (mo)-5 b(del)p Fv(,)24 b(Comm.)f(Math.)h(Ph)m(ys.)h Fh(222)p Fv(,)h(no.)d(1,)i(147{179)622 2528 y(\(2001\).)324 2731 y([D])169 b(R.)44 b(L.)h(Dobrushin:)67 b Ff(The)45 b(Gibbs)g(state)i (that)f(describ)-5 b(es)44 b(the)i(c)-5 b(o)g(existenc)g(e)45 b(of)622 2852 y(phases)36 b(for)h(a)g(thr)-5 b(e)g(e-dimensional)35 b(Ising)h(mo)-5 b(del)p Fv(.)34 b(\(In)i(Russian\))f(T)-8 b(eor.)35 b(V)-8 b(ero-)622 2972 y(jatnost.)32 b(i)g(Primenen.)g Fh(17)p Fv(,)h(619{639)e(\(1972\).)324 3176 y([DKS])48 b(R.L.)d(Dobrushin,)j(R.)c(Kotec)m(ky)j(and)e(S.)g(B.)g(Shlosman:)67 b Ff(Wul\013)47 b(c)-5 b(onstruc-)622 3296 y(tion:)58 b(a)41 b(glob)-5 b(al)41 b(shap)-5 b(e)40 b(fr)-5 b(om)41 b(lo)-5 b(c)g(al)41 b(inter)-5 b(action,)43 b Fv(AMS)d(translations)f (series,)622 3416 y(Pro)m(vidence)34 b(\(Rho)s(de)e(Island\),)h(1992.) 324 3620 y([FPS])60 b(P)-8 b(.)43 b(F)-8 b(errari,)45 b(M.)e(Praehofer)h(and)f(H.)h(Sp)s(ohn,)i Ff(Fluctuations)f(of)f(an)g (A)n(tomic)622 3740 y(L)-5 b(e)g(dge)34 b(Bor)-5 b(dering)34 b(a)h(Crystal)5 b(line)34 b(F)-7 b(ac)i(et)p Fv(,)32 b(preprin)m(t,)g(\(cond-mat/0303162\))324 3944 y([FS])126 b(P)-8 b(.)57 b(F)-8 b(errari)54 b(and)j(H.)f(Sp)s(ohn,)63 b Ff(Step)57 b(\015uctuations)g(for)g(a)f(fac)-5 b(ete)g(d)56 b(crystal)p Fv(,)622 4064 y(preprin)m(t,)33 b(\(cond-mat/0212456\).)324 4267 y([F)-8 b(rSp])49 b(J.)d(F)-8 b(r\177)-49 b(ohlic)m(h)44 b(and)i(T.)g(Sp)s(encer,)98 b Ff(The)46 b(Kosterlitz-Thouless)g(tr)-5 b(ansition)47 b(in)622 4388 y(two-dimensional)40 b(ab)-5 b(elian)40 b(spin)i(systems)f(and)g(the)h(Coulomb)f(gas)p Fv(,)h(Comm.)622 4508 y(Math.)33 b(Ph)m(ys.)h Fh(81)p Fv(,)f(no.)f(4,)h(527{602)e(\(1981\).)324 4711 y([IS])155 b(D.)38 b(Io\013e)h(and)f(R.)h(Sc)m(honmann,)h Ff(Dobrushin-Kote)-5 b(ck)o(\023)-49 b(y-Shlosman)38 b(the)-5 b(ory)41 b(up)622 4832 y(to)32 b(the)f(critic)-5 b(al)31 b(temp)-5 b(er)g(atur)g(e)7 b Fv(,)31 b(Comm.)c(Math.)j(Ph)m(ys.)g Fh(199)p Fv(,)g(117{167)e (\(1998\).)1894 5251 y(27)p eop %%Page: 28 28 28 27 bop 324 548 a Fv([MRSV])49 b(Ch.)29 b(Maes,)g(F.)f(Redig,)g(S.)f (Shlosman)g(and)h(A.)g(v)-5 b(an)27 b(Mo\013aert:)41 b Ff(Per)-5 b(c)g(olation,)622 668 y(Path)29 b(L)-5 b(ar)g(ge)28 b(Deviations)g(and)g(We)-5 b(akly)29 b(Gibbs)g(States,)g Fv(Comm.)c(Math.)i(Ph)m(ys.,)622 789 y Fh(209)p Fv(,)33 b(517-545)d(\(2000\).)324 992 y([M])155 b(S.)45 b(Miracle-Sole.)f Ff(Surfac)-5 b(e)46 b(tension,)j(step)d(fr)-5 b(e)g(e)47 b(ener)-5 b(gy)46 b(and)g(fac)-5 b(ets)46 b(in)h(the)622 1112 y(e)-5 b(quilibrium)34 b(crystal.)f Fv(J.Stat.)f(Ph)m(ys.,)j Fh(79,)j Fv(183-214)30 b(\(1995\).)324 1316 y([SS1])87 b(R.H.)38 b(Sc)m(honmann)g(and)g(S.)g(Shlosman:)52 b Ff(Constr)-5 b(aine)g(d)39 b(variational)g(pr)-5 b(oblem)622 1436 y(with)26 b(applic)-5 b(ations)26 b(to)g(the)h(Ising)f(mo)-5 b(del)p Fv(,)24 b(J.)g(Stat.)g(Ph)m(ys.)h Fh(83)p Fv(,)h(867{905)c (\(1996\).)324 1640 y([SS2])87 b(R.H.)30 b(Sc)m(honmann)h(and)f(S.)h (Shlosman:)j Ff(Complete)e(analyticity)h(for)g(2D)f(Ising)622 1760 y(c)-5 b(omplete)g(d.)31 b Fv(Comm.)h(Math.)h(Ph)m(ys.)h Fh(170)p Fv(,)f(453{482)e(\(1995\).)324 1963 y([S1])141 b(S.)49 b(Shlosman:)75 b Ff(The)56 b(Wul\013)50 b(c)-5 b(onstruction)50 b(in)f(statistic)-5 b(al)50 b(me)-5 b(chanics)48 b(and)622 2084 y(in)35 b(c)-5 b(ombinatorics,)34 b Fv(arXiv.org)e(e-Prin)m(t)g(arc)m(hiv)m(e,)i(math-ph/0010039,)d (Russ.)622 2204 y(Math.)i(Surv.,)g Fh(56)p Fv(,)g(no.)g(4,)f(709-738)f (\(2001\).)324 2408 y([S2])141 b(S.)33 b(Shlosman:)42 b Ff(Zer)-5 b(o)35 b(temp)-5 b(er)g(atur)g(e)35 b(Ising)e(crystal,)i Fv(in)d(preparation.)324 2611 y([VKer])49 b(A.)42 b(V)-8 b(ershik)42 b(and)f(S.)h(Kero)m(v.)g Ff(Asymptotic)i(of)f(the)g(lar)-5 b(gest)43 b(and)f(typic)-5 b(al)43 b(di-)622 2731 y(mensions)34 b(of)i(irr)-5 b(e)g(ducible)35 b(r)-5 b(epr)g(esentations)35 b(of)g(a)h(symmetric)f(gr)-5 b(oup.)33 b Fv(F)-8 b(unct.)622 2852 y(Anal.)32 b(Appl.,)g Fh(19)p Fv(,)h(21-31)e(\(1985\).)1894 5251 y(28)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0401070054274--