Content-Type: multipart/mixed; boundary="-------------0109051105251" This is a multi-part message in MIME format. ---------------0109051105251 Content-Type: text/plain; name="01-320.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-320.keywords" Low temperature Gibbs states, discrete spin lattice models of Kac Ising type, restricted ensembles with low density constraints, cluster expansion, contours, Pirogov-Sinai theory ---------------0109051105251 Content-Type: application/postscript; name="kps.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="kps.ps" From bovier@wias-berlin.de Wed Sep 5 17:41:41 2001 Received: from wias-berlin.de (descartes.wias-berlin.de [192.124.249.194]) by csmat.karlin.mff.cuni.cz (8.9.3/8.9.3) with ESMTP id RAA20819 for ; Wed, 5 Sep 2001 17:41:40 +0200 (MET DST) Received: from mailclient by wias-berlin.de (8.8.8/1.1.22.3/16Mar99-0205PM) id RAA0000028469; Wed, 5 Sep 2001 17:41:39 +0200 (MET DST) From: Anton Bovier Message-Id: <200109051541.RAA0000028469@wias-berlin.de> X-Mailer: exmh version 2.0.2 2/24/98 To: Milos Zahradnik cc: bovier@wias-berlin.de Subject: kps.ps In-reply-to: Your message of "Wed, 05 Sep 2001 14:34:54 +0200." <200109051234.OAA14929@csmat.karlin.mff.cuni.cz> Mime-Version: 1.0 Content-Type: multipart/mixed ; boundary="==_Exmh_3081196201500" Date: Wed, 05 Sep 2001 17:41:39 +0200 X-Mts: smtp Status: ROr This is a multipart MIME message. --==_Exmh_3081196201500 Content-Type: text/plain; charset=us-ascii --==_Exmh_3081196201500 Content-Type: application/postscript ; name="kps.ps" Content-Description: kps.ps Content-Transfer-Encoding: quoted-printable Content-Disposition: attachment; filename="kps.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radical= eye.com) %%Title: kps.dvi %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips kps -o %DVIPSParameters: dpi=3D600, compressed %DVIPSSource: TeX output 2001.08.29:1349 %%BeginProcSet: texc.pro %! 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h(desirable)e(extension)h(of)i(the)50 4158 y(Pirogo)m(v-Sinai)23 b(theory)i(to)h(a)f(reasonable)g(set)g(of)g(in)m(teractions)f(and)h (pro)m(vides)e(natural)h(criteria)g(for)h(when)50 4267 y(to)f(exp)s(ect)g(a)g(phase)f(transition.)37 b(It)24 b(should)d(b)s(e)i(observ)m(ed)h(that)g(as)g(far)f(as)h(the)g (high-temp)s(erature)e(phases)50 4376 y(are)28 b(concerned,)g(the)g (class)f(of)h(in)m(teractions)f(for)g(whic)m(h)f(uniqueness)g(prop)s (erties)f(can)j(b)s(e)f(pro)m(v)m(en)h(is)e(v)m(ery)50 4485 y(general)31 b(and)f(qualitativ)m(ely)g(optimal.)41 b(F)-8 b(or)32 b(lo)m(w)f(temp)s(eratures,)g(so)g(far,)g(the)g (existing)f(results)g(concern)50 4595 y(v)m(ery)h(restricted)f(classes) g(of)h(in)m(teractions.)159 4776 y(As)e(a)g(\014rst)f(exploration)f(in) m(to)i(this)f(direction,)g(ho)m(w)m(ev)m(er,)i(w)m(e)f(did)e(not)i (striv)m(e)g(for)f(full)f(generalit)m(y)-8 b(.)40 b(In-)50 4885 y(deed,)26 b(it)g(will)d(require)h(more)i(insigh)m(t)e(and)i 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b(us)f(men)m(tion)h(the)g(related)f(results)g(b)m(y)h(Leb) s(o)m(witz,)h(Mazel,)h(and)d(Presutti)g([LMP])h(that)g(treat)h(a)50 849 y(particular)19 b(Kac-t)m(yp)s(e)j(mo)s(del)e(in)f(con)m(tin)m (uous)i(space.)38 b(While)20 b(in)g(principle)d(this)j(pap)s(er)g (deals)g(with)g(rather)50 958 y(similar)31 b(problems,)h(there)i(are)g (substan)m(tial)e(di\013erences)h(that)h(will)c(not)k(allo)m(w)f(the)h (direct)e(application)50 1067 y(of)f(their)f(metho)s(ds)g(in)f(our)h (situation.)41 b(It)31 b(will)d(b)s(e)j(in)m(teresting)f(to)h(compare)h (b)s(oth)e(approac)m(hes,)h(also)g(in)50 1176 y(other)f(applications.) 159 1358 y(The)22 b(remainder)e(of)j(this)e(pap)s(er)g(is)g(organized)i (as)f(follo)m(ws.)37 b(In)22 b(Section)g(2)g(w)m(e)h(giv)m(e)g(a)f (precise)g(de\014nition)50 1467 y(of)42 b(the)h(class)f(of)g(mo)s(dels) f(w)m(e)i(are)g(studying)d(and)i(w)m(e)h(form)m(ulate)f(the)g(main)g (results.)75 b(In)41 b(Section)h(3)50 1576 y(w)m(e)d(explain)d(ho)m(w)j (to)g(de\014ne)e(the)i(crucial)e(notion)h(of)h Fw(c)-5 b(ontours)40 b Fz(and)d Fw(r)-5 b(e)g(gular)42 b(r)-5 b(e)g(gions)p Fz(.)66 b(Here)39 b(w)m(e)f(will)50 1685 y(encoun)m(ter)45 b(the)g(main)e(di\013erence)h(from)g(standard)g (Pirogo)m(v-Sinai)f(theory:)70 b(the)44 b(con\014gurations)g(in)50 1795 y(regular)35 b(regions)g(will)e Fw(not)k Fz(simply)c(lo)s(ok)i (lik)m(e)g(ground)g(states,)k(but)c(they)g(will)f(b)s(e)h(c)m (haracterized)i(only)50 1904 y(b)m(y)42 b(some)h(carefully)f(c)m(hosen) h(nonlo)s(cal)e(condition.)77 b(The)42 b(next)h(t)m(w)m(o)h(c)m (hapters)f(will)d(sho)m(w)j(that)g(this)50 2013 y(de\014nition)32 b(w)m(as)j(reasonable:)49 b(In)34 b(Section)g(4)h(w)m(e)g(pro)m(v)m(e)g (that)g(con)m(tours)g(satisfy)f(the)h(P)m(eierls)f(estimate,)50 2122 y(while)40 b(in)h(Section)h(5)h(w)m(e)f(sho)m(w)h(ho)m(w)f(to)h(p) s(erform)e(high-temp)s(erature)g(expansions)g(in)g(the)h(\\regular)50 2231 y(regions",)30 b(i.e.)40 b(outside)29 b(of)h(the)f(con)m(tours.)41 b(This)28 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Fz(\010)p Fw(\))p Fz(:)41 b(W)-8 b(e)31 b(assume)g(that)g(\010) 2177 1400 y Fq(i;j)2262 1386 y Fp(f)p Fr(x)2359 1400 y Fq(i)2390 1386 y Fr(;)15 b(x)2482 1400 y Fq(j)2520 1386 y Fp(g)26 b(\025)f Fz(0.)50 1620 y Fw(Note.)46 b Fz(Actually)-8 b(,)35 b(this)e(requiremen)m(t)g(can)h(b)s(e)f(greatly)i (relaxed.)51 b(W)-8 b(e)35 b(mak)m(e)g(this)e(assumption)f(only)h(to)50 1729 y(get)h(a)f(simpler)d(pro)s(of)i(for)g(the)h(P)m(eierls)e (condition)h(for)g(the)h(con)m(tours)g(\(de\014ned)f(b)s(elo)m(w\).The) g(p)s(ositivit)m(y)50 1838 y(of)h(the)g(in)m(teraction)g(\(together)h (with)e(some)h(con)m(tin)m(uit)m(y)g(-)g(see)h(b)s(elo)m(w\))e(assures) g(the)i(v)-5 b(alidit)m(y)31 b(of)i(suc)m(h)f(a)50 1947 y(condition,)21 b(but)f(it)h(is)e(of)i(course)g(not)g(necessary)-8 b(.)38 b(W)-8 b(e)22 b(k)m(eep)g(the)e(condition)g(of)h(p)s(ositivit)m (y)e(throughout)h(this)50 2056 y(pap)s(er)30 b(for)g(simplicit)m(y)-8 b(.)41 b(A)31 b(more)g(systematic)g(in)m(v)m(estigation)g(of)g(the)h(v) -5 b(alidit)m(y)29 b(of)i(the)g(P)m(eierls)f(condition)50 2166 y(for)g(long)g(range,)h(Kac)g(t)m(yp)s(e,)g(mo)s(dels)e(deserv)m (es)h(a)h(separate)h(pap)s(er.)159 2347 y(Our)i(\014rst)h(assumption)g (states)h(that)h(the)f(in)m(teraction)f(has)h(\014nite)e(range)2743 2314 y Fm(1)2820 2347 y Fr(R)j Fz(\(where)e Fr(R)i Fw(may)f Fz(b)s(e)f(a)50 2456 y(v)m(ery)c(large)f(n)m(um)m(b)s(er\).)50 2638 y FB(Assumption)35 b(1)60 b Fw(\(\014nite)33 b(r)-5 b(ange)34 b Fr(R)f Fw(anf)g(\014nite)g(varianc)-5 b(e\))p Fz(:)41 b(Denote)32 b(b)m(y)464 2847 y(\010)530 2861 y Fq(i)586 2847 y Fp(\021)25 b Fz(max)851 2861 y Fq(s;s)941 2843 y Fj(0)966 2861 y Fl(2)p Fq(S)s Fl(\002)p Fq(S)1187 2847 y Fp(j)q Fz(\010)1279 2861 y Fm(0)p Fq(;i)1366 2847 y Fp(f)p Fr(s;)15 b(s)1537 2810 y Fl(0)1563 2847 y Fp(gj)167 b Fz(and)151 b(\011)2169 2861 y Fq(i)2224 2847 y Fp(\021)25 b Fz(min)2472 2861 y Fq(s;s)2562 2843 y Fj(0)2587 2861 y Fl(2)p Fq(S)s Fl(\002)p Fq(S)2808 2847 y Fp(j)q Fz(\010)2900 2861 y Fm(0)p Fq(;i)2987 2847 y Fp(f)p Fr(s;)15 b(s)3158 2810 y Fl(0)3184 2847 y Fp(gj)h Fr(:)50 3056 y Fz(W)-8 b(e)33 b(assume)f(that)g(there)g(exists)g(0)c Fr(<)g(R)h(<)e Fp(1)32 b Fz(suc)m(h)g(that)g(\010)2171 3070 y Fq(i)2229 3056 y Fz(=3D)27 b(0,)33 b(if)e Fp(j)p Fr(i)p Fp(j)e Fr(>)f(R)q Fz(.)45 b(Moreo)m(v)m(er,)35 b(w)m(e)d(assume)50 3165 y(that)f(there)f(exists)g(a)h(\014nite)e(p)s(ositiv)m(e)h(constan)m(t)h (0)26 b Fr(<)f(D)j Fp(\024)d Fz(1)31 b(suc)m(h)f(that)h(for)f(all)f Fp(j)p Fr(i)p Fp(j)d(\024)f Fr(R)q Fz(,)1462 3374 y(\(\010)1563 3388 y Fq(i)1649 3374 y Fp(\025)p Fz(\))61 b(\011)1887 3388 y Fq(i)1972 3374 y Fp(\025)55 b Fr(D)s Fz(\010)2242 3388 y Fq(i)2272 3374 y Fr(:)1226 b Fz(\(2)p Fr(:)p Fz(3\))159 3655 y(Our)21 b(second)i(assumption)e(assures)h(a)h(su\016cien)m(tly)e (attractiv)m(e)k(in)m(teraction:)36 b(Again,)25 b(w)m(e)e(do)f(not)h (striv)m(e)50 3764 y(here)i(for)g(a)g(maximal)f(generalit)m(y)-8 b(.)40 b(W)-8 b(e)26 b(select)g(no)m(w)f(a)h(subset)e Fr(Q)h Fp(\032)g(S)32 b Fz(con)m(taining)25 b(all)f(the)h(\\appro)m (ximate)50 3873 y(minima")31 b(of)h Fr(U)10 b Fz(.)47 b(Often)33 b(\(e.g.)48 b(in)31 b(our)h(main)g(example)g(w)m(e)h(ha)m(v) m(e)h(in)d(mind)g(here)h(i.e.)47 b(in)31 b(the)i(Kac)g(Ising)50 3982 y(mo)s(del)28 b(with)f(a)j(rather)e(small)g(external)h(\014eld\),) f(this)g(set)i(of)f(\\reference)h(colours")e(will)f(b)s(e)h(tak)m(en)i (simply)50 4092 y(as)g Fr(Q)c Fp(\021)f(S)7 b Fz(\).)50 4273 y FB(Assumption)35 b(2)30 b Fw(\(str)-5 b(ong)34 b(attr)-5 b(action\))p Fz(:)44 b(F)-8 b(or)31 b Fr(q)d Fp(2)d Fr(Q)30 b Fz(and)g Fr(s)24 b Fp(2)h(S)7 b Fr(;)15 b(s)25 b Fp(6)p Fz(=3D)g Fr(q)34 b Fz(w)m(e)c(de\014ne)g(b)m(y)457 4500 y Fr(V)510 4514 y Fq(q)550 4500 y Fz(\()p Fr(s)p Fz(\))56 b(=3D)f Fr(U)10 b Fz(\()p Fr(s)p Fz(\))21 b Fp(\000)f Fr(U)10 b Fz(\()p Fr(q)s Fz(\))51 b(+)1500 4414 y Fk(X)1503 4609 y Fq(i)p Fl(2)p Fm(\003)1646 4500 y Fz(\010)1712 4514 y Fm(0)p Fq(;i)1800 4500 y Fp(f)p Fr(s;)15 b(q)s Fp(g)92 b Fz(and)151 b Fr(V)2460 4514 y Fq(q)2525 4500 y Fz(=3D)25 b(min)2773 4517 y Fq(s)p Fl(2)p Fq(S)s Fl(nf)p Fq(q)r Fl(g)3071 4500 y Fr(V)3124 4514 y Fq(q)3164 4500 y Fz(\()p Fr(s)p Fz(\))p Fr(:)221 b Fz(\(2)p Fr(:)p Fz(4\))50 4795 y(W)-8 b(e)31 b(assume)g(that)f(there)h(exists)f Fr(\034)35 b(>)25 b Fz(0)31 b(suc)m(h)f(that)h(for)f Fw(e)-5 b(ach)38 b Fr(q)29 b Fp(2)24 b Fr(Q)1484 5004 y Fz(min)1635 5018 y Fq(q)r Fl(2)p Fq(Q)1797 5004 y Fr(V)1850 5018 y Fq(q)1915 5004 y Fp(\021)h Fr(V)46 b Fp(\025)25 b Fr(\034)5 b(:)1247 b Fz(\(2)p Fr(:)p Fz(5\))p 50 5111 546 4 v 159 5172 a Fi(1)197 5198 y Fx(Again)21 b(this)i(assumption)e (is)i(not)e(strictly)i(neessary)-6 b(,)24 b(and)d(as)h(in)g(the)g (usual)g(short-range)f(situations,)j(one)e(ma)n(y)e(admitt)50 5289 y(an)25 b(additional)h(w)n(eak)f(in)n(teraction)i(that)e(is)i (rapidly)f(deca)n(ying.)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1165 b Fx(4)p eop %%Page: 5 5 5 4 bop 50 340 a Fz(W)-8 b(e)40 b(will)d(c)m(ho)s(ose)j Fr(Q)f Fz(in)f(suc)m(h)h(a)h(w)m(a)m(y)g(that)g(the)f(one-b)s(o)s(dy)f (p)s(oten)m(tial)h(v)-5 b(aries)39 b(not)g(to)s(o)h(m)m(uc)m(h)f(o)m(v) m(er)i Fr(Q)50 449 y Fz(compared)30 b(to)h(the)g(strength)f(of)h(the)f (in)m(teraction)1778 416 y Fm(2)1819 449 y Fz(,)1047 667 y Fr(Q)25 b Fp(\021)g(f)p Fr(q)k Fp(2)24 b(S)32 b Fz(:)26 b Fr(U)10 b Fz(\()p Fr(q)s Fz(\))26 b Fp(\024)f Fz(min)2037 681 y Fq(s)p Fl(2S)2190 667 y Fr(U)10 b Fz(\()p Fr(s)p Fz(\))20 b(+)g Fr(\034)10 b(=3D)p Fz(4)p Fp(g)16 b Fr(:)811 b Fz(\(2)p Fr(:)p Fz(6\))159 958 y(The)30 b(third)e(assumption)h(expresses)h(some)h(smo)s(othness)f(of)g(the)h (in)m(teraction:)50 1139 y FB(Assumption)k(3)30 b Fw(\(c)-5 b(ontinuity\))p Fz(:)42 b(F)-8 b(or)31 b Fr(x)25 b Fp(2)g Fr(X)38 b Fz(set)31 b(\(compare)g(with)e(the)i(quan)m(tit)m(y)f Fr(V)3007 1153 y Fq(q)3048 1139 y Fz(\()p Fr(x)3135 1153 y Fq(i)3165 1139 y Fz(\))h(for)f Fr(U)35 b Fp(\021)25 b Fz(0\))1382 1430 y Fr(V)1435 1444 y Fq(x)1481 1430 y Fz(\()p Fr(i;)15 b(q)s Fz(\))27 b(=3D)1789 1344 y Fk(X)1797 1540 y Fq(j)t Fl(6)p Fm(=3D)p Fq(i)1935 1430 y Fz(\010)2001 1444 y Fq(i;j)2086 1430 y Fp(f)p Fr(q)s(;)15 b(x)2267 1444 y Fq(j)2306 1430 y Fp(g)p Fr(:)1147 b Fz(\(2)p Fr(:)p Fz(7\))50 1697 y(W)-8 b(e)35 b(assume)e(that)h(there)g(exists)f(a)h (constan)m(t)h Fr(C)i(>)31 b Fz(0,)k(suc)m(h)e(that)h(for)g(an)m(y)g(v) m(ector)h Fr(k)f Fp(2)c Fz(\003,)35 b(an)m(y)f Fr(q)f Fp(2)e Fr(Q)p Fz(,)50 1806 y(and)f(an)m(y)g Fr(x)c Fp(2)e Fr(X)7 b Fz(,)1223 1962 y Fp(j)p Fr(V)1301 1976 y Fq(x)1347 1962 y Fz(\()p Fr(i)21 b Fz(+)f Fr(k)s(;)15 b(q)s Fz(\))21 b Fp(\000)f Fr(V)1859 1976 y Fq(x)1904 1962 y Fz(\()p Fr(i;)15 b(q)s Fz(\))p Fp(j)27 b(\024)2248 1901 y Fr(C)7 b(\034)40 b Fp(j)p Fr(k)s Fp(j)p 2248 1942 253 4 v 2339 2025 a Fr(R)2511 1962 y(:)987 b Fz(\(2)p Fr(:)p Fz(8\))159 2233 y(It)27 b(is)f(b)s(oth)g(natural)g(and)h(imp)s(ortan)m(t)f(that)i (the)f(con)m(tin)m(uit)m(y)g(assumption)e(is)h(related)h(to)h(the)f (fact)h(that)50 2342 y(the)i(individual)c(in)m(teractions)k(ha)m(v)m(e) i(to)f(b)s(e)f(rather)g(small.)39 b(In)30 b(fact)h(w)m(e)g(ha)m(v)m(e) 50 2524 y Fh(Lemma)38 b(2.1.)p Fw(If)32 b Fz(\010)g Fw(satis\014es)h (the)g(assumptions)i(ab)-5 b(ove,)33 b(then)1138 2742 y Fz(max)1307 2756 y Fq(q)r(;q)1399 2738 y Fj(0)1425 2756 y Fl(2S)1543 2742 y Fz(max)1712 2761 y Fq(j)t Fl(2)p Fo(Z)1843 2742 y Fs(d)18 b Fp(j)p Fz(\010)1986 2756 y Fq(i;j)2072 2742 y Fp(f)p Fr(q)s(;)d(q)2245 2705 y Fl(0)2271 2742 y Fp(g)25 b(\024)g Fr(C)t(=3DR)903 b Fz(\(2)p Fr(:)p Fz(9\))50 3196 y Fw(Pr)-5 b(o)g(of.)48 b Fz(Note)31 b(that)g(b)m(y)g (translation)e(in)m(v)-5 b(ariance,)30 b(for)g(an)m(y)h Fr(i;)15 b(k)29 b Fp(2)c Fn(Z)2403 3163 y Fq(d)2441 3196 y Fz(,)691 3427 y Fr(V)744 3441 y Fq(x)790 3427 y Fz(\()p Fr(i;)15 b(q)s Fz(\))22 b Fp(\000)d Fr(V)1140 3441 y Fq(x)1186 3427 y Fz(\()p Fr(i)i Fz(+)f Fr(k)s(;)15 b(q)s Fz(\))26 b(=3D)1655 3341 y Fk(X)1663 3537 y Fq(j)t Fl(6)p Fm(=3D)p Fq(i)1801 3427 y Fz(\010)1867 3441 y Fq(i;j)1953 3427 y Fp(f)p Fr(q)s(;)15 b(x)2134 3441 y Fq(j)2173 3427 y Fp(g)20 b(\000)2369 3341 y Fk(X)2329 3537 y Fq(j)t Fl(6)p Fm(=3D)p Fq(i)p Fm(+)p Fq(k)2556 3427 y Fz(\010)2622 3441 y Fq(i)p Fm(+)p Fq(k)r(;j)2803 3427 y Fp(f)p Fr(q)s(;)15 b(x)2984 3441 y Fq(j)3023 3427 y Fp(g)1559 3690 y Fz(=3D)1655 3604 y Fk(X)1663 3799 y Fq(j)t Fl(6)p Fm(=3D)p Fq(i)1801 3690 y Fz(\()q(\010)1903 3704 y Fq(i;j)1988 3690 y Fp(f)p Fr(q)s(;)g(x)2169 3704 y Fq(j)2208 3690 y Fp(g)21 b(\000)f Fz(\010)2431 3704 y Fq(i;j)2516 3690 y Fp(f)p Fr(q)s(;)15 b(x)2697 3704 y Fq(j)t Fm(+)p Fq(k)2832 3690 y Fp(g)p Fz(\))3477 3599 y(\(2)p Fr(:)p Fz(10\))50 4016 y(No)m(w)31 b(c)m(ho)s(ose)g(the)g(con\014guration)f Fr(x)1305 3983 y Fq(`)1370 4016 y Fz(to)h(b)s(e)1021 4288 y Fr(x)1073 4250 y Fq(`)1073 4310 y(j)1137 4288 y Fz(=3D)1233 4160 y Fk(\032)1331 4220 y Fr(q)1375 4187 y Fl(0)1401 4220 y Fr(;)106 b Fz(if)29 b Fr(j)i Fz(=3D)25 b Fr(`)20 b Fp(\000)g Fr(nk)s(;)30 b Fz(for)h(some)15 b Fr(n)25 b Fp(2)g Fn(N)2665 4234 y Fm(0)1331 4356 y Fr(q)s(;)132 b Fz(else)3477 4288 y(\(2)p Fr(:)p Fz(11\))50 4565 y(Then)29 b(b)m(y)h(\(2.10\),)616 4856 y Fr(V)669 4870 y Fq(x)715 4856 y Fz(\()p Fr(i;)15 b(q)s Fz(\))21 b Fp(\000)f Fr(V)1065 4870 y Fq(x)1111 4856 y Fz(\()p Fr(i)h Fz(+)f Fr(k)s(;)15 b(q)s Fz(\))26 b(=3D)1612 4742 y Fl(1)1582 4769 y Fk(X)1580 4962 y Fq(n)p Fm(=3D0)1732 4782 y Fk(\000)1773 4856 y Fz(\010)1839 4870 y Fq(i;`)p Fl(\000)p Fq(nk)2063 4856 y Fp(f)p Fr(q)s(;)15 b(q)2236 4818 y Fl(0)2262 4856 y Fp(g)21 b(\000)f Fz(\010)2485 4872 y Fq(i;`)p Fl(\000)p Fm(\()p Fq(n)p Fm(+1\))p Fq(k)2858 4856 y Fp(f)p Fr(q)s(;)15 b(q)3031 4818 y Fl(0)3057 4856 y Fp(g)3102 4782 y Fk(\001)1484 5085 y Fz(=3D)25 b(\010)1646 5099 y Fq(i;`)1728 5085 y Fp(f)p Fr(q)s(;)15 b(q)1901 5047 y Fl(0)1927 5085 y Fp(g)3477 4934 y Fz(\(2)p Fr(:)p Fz(12\))p 50 5202 546 4 v 159 5263 a Fi(2)197 5289 y Fx(The)26 b(precise)h(c)n(hoice)g(is)g(of)e(course)h(to)g(a)f(large)h (exten)n(t)g(arbitrary)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1165 b Fx(5)p eop %%Page: 6 6 6 5 bop 50 340 a Fz(where)39 b(w)m(e)h(used)f(that)i(the)f(last)f(sum)g (is)g(telescopic)h(and)f(that)h(\(\010)2451 354 y Fq(i;`)p Fl(\000)p Fq(nk)2676 340 y Fp(f)p Fr(q)s(;)15 b(q)2849 307 y Fl(0)2875 340 y Fp(g)40 b Fz(tends)f(to)i(zero)f(as)g Fr(n)50 449 y Fz(tends)33 b(to)i(in\014nit)m(y)-8 b(.)50 b(Cho)s(osing)32 b Fr(k)37 b Fz(to)e(b)s(e)e(a)i(unit)d(v)m(ector,)37 b(\(2.12\))f(together)f(with)e(the)h(upp)s(er)e(b)s(ound)g(on)50 558 y(the)e(left-hand)g(side)f(giv)m(en)h(b)m(y)h(\(2.8\))h(yields)c (the)j(statemen)m(t)h(of)f(the)f(lemma.)41 b Fp(})159 807 y Fz(Our)26 b(aim)h(is)f(to)j(dev)m(elop)e(the)h(Pirogo)m(v-Sinai)e (theory)i(in)e(this)g(setting)i(under)e(suitable)g(requiremen)m(ts)50 916 y(on)g(the)h(v)-5 b(alues)26 b(of)g(the)h(three)g(constan)m(ts)g Fr(D)s(;)15 b(\034)5 b(;)15 b(C)7 b Fz(,)28 b(and)e(the)g(temp)s (erature)h Fr(T)13 b Fz(.)39 b(F)-8 b(ollo)m(wing)26 b(the)g(traditions)50 1025 y(of)k(one)h(of)g(us)e(\(M.Z.\),)j(w)m(e)f (will,)d(ho)m(w)m(ev)m(er,)k(alw)m(a)m(ys)f(set)1764 1262 y Fr(T)38 b Fz(=3D)25 b(1)1481 b(\(2)p Fr(:)p Fz(13\))50 1498 y(and)29 b(th)m(us)h(incorp)s(orate)f(the)i(temp)s(erature)e(in)g (the)h(in)m(teractions.)41 b(W)-8 b(e)31 b(could)e(just)h(as)g(w)m(ell) f(set)h Fr(\034)36 b Fz(=3D)25 b(1)30 b(in)50 1607 y(Assumption)e(2)j (and)f(express)g(our)g(conditions)f(in)g(terms)h(of)g(large)h(enough)f Fr(\014)g Fp(\021)25 b Fz(1)p Fr(=3DT)44 b Fz(only)-8 b(.)50 1856 y Fw(Note.)46 b Fz(Assumption)28 b(2)i(assures)g(that)g(the)g (constan)m(t)h(con\014gurations)f(with)e(v)-5 b(alue)29 b Fr(q)f Fp(2)d Fr(Q)30 b Fz(giv)m(e)g(a)h(strong)50 1965 y(energetic)36 b(p)s(enalt)m(y)g(against)g(c)m(hanging)f(of)h(a)h (single)d(spin.)55 b(Ho)m(w)m(ev)m(er,)40 b(this)34 b(energy)i(is)f (not)h(enough)g(to)50 2074 y(comp)s(ensate)27 b(the)f(en)m(tropic)g (loss)g(o)s(ccuring)f(when)g(\014xing)g(all)h(spins)e(in)h(an)h Fr(R)q Fz(-neigh)m(b)s(orho)s(o)s(d)e(of)i(one)h(site,)50 2183 y(if)22 b Fr(R)h Fz(is)f(large)h(\(and)g(the)g(temp)s(erature)g (not)g(to)s(o)g(small\).)37 b(Th)m(us)22 b(the)h(usual)f(de\014nition)e (of)j(P)m(eierls)f(con)m(tours)50 2292 y(\(calling)31 b(an)m(y)h(spin)e(part)i(of)h(a)f(con)m(tour)h(if)e(at)i(least)f(one)h (of)f(its)g Fr(R)q Fz(-neigh)m(b)s(ors)e(do)i(not)h(agree)g(with)e (him\))50 2402 y(w)m(ould)i(not)i(allo)m(w)f(us)g(to)i(obtain)e(an)h Fr(R)q Fz(-indep)s(enden)m(t)d(constan)m(t)k(in)e(the)g(P)m(eierls)g (condition.)53 b(In)33 b(other)50 2511 y(w)m(ords,)23 b(it)e(is)f(not)i(reasonable)g(to)g(think)e(of)i(p)s(erforming)d(a)j (lo)m(w-temp)s(erature)f(expansion)g(around)f(a)i(single)50 2620 y("ground)j(state")j(con\014guration)1184 2587 y Fm(3)1224 2620 y Fz(,)f(but)f(rather)f(ab)s(out)h(some)h(more)f (relaxed,)g(non)m(trivial)e(measures)i(that)50 2729 y(are)d(in)f(some)h (sp)s(eci\014c)f(w)m(a)m(y)i(concen)m(trated)g(in)e(some)h(neigh)m(b)s (orho)s(o)s(ds)e(of)i(these)g(constan)m(t)h(con\014gurations.)50 2838 y(T)-8 b(o)31 b(mak)m(e)g(this)e(precise)h(will)e(b)s(e)h (somewhat)i(tec)m(hnical,)g(and)f(requires)e(the)j(Assumption)e(3.)50 3087 y FB(Basic)36 b(examples.)159 3338 y Fz(1\))d(In)g(the)g (classical)f(Kac-Ising)g(mo)s(del,)h(where)f Fr(S)i Fz(=3D)29 b Fr(Q)g Fz(=3D)g Fp(f\000)p Fz(1)p Fr(;)15 b Fz(1)p Fp(g)p Fz(,)36 b(\010)2682 3352 y Fq(i;j)2767 3338 y Fz(\()p Fr(x;)15 b(y)s Fz(\))31 b Fp(\021)3118 3297 y Fq(\014)p 3118 3317 45 4 v 3122 3369 a Fm(2)3174 3338 y Fr(J)3224 3352 y Fq(\015)3271 3338 y Fz(\()p Fr(i)22 b Fp(\000)g Fr(j)5 b Fz(\)\()p Fr(x)23 b Fp(\000)50 3455 y Fr(y)s Fz(\))133 3422 y Fm(2)214 3455 y Fz(and)40 b Fr(J)451 3469 y Fq(\015)498 3455 y Fz(\()p Fr(k)s Fz(\))j(=3D)f Fr(\015)826 3422 y Fq(d)869 3455 y Fr(J)9 b Fz(\()p Fr(\015)c(k)s Fz(\))42 b(with)d Fr(J)49 b Fz(e.g.)73 b(b)s(eing)39 b(the)i(indicator)e(function)g(of)i(the)g(unit)e(cub)s(e,)k(the)50 3564 y(assumptions)28 b(ab)s(o)m(v)m(e)k(are)f(satis\014ed)e(with)g Fr(R)d Fz(=3D)f Fr(\015)1775 3531 y Fl(\000)p Fm(1)1873 3564 y Fz(,)31 b Fr(D)d Fz(=3D)d(1,)31 b Fr(\034)k Fz(=3D)25 b Fr(\014)5 b(=3D)p Fz(2,)32 b Fr(C)f Fz(=3D)25 b(2)p Fr(\014)5 b(d)p Fz(.)159 3746 y(2\))21 b(Analogously)-8 b(,)22 b(in)d(the)i(ferromagnetic)f(Blume)g(Cap)s(el)f(mo)s(del)g(with)g(p)s (ositiv)m(e)g(in)m(teractions)h(\010)3450 3760 y Fq(i;j)3535 3746 y Fp(f)p Fr(x)3632 3760 y Fq(i)3663 3746 y Fr(;)15 b(x)3755 3760 y Fq(j)3794 3746 y Fp(g)p Fz(,)50 3855 y Fp(j)p Fr(i)25 b Fp(\000)f Fr(j)5 b Fp(j)36 b(\024)f Fr(R)i Fz(of)f(the)h(ab)s(o)m(v)m(e)h(Kac)f(t)m(yp)s(e)f(and)g(with)f (p)s(oten)m(tials)h Fr(U)10 b Fz(\()p Fr(s)p Fz(\))p Fr(;)15 b(s)36 b Fp(2)f(S)42 b Fz(=3D)35 b Fp(f\000)p Fz(1)p Fr(;)15 b Fz(0)p Fr(;)g Fz(1)p Fp(g)39 b Fz(the)e(ab)s(o)m(v)m(e)50 3964 y(assumptions)25 b(are)j(also)f(satis\014ed)g(at)h(small)e(enough) h(temp)s(eratures,)h(if)e(w)m(e)i(c)m(ho)s(ose)g(suitably)e(the)h (subset)50 4074 y Fr(Q)e Fp(\032)g(S)7 b Fz(:)39 b(If)28 b(all)e Fr(U)10 b Fz(\()p Fr(s)p Fz(\))29 b(are)f(roughly)f(the)h(same) g(and)f(if)g(all)g(in)m(teractions)h(\010)2588 4088 y Fq(i;j)2673 4074 y Fp(f)p Fr(x)2770 4088 y Fq(i)2801 4074 y Fr(;)15 b(x)2893 4088 y Fq(j)2931 4074 y Fp(g)29 b Fz(are)f(roughly)e(of)j(the)50 4183 y(same)i(order)e(\(for)i(an)m(y)g Fr(x)914 4197 y Fq(i)969 4183 y Fp(6)p Fz(=3D)25 b Fr(x)1117 4197 y Fq(j)1155 4183 y Fz(\))31 b(then)f(it)g(is)f(natural)h(to)h(c)m (ho)s(ose)g Fr(Q)25 b Fz(=3D)g Fp(S)7 b Fz(.)159 4364 y(3\))41 b(Ho)m(w)m(ev)m(er,)k(in)m(terpreting)39 b(sites)i(o)s(ccupied)e(b)m(y) h(0)h(as)g(\\empt)m(y")g(and)f(those)h(o)s(ccupied)f(b)m(y)g Fp(\006)p Fz(1)g(as)50 4474 y(particles)32 b(ha)m(ving)g(a)h(strong)g (fugacit)m(y)g(\()p Fr(U)10 b Fz(\(1\))1694 4422 y Fr(:)1671 4474 y Fz(=3D)29 b Fr(U)10 b Fz(\()p Fp(\000)p Fz(1\))2082 4422 y Fr(:)2059 4474 y Fz(=3D)29 b(0\);)15 b Fr(U)10 b Fz(\(0\))31 b Fr(>)e(U)10 b Fz(\()p Fp(\006)p Fz(1\)\),)35 b(if)c(w)m(e)j(ha)m(v)m(e)g(a)f(strong)50 4583 y(Kac)40 b(repulsion)e(resp.)68 b(attraction)42 b(b)s(et)m(w)m(een)e(the)g (particles)f(of)i(di\013eren)m(t)e(resp.)69 b(same)40 b(t)m(yp)s(e)g(and)g(no)50 4692 y(in)m(teraction)26 b(b)s(et)m(w)m(een) g(the)g(particles)f(and)g(the)h(empt)m(y)h(sites,)f(then)g(the)g(only)f (appropriate)g(c)m(hoice)i(of)f(the)50 4801 y(set)31 b(of)f(\\reference)i(colors")e(ma)m(y)h(b)s(e)f Fr(Q)25 b Fz(=3D)g Fp(f)p Fz(1)p Fr(;)15 b Fp(\000)p Fz(1)p Fp(g)32 b Fz(if)e Fr(U)10 b Fz(\(0\))26 b Fp(\035)f Fr(U)10 b Fz(\()p Fp(\006)p Fz(1\).)159 4983 y(W)-8 b(e)28 b(no)m(w)f(announce)g (the)g(cen)m(tral)g(result)f(on)h(the)g(lo)m(w)f(temp)s(erature)h (phases)g(for)f(the)h(class)g(of)g(mo)s(dels)50 5092 y(in)m(tro)s(duced)i(ab)s(o)m(v)m(e.)p 50 5202 546 4 v 159 5263 a Fi(3)197 5289 y Fx(Except)d(for)f Fv(very)31 b Fx(large)26 b Fg(\034)8 b Fx(,)27 b(corresp)r(onding)g(to)e(v)n(ery)h (small)g(temp)r(eratures)h(of)e(order)h Fg(R)2811 5263 y Ff(\000)p Fe(d)2903 5289 y Fx(.)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1165 b Fx(6)p eop %%Page: 7 7 7 6 bop 50 340 a Fh(Theorem)36 b(2.2.)k Fw(T)-7 b(ake)32 b(a)g(class)g(of)g(mo)-5 b(del)33 b(satisfying)g(the)f(assumptions)h Fz(0)19 b Fp(\000)f(\000)p Fz(3)32 b Fw(ab)-5 b(ove.)42 b(Then)32 b(ther)-5 b(e)50 449 y(exists)35 b(a)g(c)-5 b(onstant)37 b Fr(\034)787 463 y Fq(c)853 449 y Fr(<)30 b Fp(1)p Fw(,)35 b(dep)-5 b(ending)36 b(on)g(the)f(c)-5 b(onstants)37 b Fr(C)k Fw(and)36 b Fr(D)i Fw(fr)-5 b(om)36 b(Assumptions)g(1)g(and)f(3,)50 558 y(but)g(not)h(on)g Fr(R)q Fw(,)g(such)g(that)h(the)e(fol)5 b(lowing)37 b(is)f(true.)50 b(Ther)-5 b(e)36 b(exist)g(Lipshitz)h(c)-5 b(ontinuous)36 b(functions)g(\(of)50 667 y Fr(\014)5 b Fw(,)32 b(r)-5 b(esp.)43 b Fz(\010)p Fw(,)32 b(and)i Fr(U)10 b Fz(\()p Fr(q)849 634 y Fl(0)875 667 y Fz(\))p Fw(\))33 b Fr(h)1032 681 y Fq(q)1072 667 y Fw(,)g Fr(q)28 b Fp(2)d Fr(Q)p Fw(,)32 b(such)h(that)10 849 y(\(i\))47 b(If)33 b(for)h(some)g Fr(q)678 816 y Fl(\003)746 849 y Fp(2)26 b Fr(Q)p Fw(,)33 b Fr(h)1018 863 y Fq(q)1053 845 y Fj(\003)1122 849 y Fr(>)26 b Fz(max)1389 865 y Fq(q)r Fl(2)p Fq(Q)p Fl(n)p Fq(q)1602 847 y Fj(\003)1661 849 y Fr(h)1713 863 y Fq(q)1754 849 y Fw(,)33 b(then)g(ther)-5 b(e)35 b(exists)e(a)h(unique)e (in\014nite)h(volume)h(Gibbs)159 972 y(state)f Fr(\026)432 939 y Fq(q)467 912 y Fj(\003)510 972 y Fw(.)-18 1154 y(\(ii\))47 b(If)c(ther)-5 b(e)44 b(is)g(a)f(subset)h Fr(Q)1053 1121 y Fl(\003)1139 1154 y Fp(\032)g Fr(Q)f Fw(such)h(that)h(for)e(al)5 b(l)44 b Fr(p;)15 b(p)2212 1121 y Fl(0)2282 1154 y Fp(2)45 b Fr(Q)2460 1121 y Fl(\003)2501 1154 y Fw(,)h Fr(h)2627 1168 y Fq(p)2714 1154 y Fz(=3D)f Fr(h)2882 1168 y Fq(p)2919 1149 y Fj(0)2949 1154 y Fw(,)g(and)g(for)f (al)5 b(l)44 b(other)159 1263 y Fr(q)39 b Fp(2)e Fr(Q)p Fp(n)p Fr(Q)526 1230 y Fl(\003)568 1263 y Fw(,)j Fr(h)688 1277 y Fq(p)767 1263 y Fr(>)d(h)927 1277 y Fq(q)967 1263 y Fw(,)j(then)g(ther)-5 b(e)40 b(exist)f(exactly)g(one)h(distinct)f (extr)-5 b(emal)41 b(tr)-5 b(anslation)42 b(invariant)159 1372 y(Gibbs)32 b(state)i(for)f(e)-5 b(ach)33 b(value)g Fr(p)25 b Fp(2)g Fr(Q)1442 1339 y Fl(\003)1484 1372 y Fw(.)-46 1554 y(\(iii\))47 b(Ther)-5 b(e)34 b(is)f(a)g(\014nite)h(p)-5 b(ositive)34 b(c)-5 b(onstant)35 b Fr(c)e Fw(such)g(that)i Fp(j)p Fr(h)2068 1568 y Fq(q)2129 1554 y Fp(\000)20 b Fr(U)10 b Fz(\()p Fr(q)s Fz(\))p Fp(j)27 b(\024)f Fr(ce)2636 1521 y Fl(\000)p Fq(\034)7 b(=3Dc)2809 1554 y Fw(.)43 b(The)34 b(functions)f Fr(h)3514 1568 y Fq(q)3588 1554 y Fw(ar)-5 b(e)50 1663 y(given)32 b(in)g(terms)i(of)f(r)-5 b(apid)5 b(ly)34 b(c)-5 b(onver)g(ging)33 b(series.)50 1892 y(R)-5 b(emark.)48 b Fz(Theorem)20 b(2.2)h(is)f(the)h(k)m(ey)g(statemen)m(t)h (of)f(the)g(Pirogo)m(v-Sinai)e(theory)-8 b(.)38 b(It)21 b(implies)c(in)i(particular)50 2001 y(that)30 b(in)e(the)i Fp(j)p Fr(Q)p Fp(j)p Fz(-dimensional)e(space)i(of)g(the)f(parameters)h Fr(U)10 b Fz(\()p Fr(q)s Fz(\),)31 b(the)f(sets)g(where)f(the)h(Gibbs)e (measure)50 2110 y(is)33 b(unique)f(are)j(op)s(en)e(sets,)j(and)d(the)i (sets)f(where)g Fr(k)j Fz(Gibbs)32 b(measures)i(co-exists)h(are)g (closed,)g(Lipshitz)50 2219 y(con)m(tn)m(uous)21 b(submanifolds)d(of)j (co-dimension)e Fr(k)t Fp(\000)q Fz(1.)38 b(F)-8 b(urthermore,)23 b(all)d(exp)s(ectations)h(of)g(lo)s(cal)f(functions)50 2329 y(with)37 b(resp)s(ect)h(to)h(the)f(pure)f(states)i(men)m (tionened)f(in)f(the)h(theorem)h(can)f(b)s(e)g(computed)g(in)f(terms)h (of)50 2438 y(con)m(v)m(ergen)m(t)h(expansions.)58 b(Note)38 b(on)f(the)g(other)g(hand)e(that)j(when)d(more)i(than)g(one)g(Gibbs)e (state)j(co-)50 2547 y(exists,)i(it)e(is)f(in)g(general)h(p)s(ossible)e (that)j(there)g(exist)f(further,)h(non-translation)e(in)m(v)-5 b(arian)m(t)38 b(extremal)50 2656 y(state.)50 2885 y Fw(Note.)46 b Fz(The)40 b(statemen)m(ts)j(of)d(the)h(theorem)g(are)g (the)g(standard)e(consequences)j(of)e(the)h(Pirogo)m(v-Sinai)50 2994 y(theory)29 b(once)h(a)f(mo)s(del)f(can)i(b)s(e)e(form)m(ulated)h (in)f(terms)h(of)g(what)g(it)g(called)f(a)h(\\abstract)i(Pirogo)m(v)e (Sinai")50 3103 y(mo)s(del.)39 b(In)27 b(this)g(pap)s(er)g(w)m(e)i (will)c(sho)m(w)k(that)f(this)f(can)i(b)s(e)e(done)h(in)f(the)i(sense)f (of)g(a)h(sligh)m(tly)d(generalized)50 3212 y(sense)43 b(compared)g(to)h(the)g(standard)e(form)m(ulation,)j(namely)e(where)g (additional,)h(quic)m(kly)e(deca)m(ying,)50 3321 y(\\cluster)28 b(\014elds")f(pro)s(duce)g(a)h(w)m(eak)h(in)m(teraction)f(b)s(et)m(w)m (een)h(con)m(tours.)40 b(The)28 b(reform)m(ulation)f(of)h(Pirogo)m(v{) 50 3431 y(Sinai)33 b(theory)j(in)f(that)h(class)g(is)e(done)i(in)e(a)i (companion)f(pap)s(er)g([Z3].)57 b(The)35 b(pro)s(of)g(of)h(the)g (uniqueness)50 3540 y(result)29 b(\(i\))h(will)e(b)s(e)i(giv)m(en)g(in) f([BZ3].)159 3721 y(The)g(aim)h(of)g(this)f(pap)s(er)f(is)h(to)i (prepare)e(the)i(ground)d(for)i(the)g(application)f(of)h(metho)s(ds)f (of)h([Z3],)h(i.e.)50 3831 y(to)37 b(reform)m(ulate)f(our)g(mo)s(del)f (suc)m(h)h(that)h(its)f(partition)f(functions)g(are)i(expressed)f(in)f (a)i(w)m(a)m(y)g(to)g(allo)m(w)50 3940 y(to)d(apply)f(the)h(results)e (of)i([Z3].)51 b(Emphasis)32 b(is)h(put)g(on)g(sho)m(wing)g(that)h(all) f(estimates)h(in)f(the)h(resulting)50 4049 y(abstrat)d(mo)s(del)e(are)i Fw(uniform)63 b Fz(in)29 b(the)i(range)g(of)f(the)h(in)m(teraction,)f Fr(R)q Fz(.)159 4231 y(Moreo)m(v)m(er,)i(w)m(e)f(dev)m(ote)h(a)e(great) h(care)g(to)g(the)f(construction)g(of)g(cluster)f(expansions.)40 b(These)30 b(expan-)50 4340 y(sions)f(are)h(absolutely)f(crucial)g(and) g(giv)m(e)i(in)d(fact)j(m)m(uc)m(h)f(more)h(information)d(than)i(just)f (the)h(condensed)50 4449 y(statemen)m(t)i(of)f(the)g(theorem.)42 b(Rather,)32 b(they)e(allo)m(w,)h(in)e(principle)f(and)i(ev)m(en)h(in)f (practice,)h(to)g(compute)50 4558 y(all)36 b(ph)m(ysical)g(quan)m (tities)g(in)g(terms)h(of)g(rapidly)e(con)m(v)m(erging)j(series.)61 b(W)-8 b(e)38 b(emphasize)f(this)f(p)s(oin)m(t)g(also)50 4667 y(in)30 b(view)h(of)g(applications)f(to)i(disordered)d(mo)s(dels)i (where)g(renormalization)f(group)h(metho)s(de)g(will)e(rely)50 4776 y(hea)m(vily)g(on)i(the)f(a)m(v)-5 b(ailabilit)m(y)29 b(of)h(con)m(v)m(ergen)m(t)j(expansions.)267 5053 y FA(3.)51 b Fr(q)s FA({points,)32 b(contours)g(and)g(st)-6 b(ars.)49 b(Ref)n(ormula)-6 b(tion)31 b(of)i(the)g(Hamil)-6 b(tonian.)50 5289 y Fz(This)36 b(section)j(in)m(tro)s(duces)e(the)i(crucial)e (notions)h(of)h Fr(q)s Fz(-correct)h(p)s(oin)m(ts)d(and)h(con)m(tours.) 65 b(A)m(t)40 b(the)e(same)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1165 b Fx(7)p eop %%Page: 8 8 8 7 bop 50 340 a Fz(time)29 b(w)m(e)h(will)c(reform)m(ulate)k(the)f (Hamiltonian)f(in)g(a)i(w)m(a)m(y)g(that)g(is)e(adapted)i(to)g(these)g (new)f(notions)f(and)50 449 y(that)j(will)d(b)s(e)h Fw(use)-5 b(d)33 b(systematic)-5 b(al)5 b(ly)41 b Fz(b)s(elo)m(w.)50 744 y FB(3.1.)47 b(The)34 b(cleaning)i(of)f(a)g(con\014guration.)50 925 y Fz(The)e(basic)f(notion)h(of)h(the)f(Pirogo)m(v-Sinai)f(theory)i (is)e(that)i(of)g(a)f(con)m(tour.)51 b(Coun)m(tours)32 b(\(of)i(a)g(con\014gu-)50 1035 y(ration\))e(are)h(connected)h(regions) e(in)f(space)i(where)f(the)h(con\014guaration)f(has)g(lo)s(cally)f(a)i ("high")f(energy)50 1144 y(densit)m(y)d(so)i(that)g(one)g(w)m(ould)e (exp)s(ect)i(that)g(at)g(lo)m(w)f(temp)s(eratures)g(the)h(app)s (earence)f(of)h(large)f(con)m(tours)50 1253 y(should)h(b)s(e)h("unlik)m (ely")f(with)g(rep)s(ect)i(to)g(the)g(Gibbs)e(measure.)48 b(The)32 b(rest)h(of)g(\003)f(is)g(then)h(o)s(ccupied,)f(for)50 1362 y(an)m(y)c(giv)m(en)g(con\014guration,)g(restricted)g(ensem)m (bles)f(of)h(almost)g Fr(q)j Fz(con\014gurations.)39 b(The)28 b(the)g(energy)g(of)g(a)50 1471 y(con\014guration)j(is)h (expressed)f(as)i(a)f(sum)f(of)i(the)f(energies)g(of)g(con)m(tours)h (\(the)g(\\energy)g(barrier")d(terms\))50 1580 y(and)g(energies)g(of)g (the)h(\(almost)g(constan)m(t\))h(con\014gurations)d(around)h(them.)159 1762 y(The)f(aim)h(is)f(to)i(sho)m(w)f(that)g(under)f(suitable)f (conditions)h(the)h(Gibbs)e(measures)i(is)f(concen)m(trated)j(on)50 1871 y(con\014gurations)37 b(that)i(are)f(\\mostly)h(constan)m(t")g (and)f(moreo)m(v)m(er)i(suc)m(h)d(that)i(the)f(con)m(tours)h(form)f (only)50 1980 y(small)29 b(and)g(rare)i(islands)d(in)h(suc)m(h)h(a)h({) f(predominan)m(tly)f(constan)m(t)i({)g(con\014guration.)159 2162 y(In)e(the)h(standard)g(Pirogo)m(v)g(Sinai)e(theory)i(dev)m(elop)s (ed)f(for)h(short)g(range)g(mo)s(dels,)f(con)m(tours)i(are)f(con-)50 2271 y(nected)36 b(comp)s(onen)m(ts)g(of)g(the)g(set)h(of)f (\\incorrect")h(p)s(oin)m(ts.)56 b(A)36 b(p)s(oin)m(t)f Fr(i)h Fz(is)f(called)g Fr(q)k Fz(correct,)g(for)c(some)50 2380 y Fr(q)e Fp(2)c Fr(Q)p Fz(,)34 b(if)e(the)h(con\014guration)g(has) g(v)-5 b(alue)32 b Fr(q)k Fz(an)m(ywhere)d(at)h(the)f(distance)g Fp(\024)d Fr(R)j Fz(from)g Fr(i)p Fz(,)h(and)f(often)h(the)50 2489 y(c)m(hoice)d(of)g Fr(Q)25 b Fp(\032)g Fr(S)35 b Fz(is)29 b(made)i(corresp)s(onding)d(only)i(to)h(the)f(minima)f(of)h Fr(U)10 b Fz(.)159 2671 y(The)30 b Fr(q)s Fz({lik)m(e)h (con\014gurations)f(on)g(the)h(whole)f(\003)h(are)g(th)m(us)g(simply)d (constan)m(t)k(\(or)f(in)f(a)h(suitably)e(more)50 2780 y(general)39 b(setting)g(p)s(erio)s(dic\))d(con\014gurations)j(that)g (corresp)s(ond)f(to)h(the)g(lo)s(cal)g(grounds)e(states)j(of)g(the)50 2889 y(Hamiltonian.)46 b(Regions)33 b(where)f(the)h(con\014gurations)f (are)i(ev)m(erywhere)f(non-constan)m(t)h(\(when)e(lo)s(oking)50 2998 y(up)21 b(to)j(a)f(distance)f Fr(R)q Fz(\))h(are)g(then)g (included)d(in)m(to)i Fw(c)-5 b(ontours)p Fz(.)40 b(The)22 b(imp)s(ortan)m(t)g(p)s(oin)m(t)g(is)g(that)h Fr(q)s Fz(-lik)m(e)f(regions)50 3107 y(with)31 b(di\013eren)m(t)g(v)-5 b(alues)31 b(of)i Fr(q)h Fz(m)m(ust)e(b)s(e)g(separated)g(b)m(y)g(con)m (tours,)h(so)g(that)f(if)f(one)i(can)f(sho)m(w)g(that)h(large)50 3216 y(con)m(tours)i(are)g(indead)e(higly)f(unlik)m(ely)-8 b(,)34 b(one)h(can)f(establish)f(the)i(desired)e(result)g(that)i(Gibbs) e(measure)50 3325 y(are)c(concen)m(trated)i(on)e(essen)m(tilly)f (constan)m(t)j(con\014gurations.)39 b(A)30 b(su\016cien)m(t)e (condition)g(for)h(this)f(to)i(hold)50 3435 y(is)e(a)i(so-called)f(P)m (eirls)f(estimate)i(that)f(relates)h(the)f(energy)h(of)f(a)h(con)m (tour)g(to)f(the)h(geometric)g(size)f(of)h(the)50 3544 y(v)m(olume)e(of)h(its)f(supp)s(ort.)38 b(In)28 b(short)h(range)g(mo)s (del)e(lik)m(e)h(the)h(Ising)e(mo)s(dels)g(suc)m(h)i(an)f(estimate)i (is)d(readily)50 3653 y(pro)m(v)m(en)h(with)e(a)i(prop)s(ortinalit)m(y) e(constan)m(t)j(\(the)f("P)m(eierls)f(constan)m(t"\))j(of)e(order)f(of) h(the)g(strength)g(of)g(the)50 3762 y(nearest)j(neigh)m(b)s(or)e(in)m (teraction.)159 3944 y(Ho)m(w)m(ev)m(er)f(in)d(the)h(situation)f(of)i (Kac)f(t)m(yp)s(e)h(mo)s(dels)d(that)j(w)m(e)g(are)f(in)m(terested)g (in,)g(a)h(direct)e(application)50 4053 y(of)k(the)g(ab)s(o)m(v)m(e)h (pro)s(cedure)e(w)m(ould)g(lead)g(to)i(estimates)f(for)g(the)g(P)m (eierls)f(constan)m(t)i(of)f(the)g(order)g(of)g Fr(\034)10 b(R)3611 4020 y Fl(\000)p Fq(d)50 4162 y Fz(and)31 b(whence)g(our)h (estimate)g(on)f(the)h(critical)f(temp)s(erature)g(w)m(ould)g(get)i (arbitrarily)c(bad)i(as)h(the)f(range)50 4271 y Fr(R)k Fz(of)g(the)g(in)m(teraction)f(increases.)53 b(The)34 b(p)s(oin)m(t)g(here)g(is)g(that)h(the)g(in)m(teractions)f(are)h(lo)s (cally)f(to)s(o)h(w)m(eak)50 4380 y(to)f(enforce)g(constan)m(t)g (con\014gurations)f(as)g(\\most)h(lik)m(ely")e(ones,)j(and)e(that)g(it) g(is)f(necessary)i(in)e(the)i(the)50 4489 y(de\014nition)29 b(of)j(the)h(restricted)e(ensem)m(bles)g(\(in)g(the)h(complemen)m(ts)g (of)g(con)m(tours\)to)i(tak)m(e)g(also)d(the)h(lo)s(cal)50 4598 y(en)m(trop)m(y)f(in)m(to)f(accoun)m(t.)159 4780 y(This)f(will)f(imply)g(that)j(the)g(collection)f(of)h(the)g(allo)m(w)m (ed)f(con\014gurations)g(in)f(the)i(\\restricted)g(ensem-)50 4889 y(bles")40 b(will)e(b)s(e)h(m)m(uc)m(h)i(larger,)i(c)m (haracterized)f(b)m(y)e(some)h(non-lo)s(cal)f(condition)f(to)i(b)s(e)f (\\)p Fr(q)s Fz(-lik)m(e")h(in)e(an)50 4998 y(a)m(v)m(eraged)h(sense.) 63 b(Con)m(tours)38 b(will)d(corresp)s(ondingly)g(ha)m(v)m(e)40 b(to)e(b)s(e)g(de\014ned)e(in)h(a)h(rather)g(complicated)50 5107 y(w)m(a)m(y)-8 b(,)32 b(as)e(will)e(b)s(e)i(detailed)f(b)s(elo)m (w.)159 5289 y(Th)m(us,)42 b(while)d(the)h(general)h(philosoph)m(y)d (of)i(the)h(Pirogo)m(v-Sinai)e(theory)i(will)d(b)s(e)i(unc)m(hanged,)j (the)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1165 b Fx(8)p eop %%Page: 9 9 9 8 bop 50 340 a Fz(details)32 b(of)i(the)g(implemen)m(tation,)f(and)g (in)f(particular)g(the)h(expansion)g(tec)m(hniques)g(used)f(will)f(ha)m (v)m(e)k(to)50 449 y(b)s(e)30 b(adopted)g(to)h(this)e(more)i (complicated)f(situation.)159 631 y(The)23 b(de\014nition)f(of)i(the)h Fr(q)s Fz(-lik)m(e)e(con\014gurations)g(b)s(egins)g(with)g(a)h (necessary)h(criterion)d(that)j(establishes)50 740 y(whether)30 b(a)g(con\014guration)g Fr(x)g Fz(could)g(b)s(e)f(considered)h Fr(q)s Fz(-lik)m(e)g(at)h(a)f(p)s(oin)m(t)g Fr(i)p Fz(:)50 961 y FB(De\014nition)35 b(of)g(a)g Fr(q)i FB({)f(p)s(oin)m(t.)52 b Fw(F)-7 b(or)34 b Fr(s;)15 b(s)1524 928 y Fl(0)1574 961 y Fp(2)25 b Fr(S)37 b Fw(and)d Fr(q)28 b Fp(2)d Fr(S)37 b Fw(and)d Fr(i;)15 b(j)32 b Fp(2)25 b Fz(\003)32 b Fw(denote)i(by)910 1112 y Fr(W)1009 1069 y Fq(q)996 1138 y(i;j)1082 1112 y Fp(f)p Fr(s;)15 b(s)1253 1075 y Fl(0)1278 1112 y Fp(g)26 b Fz(=3D)f(\010)1511 1126 y Fq(i;j)1596 1112 y Fp(f)p Fr(s;)15 b(s)1767 1075 y Fl(0)1793 1112 y Fp(g)21 b(\000)e Fz(\010)2015 1126 y Fq(i;j)2101 1112 y Fp(f)p Fr(s;)c(q)s Fp(g)21 b(\000)f Fz(\010)2496 1126 y Fq(i;j)2581 1112 y Fp(f)p Fr(q)s(;)15 b(s)2753 1075 y Fl(0)2779 1112 y Fp(g)p Fr(:)674 b Fz(\(3)p Fr(:)p Fz(1\))50 1275 y Fw(L)-5 b(et)34 b Fr(x)26 b Fp(2)h Fr(X)41 b Fw(b)-5 b(e)33 b(some)h(c)-5 b(on\014gur)g(ation.)47 b(Then)34 b Fr(i)g Fw(is)f(c)-5 b(al)5 b(le)-5 b(d)35 b(a)f Fr(q)i Fw({)e(p)-5 b(oint)35 b(of)e Fr(x)h Fw(and)g(we)g(write)3252 1242 y Fm(4)3327 1275 y Fr(c)3366 1289 y Fq(x)3412 1275 y Fz(\()p Fr(i)p Fz(\))28 b(=3D)e Fr(q)s Fw(,)50 1384 y(if)32 b(for)h(any)h Fr(s)24 b Fp(6)p Fz(=3D)h Fr(q)1357 1420 y Fk(X)1405 1614 y Fq(j)1536 1506 y Fp(j)p Fr(W)1660 1463 y Fq(q)1647 1532 y(i;j)1732 1506 y Fp(f)p Fr(s;)15 b(x)1912 1520 y Fq(j)1951 1506 y Fp(gj)59 b(\024)e 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y(of)i(its)g(p)s(oin)m (ts)f(is)h(a)g(distance)g(farther)g(than)g Fr(R)h Fz(a)m(w)m(a)m(y)h (from)e(a)2197 3556 y(~)2193 3580 y Fr(\016)t Fz(-correct)i(p)s(oin)m (t.)39 b(This)27 b(construction)i(will)50 3689 y(lo)s(ok)37 b(at)i(\014rst)e(glance)h(rather)g(surprising.)59 b(In)37 b(particular,)h(the)g(reader)g(ma)m(y)g(w)m(onder)f(what)h(the)g(role) 50 3801 y(of)g Fr(\016)k Fz(and)431 3777 y(~)427 3801 y Fr(\016)g Fz(in)37 b(the)h(construction)f(is.)63 b(This)36 b(will)g(only)h(b)s(ecome)h(clear)g(once)h(the)f(expansions)f(in)g(the) 50 3910 y(complemen)m(ts)30 b(of)h(this)e(set)i(are)g(discussed.)159 4092 y(No)m(w)g(w)m(e)g(can)f(\014nally)f(giv)m(e)i(the)f(de\014nition) e(of)j(con)m(tours.)50 4325 y FB(De\014nition)37 b(of)f(a)h(con)m (tour.)53 b Fw(L)-5 b(et)34 b FB(\000)p 1335 4340 63 4 v 34 w Fw(b)-5 b(e)34 b(a)g(c)-5 b(onne)g(cte)g(d)35 b(c)-5 b(omp)g(onent)37 b(of)d(the)g(set)2914 4302 y Fk(e)2895 4325 y Fr(B)5 b Fz(\()p Fr(x)p Fz(\))p Fw(.)46 b(Then)34 b(the)g(p)-5 b(air)50 4434 y FB(\000)25 b Fz(=3D)g(\()p FB(\000)p 269 4449 V Fr(;)15 b(x)424 4448 y Fd(\000)p 424 4463 54 4 v 498 4434 a Fp([)26 b Fz(\026)-51 b Fr(x)631 4448 y Fd(\000)p 631 4463 50 4 v -22 x Fs(c)719 4434 y Fz(\))33 b Fw(is)g(c)-5 b(al)5 b(le)-5 b(d)33 b(a)g(c)-5 b(ontour)67 b(of)33 b Fr(x)p Fw(.)50 4661 y Fz(The)39 b(set)i FB(\000)p 398 4676 63 4 v 40 w Fz(is)e(usually)f(called)i(the)g Fw(supp)-5 b(ort)42 b Fz(of)f(the)f(con)m(tour)h FB(\000)p Fz(.)70 b(It)40 b(will)e(b)s(e)h(useful)g(to)i(distinguish)50 4770 y(b)s(et)m(w)m(een)33 b(the)f Fw(c)-5 b(or)g(e)34 b FB(\000)p 753 4785 V -40 x Fl(\017)886 4770 y Fp(\021)28 b FB(\000)p 985 4785 V 22 w Fp(\\)21 b Fr(B)1221 4784 y Fq(\016)1261 4770 y Fz(\()p Fr(x)p Fz(\),)34 b(the)e Fw(extende)-5 b(d)36 b(c)-5 b(or)g(e)33 b Fz(\()p FB(\000)p 2198 4785 V Fz(\))2296 4737 y Fl(\017)q Fq(R)2422 4770 y Fp(\021)c FB(\000)p 2522 4785 V 21 w Fp(\\)21 b Fr(B)2762 4737 y Fq(R)2757 4796 y(\016)2821 4770 y Fz(\()p Fr(x)p Fz(\),)34 b(and)e(the)g Fw(b)-5 b(elts)33 b 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Fx(substan)n(tially\))50 5289 y(should)26 b(not)f(b)r(e)h(confused)g(with)g(another)g(notation)f Fb(\000)d Fx(=3D)f Fb(\000)1831 5263 y Fe(q)1895 5289 y Fx(marking)k(con)n(tours)h Fb(\000)g Fx(ha)n(ving)f(the)h(external)g (colour)g Fg(q)s Fx(.)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(11)p eop %%Page: 12 12 12 11 bop 159 340 a Fz(Finally)-8 b(,)31 b(it)g(will)e(b)s(e)i(useful)g (to)h(in)m(tro)s(duce)f(the)h Fr(q)s Fz(-regular)f(regions)g(\(the)i (decomp)s(osition,)e(according)50 449 y(to)g(\\colours",)g(of)f(the)h (set)g(\()p Fp([)p FB(\000)p 1084 464 63 4 v Fz(\))1182 416 y Fq(c)1249 449 y Fz(\))1367 657 y Fk(e)1360 680 y Fz(\003)1423 642 y Fq(q)1489 680 y Fp(\021)1591 657 y Fk(e)1585 680 y Fz(\003)1648 694 y Fq(q)1688 680 y Fz(\()p Fr(x)p Fz(\))26 b Fp(\021)f Fz(\003)1995 694 y Fq(q)2035 680 y Fz(\()p Fr(x)p Fz(\))p Fp(n)2222 657 y Fk(e)2202 680 y Fr(B)6 b Fz(\()p Fr(x)p Fz(\))1078 b(\(3)p Fr(:)p Fz(12\))50 899 y(whose)30 b(complemen)m(t)h(will)c(b)s 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4710 y Fr(G)p Fw(.)50 4951 y(Pr)-5 b(o)g(of.)48 b Fz(By)30 b(de\014nition,)1013 5189 y Fr(E)5 b Fz(\()p FB(\000)p Fz(\))26 b(=3D)1340 5103 y Fk(X)1344 5298 y Fq(i)p Fl(2)p Fd(\000)p 1419 5313 50 4 v 1487 5189 a Fr(V)1540 5203 y Fq(x)1581 5212 y Fa(\000)1626 5203 y Fl([)t Fm(\026)-40 b Fq(x)1716 5214 y Fa(\000)p 1716 5229 41 4 v -13 x Fs(c)1801 5189 y Fz(\()p Fr(i;)15 b(x)1959 5203 y Fq(i)1990 5189 y Fz(\))20 b(+)2136 5103 y Fk(X)2140 5298 y Fq(i)p Fl(2)p Fd(\000)p 2215 5313 50 4 v 2268 5189 a Fz(\()p Fr(U)10 b Fz(\()p Fr(x)2462 5203 y Fq(i)2493 5189 y Fz(\))20 b Fp(\000)g Fr(U)10 b Fz(\))3523 5232 y(\(4)p Fr(:)p Fz(6\))50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(13)p eop %%Page: 14 14 14 13 bop 50 340 a Fz(where)25 b Fr(V)361 354 y Fq(x)407 340 y Fz(\()p Fr(i;)15 b(q)s Fz(\))27 b(w)m(as)g(de\014ned)e(in)f (\(2.7\).)41 b(Let)27 b(us)e(assume)h(that)h(all)d(p)s(oin)m(ts)h Fr(i)i Fz(in)2792 317 y Fk(e)2774 340 y Fr(G)f Fz(are)g(actually)g (incorrect)50 449 y(w.r.t.)40 b(all)29 b(colors)h(in)e Fp(S)7 b Fz(.)41 b(Then)28 b(it)i(is)f(enough)g(to)i(b)s(ound)c(the)j (righ)m(t)g(hand)e(side)h(of)h(\(4.6\))i(from)d(b)s(elo)m(w)g(b)m(y) 1489 674 y Fp(\025)1585 588 y Fk(X)1589 783 y Fq(i)p Fl(2)p Fd(\000)p 1664 798 50 4 v 1732 674 a Fr(V)1785 688 y Fq(x)1826 697 y Fa(\000)1871 688 y Fl([)t Fm(\026)-40 b Fq(x)1961 699 y Fa(\000)p 1961 714 41 4 v -13 x Fs(c)2046 674 y Fz(\()p Fr(i;)15 b(x)2204 688 y Fq(i)2235 674 y Fz(\))1253 b(\(4)p Fr(:)p Fz(7\))50 1006 y(Using)31 b(the)h(upp)s(er)e (b)s(ound)f(of)j(Assumption)e(1',)j(together)h(with)c(the)i (de\014nition)e(of)3002 983 y Fk(e)2984 1006 y Fr(G)h Fz(one)h(sees)g(readily)50 1121 y(that)f(for)f(all)f Fr(i)d Fp(2)673 1098 y Fk(e)655 1121 y Fr(G)j Fz(for)i(an)m(y)f Fr(q)e Fp(2)d(S)7 b Fz(,)1500 1273 y Fr(V)1553 1287 y Fq(x)1594 1296 y Fa(\000)1639 1287 y Fl([)t Fm(\026)-40 b Fq(x)1729 1298 y Fa(\000)p 1729 1313 V -13 x Fs(c)1813 1273 y Fz(\()p Fr(i;)15 b(q)s Fz(\))27 b Fp(\025)2132 1211 y Fr(V)20 b(\016)p 2132 1252 118 4 v 2139 1335 a(D)2217 1309 y Fl(0)3523 1273 y Fz(\(4)p Fr(:)p Fz(8\))50 1466 y(Note)37 b(that)g(this)e(estimate)i(uses)f(in)e(a)j(crucial)e(w)m(a)m (y)i(the)f(fact)h(that)g(correctness)g(of)f(a)h(p)s(oin)m(t)e(do)s(es)g (not)50 1575 y(dep)s(end)28 b(on)j(the)f(v)-5 b(alues)30 b(of)h(the)f(spin)f(at)i(this)e(p)s(oin)m(t.)159 1757 y(Next)i(note)g(that)g(b)m(y)f(the)h(con)m(tin)m(uit)m(y)f(Assumption)f (3,)1023 2011 y Fr(V)1076 2025 y Fq(x)1117 2034 y Fa(\000)1162 2025 y Fl([)t Fm(\026)-40 b Fq(x)1252 2036 y Fa(\000)p 1252 2051 41 4 v -13 x Fs(c)1336 2011 y Fz(\()p Fr(i;)15 b(x)1494 2025 y Fq(i)1526 2011 y Fz(\))25 b Fp(\025)g Fr(V)1735 2025 y Fq(x)1776 2034 y Fa(\000)1822 2025 y Fl([)t Fm(\026)-40 b Fq(x)1912 2036 y Fa(\000)p 1912 2051 V -13 x Fs(c)1996 2011 y Fz(\()p Fr(i)21 b Fz(+)f Fr(k)s(;)15 b(x)2316 2025 y Fq(i)2347 2011 y Fz(\))20 b Fp(\000)2504 1950 y Fr(C)7 b Fp(j)p Fr(k)s Fp(j)p Fr(\034)p 2504 1990 223 4 v 2580 2074 a(R)3523 2011 y Fz(\(4)p Fr(:)p 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a(D)2647 2670 y Fl(0)2710 2634 y Fp(\000)2812 2572 y Fr(C)g(K)g(\034)p 2812 2613 206 4 v 2879 2696 a(R)3477 2634 y Fz(\(4)p Fr(:)p Fz(10\))50 2866 y(F)-8 b(rom)30 b(here)g(the)h(lemma)e(follo)m(ws)h(immediately)e (under)h(the)h(ab)s(o)m(v)m(e)h(assumption.)39 b(No)m(w,)32 b(if)d(some)h(of)h(the)50 2976 y(p)s(oin)m(ts)e Fr(i)h Fz(are)h(incorrect)f(b)s(ecause)g Fr(c)1290 2990 y Fq(i)1321 2976 y Fz(\()p Fr(x)p Fz(\))c Fp(2)f(S)7 b(n)p Fr(Q)p Fz(,)30 b(then)g(this)g(implies)d(that)k(there)f(is)g(a)g(large)h (fraction)f(of)50 3085 y(p)s(oin)m(ts)j(in)f(the)i Fr(R)q Fz(-neigh)m(b)s(orho)s(o)s(d)e(of)i(these)g(p)s(oin)m(ts)f(that)h(ha)m (v)m(e)h Fr(U)10 b Fz(\()p Fr(x)2463 3099 y Fq(i)2494 3085 y Fz(\))31 b Fp(\025)g Fr(U)h Fz(+)23 b Fr(\034)10 b(=3D)p Fz(4.)52 b(But)34 b(this)f(implies)50 3194 y(an)d(ev)m(en)h (stronger)g(excess)g(energy)g(as)f(w)m(as)h(obtained)f(ab)s(o)m(v)m(e.) 42 b Fp(})159 3428 y Fz(W)-8 b(e)26 b(are)g(no)m(w)f(ready)g(to)h 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Fr(R)q Fz(\))h(carries)e(con\014gurations)50 3528 y(that)26 b(are)g(ev)m(en)595 3504 y(~)591 3528 y Fr(\016)t Fz(-correct)h(if)d (\\lo)s(ok)m(ed)i(from)g(the)f(p)s(oin)m(t)g(of)g(view)g(of)h(the)g (cleaned)f(con\014guration)g(outside")50 3643 y(and)30 b(th)m(us)g(ha)m(v)m(e)h(an)f(ev)m(en)i(lo)m(w)m(er)e(densit)m(y)g(of)g (stars)h(then)f(otherwise)g(allo)m(w)m(ed)g(in)2890 3620 y Fk(e)2884 3643 y Fz(\003)2947 3657 y Fq(q)2987 3643 y Fz(.)159 3825 y(The)k(only)g(obstacle)h(w)m(e)h(will)c(encoun)m(ter)j (is)f(the)h(presence)f(of)h(the)g(densit)m(y)f(constrain)m(t.)54 b(This)33 b(con-)50 3934 y(strain)m(t)c(in)f(crucial)h(on)g(the)h(one)g (hand)f(since)f(it)i(ensures)e(that)i(the)g(in)m(teraction)g(is)e(w)m (eak)j(and)e(that)h(th)m(us)50 4043 y(a)k(high-temp)s(erature)e (expansion)h(ma)m(y)h(con)m(v)m(erge.)53 b(On)33 b(the)h(other)g(hand,) g(this)e(constrain)m(t)i(imp)s(oses)e(a)50 4153 y(non-lo)s(cal)c(in)m (teraction)h(b)s(et)m(w)m(een)h(the)g(high-temp)s(erature)e(p)s 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y(expansion)23 b(of)i(this)f(tree)i(ensem)m(ble)e(ev)m(en)h(if)f(there)h(w)m(ere)g(no) g(densit)m(y)f(constrain)m(ts)h(\(i.e.)39 b(in)24 b(this)f(ensem)m(ble) p 50 5015 546 4 v 159 5081 a Fi(7)197 5107 y Fx(Emphasise)k(that)g (while)958 5088 y(~)954 5107 y Fg(\016)j Fx(pla)n(y)n(ed)d(a)g(crucial) h(role)g(in)g(the)f(de\014nition)h(of)e(con)n(tours)i(the)f (de\014nition)h(of)e(the)i(ensem)n(ble)50 5198 y(o)n(v)n(er)i(whic)n(h) h(w)n(e)g(sum)g(in)g(\(5.2\))f(is)i(actually)f(tak)n(en)f(do)r(es)i (not)e(dep)r(end)h(on)2326 5179 y(~)2323 5198 y Fg(\016)i Fx(once)e(the)g(collection)i(of)d(con)n(tours)h Fu(f)p Fb(\000)3560 5209 y Fe(i)3588 5198 y Fu(g)g Fx(is)50 5289 y(\014xed,)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)h(13:49) 1146 b Fx(15)p eop %%Page: 16 16 16 15 bop 50 340 a Fz(the)26 b(lo)m(w)g(densit)m(y)g(constrain)m(t)g(w) m(ould)f(b)s(e)g(automatically)h(resp)s(ected)g(with)f(o)m(v)m (erwhelming)h(probabilit)m(y\).)50 449 y(A)m(t)j(this)e(p)s(oin)m(t)g (the)h(constrain)m(t)h(is)e(th)m(us)h(a)g(pure)f(n)m(uisance,)h(and)g (w)m(e)g(w)m(ould)f(lik)m(e)h(to)h(get)g(rid)d(of)j(it.)39 b(T)-8 b(o)29 b(do)50 558 y(so)34 b(w)m(e)h(actually)f(treat)i(it)e(as) g(a)h(p)s(erturbation)d(and)i(again)g(expand)g(it)g(in)f(the)h("F.K.)i (w)m(a)m(y".)54 b(This)32 b(will)50 667 y(pro)s(duce)27 b(a)i(new)f(class)g(of)h(ob)5 b(jects)29 b(\(high)f(densit)m(y)g (graphs\))g(whic)m(h)f(w)m(e)i(will)d(call)i(\\galaxies")h(and)f(whic)m (h)50 776 y(in)h(man)m(y)h(w)m(a)m(ys)i(share)e(the)g(prop)s(erties)f (of)i(the)f(original)f(con)m(tours.)159 958 y(After)k(these)g (preparations)e(the)i(partition)e(function)h(is)f(reform)m(ulated)i(in) e(the)i(form)f(of)h(a)g(standard)50 1067 y(p)s(olymer)c(mo)s(del)g(and) h(con)m(v)m(en)m(tional)h(tec)m(hniques)f(can)g(b)s(e)g(used)g(to)h (deal)f(with)f(it.)159 1249 y(T)-8 b(o)38 b(simplify)d(notation)j(from) g(the)g(somewhat)h(gruesome)f(lo)s(oking)f(\(5.2\),)42 b(let)c Fr(M)48 b Fz(b)s(e)37 b(an)m(y)i(set)f(and)50 1358 y(let)c Fr(x)237 1372 y Fq(M)313 1354 y Fs(c)387 1358 y Fz(b)s(e)g(a)h(b)s(oundary)d(condition)i(of)g(the)h(t)m(yp)s(e)g (that)g(can)g(arise)f(in)g(our)g(situation,)h(i.e.)53 b(suc)m(h)34 b(that)50 1467 y Fr(M)21 b Fp(\\)11 b Fr(B)303 1478 y Fm(~)300 1496 y Fq(\016)340 1467 y Fz(\()p Fr(q)416 1481 y Fq(M)508 1467 y Fp([)g Fr(x)632 1481 y Fq(M)708 1463 y Fs(c)746 1467 y Fz(\))26 b(=3D)f Fp(;)p Fz(.)40 b(Using)25 b(Prop)s(osition)f(3.4,)k(w)m(e)e(can)g(write)f(the)h (partition)f(function)g(of)h(in)m(terest)50 1576 y(is)j(\(up)h(to)h(an) f(o)m(v)m(erall)h(factor)g(exp\()p Fr(U)10 b Fz(\()p Fr(q)s Fz(\))p Fp(j)p Fr(M)g Fp(j)p Fz(\)\))203 1821 y Fr(Z)272 1777 y Fq(q)265 1848 y(M)346 1821 y Fz(\()p Fr(x)433 1835 y Fq(M)509 1817 y Fs(c)548 1821 y Fz(\))25 b(=3D)766 1735 y Fk(X)694 1936 y Fq(x)735 1945 y Fs(M)803 1936 y Fl(2S)902 1917 y Fs(M)984 1821 y Fz(1)-23 b(I)1040 1838 y Fq(M)7 b Fl(\032)p Fq(B)1231 1816 y Fs(c)1228 1862 y(\016)1265 1838 y Fm(\()p Fq(x)p Fm(\))1383 1821 y Fz(exp\()p Fp(\000)p Fr(H)1711 1777 y Fq(q)1704 1848 y(M)1784 1821 y Fz(\()p Fr(x)p Fz(\)\))704 2086 y(=3D)872 1999 y Fk(X)800 2200 y Fq(x)841 2209 y Fs(M)909 2200 y Fl(2S)1008 2182 y Fs(M)1091 2086 y Fz(1)-23 b(I)1146 2102 y Fq(M)7 b Fl(\032)p Fq(B)1337 2080 y Fs(c)1334 2126 y(\016)1371 2102 y Fm(\()p Fq(x)p Fm(\))1506 1999 y Fk(Y)1489 2194 y Fq(i)p Fl(2)p Fq(M)1655 2086 y Fr(e)1697 2048 y Fl(\000)p Fm(\()p Fq(V)1825 2057 y Fs(q)1862 2048 y Fm(\()p Fq(x)1931 2057 y Fs(i)1960 2048 y Fm(\)+)p Fq(U)g Fm(\()p Fq(x)2170 2057 y Fs(i)2199 2048 y Fm(\))p Fl(\000)p Fq(U)g Fm(\()p Fq(q)r Fm(\)\))2610 1999 y Fk(Y)2481 2198 y Fl(f)p Fq(i;j)t Fl(g\\)p Fq(M)g Fl(6)p Fm(=3D)p Fl(;)2870 2086 y Fr(e)2912 2044 y Fl(\000)p Fq(W)3047 2013 y Fs(q)3037 2067 y(i;j)3117 2044 y Fm(\()p Fq(x)3186 2053 y Fs(i)3215 2044 y Fq(;x)3278 2053 y Fs(j)3312 2044 y Fm(\))3345 2086 y Fr(:)3523 1997 y Fz(\(5)p Fr(:)p Fz(3\))50 2418 y(where)30 b Fr(B)387 2385 y Fq(c)382 2444 y(\016)423 2418 y Fz(\()p Fr(x)p Fz(\))c Fp(\021)f Fz(\003)20 b Fp(n)h Fr(B)885 2432 y Fq(\016)925 2418 y Fz(\()p Fr(x)p Fz(\))31 b(and)f(it)g(is)f(understo)s(o)s(d)g(that)h Fr(x)c Fz(=3D)f Fr(x)2335 2432 y Fq(M)2436 2418 y Fp([)19 b Fr(q)2557 2432 y Fq(M)2633 2414 y Fs(c)2672 2418 y Fz(.)50 2660 y FB(5.1.)47 b(Construction)35 b(of)g(the)g(p)s(olymer)f (represen)m(tation)h(in)g(the)f(v)m(olume)h Fr(M)10 b FB(.)159 2902 y Fz(W)-8 b(e)30 b(b)s(egin)d(the)i(program)f(outlined)f (ab)s(o)m(v)m(e)j(with)d(the)i(\014rst)f(step,)i(the)e(high-temp)s (erature)g(expansion)50 3011 y(of)i(the)h(in)m(teraction)f(b)s(et)m(w)m (een)h(stars.)159 3193 y(F)-8 b(or)31 b Fr(i)26 b Fp(6)p Fz(=3D)f Fr(j)36 b Fz(write)29 b(the)i(term)f Fr(e)1196 3152 y Fl(\000)p Fq(W)1331 3121 y Fs(q)1321 3174 y(i;j)1401 3152 y Fl(f)p Fq(x)1479 3161 y Fs(i)1508 3152 y Fq(;x)1571 3161 y Fs(j)1605 3152 y Fl(g)1677 3193 y Fz(as)325 3420 y(exp\()p Fp(\000)p Fr(W)669 3376 y Fq(q)656 3445 y(i;j)742 3420 y Fp(f)p Fr(x)839 3434 y Fq(i)869 3420 y Fr(;)15 b(x)961 3434 y Fq(j)1000 3420 y Fp(g)p Fz(\))26 b(=3D)f(1)c(+)f Fr(w)1426 3376 y Fq(q)1424 3445 y(i;j)1509 3420 y Fp(f)p Fr(x)1606 3434 y Fq(i)1637 3420 y Fr(;)15 b(x)1729 3434 y Fq(j)1768 3420 y Fp(g)25 b(\021)g Fz(1)c(+)f Fr(w)2156 3434 y Fq(b)2374 3420 y Fz(where)151 b Fr(b)26 b Fz(=3D)f Fp(f)p Fr(x)3016 3434 y Fq(i)3046 3420 y Fr(;)15 b(x)3138 3434 y Fq(j)3177 3420 y Fp(g)p Fr(:)276 b Fz(\(5)p Fr(:)p Fz(4\))50 3647 y(W)-8 b(e)31 b(also)g(write)1126 3765 y Fr(v)s Fz(\()p Fr(x)1260 3779 y Fq(i)1291 3765 y Fz(\))26 b(=3D)f(exp\()p Fp(\000)p Fr(U)10 b Fz(\()p Fr(x)1852 3779 y Fq(i)1882 3765 y Fz(\))21 b(+)f Fr(U)10 b Fz(\()p Fr(q)s Fz(\))21 b Fp(\000)e Fr(V)2379 3779 y Fq(q)2420 3765 y Fz(\()p Fr(x)2507 3779 y Fq(i)2537 3765 y Fz(\)\))p Fr(:)891 b Fz(\(5)p Fr(:)p Fz(5\))50 3946 y(Note)46 b(that)f(if)e Fr(x)644 3960 y Fq(i)724 3946 y Fz(=3D)48 b Fr(q)s Fz(,)g(then)d Fr(w)1249 3902 y Fq(q)1247 3971 y(i;j)1332 3946 y Fp(f)p Fr(x)1429 3960 y Fq(i)1460 3946 y Fr(;)15 b(x)1552 3960 y Fq(j)1591 3946 y Fp(g)49 b Fz(=3D)g(0)44 b(and)g Fr(v)s Fz(\()p Fr(x)2219 3960 y Fq(i)2250 3946 y Fz(\))50 b(=3D)e(1.)84 b(Recall)44 b(that)h(w)m(e)g(assumed)f(in)50 4061 y(Assumption)35 b(2)j(and)f(3)g(\(\(2.6\))j(and)c(\(2.5\)\))k(that)e(w)m(e)f(ha)m(v)m (e)i(b)s(oth)d(the)i(b)s(ound)d(for)i(the)g(\\oscillation)g(of)50 4170 y Fr(U)10 b Fz("as)35 b(w)m(ell)e(as)i(a)g(lo)m(w)m(er)g(b)s(ound) d(for)i Fr(V)1389 4184 y Fq(q)1430 4170 y Fz(\()p Fr(x)1517 4184 y Fq(i)1547 4170 y Fz(\).)54 b(W)-8 b(e)35 b(ma)m(y)g(summarize)f (these)h(t)m(w)m(o)h(b)s(ounds)c(b)m(y)i(assuming)50 4279 y(that)d(for)f(an)m(y)h Fr(x)610 4293 y Fq(i)665 4279 y Fp(6)p Fz(=3D)25 b Fr(q)1173 4396 y(V)1226 4410 y Fq(q)1266 4396 y Fz(\()p Fr(x)1353 4410 y Fq(i)1384 4396 y Fz(\))20 b(+)g Fr(U)10 b Fz(\()p Fr(x)1689 4410 y Fq(i)1720 4396 y Fz(\))20 b Fp(\000)g Fr(U)10 b Fz(\()p Fr(q)s Fz(\))26 b Fr(>)f Fz(3)p Fr(=3D)p Fz(4)32 b Fr(V)2394 4410 y Fq(q)2434 4396 y Fz(\()p Fr(x)2521 4410 y Fq(i)2551 4396 y Fz(\))937 b(\(5)p Fr(:)p Fz(6\))159 4650 y(Already)27 b(at)h(this)f(p)s(oin)m(t)g(it)g(is)g(reasonable)h(to)g(form)m(ulate)g (our)f(later)h(requiremen)m(ts)f(on)g(the)h(fugacities)50 4759 y Fr(V)103 4773 y Fq(q)143 4759 y Fz(\()p Fr(x)230 4773 y Fq(i)261 4759 y Fz(\))f({)g(whic)m(h)e(will)g(ha)m(v)m(e)j(to)f (\\b)s(eat")h(b)s(oth)e(the)h(repulsiv)m(e)e(energies)i(of)g(the)g(b)s (ond)e(terms)i Fr(W)3334 4716 y Fq(q)3321 4785 y(i;j)3406 4759 y Fp(f)p Fr(x)3503 4773 y Fq(i)3534 4759 y Fr(;)15 b(x)3626 4773 y Fq(j)3664 4759 y Fp(g)50 4874 y Fz(\(whose)30 b(sum)g(o)m(v)m(er)h Fr(j)36 b Fz(ma)m(y)31 b(substan)m(tially)e(lo)m (w)m(er)h Fr(V)1837 4888 y Fq(q)1878 4874 y Fz(\()p Fr(x)1965 4888 y Fq(i)1995 4874 y Fz(\)\))h(as)g(w)m(ell)e(as)i(the)f(en)m(trop)m (y)-8 b(.)159 5056 y(In)30 b(order)f(to)j(simplify)26 b(the)31 b(notation)g(let)f(us)g(in)m(tro)s(duce)f(the)i(mo)s(di\014ed) d(p)s(oten)m(tial)976 5266 y Fk(e)964 5289 y Fr(V)1017 5303 y Fq(q)1057 5289 y Fz(\()p Fr(x)1144 5303 y Fq(i)1175 5289 y Fz(\))d(=3D)g(\(1)p Fr(=3D)p Fz(2)d Fp(\000)e Fr(\016)s Fz(\))32 b Fr(V)1777 5303 y Fq(q)1817 5289 y Fz(\()p Fr(x)1904 5303 y Fq(i)1935 5289 y Fz(\))20 b(+)g Fr(U)10 b Fz(\()p Fr(x)2240 5303 y Fq(i)2271 5289 y Fz(\))20 b Fp(\000)g Fr(U)10 b Fz(\()p Fr(q)s Fz(\))26 b Fr(>)f Fz(0)p Fr(:)728 b Fz(\(5)p Fr(:)p Fz(7\))50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(16)p eop %%Page: 17 17 17 16 bop 50 340 a Fh(Lemma)38 b(5.1.)p Fw(Ther)-5 b(e)33 b(exists)g Fr(\034)1155 307 y Fl(\003)1222 340 y Fr(>)25 b Fz(0)33 b Fw(such)f(that)i(for)f(e)-5 b(ach)34 b Fr(i)f Fw(and)g(e)-5 b(ach)34 b Fr(x)2635 354 y Fq(i)2690 340 y Fp(2)25 b Fr(S)1118 501 y Fk(X)1166 695 y Fq(j)1284 501 y Fk(X)1264 696 y Fq(x)1305 705 y Fs(j)1339 696 y Fl(2)p Fq(S)1451 587 y Fr(w)1518 544 y Fq(q)1516 613 y(i;j)1602 587 y Fp(f)p Fr(x)1699 601 y Fq(i)1730 587 y Fr(;)15 b(x)1822 601 y Fq(j)1860 587 y Fp(g)33 b Fr(e)1980 550 y Fl(\000)2041 546 y Fk(e)2037 550 y Fq(V)2080 559 y Fs(q)2117 550 y Fm(\()p Fq(x)2186 559 y Fs(i)2215 550 y Fm(\))2306 587 y Fp(\024)57 b Fr(e)2476 550 y Fl(\000)p Fq(\034)2574 522 y Fj(\003)2617 587 y Fr(:)881 b Fz(\(5)p Fr(:)p Fz(8\))50 914 y Fw(Mor)-5 b(e)g(over,)33 b(for)h(lar)-5 b(ge)33 b(enough)g Fr(\034)10 b Fw(,)33 b Fr(\034)1312 881 y Fl(\003)1379 914 y Fr(>)25 b(\034)10 b(=3D)p Fz(3)p Fw(.)50 1156 y(Pr)-5 b(o)g(of.)48 b Fz(By)40 b(Lemma)h(2.1)g(w)m(e)g (ha)m(v)m(e)h(that)f Fp(j)p Fr(w)1642 1112 y Fq(q)1640 1181 y(i;j)1726 1156 y Fp(f)p Fr(x)1823 1170 y Fq(i)1853 1156 y Fr(;)15 b(x)1945 1170 y Fq(j)1984 1156 y Fp(gj)43 b(\024)e Fr(W)2308 1112 y Fq(q)2295 1181 y(i;j)2381 1156 y Fz(\()p Fr(x)2468 1170 y Fq(i)2498 1156 y Fr(;)15 b(x)2590 1170 y Fq(j)2629 1156 y Fz(\))p Fr(e)2706 1123 y Fm(2)p Fq(C)r(=3DR)2895 1156 y Fz(.)70 b(Then,)42 b(taking)f(in)m(to)50 1266 y(accoun)m(t)32 b(Assumption)c(1',)j(w)m(e)g(see)g(that)276 1462 y Fk(X)324 1655 y Fq(j)443 1462 y Fk(X)422 1657 y Fq(x)463 1666 y Fs(j)497 1657 y Fl(2)p Fq(S)609 1548 y Fr(w)676 1505 y Fq(q)674 1574 y(i;j)760 1548 y Fp(f)p 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Fp(\032)h Fr(G)d Fz(suc)m(h)h(that)g Fr(T)p 2238 1765 66 4 v 2304 1773 a Fq(G)2391 1750 y Fz(=3D)i Fr(G)p 2487 1765 72 4 v -1 w Fz(,)g(all)d(connected)h(comp)s(onen)m(ts)h(of)50 1859 y Fr(T)103 1873 y Fq(G)189 1859 y Fz(are)h Fw(tr)-5 b(e)g(es)521 1826 y Fm(8)563 1859 y Fz(,)26 b(and,)f(for)g(all)e(p)s (oin)m(ts)h Fr(i)h Fp(2)g Fr(G)p 1475 1874 V Fz(,)h(the)f(v)-5 b(alues)24 b(of)g(the)h(spins)e(\(v)m(ertices)i(of)g Fr(G)f Fz(resp.)38 b Fr(T)3316 1873 y Fq(G)3378 1859 y Fz(\))25 b(are)g(the)50 1969 y(same:)45 b Fr(x)369 1983 y Fq(i)400 1969 y Fz(\()p Fr(T)488 1983 y Fq(G)549 1969 y Fz(\))29 b(=3D)g Fr(x)765 1983 y Fq(i)795 1969 y Fz(\()p Fr(G)p Fz(\).)48 b(It)32 b(is)g(imp)s(ortan)m(t)g(to)h(note)g (that)g(b)m(y)g(this)e(mapping)g(the)i(spin)d(con\014guration)50 2078 y(asso)s(ciated)c(to)g Fr(G)f Fz(is)g(en)m(tirely)f(determined)g (b)m(y)i Fr(T)1736 2092 y Fq(G)1797 2078 y Fz(.)39 b(In)25 b(particular,)g(a)h(graph)f Fr(G)g Fz(o)s(ccurs)g(in)f(the)i(partition) 50 2187 y(sum)j Fr(Z)311 2143 y Fq(q)304 2213 y(M)385 2187 y Fz(\()p Fr(x)472 2201 y Fq(M)548 2182 y Fs(c)587 2187 y Fz(\))i(if)e(and)h(only)f(if)h Fr(T)1249 2201 y Fq(G)1340 2187 y Fz(o)s(ccurs.)159 2369 y(F)-8 b(or)31 b(an)m(y)g(forest)g Fr(T)13 b Fz(,)30 b(denote)h(b)m(y)46 b Fk(e)-67 b Fr(w)1350 2383 y Fq(T)1438 2369 y Fz(the)30 b(quan)m(tit)m(y)1550 2615 y Fk(e)-67 b Fr(w)1599 2629 y Fq(T)1682 2615 y Fz(=3D)1853 2528 y Fk(X)1777 2723 y Fq(G)p Fm(:)p Fq(T)1898 2732 y Fs(G)1951 2723 y Fm(=3D)p Fq(T)2074 2615 y Fr(w)2139 2629 y Fq(G)2200 2615 y Fr(:)1252 b Fz(\(5)p Fr(:)p Fz(15\))50 2943 y(Note)32 b(that)e(if)g(the)g(forest) h Fr(T)38 b Fz(=3D)25 b Fp(f)p Fr(t)1222 2957 y Fm(1)1263 2943 y Fr(;)15 b(:)g(:)g(:)i(;)e(t)1498 2957 y Fq(n)1547 2943 y Fp(g)31 b Fz(where)f Fr(t)1919 2957 y Fq(l)1977 2943 y Fz(are)h(connected)g(trees,)g(then)1609 3249 y Fk(e)-67 b Fr(w)1658 3263 y Fq(T)1740 3249 y Fz(=3D)1872 3135 y Fq(n)1836 3163 y Fk(Y)1837 3358 y Fq(l)p Fm(=3D1)1984 3249 y Fk(e)g Fr(w)r Fz(\()p Fr(t)2103 3263 y Fq(l)2131 3249 y Fz(\))1311 b(\(5)p Fr(:)p Fz(16\))50 3647 y Fh(Lemma)38 b(5.2.)p Fw(F)-7 b(or)33 b(any)g(for)-5 b(est)34 b Fr(T)45 b Fw(which)34 b(is)e Fr(\016)37 b Fw(c)-5 b(orr)g(e)g(ct)1988 3614 y Fm(9)2063 3647 y Fw(we)32 b(have)h(the)g(fol)5 b(lowing)34 b(b)-5 b(ound:)1143 3906 y Fp(j)17 b Fk(e)-68 b Fr(w)1233 3920 y Fq(T)1291 3906 y Fp(j)58 b(\024)1479 3819 y Fk(Y)1470 4015 y Fq(b)p Fl(2)p Fq(T)1619 3906 y Fp(j)p Fr(w)1711 3862 y Fq(q)1709 3931 y(i;j)1795 3906 y Fp(f)p Fr(x)1892 3920 y Fq(i)1922 3906 y Fr(;)15 b(x)2014 3920 y Fq(j)2053 3906 y Fp(gj)2145 3819 y Fk(Y)2138 4014 y Fq(i)p Fl(2)p Fq(T)p 2213 4029 53 4 v 2281 3906 a Fr(e)2323 3868 y Fl(\000)2384 3864 y Fk(e)2380 3868 y Fq(V)2423 3877 y Fs(q)2460 3868 y Fm(\()p Fq(x)2529 3877 y Fs(i)2558 3868 y Fm(\))2591 3906 y Fr(:)861 b Fz(\(5)p Fr(:)p Fz(17\))50 4421 y Fw(Pr)-5 b(o)g(of.)48 b Fz(Recall)31 b(that)i(since)e(only)g (suc)m(h)g(forests)h(can)g(con)m(tribute)g(in)e(the)i(partition)f (function)f(for)i(whic)m(h)50 4530 y(all)d(p)s(oin)m(ts)g Fr(i)i Fz(are)g Fr(\016)s Fz(-correct,)i(w)m(e)d(ha)m(v)m(e)i(that)f (for)f(an)m(y)h Fr(i)25 b Fp(2)g Fr(T)p 2059 4545 66 4 v 13 w Fz(,)1348 4690 y Fk(X)1256 4890 y Fq(j)t Fm(:)p Fl(f)p Fq(i;j)t Fl(g2)p Fq(G)1587 4777 y Fp(j)p Fr(W)1711 4733 y Fq(q)1698 4802 y(i;j)1783 4777 y Fz(\()p Fr(x)1870 4791 y Fq(i)1901 4777 y Fr(;)15 b(x)1993 4791 y Fq(j)2032 4777 y Fz(\))p Fp(j)26 b(\024)f Fr(\016)s(V)2310 4791 y Fq(q)2351 4777 y Fz(\()p Fr(x)2438 4791 y Fq(i)2468 4777 y Fz(\))974 b(\(5)p Fr(:)p Fz(18\))p 50 5020 546 4 v 159 5082 a Fi(8)197 5107 y Fx(W)-6 b(e)26 b(consider)h(a)e(single)i (p)r(oin)n(t)f(also)g(as)g(a)f(tree.)159 5172 y Fi(9)197 5198 y Fx(In)d(other)g(w)n(ords,)i(whic)n(h)e(o)r(ccurs)i(as)e(a)g (subgraph)g(of)g(some)g(graph)g(index)h(in)f(the)h(sum)f (\(5.3\),\(5.4\))h(o)n(v)n(er)f(the)h(restricted)50 5289 y(ensem)n(bles)k(of)e(graphs)h(in)g(the)g(partition)g(functions)g Fg(Z)1682 5253 y Fe(q)1677 5314 y(M)1751 5289 y Fx(.)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(18)p eop %%Page: 19 19 19 18 bop 50 340 a Fz(Next)31 b(observ)m(e)g(that)g(since)f(for)g(an)m (y)h Fr(z)e Fp(2)c Fn(R)s Fz(,)36 b(1)21 b(+)f Fp(j)p Fr(e)1829 307 y Fq(z)1891 340 y Fp(\000)g Fz(1)p Fp(j)26 b(\024)f Fr(e)2216 307 y Fl(j)p Fq(z)s Fl(j)376 601 y Fk(e)-67 b Fr(w)425 615 y Fq(T)508 601 y Fp(\024)679 514 y Fk(X)604 709 y Fq(G)p Fm(:)p Fq(T)725 718 y Fs(G)777 709 y Fm(=3D)p Fq(T)900 601 y Fp(j)p Fr(w)990 615 y Fq(G)1052 601 y Fp(j)25 b(\024)1204 514 y Fk(Y)1198 709 y Fq(i)p Fl(2)p Fq(T)p 1273 724 53 4 v 1341 601 a Fr(v)1385 615 y Fq(i)1415 601 y Fz(\()p Fr(x)1502 615 y Fq(i)1533 601 y Fz(\))1592 514 y Fk(Y)1583 710 y Fq(b)p Fl(2)p Fq(T)1732 601 y Fp(j)p Fr(w)1824 557 y Fq(q)1822 626 y(i;j)1908 601 y Fp(f)p Fr(x)2005 615 y Fq(i)2036 601 y Fr(;)15 b(x)2128 615 y Fq(j)2166 601 y Fp(gj)2286 514 y Fk(Y)2251 709 y Fq(i;j)t Fl(2)p Fq(T)p 2382 724 V 2435 601 a Fz(\(1)21 b(+)f Fp(j)p Fr(w)2719 557 y Fq(q)2717 626 y(i;j)2803 601 y Fp(f)p Fr(x)2900 615 y Fq(i)2931 601 y Fr(;)15 b(x)3023 615 y Fq(j)3061 601 y Fp(gj)p Fz(\))1102 868 y Fp(\024)1204 782 y Fk(Y)1198 977 y Fq(i)p Fl(2)p Fq(T)p 1273 992 V 1341 868 a Fr(v)1385 882 y Fq(i)1415 868 y Fz(\()p Fr(x)1502 882 y Fq(i)1533 868 y Fz(\))1592 782 y Fk(Y)1583 978 y Fq(b)p Fl(2)p Fq(T)1732 868 y Fp(j)p Fr(w)1824 825 y Fq(q)1822 894 y(i;j)1908 868 y Fp(f)p Fr(x)2005 882 y Fq(i)2036 868 y Fr(;)g(x)2128 882 y Fq(j)2166 868 y Fp(gj)2286 782 y Fk(Y)2251 977 y Fq(i;j)t Fl(2)p Fq(T)p 2382 992 V 2450 868 a Fr(e)2492 831 y Fl(j)p Fq(W)10 b(ij)t Fl(j)1102 1136 y Fp(\024)1204 1050 y Fk(Y)1198 1245 y Fq(i)p Fl(2)p Fq(T)p 1273 1260 V 1341 1136 a Fr(v)1385 1150 y Fq(i)1415 1136 y Fz(\()p Fr(x)1502 1150 y Fq(i)1533 1136 y Fz(\))1592 1050 y Fk(Y)1583 1245 y Fq(b)p Fl(2)p Fq(T)1732 1136 y Fp(j)p Fr(w)1824 1092 y Fq(q)1822 1161 y(i;j)1908 1136 y Fp(f)p Fr(x)2005 1150 y Fq(i)2036 1136 y Fr(;)15 b(x)2128 1150 y Fq(j)2166 1136 y Fp(gj)2258 1050 y Fk(Y)2251 1245 y Fq(i)p Fl(2)p Fq(T)p 2326 1260 V 2395 1136 a Fr(e)2437 1099 y Fq(\016)r(V)2515 1108 y Fs(q)2552 1099 y Fm(\()p Fq(x)2621 1108 y Fs(i)2650 1099 y Fm(\))3477 911 y Fz(\(5)p Fr(:)p Fz(19\))50 1483 y(Using)29 b(\(5.6\),)k(this)c(yields)g(the)h(Lemma.)848 3738 y @beginspecial 0 @llx 0 @lly 238 @urx 228 @ury 2380 @rwi @setspecial %%BeginDocument: kps5.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: kps5.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta2 %%CreationDate: Fri Aug 3 09:36:07 2001 %%For: bovier@hilbert.wias-berlin.de (Anton Bovier,403,547,) %%Orientation: Portrait %%BoundingBox: 0 0 238 228 %%Pages: 0 %%BeginSetup %%IncludeFeature: *PageSize Letter %%EndSetup %%Magnification: 0.50 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -35.0 245.0 translate 1 -1 scale =2E9 .9 scale % to make patterns same scale as in xfig % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=3Dqw-pw nh=3Dqh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) =3D cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manu= al % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n 0 8182 m 0 0 l 9112 0 l 9112 8182 l cp clip 0.03000 0.03000 sc % Polyline 225.000 slw n 3638 1675 m 3615 1679 l 3594 1683 l 3572 1687 l 3550 1692 l 3528 1697 l= 3507 1703 l 3485 1709 l 3463 1715 l 3442 1722 l 3421 1728 l 3401 1736 l 3381 1743 l 3362 1750 l 3343 1757 l 3326 1765 l 3309 1772 l 3293 1779 l 3278 1786 l 3264 1793 l 3250 1800 l 3233 1809 l 3216 1817 l 3200 1827 l 3183 1836 l 3166 1846 l 3149 1857 l 3133 1868 l 3116 1880 l 3100 1892 l 3084 1905 l 3069 1918 l 3054 1931 l 3040 1945 l 3026 1959 l 3013 1973 l 3000 1988 l 2989 2001 l 2977 2016 l 2965 2031 l 2953 2047 l 2941 2064 l 2928 2082 l 2915 2101 l 2902 2120 l 2889 2139 l 2876 2159 l 2864 2178 l 2852 2197 l 2840 2216 l 2828 2234 l 2818 2252 l 2807 2268 l 2797 2284 l 2788 2300 l 2777 2317 l 2766 2334 l 2756 2351 l 2745 2368 l 2735 2385 l 2724 2402 l 2714 2420 l 2703 2438 l 2694 2455 l 2684 2473 l 2675 2490 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l 5313 3338 l 5308 3325 l 5304 3312 l 5301 3299 l 5299 3284 l 5298 3269 l 5297 3252 l 5298 3235 l 5301 3217 l 5304 3198 l 5308 3179 l 5314 3159 l 5321 3139 l 5329 3118 l 5338 3098 l 5348 3077 l 5359 3057 l 5371 3036 l 5384 3016 l 5398 2995 l 5413 2975 l 5425 2959 l 5438 2943 l 5452 2926 l 5466 2909 l 5482 2891 l 5498 2873 l 5515 2854 l 5533 2834 l 5551 2814 l 5570 2794 l 5590 2772 l 5610 2751 l 5630 2729 l 5651 2707 l 5671 2684 l 5692 2661 l 5712 2639 l 5732 2616 l 5752 2594 l 5771 2571 l 5790 2549 l 5808 2526 l 5825 2504 l 5842 2482 l 5859 2460 l 5875 2438 l 5891 2415 l 5906 2392 l 5921 2368 l 5936 2344 l 5951 2319 l 5965 2294 l 5979 2268 l 5993 2242 l 6006 2215 l 6019 2188 l 6031 2160 l 6043 2133 l 6054 2105 l 6065 2078 l 6074 2051 l 6083 2024 l 6091 1998 l 6099 1973 l 6105 1948 l 6110 1925 l 6115 1902 l 6119 1880 l 6121 1859 l 6123 1838 l 6125 1819 l 6125 1800 l 6124 1776 l 6122 1754 l 6119 1731 l 6114 1710 l 6108 1689 l 6101 1668 l 6092 1648 l 6082 1629 l 6070 1610 l 6058 1592 l 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1662 l 3730 1664 l 3710 1666 l 3691 1668 l 3673 1670 l 3655 1672 l cp gs col2 0.05 tnt ef gr gs col0 s gr = % Polyline n 6913 5863 m 6895 5860 l 6878 5858 l 6861 5857 l 6843 5855 l 6825 5854 l= 6806 5854 l 6787 5853 l 6769 5854 l 6750 5854 l 6732 5855 l 6714 5857 l 6697 5858 l 6680 5860 l 6663 5863 l 6647 5865 l 6632 5867 l 6615 5870 l 6599 5874 l 6581 5878 l 6564 5882 l 6546 5887 l 6528 5893 l 6511 5899 l 6494 5905 l 6478 5912 l 6463 5919 l 6449 5926 l 6436 5934 l 6424 5942 l 6413 5950 l 6401 5960 l 6390 5971 l 6379 5983 l 6370 5996 l 6360 6010 l 6352 6024 l 6344 6039 l 6338 6054 l 6332 6069 l 6326 6084 l 6322 6098 l 6318 6112 l 6315 6125 l 6313 6138 l 6310 6152 l 6308 6166 l 6306 6180 l 6305 6195 l 6304 6210 l 6303 6225 l 6304 6240 l 6305 6255 l 6306 6270 l 6308 6284 l 6310 6298 l 6313 6313 l 6315 6325 l 6318 6338 l 6322 6352 l 6326 6366 l 6332 6381 l 6338 6397 l 6344 6413 l 6352 6428 l 6360 6444 l 6370 6459 l 6379 6473 l 6390 6487 l 6401 6500 l 6413 6513 l 6424 6523 l 6436 6534 l 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6175 l 7853 6169 l 7856 6163 l 7858 6157 l 7860 6152 l 7861 6146 l 7862 6141 l 7862 6135 l 7862 6130 l 7861 6125 l 7859 6119 l 7856 6114 l 7852 6109 l 7847 6104 l 7842 6099 l 7835 6095 l 7828 6090 l 7819 6085 l 7810 6081 l 7800 6076 l 7788 6072 l 7776 6067 l 7763 6063 l 7748 6058 l 7732 6053 l 7714 6048 l 7696 6042 l 7675 6037 l 7654 6031 l 7631 6025 l 7607 6019 l 7582 6013 l 7557 6006 l 7531 6000 l 7504 5994 l 7477 5987 l 7451 5981 l 7425 5975 l 7399 5969 l 7374 5963 l 7350 5958 l 7327 5952 l 7305 5947 l 7283 5942 l 7263 5938 l 7238 5932 l 7214 5926 l 7191 5921 l 7168 5916 l 7146 5910 l 7124 5905 l 7102 5900 l 7081 5896 l 7060 5891 l 7041 5887 l 7022 5883 l 7003 5879 l 6986 5875 l 6970 5872 l 6955 5869 l 6940 5867 l 6926 5865 l cp gs col2 0.05 tnt ef gr gs col0 s gr = % Polyline n 7475 2275 m 7469 2295 l 7463 2316 l 7457 2337 l 7450 2358 l 7443 2381 l= 7435 2403 l 7427 2427 l 7419 2451 l 7410 2475 l 7401 2500 l 7392 2525 l 7382 2550 l 7373 2575 l 7363 2600 l 7353 2625 l 7343 2649 l 7332 2673 l 7322 2697 l 7312 2719 l 7302 2742 l 7292 2763 l 7282 2784 l 7272 2805 l 7263 2825 l 7251 2847 l 7240 2868 l 7229 2890 l 7217 2912 l 7205 2935 l 7193 2957 l 7180 2980 l 7168 3003 l 7155 3026 l 7143 3050 l 7130 3073 l 7118 3097 l 7106 3120 l 7095 3143 l 7084 3165 l 7074 3187 l 7064 3209 l 7055 3230 l 7046 3251 l 7039 3272 l 7032 3292 l 7025 3313 l 7019 3333 l 7014 3354 l 7009 3375 l 7004 3396 l 7000 3419 l 6997 3441 l 6994 3465 l 6991 3488 l 6989 3512 l 6988 3536 l 6987 3561 l 6987 3585 l 6987 3609 l 6988 3633 l 6990 3656 l 6992 3678 l 6994 3700 l 6997 3721 l 7000 3742 l 7004 3762 l 7008 3781 l 7013 3800 l 7018 3820 l 7024 3841 l 7030 3861 l 7038 3882 l 7046 3902 l 7054 3923 l 7063 3943 l 7073 3964 l 7084 3984 l 7094 4004 l 7106 4023 l 7117 4042 l 7129 4060 l 7141 4077 l 7153 4094 l 7165 4109 l 7177 4124 l 7189 4137 l 7201 4150 l 7213 4163 l 7228 4177 l 7243 4191 l 7260 4205 l 7277 4218 l 7295 4230 l 7313 4241 l 7332 4252 l 7352 4261 l 7371 4269 l 7391 4276 l 7410 4282 l 7429 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l 7875 2735 l 7869 2716 l 7863 2698 l 7856 2681 l 7850 2664 l 7844 2649 l 7838 2634 l 7831 2619 l 7825 2606 l 7819 2593 l 7812 2581 l 7806 2570 l 7800 2559 l 7794 2548 l 7788 2538 l 7781 2527 l 7774 2515 l 7767 2503 l 7759 2490 l 7751 2476 l 7742 2461 l 7734 2446 l 7725 2429 l 7716 2411 l 7708 2393 l 7699 2375 l 7691 2356 l 7683 2336 l 7676 2316 l 7669 2296 l 7663 2275 l 7657 2255 l 7651 2235 l 7646 2214 l 7641 2192 l 7635 2169 l 7630 2146 l 7625 2123 l 7619 2100 l 7614 2078 l 7609 2057 l 7604 2038 l 7600 2020 l 7595 2004 l 7591 1991 l 7587 1980 l 7583 1972 l 7579 1966 l 7575 1963 l 7571 1962 l 7567 1964 l 7563 1969 l 7559 1976 l 7555 1986 l 7550 1999 l 7545 2014 l 7540 2031 l 7534 2051 l 7528 2072 l 7522 2095 l 7516 2119 l 7510 2144 l 7503 2170 l 7497 2196 l 7490 2222 l 7482 2248 l cp gs col4 0.05 tnt ef g= r gs col0 s gr = % Polyline n 4938 5313 m 4920 5315 l 4903 5318 l 4884 5323 l 4866 5329 l 4847 5335 l= 4828 5343 l 4809 5352 l 4790 5362 l 4771 5373 l 4753 5384 l 4736 5396 l 4720 5409 l 4705 5422 l 4691 5435 l 4679 5448 l 4668 5461 l 4658 5474 l 4650 5488 l 4643 5501 l 4637 5515 l 4632 5529 l 4629 5544 l 4626 5559 l 4624 5575 l 4624 5591 l 4625 5607 l 4626 5623 l 4629 5639 l 4633 5655 l 4637 5671 l 4642 5685 l 4648 5700 l 4654 5713 l 4661 5726 l 4668 5738 l 4675 5750 l 4684 5763 l 4693 5775 l 4704 5787 l 4715 5799 l 4727 5811 l 4740 5822 l 4753 5833 l 4768 5843 l 4782 5853 l 4797 5862 l 4812 5870 l 4828 5878 l 4843 5884 l 4858 5890 l 4873 5895 l 4888 5900 l 4903 5904 l 4918 5908 l 4935 5911 l 4951 5913 l 4969 5915 l 4987 5917 l 5005 5918 l 5023 5918 l 5041 5918 l 5059 5917 l 5076 5915 l 5093 5913 l 5108 5911 l 5123 5908 l 5137 5904 l 5150 5900 l 5163 5895 l 5175 5890 l 5187 5884 l 5199 5877 l 5210 5870 l 5221 5861 l 5232 5852 l 5241 5841 l 5250 5830 l 5259 5818 l 5266 5806 l 5272 5793 l 5277 5780 l 5282 5766 l 5285 5752 l 5288 5738 l 5289 5722 l 5290 5706 l 5290 5688 l 5290 5670 l 5288 5650 l 5285 5630 l 5282 5609 l 5277 5588 l 5272 5566 l 5265 5545 l 5258 5525 l 5250 5505 l 5242 5487 l 5233 5469 l 5223 5453 l 5213 5438 l 5203 5424 l 5192 5412 l 5181 5400 l 5168 5388 l 5155 5377 l 5141 5366 l 5125 5356 l 5110 5347 l 5093 5339 l 5076 5332 l 5059 5325 l 5041 5320 l 5024 5316 l 5006 5313 l 4989 5311 l 4972 5311 l 4955 5311 l cp gs col1 0.05 tnt ef gr gs col0 s gr = % Polyline n 4825 3013 m 4818 2994 l 4810 2977 l 4801 2960 l 4791 2944 l 4781 2929 l= 4770 2915 l 4757 2901 l 4744 2889 l 4731 2877 l 4716 2866 l 4702 2855 l 4686 2846 l 4671 2838 l 4655 2830 l 4640 2823 l 4624 2817 l 4609 2812 l 4593 2808 l 4578 2804 l 4563 2800 l 4545 2796 l 4527 2793 l 4509 2790 l 4490 2788 l 4471 2786 l 4451 2785 l 4431 2784 l 4411 2783 l 4391 2784 l 4371 2785 l 4352 2786 l 4333 2788 l 4315 2791 l 4297 2794 l 4281 2798 l 4266 2802 l 4251 2807 l 4238 2813 l 4223 2819 l 4209 2827 l 4195 2836 l 4182 2846 l 4168 2856 l 4156 2868 l 4144 2881 l 4132 2894 l 4121 2909 l 4110 2923 l 4101 2938 l 4092 2953 l 4084 2968 l 4076 2983 l 4069 2998 l 4063 3013 l 4056 3028 l 4050 3043 l 4044 3060 l 4038 3077 l 4033 3094 l 4027 3113 l 4022 3131 l 4018 3150 l 4014 3169 l 4010 3187 l 4007 3206 l 4004 3223 l 4002 3240 l 4001 3257 l 4000 3272 l 4000 3288 l 4000 3305 l 4001 3322 l 4003 3339 l 4006 3356 l 4010 3374 l 4014 3392 l 4019 3409 l 4026 3426 l 4032 3443 l 4040 3458 l 4048 3473 l 4057 3487 l 4065 3500 l 4075 3513 l 4085 3525 l 4096 3537 l 4108 3549 l 4121 3560 l 4135 3572 l 4150 3584 l 4166 3595 l 4182 3605 l 4198 3615 l 4214 3624 l 4230 3632 l 4245 3639 l 4260 3645 l 4275 3650 l 4290 3655 l 4305 3659 l 4321 3662 l 4337 3665 l 4354 3667 l 4372 3669 l 4391 3670 l 4410 3670 l 4429 3670 l 4448 3669 l 4467 3668 l 4487 3667 l 4506 3665 l 4525 3663 l 4539 3661 l 4553 3659 l 4568 3656 l 4583 3653 l 4599 3650 l 4616 3646 l 4632 3641 l 4649 3636 l 4666 3629 l 4682 3622 l 4699 3614 l 4715 3605 l 4730 3595 l 4744 3584 l 4758 3572 l 4771 3560 l 4783 3546 l 4794 3532 l 4803 3517 l 4813 3500 l 4820 3484 l 4827 3466 l 4833 3448 l 4839 3428 l 4844 3407 l 4848 3385 l 4852 3361 l 4855 3337 l 4857 3312 l 4859 3287 l 4859 3261 l 4860 3235 l 4859 3209 l 4858 3184 l 4856 3159 l 4853 3135 l 4850 3112 l 4846 3090 l 4841 3069 l 4836 3049 l 4831 3030 l cp gs col1 0.05 tnt ef gr gs col0 s = gr = % Polyline n 3313 2350 m 3305 2360 l 3298 2371 l 3292 2383 l 3285 2396 l 3279 2410 l= 3274 2425 l 3269 2441 l 3266 2457 l 3263 2473 l 3261 2489 l 3260 2505 l 3260 2520 l 3261 2535 l 3263 2550 l 3265 2565 l 3270 2580 l 3275 2595 l 3281 2610 l 3288 2626 l 3296 2641 l 3305 2656 l 3315 2669 l 3325 2682 l 3335 2694 l 3345 2704 l 3355 2712 l 3365 2719 l 3375 2725 l 3382 2728 l 3389 2731 l 3396 2732 l 3403 2733 l 3410 2734 l 3417 2733 l 3424 2732 l 3431 2729 l 3438 2726 l 3445 2722 l 3451 2717 l 3457 2711 l 3463 2704 l 3468 2697 l 3472 2689 l 3476 2680 l 3480 2670 l 3483 2660 l 3486 2649 l 3488 2638 l 3489 2622 l 3490 2605 l 3491 2586 l 3490 2566 l 3489 2545 l 3487 2523 l 3484 2501 l 3481 2478 l 3477 2456 l 3472 2435 l 3467 2414 l 3462 2395 l 3456 2378 l 3450 2363 l 3444 2349 l 3438 2338 l 3432 2329 l 3427 2323 l 3421 2317 l 3415 2312 l 3409 2308 l 3402 2305 l 3396 2303 l 3389 2302 l 3382 2302 l 3375 2303 l 3368 2305 l 3361 2308 l 3354 2311 l 3348 2315 l 3341 2320 l 3335 2325 l 3329 2331 l 3323 2337 l 3318 2343 l cp gs col0 s gr = % Polyline n 1850 1413 m 1840 1428 l 1831 1443 l 1822 1459 l 1814 1474 l 1806 1490 l= 1800 1506 l 1794 1522 l 1789 1537 l 1785 1553 l 1782 1568 l 1780 1583 l 1779 1597 l 1779 1611 l 1781 1624 l 1783 1637 l 1787 1648 l 1792 1659 l 1798 1669 l 1804 1679 l 1813 1688 l 1823 1697 l 1835 1705 l 1848 1714 l 1863 1721 l 1880 1729 l 1898 1735 l 1918 1741 l 1939 1746 l 1960 1750 l 1983 1754 l 2006 1756 l 2029 1758 l 2052 1759 l 2075 1759 l 2098 1758 l 2120 1756 l 2141 1753 l 2163 1750 l 2183 1746 l 2204 1741 l 2225 1734 l 2247 1727 l 2268 1719 l 2289 1710 l 2311 1699 l 2332 1688 l 2352 1675 l 2372 1662 l 2391 1648 l 2408 1633 l 2425 1618 l 2440 1603 l 2454 1587 l 2466 1571 l 2477 1554 l 2488 1538 l 2496 1522 l 2503 1505 l 2510 1488 l 2516 1471 l 2521 1452 l 2525 1433 l 2528 1414 l 2531 1394 l 2532 1374 l 2533 1353 l 2532 1334 l 2531 1314 l 2528 1295 l 2525 1277 l 2521 1260 l 2516 1243 l 2510 1228 l 2503 1213 l 2496 1200 l 2488 1188 l 2478 1176 l 2469 1165 l 2458 1155 l 2446 1145 l 2433 1137 l 2419 1129 l 2404 1122 l 2388 1116 l 2371 1111 l 2354 1107 l 2336 1104 l 2318 1102 l 2300 1101 l 2281 1101 l 2263 1103 l 2245 1105 l 2227 1109 l 2210 1113 l 2192 1119 l 2175 1125 l 2158 1132 l 2140 1141 l 2123 1150 l 2105 1161 l 2087 1173 l 2068 1186 l 2050 1199 l 2031 1214 l 2013 1230 l 1994 1246 l 1977 1263 l 1959 1280 l 1943 1297 l 1927 1314 l 1912 1332 l 1898 1348 l 1884 1365 l 1872 1381 l 1861 1397 l cp gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 118.60 73.40] PATmp PATsp ef gr PATusp gs c= ol0 s gr = % Polyline 30.000 slw n 1200 600 m 9000 600 l 9000 7275 l 1200 7275 l cp gs col0 s gr = % Polyline 225.000 slw n 8288 1200 m 8297 1219 l 8308 1236 l 8319 1253 l 8332 1269 l 8345 1284 l= 8359 1299 l 8374 1313 l 8389 1326 l 8405 1338 l 8422 1349 l 8438 1360 l 8455 1370 l 8471 1378 l 8487 1386 l 8504 1394 l 8519 1400 l 8535 1407 l 8550 1413 l 8567 1419 l 8585 1425 l 8603 1431 l 8621 1437 l 8640 1443 l 8658 1448 l 8678 1454 l 8697 1459 l 8715 1464 l 8733 1468 l 8751 1473 l 8767 1476 l 8783 1480 l 8798 1483 l 8812 1485 l 8825 1488 l 8831 1488 l 8836 1489 l 8842 1490 l 8847 1491 l 8852 1491 l 8857 1492 l 8862 1492 l 8867 1492 l 8872 1492 l 8876 1492 l 8880 1491 l 8884 1490 l 8887 1489 l 8890 1488 l 8893 1486 l 8896 1484 l 8897 1482 l 8899 1479 l 8900 1476 l 8901 1473 l 8901 1469 l 8900 1465 l 8899 1460 l 8898 1455 l 8896 1450 l 8894 1444 l 8890 1438 l 8887 1431 l 8883 1424 l 8878 1417 l 8873 1409 l 8867 1400 l 8860 1392 l 8853 1382 l 8846 1373 l 8838 1363 l 8825 1347 l 8811 1331 l 8796 1314 l 8780 1296 l 8762 1276 l 8743 1256 l 8723 1235 l 8702 1214 l 8680 1191 l 8657 1169 l 8634 1146 l 8610 1124 l 8586 1101 l 8562 1079 l 8538 1058 l 8514 1038 l 8491 1019 l 8468 1001 l 8447 984 l 8426 969 l 8406 955 l 8387 943 l 8370 933 l 8354 924 l 8339 917 l 8325 913 l 8311 909 l 8299 908 l 8288 909 l 8278 912 l 8269 917 l 8261 924 l 8255 932 l 8249 942 l 8245 954 l 8242 967 l 8240 982 l 8238 998 l 8238 1014 l 8239 1031 l 8241 1049 l 8244 1067 l 8247 1085 l 8252 1103 l 8256 1120 l 8262 1137 l 8268 1154 l 8274 1170 l 8281 1185 l cp gs /PC [[0.00 0.00 0.= 00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 549.20 60.53] PATmp PATsp ef gr PATusp gs c= ol0 s gr = /Times-Bold ff 240.00 scf sf 2475 8100 m gs 1 -1 sc (Contours are depicted as black regions) col0 sh gr % Polyline n 4988 1975 m 4973 1984 l 4959 1993 l 4945 2004 l 4932 2015 l 4920 2027 l= 4908 2039 l 4897 2052 l 4887 2066 l 4878 2080 l 4871 2094 l 4864 2108 l 4859 2122 l 4855 2136 l 4852 2149 l 4851 2162 l 4850 2175 l 4851 2188 l 4852 2201 l 4855 2214 l 4859 2228 l 4864 2242 l 4871 2257 l 4878 2271 l 4887 2286 l 4897 2300 l 4908 2314 l 4920 2328 l 4932 2341 l 4945 2353 l 4959 2365 l 4973 2377 l 4988 2388 l 5001 2397 l 5016 2406 l 5031 2416 l 5047 2425 l 5064 2434 l 5082 2442 l 5101 2451 l 5120 2459 l 5139 2466 l 5159 2473 l 5178 2479 l 5197 2485 l 5216 2490 l 5234 2493 l 5252 2496 l 5268 2498 l 5284 2500 l 5300 2500 l 5314 2500 l 5327 2499 l 5340 2497 l 5353 2494 l 5366 2491 l 5378 2486 l 5390 2481 l 5402 2475 l 5413 2468 l 5424 2459 l 5433 2450 l 5442 2441 l 5450 2430 l 5457 2419 l 5462 2407 l 5467 2394 l 5470 2381 l 5473 2367 l 5474 2353 l 5475 2338 l 5474 2322 l 5473 2305 l 5470 2287 l 5467 2268 l 5462 2249 l 5457 2228 l 5450 2207 l 5442 2186 l 5433 2164 l 5424 2143 l 5413 2121 l 5402 2101 l 5390 2081 l 5378 2062 l 5366 2044 l 5353 2027 l 5340 2012 l 5327 1998 l 5314 1986 l 5300 1975 l 5284 1964 l 5268 1955 l 5252 1947 l 5234 1941 l 5216 1936 l 5197 1933 l 5178 1930 l 5159 1929 l 5139 1930 l 5120 1931 l 5101 1934 l 5082 1938 l 5064 1943 l 5047 1948 l 5031 1954 l 5016 1961 l 5001 1968 l cp gs col7 1.00 shd ef gr gs col0 s gr = % Polyline n 2588 6025 m 2586 6035 l 2585 6046 l 2585 6058 l 2586 6071 l 2588 6084 l= 2592 6098 l 2596 6113 l 2602 6127 l 2608 6141 l 2615 6154 l 2623 6167 l 2632 6179 l 2640 6190 l 2650 6200 l 2662 6211 l 2675 6221 l 2689 6231 l 2704 6240 l 2720 6247 l 2736 6254 l 2751 6259 l 2766 6263 l 2780 6265 l 2792 6265 l 2803 6265 l 2813 6263 l 2819 6260 l 2825 6255 l 2830 6250 l 2835 6244 l 2838 6236 l 2840 6228 l 2841 6218 l 2842 6208 l 2841 6197 l 2839 6185 l 2837 6173 l 2833 6162 l 2830 6150 l 2825 6138 l 2820 6125 l 2814 6112 l 2807 6099 l 2799 6085 l 2791 6071 l 2781 6057 l 2772 6043 l 2761 6029 l 2751 6017 l 2740 6006 l 2730 5996 l 2720 5988 l 2710 5981 l 2700 5975 l 2691 5971 l 2683 5968 l 2674 5967 l 2665 5966 l 2656 5966 l 2647 5967 l 2639 5970 l 2630 5973 l 2622 5977 l 2615 5982 l 2608 5988 l 2603 5994 l 2598 6001 l 2593 6009 l 2590 6017 l cp gs col7 1.00 shd ef g= r gs col0 s gr = % Polyline n 2700 2325 m 2700 2324 l 2700 2320 l 2700 2310 l 2700 2295 l 2700 2277 l= 2700 2257 l 2700 2239 l 2700 2221 l 2700 2205 l 2700 2191 l 2700 2177 l 2700 2163 l 2700 2150 l 2700 2137 l 2700 2123 l 2700 2109 l 2701 2093 l 2701 2077 l 2702 2061 l 2703 2044 l 2704 2028 l 2705 2011 l 2706 1995 l 2708 1980 l 2710 1965 l 2713 1950 l 2715 1935 l 2718 1920 l 2722 1905 l 2726 1888 l 2732 1872 l 2738 1855 l 2744 1837 l 2752 1820 l 2760 1803 l 2770 1787 l 2779 1770 l 2790 1755 l 2801 1740 l 2813 1725 l 2824 1712 l 2836 1699 l 2849 1686 l 2863 1672 l 2878 1658 l 2894 1645 l 2910 1631 l 2927 1618 l 2944 1605 l 2961 1592 l 2977 1581 l 2993 1570 l 3008 1561 l 3023 1552 l 3037 1544 l 3050 1538 l 3067 1530 l 3083 1523 l 3100 1517 l 3116 1512 l 3132 1508 l 3148 1505 l 3164 1503 l 3180 1502 l 3195 1501 l 3209 1500 l 3223 1500 l 3238 1500 l 3252 1500 l 3267 1500 l 3284 1500 l 3301 1501 l 3318 1502 l 3336 1503 l 3353 1505 l 3370 1508 l 3385 1511 l 3400 1515 l 3413 1520 l 3425 1525 l 3436 1531 l 3446 1539 l 3456 1548 l 3466 1559 l 3477 1573 l 3488 1589 l 3499 1606 l 3509 1623 l 3517 1636 l 3522 1645 l 3525 1649 l 3525 1650 l gs 0.00 setgray ef gr gs col0 s gr = % Polyline n 6825 5775 m 6828 5773 l 6835 5768 l 6846 5760 l 6862 5748 l 6881 5735 l= 6902 5719 l 6923 5703 l 6943 5687 l 6962 5672 l 6980 5658 l 6996 5644 l 7010 5630 l 7024 5616 l 7037 5602 l 7050 5588 l 7061 5574 l 7073 5559 l 7085 5544 l 7097 5528 l 7110 5511 l 7123 5493 l 7137 5474 l 7150 5455 l 7164 5436 l 7178 5416 l 7192 5397 l 7206 5378 l 7220 5359 l 7234 5341 l 7248 5323 l 7261 5307 l 7274 5291 l 7288 5275 l 7301 5260 l 7315 5245 l 7329 5230 l 7344 5216 l 7359 5201 l 7375 5187 l 7391 5173 l 7407 5159 l 7423 5146 l 7439 5134 l 7455 5122 l 7471 5111 l 7485 5101 l 7500 5091 l 7513 5083 l 7526 5075 l 7538 5069 l 7550 5063 l 7563 5056 l 7575 5051 l 7587 5047 l 7599 5043 l 7611 5040 l 7623 5038 l 7634 5037 l 7645 5037 l 7655 5038 l 7665 5041 l 7675 5044 l 7683 5048 l 7691 5054 l 7699 5060 l 7706 5067 l 7713 5075 l 7720 5086 l 7727 5098 l 7734 5111 l 7740 5126 l 7747 5143 l 7753 5161 l 7759 5180 l 7765 5200 l 7770 5220 l 7774 5239 l 7778 5258 l 7782 5277 l 7785 5295 l 7788 5313 l 7790 5330 l 7792 5347 l 7793 5364 l 7794 5381 l 7795 5399 l 7795 5417 l 7795 5434 l 7794 5451 l 7792 5468 l 7790 5483 l 7787 5498 l 7784 5512 l 7780 5525 l 7775 5538 l 7769 5552 l 7762 5566 l 7753 5580 l 7745 5594 l 7735 5608 l 7725 5622 l 7715 5635 l 7705 5647 l 7697 5659 l 7688 5669 l 7681 5679 l 7675 5688 l 7669 5698 l 7663 5707 l 7659 5717 l 7655 5727 l 7652 5738 l 7649 5748 l 7646 5758 l 7643 5768 l 7641 5777 l 7638 5788 l 7634 5798 l 7629 5810 l 7624 5822 l 7618 5835 l 7611 5848 l 7604 5861 l 7596 5873 l 7589 5883 l 7582 5892 l 7575 5900 l 7566 5908 l 7557 5913 l 7546 5917 l 7533 5921 l 7519 5923 l 7507 5924 l 7501 5925 l 7500 5925 l gs 0.00 setg= ray ef gr gs col0 s gr = % Polyline n 2625 2700 m 2625 2703 l 2625 2710 l 2626 2722 l 2626 2739 l 2627 2759 l= 2628 2782 l 2629 2805 l 2631 2827 l 2633 2849 l 2635 2869 l 2637 2888 l 2639 2907 l 2642 2925 l 2646 2944 l 2650 2963 l 2654 2980 l 2658 2998 l 2663 3017 l 2669 3037 l 2674 3058 l 2680 3080 l 2687 3102 l 2693 3124 l 2700 3147 l 2707 3169 l 2713 3190 l 2720 3211 l 2726 3231 l 2731 3250 l 2737 3267 l 2742 3284 l 2746 3299 l 2750 3313 l 2755 3332 l 2760 3349 l 2764 3365 l 2767 3380 l 2770 3394 l 2772 3407 l 2773 3420 l 2774 3431 l 2775 3442 l 2775 3453 l 2775 3464 l 2775 3475 l 2775 3487 l 2775 3500 l 2775 3515 l 2774 3530 l 2774 3547 l 2773 3564 l 2772 3582 l 2771 3599 l 2769 3616 l 2767 3633 l 2765 3648 l 2763 3663 l 2759 3677 l 2756 3691 l 2751 3705 l 2745 3721 l 2738 3739 l 2731 3757 l 2722 3777 l 2714 3795 l 2707 3810 l 2702 3820 l 2700 3824 l 2700 3825 l gs 0.00 setgray ef gr gs= col0 s gr = % Polyline n 2350 4975 m 2343 4989 l 2337 5003 l 2331 5018 l 2324 5034 l 2318 5051 l= 2312 5068 l 2306 5087 l 2301 5106 l 2295 5126 l 2289 5146 l 2284 5167 l 2278 5188 l 2273 5210 l 2268 5232 l 2263 5253 l 2258 5275 l 2253 5297 l 2248 5319 l 2243 5340 l 2238 5363 l 2233 5381 l 2229 5401 l 2224 5421 l 2220 5441 l 2215 5463 l 2210 5485 l 2206 5508 l 2201 5532 l 2197 5556 l 2193 5582 l 2189 5607 l 2185 5634 l 2182 5660 l 2180 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l 3546 6733 l 3552 6724 l 3557 6715 l 3562 6705 l 3565 6694 l 3566 6682 l 3567 6668 l 3566 6653 l 3564 6637 l 3560 6619 l 3555 6600 l 3548 6579 l 3540 6556 l 3530 6532 l 3519 6506 l 3507 6478 l 3493 6449 l 3478 6418 l 3462 6385 l 3444 6350 l 3425 6313 l 3411 6286 l 3397 6259 l 3382 6230 l 3366 6200 l 3349 6168 l 3331 6136 l 3313 6102 l 3294 6067 l 3274 6031 l 3253 5993 l 3232 5955 l 3210 5916 l 3188 5876 l 3165 5836 l 3142 5795 l 3118 5754 l 3094 5713 l 3070 5671 l 3046 5630 l 3021 5589 l 2997 5549 l 2973 5509 l 2950 5470 l 2926 5432 l 2903 5395 l 2881 5359 l 2859 5324 l 2837 5290 l 2816 5258 l 2796 5228 l 2776 5198 l 2757 5171 l 2739 5145 l 2721 5120 l 2704 5097 l 2688 5075 l 2670 5052 l 2652 5031 l 2636 5012 l 2619 4994 l 2604 4978 l 2588 4963 l 2574 4949 l 2559 4937 l 2545 4926 l 2532 4917 l 2519 4909 l 2506 4903 l 2494 4898 l 2482 4894 l 2471 4891 l 2460 4889 l 2450 4889 l 2440 4890 l 2431 4892 l 2422 4895 l 2414 4898 l 2406 4903 l 2399 4908 l 2392 4914 l 2386 4920 l 2380 4927 l 2374 4934 l 2369 4942 l 2364 4950 l 2359 4958 l 2354 4966 l cp gs col4 0.05 tnt ef gr gs col0 s gr = /Times-Bold ff 240.00 scf sf 2175 7875 m gs 1 -1 sc (A collection of contours in a model with 4 colors) col0 sh g= r $F2psEnd rs end %%EndDocument @endspecial 50 3991 a FB(5.3.)47 b(A)35 b(new)f(represen)m(tation)h (of)g(the)f(partition)h(function)g(\(5.3\).)159 4244 y Fz(W)-8 b(e)31 b(can)g(no)m(w)g(represen)m(t)f(the)h(partition)e (function)g(\(5.3\)as)278 4504 y Fr(Z)347 4460 y Fq(q)340 4531 y(M)421 4504 y Fz(\()p Fr(x)508 4518 y Fq(M)584 4500 y Fs(c)623 4504 y Fz(\))d(=3D)936 4418 y Fk(X)780 4617 y Fq(T)10 b Fl(2G)t Fm(\()p Fq(M)s(;x)1093 4628 y Fs(M)1156 4615 y(c)1194 4617 y Fm(\))1238 4504 y Fz(1)-23 b(I)1293 4521 y Fq(B)1348 4531 y Fs(\016)1384 4521 y Fm(\()p Fq(x)1453 4530 y Fs(M)1522 4521 y Fl([)p Fq(x)1612 4532 y Fs(M)1675 4519 y(c)1714 4521 y Fm(\))p Fl(\032)p Fq(M)1875 4502 y Fs(c)1960 4504 y Fk(e)-67 b Fr(w)2009 4518 y Fq(T)2092 4504 y Fz(=3D)2374 4418 y Fk(X)2218 4617 y 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b(should)f(b)s(e)h(also)h(noted)50 4545 y(that)j(for)g(small)e Fr(\016)32 b Fz(the)c(quan)m(tit)m(y)g(in)f (\(5.28\))j(is)d(prop)s(ortional)e(to)k(the)f(energy)g(of)g(the)g(core) h(of)f(a\\con)m(tour")50 4654 y Fr(P)121 4621 y Fm(0)190 4654 y Fz(and)f(the)h(constan)m(t)h Fr(\034)932 4621 y Fq(c)969 4654 y Fr(ir)s(c)g Fz(is)e(essen)m(tially)g(the)h(same)g(as) g Fr(\034)2172 4621 y Fl(\017)2242 4654 y Fz(in)f(the)h(pro)s(of)f(of)h (P)m(eierls)f(condition)g(\(4.3\))50 4763 y(\(if)j(w)m(e)g(consider)g (there)g(the)h(sp)s(ecial)e(case)i(of)g(a)f(con)m(tour)i FB(\000)e Fz(ha)m(ving)g(an)g(empt)m(y)h(core)g FB(\000)3090 4730 y Fl(\017)3157 4763 y Fz(=3D)25 b Fp(;)p Fz(\).)50 4991 y Fw(Pr)-5 b(o)g(of.)48 b Fz(The)30 b(idea)f(is)h(v)m(ery)h (similar)c(to)k(that)g(of)g(the)g(pro)s(of)e(of)i(P)m(eierls)e (condition.)39 b(By)31 b(de\014nition,)1133 5096 y Fk(X)1115 5298 y Fq(i)p Fl(2)p Fq(P)p 1190 5313 57 4 v 1246 5277 a Ft(0)1296 5183 y Fr(V)1349 5197 y Fq(q)1390 5183 y Fz(\()p Fr(x)1477 5197 y Fq(i)1507 5183 y Fz(\))26 b(=3D)1827 5096 y Fk(X)1664 5298 y Fq(i)p Fl(2)p Fq(P)p 1739 5313 V 1795 5277 a Ft(0)1830 5298 y Fq(;j)t Fl(2)p Fm(\003)p Fq(;j)t Fl(6)p Fm(=3D)p Fq(i)2136 5183 y Fz(\010)2202 5199 y Fl(f)p Fq(i;j)t Fl(g)2362 5183 y Fp(f)p Fr(x)2459 5197 y Fq(i)2490 5183 y Fr(;)15 b(q)s Fp(g)p Fr(:)833 b Fz(\(5)p Fr(:)p Fz(29\))50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(22)p eop %%Page: 23 23 23 22 bop 50 340 a Fz(Using)35 b(the)h(symmetry)g(of)g(\010)g(with)f (resp)s(ect)h(to)h(the)f(in)m(terc)m(hange)h(of)f(the)h(t)m(w)m(o)g (argumen)m(ts)f(and)g(in)m(ter-)50 449 y(c)m(hanging)30 b(the)h(order)f(of)g(summation)f(w)m(e)i(get)h(that)1074 573 y Fk(X)994 772 y Fq(i)p Fl(2)p Fq(P)1125 753 y Ft(0)1160 772 y Fl(\\)p Fq(M)1300 659 y Fr(V)1353 673 y Fq(q)1394 659 y Fz(\()p Fr(x)1481 673 y Fq(i)1511 659 y Fz(\))26 b(=3D)1891 573 y Fk(X)1668 772 y Fq(i)p Fl(2)p Fq(P)1799 753 y Ft(0)1834 772 y Fl(\\)p Fq(M)s(;j)t Fl(2)p Fm(\003)p Fq(;j)t Fl(6)p Fm(=3D)p Fq(i)2261 659 y Fz(\010)2327 675 y Fl(f)p Fq(j;i)p Fl(g)2483 659 y Fp(f)p Fr(q)s(;)15 b(x)2664 673 y Fq(i)2695 659 y Fp(g)p Fr(:)712 b Fz(\(5)p Fr(:)p Fz(30\))50 961 y(But)34 b(if)e Fr(x)372 975 y Fq(i)433 961 y Fz(=3D)f Fr(q)s Fz(,)j(\010)704 975 y Fq(ij)768 961 y Fz(\()p Fr(x)855 975 y Fq(i)885 961 y Fr(;)15 b(q)s Fz(\))32 b(=3D)e(0)h(=3D)g Fr(W)1414 917 y Fq(q)1401 986 y(ij)1465 961 y Fz(\()p Fr(x)1552 975 y Fq(i)1582 961 y Fr(;)15 b(x)1674 975 y Fq(j)1713 961 y Fz(\),)35 b(while)d(if)g Fr(x)2187 975 y Fq(i)2248 961 y Fp(6)p Fz(=3D)f Fr(q)s Fz(,)j(b)m(y)g(Assumption)e(1',)j(\010)3286 975 y Fq(i;j)3371 961 y Fz(\()p Fr(q)s(;)15 b(x)3542 975 y Fq(i)3573 961 y Fz(\))31 b(=3D)50 1085 y(\010)116 1099 y Fq(i;j)201 1085 y Fz(\()p Fr(x)288 1099 y Fq(i)319 1085 y Fr(;)15 b(q)s Fz(\))47 b Fp(\025)643 1049 y Fm(1)p 613 1064 96 4 v 613 1116 a Fq(D)674 1098 y Fj(0)672 1134 y Fs(q)719 1085 y Fr(W)818 1041 y Fq(q)805 1110 y(i;j)891 1085 y Fz(\()p Fr(x)978 1099 y Fq(i)1009 1085 y Fr(;)15 b(s)1092 1099 y Fq(j)1130 1085 y Fz(\))43 b(for)g(an)m(y)h(c)m(hoice)g(of)f Fr(s)1987 1099 y Fq(j)2072 1085 y Fp(6)p Fz(=3D)j Fr(q)s Fz(.)79 b(It)43 b(is)f(imp)s(ortan)m(t)g(to)i(note)g(here)f(that)50 1215 y(the)e(de\014nition)d(of)j(a)g(galaxy)g(dep)s(ends)e(on)i(a)g(b)s (oundary)d(condition)i Fr(x)2595 1229 y Fq(M)2671 1211 y Fs(c)2709 1215 y Fz(,)k(but)c Fr(x)3007 1229 y Fq(M)3128 1215 y Fz(will)e(b)s(e)i(mostly)50 1324 y(resp)s(onsible)25 b(for)k(a)g(p)s(ossible)d Fr(\016)32 b Fz(incorrectness)d(of)f(an)m(y)h Fr(i)d Fp(2)f Fr(M)38 b Fz(!)j(Namely)28 b(if)g Fr(i)h Fz(is)e(not)i(a)g Fr(\016)s Fz({)h(correct)g(p)s(oin)m(t)50 1436 y(of)h Fr(x)206 1450 y Fm(\003)291 1436 y Fz(then)g Fr(i)g Fz(is)f(\()p Fr(\016)25 b Fp(\000)849 1412 y Fz(~)845 1436 y Fr(\016)s Fz(\)-)32 b(incorrect)f(p)s(oin)m(t)f(of)h(the)h (con\014guration)e Fr(x)2468 1450 y Fq(M)2569 1436 y Fp(\\)21 b Fr(q)2692 1450 y Fq(M)2768 1431 y Fs(c)2806 1436 y Fz(.)43 b(So)30 b(w)m(e)i(ha)m(v)m(e)g(the)f(lo)m(w)m(er)50 1545 y(b)s(ound)881 1619 y Fk(X)816 1815 y Fq(i)p Fl(2)p Fm(\003)p Fq(;i)p Fl(6)p Fm(=3D)p Fq(j)1092 1705 y Fz(\010)1158 1719 y Fq(i:j)1243 1705 y Fp(f)p Fr(q)s(;)15 b(x)1424 1719 y Fq(i)1455 1705 y Fp(g)26 b(\025)1668 1644 y Fz(1)p 1633 1684 116 4 v 1633 1768 a Fr(D)1711 1741 y Fl(0)1708 1790 y Fq(q)1774 1619 y Fk(X)1778 1814 y Fq(i)p Fl(2)p Fm(\003)1921 1705 y Fp(j)p Fr(W)2045 1661 y Fq(q)2032 1731 y(j;i)2114 1705 y Fz(\()p Fr(s)2192 1719 y Fq(j)2230 1705 y Fr(;)15 b(x)2322 1719 y Fq(i)2352 1705 y Fz(\))p Fp(j)26 b(\025)2545 1644 y Fz(\()p Fr(\016)f Fp(\000)2737 1620 y Fk(e)2736 1644 y Fr(\016)s Fz(\))p Fr(V)2867 1658 y Fq(q)p 2545 1684 363 4 v 2669 1768 a Fr(D)2747 1741 y Fl(0)2744 1790 y Fq(q)2919 1705 y Fr(:)533 b Fz(\(5)p Fr(:)p Fz(31\))50 1962 y(Since)39 b Fr(\016)s Fz(-incorrect)h(p)s(oin)m (ts)f(of)h(the)g(galaxy)g Fr(P)1689 1929 y Fl(\003)1770 1962 y Fz(are)g(guaran)m(teed)h(only)e(in)f(the)i(p)s(ossibly)d(v)m (ery)j(sparse)50 2071 y(subset)27 b Fr(M)414 2085 y Fm(1)483 2071 y Fz(of)h Fr(P)655 2038 y Fm(0)696 2071 y Fz(,)h(w)m(e)f(will)e (ha)m(v)m(e)j(to)f(use)g(the)g(con)m(tin)m(uit)m(y)g(Assumption)e(3,)k (just)d(as)h(in)f(the)h(pro)s(of)f(of)i(the)50 2202 y(P)m(eierls)j (condition.)49 b(Let)34 b(us)f(\014x)g(a)h(p)s(oin)m(t)f Fr(j)j Fp(2)30 b Fr(M)1779 2216 y Fm(1)1843 2202 y Fp(\\)21 b Fr(P)p 1925 2217 71 4 v 1996 2162 a Fl(\003)2038 2202 y Fz(,)35 b(and)e(consider,)g(for)h Fp(j)p Fr(k)s Fp(j)d(\024)f Fr(R)3106 2169 y Fl(\003)3105 2224 y Fq(q)3178 2202 y Fp(\021)3290 2158 y Fm(\()p Fq(\016)r Fl(\000)3413 2141 y Fm(~)3410 2158 y Fq(\016)s Fm(\))p 3290 2181 185 4 v 3307 2233 a Fq(D)3368 2215 y Fj(0)3366 2251 y Fs(q)3402 2233 y Fq(C)3519 2202 y Fz(\(it)k(is)50 2332 y(de\014ned)26 b(analogously)g(as)h(in)f(Lemma)h(4.2,)i(the)e(con)m(tribution)f(from)g (a)h(p)s(oin)m(t)f Fr(j)19 b Fz(+)13 b Fr(k)s Fz(.)40 b(W)-8 b(e)28 b(ha)m(v)m(e)g(the)f(lo)m(w)m(er)50 2442 y(b)s(ound)1063 2494 y Fk(X)1066 2689 y Fq(i)p Fl(2)p Fm(\003)1209 2580 y Fz(\010)1275 2594 y Fq(j)t Fm(+)p Fq(k)r(;i)1457 2580 y Fz(\()p Fr(q)s(;)15 b(x)1628 2594 y Fq(i)1659 2580 y Fz(\))25 b Fp(\025)1815 2494 y Fk(X)1819 2689 y Fq(i)p Fl(2)p Fm(\003)1962 2580 y Fz(\010)2028 2594 y Fq(j;i)2109 2580 y Fp(f)p Fr(q)s(;)15 b(x)2290 2594 y Fq(i)2321 2580 y Fp(g)20 b(\000)2488 2519 y Fr(C)7 b Fp(j)p Fr(k)s Fp(j)p 2488 2559 173 4 v 2540 2643 a Fr(R)2671 2580 y(:)781 b Fz(\(5)p Fr(:)p Fz(32\))50 2834 y(for)31 b(it.)44 b(As)31 b(in)f(the)i(deriv)-5 b(ation)30 b(of)h(\(4.11\),)k(it)c(follo)m(ws)f(no)m(w)i(that)g(the)f(sum)g(o)m(v) m(er)i(the)e Fr(R)3096 2801 y Fl(\003)3169 2834 y Fz(neigh)m(b)s(orho)s (o)s(d)50 2951 y(of)f(suc)m(h)g(a)h(\()p Fr(\016)25 b Fp(\000)629 2927 y Fz(~)624 2951 y Fr(\016)t Fz(\))31 b(incorrect)f(p)s(oin)m(t)g Fr(j)36 b Fz(of)30 b Fr(x)1579 2965 y Fq(M)1680 2951 y Fp([)20 b Fr(q)1802 2965 y Fq(M)1878 2946 y Fs(c)1946 2951 y Fz(is)30 b(not)g(smaller)f(than)1286 3143 y Fk(X)1332 3339 y Fq(k)1433 3074 y Fk( )1505 3143 y(X)1508 3338 y Fq(i)p Fl(2)p Fm(\003)1651 3230 y Fz(\010)1717 3244 y Fq(j;i)1799 3230 y Fp(f)p Fr(q)s(;)15 b(x)1980 3244 y Fq(i)2010 3230 y Fp(g)21 b(\000)2178 3168 y Fr(C)7 b Fp(j)p Fr(k)s Fp(j)p 2178 3209 V 2229 3292 a Fr(R)2361 3074 y Fk(!)2448 3230 y Fr(:)1004 b Fz(\(5)p Fr(:)p Fz(33\))50 3525 y(Summing)29 b(o)m(v)m(er)34 b(the)e(union)e Fr(J)37 b Fp(\032)28 b Fr(M)42 b Fz(of)32 b(all)f(suc)m(h)h(incorrect)g Fr(j)h Fp(2)28 b Fr(M)42 b Fz(and)31 b(noticing)h(that)g Fr(J)3276 3492 y Fq(R)3364 3525 y Fp(\033)c Fr(P)3534 3492 y Fm(0)3606 3525 y Fz(w)m(e)50 3634 y(\014nally)g(ha)m(v)m(e)955 3756 y Fk(X)954 3951 y Fq(j)t Fl(2)p Fm(\003)1182 3756 y Fk(X)1102 3955 y Fq(i)p Fl(2)p Fq(P)1233 3937 y Ft(0)1268 3955 y Fl(\\)p Fq(M)1409 3842 y Fz(\010)1475 3856 y Fq(j;i)1556 3842 y Fp(f)p Fr(q)s(;)15 b(x)1737 3856 y Fq(i)1768 3842 y Fp(g)26 b(\025)e Fr(V)1987 3856 y Fq(q)2043 3842 y Fp(j)p Fr(P)2139 3805 y Fm(0)2180 3842 y Fp(j)15 b Fr(C)2285 3856 y Fq(d)2358 3687 y Fk( )2441 3781 y Fr(\016)24 b Fp(\000)2600 3757 y Fz(~)2596 3781 y Fr(\016)p 2441 3821 199 4 v 2447 3904 a(D)2525 3878 y Fl(0)2522 3927 y Fq(q)2562 3904 y Fr(C)2651 3687 y Fk(!)2722 3705 y Fq(d)2780 3842 y Fr(:)672 b Fz(\(5)p Fr(:)p Fz(34\))50 4099 y(This)28 b(pro)m(v)m(es)j(the)g(lemma.)p Fp(})50 4330 y FB(5.7.)47 b(Summing)34 b(o)m(v)m(er)h(pre-galaxies.)47 b(P)m(eierls)35 b(b)s(ounds)h(for)f(galaxies.)159 4562 y Fz(Based)41 b(on)f(the)h(previous)e(lemma,)k(w)m(e)e(can)g(no)m(w)f(estimate)h (rather)g(easily)e(also)i(the)g(activities)f(of)50 4671 y(galaxies,)c(b)m(y)f(\014xing)e(the)i(legs)g(and)f(summing)e(o)m(v)m (er)37 b(all)c(p)s(ossible)f(sk)m(eletons)k(of)f(a)g(giv)m(en)f(galaxy) -8 b(.)55 b(One)50 4780 y(should)35 b(remark)j(that)g(inside)d(the)j(b) s(o)s(dy)e(of)h(a)h(galaxy)-8 b(,)41 b(a)d(sligh)m(tly)e(di\013eren)m (t)h(t)m(yp)s(e)g(of)h(estimateof)h(the)50 4889 y(b)s(ond)29 b(w)m(eigh)m(ts)h(forming)f(the)i(sk)m(eleton)g(of)g(this)e(b)s(o)s (dy\))g(will)f(b)s(e)h(used)h(than)g(for)g(the)h(legs.)159 5071 y(Let)26 b(us)g(denote)g(b)m(y)g Fr(L)g Fz(the)g(union)e(of)i(all) f(legs)h(of)g(the)g(galaxy)h Fr(P)2328 5038 y Fl(\003)2370 5071 y Fz(.)39 b(Recall)25 b(that)i Fr(P)2967 5038 y Fm(0)3034 5071 y Fz(denotes)f(the)g(b)s(o)s(dy)50 5180 y(of)35 b Fr(P)229 5147 y Fl(\003)306 5180 y Fz(and)g(that)g(the)h(w)m (eigh)m(ts)51 b Fk(e)-67 b Fr(w)s Fz(\()p Fr(P)1353 5147 y Fl(\003)1395 5180 y Fz(\))36 b(is)e(giv)m(en)h(b)m(y)g(the)g(sum)g (\(5.26\),\(5.25\))40 b(where)50 b Fk(e)-67 b Fr(w)3151 5194 y Fq(T)3244 5180 y Fz(satis\014es)35 b(the)50 5289 y(b)s(ound)28 b(\(5.17\).)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)k(13:49)1146 b Fx(23)p eop %%Page: 24 24 24 23 bop 159 340 a Fz(Recall)38 b(that)h(b)m(y)f(re-summing)f(o)m(v)m (er)j(graphs)d Fr(G)h Fz(corresp)s(onding)e(to)k(the)e(same)h(trees)g Fr(T)3289 354 y Fq(G)3388 340 y Fz(w)m(e)g(w)m(ere)50 449 y(left)k(in)g(the)h(situation)e(where)h(the)h(tree)h(w)m(eigh)m(ts) f(can)g(b)s(e)f(imagined)f(\(5.17\))k(as)e(pro)s(ducts)e(of)i(b)s(ond) 50 584 y(w)m(eigh)m(ts)21 b Fr(w)431 540 y Fq(q)429 609 y(i;j)514 584 y Fp(f)p Fr(x)611 598 y Fq(i)642 584 y Fr(;)15 b(x)734 598 y Fq(j)773 584 y Fp(g)20 b Fz(and)g(\\remaining")g (v)m(ertex)h(w)m(eigh)m(ts)g Fr(e)2132 551 y Fl(\000)2193 547 y Fk(e)2189 551 y Fq(V)2232 560 y Fs(q)2268 551 y Fm(\()p Fq(x)2337 560 y Fs(i)2367 551 y Fm(\))2400 584 y Fz(.)37 b(Let)21 b(us)e(decomp)s(ose)i(the)g(remaining)50 694 y(fugacit)m(y)31 b(on)f(the)h(v)m(ertices)g(of)f(trees)h(as)g (follo)m(ws)1384 962 y Fk(e)1373 985 y Fr(V)1426 999 y Fq(q)1466 985 y Fz(\()p Fr(x)1553 999 y Fq(i)1583 985 y Fz(\))26 b(=3D)f Fr(V)1813 947 y Fl(\016)1793 1007 y Fq(q)1855 985 y Fz(\()p Fr(x)1942 999 y Fq(i)1973 985 y Fz(\))20 b(+)g Fr(V)2192 947 y Fl(\017)2172 1007 y Fq(q)2234 985 y Fz(\()p Fr(x)2321 999 y Fq(i)2352 985 y Fz(\))1090 b(\(5)p Fr(:)p Fz(35\))50 1171 y(where)30 b(w)m(e)h(mak)m(e)g(the)f(somewhat)h(non-optimal)e(c)m(hoice)j Fr(V)2119 1138 y Fl(\016)2099 1193 y Fq(q)2161 1171 y Fz(\()p Fr(x)2248 1185 y Fq(i)2278 1171 y Fz(\))26 b(=3D)f Fr(V)2508 1138 y Fl(\017)2488 1193 y Fq(q)2550 1171 y Fz(\()p Fr(x)2637 1185 y Fq(i)2667 1171 y Fz(\))h(=3D)2835 1135 y Fm(1)p 2835 1150 37 4 v 2835 1202 a(2)2893 1148 y Fk(e)2882 1171 y Fr(V)2935 1185 y Fq(q)2975 1171 y Fz(\()p Fr(x)3062 1185 y Fq(i)3093 1171 y Fz(\).)1299 1375 y Fk(X)1348 1569 y Fq(j)1446 1462 y Fr(w)1513 1418 y Fq(q)1511 1487 y(i;j)1596 1462 y Fp(f)p Fr(x)1693 1476 y Fq(i)1724 1462 y Fr(;)15 b(x)1816 1476 y Fq(j)1855 1462 y Fp(g)30 b Fr(e)1972 1424 y Fl(\000)p Fq(V)2088 1397 y Fj(\017)2072 1442 y Fs(q)2126 1424 y Fm(\()p Fq(x)2195 1433 y Fs(j)2230 1424 y Fm(\))2288 1462 y Fp(\024)55 b Fr(\020)1023 b Fz(\(5)p Fr(:)p Fz(36\))50 1734 y(with)29 b Fr(\020)j Fp(\024)25 b Fr(e)467 1701 y Fl(\000)p Fq(\034)7 b(=3D)p Fm(4)673 1734 y Fz(for)30 b Fr(\034)40 b Fz(large)31 b(enough,)f(whic)m(h)f(is)h(easy)h(to)g(see)g(from)f(\(5.9\).)159 1916 y(In)22 b(the)h(sequel)e(it)i(will)d(turn)h(out)i(con)m(v)m(enien) m(t)h(to)f(forget)g(that)h(the)e(summations)g(are)h(only)e(o)m(v)m(er)j (forests,)50 2025 y(and)30 b(to)h(retain)f(only)f(the)i(b)s(ound)d (\(5.36\).)159 2207 y(F)-8 b(or)37 b(the)g(b)s(onds)d(outside)i(the)h (b)s(o)s(dy)e(of)h(a)h(galaxy)g(w)m(e)g(will)d(not)i(need)g(an)m(y)h (decomp)s(osition)e(\(5.35\);)50 2327 y(in)27 b(these)j(cases)g(w)m(e)f (incorp)s(orate)f(the)h(whole)f(remaining)g(fugacit)m(y)2416 2304 y Fk(e)2405 2327 y Fr(V)2458 2341 y Fq(q)2498 2327 y Fz(\()p Fr(x)2585 2341 y Fq(i)2616 2327 y Fz(\))h(in)m(to)g Fr(w)2930 2284 y Fq(q)2928 2353 y(i;j)3014 2327 y Fp(f)p Fr(x)3111 2341 y Fq(i)3141 2327 y Fr(;)15 b(x)3233 2341 y Fq(j)3272 2327 y Fp(g)p Fz(;)30 b(this)e(case)50 2442 y(deserv)m(es)j(a)f(new)g(notation)h(in)e(whic)m(h)g(the)i(b)s(ound)d (\(5.17\))33 b(will)27 b(b)s(e)j(written)f(in)g(the)i(follo)m(wing)e (form:)1591 2656 y Fp(j)16 b Fk(e)-67 b Fr(w)1681 2670 y Fq(T)1739 2656 y Fp(j)25 b(\024)1885 2570 y Fk(Y)1927 2765 y Fq(b)2016 2656 y Fp(j)17 b Fk(e)-68 b Fr(w)2106 2670 y Fq(b)2143 2656 y Fp(j)1309 b Fz(\(5)p Fr(:)p Fz(37\))50 2938 y(where)30 b(the)g(newly)f(de\014ned)g(b)s(ond)g(activities)46 b Fk(e)-67 b Fr(w)1724 2952 y Fq(b)1761 2938 y Fr(;)15 b(b)25 b Fz(=3D)g Fp(f)p Fr(x)2058 2952 y Fq(i)2089 2938 y Fr(;)15 b(x)2181 2952 y Fq(j)2220 2938 y Fp(g)30 b Fz(\(smaller)g(than)g(that)h(of)f(\(5.36\)\))1549 3168 y Fk(e)-67 b Fr(w)1598 3182 y Fq(b)1659 3168 y Fz(=3D)56 b Fr(w)1851 3182 y Fq(b)1918 3168 y Fr(e)1960 3130 y Fl(\000)2021 3126 y Fk(e)2017 3130 y Fq(V)2060 3139 y Fs(q)2096 3130 y Fm(\()p Fq(x)2165 3139 y Fs(i)2194 3130 y Fm(\))3477 3168 y Fz(\(5)p Fr(:)p Fz(38\))50 3363 y(satisfy)30 b(a)g(b)s(ound,)f(with)g Fr(")i Fz(small)e(enough)h(of)g(the)h(order)f Fr(")25 b Fz(=3D)g Fr(e)2249 3330 y Fl(\000)p Fq(K)t(\034)2448 3363 y Fz(where)30 b Fr(K)i Fz(=3D)25 b Fr(K)7 b Fz(\()p Fr(\016)s Fz(\))1661 3491 y Fk(X)1652 3686 y Fq(b)p Fl(3)p Fq(x)1774 3695 y Fs(i)1834 3577 y Fk(e)-67 b Fr(w)1883 3591 y Fq(b)1945 3577 y Fp(\024)24 b Fr(":)1370 b Fz(\(5)p Fr(:)p Fz(39\))159 3936 y(Then)29 b(w)m(e)i(ha)m(v)m(e)h(the)e(follo)m (wing)f(result.)1408 4230 y Fr(\034)1458 4193 y Fl(\016)1448 4253 y Fq(q)1525 4230 y Fz(=3D)c Fr(C)1686 4244 y Fq(d)1744 4230 y Fr(V)1817 4193 y Fl(\016)1797 4253 y Fq(q)1904 4075 y Fk( )1987 4169 y Fr(\016)f Fp(\000)2146 4145 y Fz(~)2142 4169 y Fr(\016)p 1987 4209 199 4 v 1993 4293 a(C)7 b(D)2143 4266 y Fl(0)2140 4315 y Fq(q)2197 4075 y Fk(!)2269 4094 y Fq(d)2326 4230 y Fr(:)1126 b Fz(\(5)p Fr(:)p Fz(40\))50 4581 y Fh(Pr)n(oposition)33 b(5.4.)p Fw(The)c(galaxy)h(activities)f(c)-5 b(an)30 b(b)-5 b(e)29 b(estimate)-5 b(d)31 b(as)f(fol)5 b(lows)30 b(\()p Fr(L)f Fw(denotes)h(the)g(union)f(of)50 4690 y(al)5 b(l)33 b(le)-5 b(gs)33 b(of)g Fr(P)530 4657 y Fl(\003)571 4690 y Fw(\))1304 4817 y Fp(j)17 b Fk(e)-68 b Fr(w)s Fz(\()p Fr(P)1503 4779 y Fl(\003)1545 4817 y Fz(\))p Fp(j)59 b(\024)e Fr(e)1834 4779 y Fl(\000)p Fq(\034)1932 4752 y Fj(\003)1925 4798 y Fs(q)1970 4779 y Fl(j)p Fq(P)2048 4752 y Ft(0)2083 4779 y Fl(j)2164 4731 y Fk(Y)2157 4926 y Fq(b)p Fl(2)p Fq(L)2303 4817 y Fp(j)16 b Fk(e)-67 b Fr(w)2393 4831 y Fq(b)2430 4817 y Fp(j)1022 b Fz(\(5)p Fr(:)p Fz(41\))50 5071 y Fw(wher)-5 b(e)50 b Fk(e)-67 b Fr(w)372 5085 y Fq(b)441 5071 y Fw(ar)-5 b(e)33 b(b)-5 b(ond)34 b(weights)f(fr)-5 b(om)34 b(\(5.38\))g(and)g(the)f(c)-5 b(onstant)34 b Fr(\034)2356 5038 y Fl(\003)2346 5094 y Fq(q)2431 5071 y Fw(satis\014es)f(the)g(b)-5 b(ound)1427 5289 y Fr(e)1469 5252 y Fl(\000)p Fq(\034)1567 5224 y Fj(\003)1560 5270 y Fs(q)1634 5289 y Fz(=3D)25 b(4)48 b(\()p Fr(e)1900 5252 y Fl(\000)p Fq(\034)1998 5224 y Fj(\016)1991 5270 y Fs(q)2037 5252 y Fm(+)p Fq(\020)2134 5289 y Fp(j)p Fr(S)5 b Fp(j)p Fz(\))2280 5252 y Fq(\013)3477 5289 y Fz(\(5)p Fr(:)p Fz(42\))50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(24)p eop %%Page: 25 25 25 24 bop 50 375 a Fw(wher)-5 b(e)34 b(\(se)-5 b(e)32 b(\(5.28\)\))j Fr(\013)25 b Fz(=3D)58 b Fr(C)1081 389 y Fq(d)1171 274 y Fk(\020)1248 340 y Fq(\016)r Fl(\000)1343 323 y Fm(~)1340 340 y Fq(\016)p 1236 355 153 4 v 1236 407 a(C)t(D)1353 388 y Fj(0)1351 425 y Fs(q)1399 274 y Fk(\021)1454 293 y Fq(d)1496 375 y Fw(.)50 641 y(Note.)46 b Fz(Let)37 b(us)f(emphasise)f(that)i(the)g(estimate)g(\(5.37\),)j (used)c(b)s(elo)m(w)f(for)h(the)h(constellations,)g(is)f(no)m(w)50 750 y(just)30 b(a)g(sp)s(ecial)f(case)j(of)e(the)h(estimate)g (\(5.41\),)i(for)d(an)g(empt)m(y)h(b)s(o)s(dy)e Fr(P)2550 717 y Fm(0)2616 750 y Fz(=3D)c Fp(;)p Fz(.)50 988 y Fw(Pr)-5 b(o)g(of.)48 b Fz(W)-8 b(e)32 b(noted)f(already)g(that)g(our)g(new)f (form)m(ulation)g(\(5.37\))j(with)d(\(5.39\))j(of)e(the)g(b)s(ounds)e (for)i(tree)50 1097 y(w)m(eigh)m(ts)37 b(will)e(allo)m(w)h(to)i(forget) g(the)g(restriction)e(that)i(our)e(summation)g(w)m(as)i(o)m(v)m(er)g (trees)g(rather)f(than)50 1207 y(general)g(graphs.)60 b(Let)38 b(us)e(consider)g(a)i(\014xed)e(set)i Fr(P)p 1831 1222 71 4 v 1902 1167 a Fl(\016)1980 1207 y Fz(\(the)g(b)s(o)s(dy) e(of)h Fr(P)2597 1174 y Fm(0)2638 1207 y Fz(\))g(together)i(with)c(a)j (\014xed)e(set)50 1316 y(\(sk)m(eleton\))i Fr(P)547 1283 y Fl(\017)624 1316 y Fz(=3D)d Fr(Y)56 b Fz(of)37 b(cardinalit)m(y)d Fr(n)i Fz(of)h(stars)f Fr(x)1886 1330 y Fq(i)1917 1316 y Fz(.)58 b(W)-8 b(e)38 b(w)m(an)m(t)f(\014rst)f(to)h(p)s(erform)e(the) i(sum)e(o)m(v)m(er)j(all)50 1425 y(forests)30 b(\(more)h(generally)-8 b(,)30 b(graphs\))f(ha)m(ving)h Fr(Y)50 b Fz(as)30 b(their)f(supp)s (ort)g(and)g(then)h(to)h(sum)e(o)m(v)m(er)i(all)e(p)s(ossible)50 1534 y Fr(Y)20 b Fz(.)40 b(Denote)32 b(b)m(y)1522 1659 y Fr(\032)p Fz(\()p Fr(Y)21 b Fz(\))k Fp(\021)1889 1573 y Fk(X)1834 1768 y Fq(T)10 b Fm(:)p Fq(T)p 1908 1783 53 4 v 9 w Fm(=3D)p Fq(Y)2106 1659 y Fk(e)-67 b Fr(w)2155 1673 y Fq(T)2212 1659 y Fr(:)1240 b Fz(\(5)p Fr(:)p Fz(43\))50 1939 y(T)-8 b(o)42 b(estimate)h Fr(\032)p Fz(\()p Fr(Y)20 b Fz(\),)45 b(w)m(e)d(note)h(that)f(from)f(eac)m(h)i(p)s(oin)m(t)e Fr(i)j Fp(2)g Fr(Y)20 b Fz(,)45 b(links)40 b(ma)m(y)i(emanate)h(to)f(p) s(oin)m(ts)f(at)50 2048 y(distance)25 b(less)g(than)h Fr(R)q Fz(.)39 b(Ho)m(w)m(ev)m(er,)29 b(the)d(sum)f(of)h(w)m(eigh)m(ts) g(of)g(these)g(links)d(satis\014es)j(\(5.36\))i(\(if)d(the)h Fr(V)3515 2015 y Fl(\016)3495 2070 y Fq(q)3557 2048 y Fz(\()p Fr(x)3644 2062 y Fq(i)3674 2048 y Fz(\))50 2160 y(part)k(of)h(the)f(fugacit)m(y)h(is)f(\\assigned)g(to)h(v)m(ertices)g (only"\))f(and)g(so)779 2391 y Fp(j)p Fr(\032)p Fz(\()p Fr(Y)20 b Fz(\))p Fp(j)26 b(\024)f Fr(e)1183 2354 y Fl(\000)p Fq(V)1299 2326 y Fj(\016)1283 2372 y Fs(q)1337 2354 y Fl(j)p Fq(Y)15 b Fl(j)1473 2391 y Fr(e)1515 2354 y Fq(\020)t Fl(j)p Fq(Y)h Fl(j)1810 2391 y Fz(where)151 b Fr(V)2268 2354 y Fl(\016)2247 2414 y Fq(q)2335 2391 y Fz(=3D)25 b(min)2582 2405 y Fq(s)p Fl(6)p Fm(=3D)p Fq(q)2728 2391 y Fr(V)2801 2354 y Fl(\016)2781 2414 y Fq(q)2843 2391 y Fz(\()p Fr(s)p Fz(\))p Fr(:)496 b Fz(\(5)p Fr(:)p Fz(44\))50 2644 y(The)41 b(exp)s(onen)m(tial)f(factor)i Fr(e)1059 2611 y Fq(\020)t Fl(j)p Fq(Y)16 b Fl(j)1246 2644 y Fz(=3D)43 b(1)28 b(+)f Fr(\020)34 b Fz(+)1714 2604 y Fq(\020)1751 2576 y Ft(2)p 1714 2624 73 4 v 1733 2676 a Fm(2)1813 2644 y Fr(:)15 b(:)g(:)58 b Fz(\(compare)42 b(\(5.36\))h(\))f(app)s(ears)e(here)h(for) g(similar)50 2754 y(reasons)32 b(as)g(in)f(\(5.19\)\);)36 b(to)c(estimate)h(\(when)f(taking)g(pro)s(duct)e(o)m(v)m(er)k(all)d Fr(x)2672 2768 y Fq(i)2730 2754 y Fp(2)c Fr(Y)20 b Fz(\))33 b(the)f(con)m(tribution)f(of)50 2863 y(all)h(p)s(ossible)g(graphs)h (\(not)h(only)f(trees\))h(on)g Fr(Y)20 b Fz(.)50 b(W)-8 b(e)35 b(actually)e(use)h(sligh)m(tly)d(less)i(precise)g(b)s(ounds)f (here)50 2972 y(than)e(in)f(\(5.19\))k(and)c(the)i(relativ)m(e)f (smallness)f(of)h Fr(\020)37 b Fz(w.r.)j(to)32 b Fr(V)2245 2939 y Fl(\016)2225 2994 y Fq(q)2317 2972 y Fz(is)d(imp)s(ortan)m(t.) 159 3154 y(T)-8 b(o)35 b(\014nish)d(our)i(estimate)h(the)f(w)m(eigh)m (t)h(of)g(galaxies,)g(it)f(remains)g(to)h(sum)e(o)m(v)m(er)j(all)d(p)s (ossible)f(c)m(hoices)50 3263 y(of)i(the)g(set)h Fr(Y)20 b Fz(,)35 b(of)g(stars,)g(and)e(o)m(v)m(er)j(the)e(sets)h Fr(M)1761 3277 y Fm(1)1836 3263 y Fz(suc)m(h)f(that)g Fr(M)2343 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Fq(n)2832 340 y Fp(g)40 b Fw(of)h(c)-5 b(ontours)41 b(and)g(the)50 449 y(sum)33 b(is)h(taken)f(over)h(al)5 b(l)34 b(c)-5 b(omp)g(atible)36 b(c)-5 b(ol)5 b(le)-5 b(ctions)34 b(as)g(de\014ne)-5 b(d)35 b(by)e(\(5.47\),)i(\(5.48\).)45 b(We)33 b(normalize)-5 b(d)36 b(the)50 558 y(p)-5 b(otential)39 b(such)f(that)g Fz(min)14 b Fr(U)c Fz(\()p Fr(q)s Fz(\))34 b(=3D)g(0)p Fw(.)56 b(The)38 b(weights)54 b Fk(e)-67 b Fr(w)2028 572 y Fd(\000)2081 558 y Fz(\()p Fr(P)2187 525 y Fl(\003)2174 582 y Fq(i)2230 558 y Fz(\))37 b Fw(and)54 b Fk(e)-67 b Fr(w)2548 572 y Fq(t)2618 558 y Fw(satisfy)38 b(the)g(b)-5 b(ounds)38 b(state)-5 b(d)39 b(in)50 667 y(L)-5 b(emma)39 b(5.2)g(and)g(Pr)-5 b(op)g(osition)40 b(5.4,)g(r)-5 b(esp)g(e)g (ctively,)41 b(and)e Fr(E)5 b Fz(\()p FB(\000)p Fz(\))38 b Fw(satisfy)h(the)f(Peierls)g(c)-5 b(ondition)40 b(\(4.3\).)50 776 y(Mor)-5 b(e)g(over,)49 b(the)c(weights)h(of)f(galaxies)61 b Fk(e)-67 b Fr(w)1522 790 y Fd(\000)1576 776 y Fz(\()p Fr(P)1682 743 y Fl(\003)1724 776 y Fz(\))48 b Fp(\021)66 b Fz(~)-64 b Fr(w)s Fz(\()p Fr(P)2099 743 y Fl(\003)2141 776 y Fz(\))45 b Fw(dep)-5 b(end)46 b(on)g(the)f(c)-5 b(ol)5 b(le)-5 b(ction)46 b(of)f(c)-5 b(ontours)50 897 y FB(\000)25 b Fp(\021)g(f)p FB(\000)342 911 y Fm(1)383 897 y Fr(;)15 b(:)g(:)g(:)i FB(\000)608 911 y Fq(n)657 897 y Fp(g)29 b Fw(only)g(thr)-5 b(ough)30 b(the)f(value)g(of)g Fr(q)i Fw(if)d(their)h(supp)-5 b(ort)31 b(lies)d(in)2686 874 y Fk(e)2680 897 y Fz(\003)2743 911 y Fq(q)2783 897 y Fw(,)h(exc)-5 b(ept)29 b(if)f Fr(P)13 b Fp(\003)29 b Fw(interse)-5 b(cts)50 1006 y(the)30 b(supp)-5 b(ort)32 b(of)e FB(\000)p Fw(,)h(when)f(it)g(also)h(may)g(dep)-5 b(end)32 b(on)e(the)g(spin)h(c)-5 b(on\014gur)g(ation)32 b(on)e(the)h Fr(R)q Fw(-b)-5 b(oundary)31 b(of)f FB(\000)p 3619 1021 63 4 v Fw(.)50 1323 y FB(5.9.)47 b(Ma)m(y)m(er)35 b(expansion)h(of)f(the)f(p)s(olymers.)159 1640 y Fz(The)22 b(main)g(step)h(no)m(w)g(is)f(to)h(p)s(erform)f(a)h(Ma)m(y)m(er)i (expansion)c(for)i(the)g(sum)f(o)m(v)m(er)i(the)f(constellations)f(and) 50 1750 y(galaxies.)53 b(T)-8 b(o)34 b(simplify)d(the)k(language,)h(w)m (e)f(will)d(call)h(galaxies)i(and)f(constellations)f(indiscriminately) 50 1859 y Fw(p)-5 b(olymers)44 b Fz(and)33 b(denote)i(them)f(b)m(y)g Fr(P)13 b Fz(.)52 b(The)33 b(purp)s(ose)g(of)h(the)g(Ma)m(y)m(er)i (expansion)d(is)g(to)i(write)e(the)i(sum)50 1968 y(o)m(v)m(er)h(these)f (p)s(olymers)e(in)g(the)i(presence)g(of)f(a)h(\014xed)f(collection)h (of)f(con)m(tours)i(as)e(an)h(exp)s(onen)m(tial)f(of)g(a)50 2077 y(sum)i(of)i(new)f(ob)5 b(jects,)40 b(called)d('clusters'.)61 b(This)36 b(sum)g(o)m(v)m(er)j(clusters)d(can)i(then)f(b)s(e)g(in)m (terpreted)g(as)g(a)50 2186 y('free)32 b(energy')g(\(dep)s(ending)d(on) i(the)h(colour)f(of)g(the)h(set)g(in)e(whic)m(h)g(the)i(p)s(olymers)e (used)g(to)i(liv)m(e\),)g(plus)d(a)50 2295 y(correction)h(to)h(the)f (con)m(tour)g(energies.)40 b(In)29 b(this)g(form)g(the)h(partition)f (function)f(is)h(v)m(ery)h(similar)e(to)i(that)50 2404 y(of)d(what)g(is)f(called)g(an)h('abstract)h(Pirogo)m(v-Sinai)d(mo)s (del',)i(the)g(only)f(di\013erence)h(b)s(eing)e(that)j(the)f(ground)50 2513 y(state)42 b(energies)f(of)g(the)g(di\013eren)m(t)f(colors)h(are)g (replaced)f(b)m(y)h(free)g(energy)g(functions)e(and)h(that)i(there)50 2622 y(are)g(some)g(non-lo)s(cal)f(in)m(teractions)g(b)s(et)m(w)m(een)h (con)m(tours)h(due)e(to)h(the)g(in)m(teractions)f(of)h(clusters)f(with) 50 2731 y(the)d(con)m(tours.)65 b(In)38 b([Z3])g(it)g(is)g(sho)m(wn)f (that)i(the)f(Pirogo)m(v-Sinai)f(theory)i(can)g(b)s(e)e(dev)m(elop)s (ed)h(without)50 2840 y(problems)28 b(in)i(this)f(general)h(con)m (text.)159 3022 y(P)m(erforming)k(the)i(Ma)m(y)m(er)h(expansion)d(as)i (indicated)e(ab)s(o)m(v)m(e)j(is)d(actually)h(a)h(v)m(ery)g(standard)e (exercise)50 3131 y(and)39 b(it)g(is)g(w)m(ell-kno)m(wn)g(that)h(this)e (expansions)h(con)m(v)m(erges)j(under)c(the)i(conditions)e(w)m(e)i(ha)m (v)m(e)h(for)f(the)50 3240 y(p)s(olymer)34 b(activities,)j(pro)m(vided) e Fr(\034)46 b Fz(is)35 b(large)h(enough.)58 b(The)35 b(p)s(ossibly)f(simplest)g(w)m(a)m(y)j(of)f(seeing)g(this)f(is)50 3350 y(b)m(y)f(using)g(a)h(v)m(ersion)f(of)h(the)f(Kotec)m(k)q(\023)-47 b(y-Preiss-Dobrushin)35 b([KP)-8 b(,Do])36 b(metho)s(d)e(as)h(giv)m(en) g(in)e([BZ2])j(\(see)50 3459 y(also)29 b([NOZ]\).)h(In)e(fact)i(all)e (w)m(e)i(need)e(is)h(part)g(of)g(the)g(Theorem)g(2.2)h(of)g(that)f (article,)h(that)f(w)m(e)h(rephrased)50 3568 y(sligh)m(tly)f(for)h(our) g(purp)s(oses.)159 3750 y(Let)h Fp(P)39 b Fz(b)s(e)30 b(a)h(collection)f(of)h(p)s(olymers,)e(and)h(let)h Fr(\023)g Fz(b)s(e)f(a)h(binary)e(relation)h(on)h Fp(P)38 b Fz(\(called)30 b('incompati-)50 3859 y(bilit)m(y'\),)c(and)g(let)g Fr(c)h Fz(denote)g(its)e(logical)h(con)m(v)m(erse)i(\(i.e.)40 b Fr(P)28 b(c)15 b(P)2186 3826 y Fl(0)2238 3859 y Fz(unless)25 b Fr(P)j(\023)15 b(P)e Fz(\).)40 b(The)26 b(relation)f Fp(\032)h Fz(induces)50 3968 y(a)37 b(natural)f(graph)h(structure)g(on) f(an)m(y)i(set)g(made)f(of)g(elemen)m(ts)g(of)h Fp(P)7 b Fz(.)61 b(W)-8 b(e)38 b(will)d(alw)m(a)m(ys)i(assume)g(that)50 4077 y Fr(P)28 b(\023)15 b(P)e Fz(,)31 b(for)f(all)f Fr(P)38 b Fp(2)25 b(P)7 b Fz(.)159 4259 y(Sa)m(y)41 b(that)g(a)g(set)g (of)f(elemen)m(ts)h(of)g Fp(P)48 b Fz(\(with)40 b(m)m(ultiplicit)m(y\)) e(is)h(compatible,)k(if)c(the)i(corresp)s(onding)50 4368 y(graph)28 b(has)h(no)f(edges,)i(and)f(call)f(it)g(a)h Fw(cluster)h Fz(if)d(its)i(graph)f(is)g(connected.)41 b(Let)29 b Fp(C)5 b Fz(\()p Fp(P)i Fz(\))31 b(denote)e(the)g(set)h(of) 50 4477 y(clusters)g(\(all)h(connected)h(graphs)f(on)g Fp(P)1460 4444 y Fo(N)1512 4477 y Fz(\),)h(and)f Fp(F)9 b Fz(\()p Fp(P)e Fz(\))33 b(denote)f(the)f(set)h(of)g(all)e(compatible) h(subsets)g(of)50 4586 y Fp(P)7 b Fz(.)159 4768 y(Let)31 b Fr(w)d Fz(:)d Fp(P)33 b(!)25 b Fn(R)39 b Fz(b)s(e)30 b(some)h(function)e(on)h Fp(P)7 b Fz(.)50 4950 y Fh(Theorem)39 b(5.6.)44 b Fz([BZ2])p Fw(Assume)34 b(that)h(for)g(some)f(function)g Fr(a)28 b Fz(:)f Fp(P)35 b(!)28 b Fn(R)2641 4917 y Fm(+)2708 4950 y Fw(,)34 b(for)g(some)g Fz(0)28 b Fr(<)f(\016)32 b(<)27 b Fz(1)p Fw(,)34 b(for)50 5059 y(al)5 b(l)33 b Fr(P)38 b Fp(2)25 b(P)1568 5289 y(j)p Fr(w)r Fz(\()p Fr(P)13 b Fz(\))p Fp(j)p Fr(e)1868 5252 y Fq(a)p Fm(\()p Fq(P)d Fm(\))2052 5289 y Fp(\024)25 b Fr(\016)1289 b Fz(\(5)p Fr(:)p Fz(50\))50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(27)p eop %%Page: 28 28 28 27 bop 50 340 a Fw(and)33 b(for)h(any)f Fr(Q)25 b Fp(2)g(P)7 b Fw(,)1307 437 y Fk(X)1145 636 y Fq(P)j Fl(2P)5 b(n)p Fq(Q;)12 b(P)23 b(\023)12 b(Q)1616 523 y Fp(j)p Fr(w)r Fz(\()p Fr(P)h Fz(\))p Fp(j)p Fr(e)1916 486 y Fq(a)p Fm(\()p Fq(P)d Fm(\)+)p Fq(d)p Fm(\()p Fq(P)g Fm(\))2306 523 y Fp(\024)2413 462 y Fr(a)p Fz(\()p Fr(Q)p Fz(\))p 2413 502 191 4 v 2420 586 a Fr(L)p Fz(\()p Fr(\016)s Fz(\))3477 523 y(\(5)p Fr(:)p Fz(51\))50 852 y Fw(wher)-5 b(e)34 b Fr(L)p Fz(\()p Fr(\016)s Fz(\))26 b(=3D)615 808 y Fl(\000)12 b Fm(ln)q(\(1)p Fl(\000)p Fq(\016)r Fm(\))p 615 832 316 4 v 755 884 a Fq(\016)974 852 y Fw(and)33 b(wher)-5 b(e)34 b Fr(d)26 b Fz(:)f Fp(P)33 b(!)25 b Fn(R)1802 819 y Fm(+)1901 852 y Fw(is)32 b(another)j(function)d(of)h Fr(P)13 b Fw(.)42 b(Then)1192 1124 y Fz(ln)1355 1038 y Fk(X)1283 1237 y Fq(F)10 b Fl(2F)c Fm(\()p Fl(P)f Fm(\))1597 1038 y Fk(Y)1575 1233 y Fq(P)10 b Fl(2)p Fq(F)1751 1124 y Fr(w)r Fz(\()p Fr(P)j Fz(\))26 b(=3D)2147 1038 y Fk(X)2081 1237 y Fq(C)t Fl(2C)s Fm(\()p Fl(P)5 b Fm(\))2374 1124 y Fk(b)-67 b Fr(w)s Fz(\()p Fr(C)7 b Fz(\))909 b(\(5)p Fr(:)p Fz(52\))50 1475 y Fw(wher)-5 b(e)50 b Fk(b)-67 b Fr(w)r Fz(\()p Fr(C)7 b Fz(\))33 b Fw(satis\014es)g(for)g(al)5 b(l)34 b Fr(Q)25 b Fp(2)g(P)40 b Fw(the)33 b(b)-5 b(ound)1421 1652 y Fk(X)1256 1851 y Fq(C)t Fl(2C)s Fm(\()p Fl(P)5 b Fm(\))p Fq(;)14 b(C)j(\023)c(Q)1733 1738 y Fp(j)j Fk(b)-67 b Fr(w)s Fz(\()p Fr(C)7 b Fz(\))p Fp(j)p Fr(e)2035 1701 y Fq(d)p Fm(\()p Fq(C)t Fm(\))2217 1738 y Fp(\024)25 b Fr(a)p Fz(\()p Fr(Q)p Fz(\))974 b(\(5)p Fr(:)p Fz(53\))50 2090 y Fw(wher)-5 b(e)34 b Fr(d)p Fz(\()p Fr(C)7 b Fz(\))25 b(=3D)617 2021 y Fk(P)713 2117 y Fq(P)10 b Fl(2)p Fq(C)895 2090 y Fr(d)p Fz(\()p Fr(P)j Fz(\))33 b Fw(me)-5 b(asur)g(es)34 b(the)f(de)-5 b(c)g(ay)34 b(of)49 b Fk(b)-67 b Fr(w)s Fz(\()p Fr(C)7 b Fz(\))p Fw(.)159 2271 y Fz(W)-8 b(e)33 b(will)c(use)j(this)f(theorem)h(with)f Fp(P)40 b Fz(b)s(e)31 b(the)h(collection)g(of)g(all)f(trees)i(and)e(galaxies)h(that)h(can)f (exist)50 2380 y(in)39 b(the)h(presence)h(of)f(a)h(giv)m(en)g (con\014guration)e(of)i(con)m(tours)g FB(\000)p Fz(,)i(and)d(with)f Fr(w)r Fz(\()p Fr(P)13 b Fz(\))41 b(the)g(corresp)s(onding)50 2490 y(activities)54 b Fk(e)-67 b Fr(w)510 2504 y Fq(t)581 2490 y Fz(and)55 b Fk(e)-67 b Fr(w)832 2504 y Fd(\000)885 2490 y Fz(\()p Fr(P)991 2457 y Fl(\003)1034 2490 y Fz(\))39 b(dep)s(ending)d(on)j(whether)f Fr(P)52 b Fz(is)38 b(a)i(tree)f(or)g(a) h(galaxy)-8 b(.)67 b(As)39 b(the)g(function)50 2599 y Fr(a)p Fz(\()p Fr(P)13 b Fz(\))39 b(w)m(e)h(will)c(simply)g(use)j Fr(a)p Fp(j)p Fr(t)p Fp(j)g Fz(resp.)66 b Fr(b)p Fp(j)p Fr(P)p 1554 2614 71 4 v 1625 2559 a Fl(\003)1667 2599 y Fp(j)39 b Fz(with)f(suitable)f(constan)m(ts)j Fr(a)f Fz(and)f Fr(b)p Fz(.)66 b(Since)38 b(activities)50 2708 y(are)f(deca)m(ying)g(exp)s(onen)m(tially)e(in)g(these)i(same)g(quan)m (tities)g(with)e(a)i(rate)g(con)m(trollable)f(b)m(y)h Fr(\034)10 b Fz(,)38 b(and)f(the)50 2817 y(n)m(um)m(b)s(er)31 b(of)h(these)h(ob)5 b(jects)33 b(of)f(giv)m(en)g(size)g(is)g(at)h(most) f(exp)s(onen)m(tially)f(increasing)f(with)h(a)i(rate)g(that)g(is)50 2926 y(indep)s(enden)m(t)g(of)i(b)s(oth)g Fr(\034)45 b Fz(and)35 b Fr(R)q Fz(,)h(it)f(is)g(an)g(elemen)m(tary)h(exercise)f (that)h(the)g(h)m(yp)s(othesis)e(of)h(Theorem)50 3035 y(5.6)c(are)g(satis\014ed)f(if)f Fr(\034)40 b Fz(is)30 b(large)g(enough.)159 3217 y(Next)38 b(w)m(e)g(need)f(to)g(understand)f (a)h(little)f(more)i(ab)s(out)f(the)g(geometric)h(structure)f(of)g(the) h(clusters)50 3326 y(obtained)33 b(b)m(y)g(computing)g(the)g(logarithm) g(of)g(the)h(sum)f(o)m(v)m(er)h(all)f(p)s(olymers)e(existing)i(in)f (the)i(presence)50 3441 y(of)g FB(\000)p Fz(.)53 b(Recall)34 b(the)h(the)g(only)e(p)s(ossible)f(p)s(olymers)h(are)i(whose)f(whose)h (supp)s(ort)d(is)i(in)f(the)i(regions)3568 3418 y Fk(e)3562 3441 y Fz(\003)3625 3408 y Fq(R)3625 3464 y(q)3685 3441 y Fz(.)50 3550 y(Therefore)44 b(it)g(is)g(geometrically)g(imp)s (ossible)d(that)k(p)s(olymers)e(whose)i(supp)s(ort)d(in)m(tersects)j (di\013eren)m(t)50 3659 y(connected)28 b(comp)s(onen)m(ts)g(of)g(the)g (complemen)m(t)g(of)g FB(\000)p 1830 3674 63 4 v 28 w Fz(are)g(incompatible,)f(implying)d(that)29 b(the)f(sum)e(o)m(v)m(er)50 3769 y(all)i(clusters)h(can)h(b)s(e)f(split)e(in)m(to)j(a)g(sum)f(o)m (v)m(er)h(con)m(tributions)e(from)h(clusters)g(whose)h(supp)s(ort)d(in) m(tersects)50 3878 y(di\013eren)m(t)34 b(connected)j(comp)s(onen)m(ts)e (of)g(the)h(complemen)m(t)f(of)g FB(\000)p 2238 3893 V Fz(.)55 b(Next)36 b(w)m(e)g(observ)m(e)g(that)g(if)e(a)h(cluster)50 3987 y(do)s(es)j(not)h(in)m(tersect)h(the)f(supp)s(ort)e(of)i FB(\000)p Fz(,)h(then)f(its)f(w)m(eigh)m(t)h(is)f(indep)s(enden)m(t)f (of)i(the)g(the)g(con)m(tours)g FB(\000)50 4096 y Fz(except)27 b(that)f(it)f(dep)s(ends)f(on)i(the)g(color)g(of)g(the)g(region)g(it)f (is)g(supp)s(orted)f(in.)38 b(Since)25 b(in)f(all)h(other)h(resp)s (ects)50 4205 y(cluster)36 b(w)m(eigh)m(ts)i(are)f(translationally)e (in)m(v)-5 b(arian)m(t,)39 b(w)m(e)e(can)h(in)m(tro)s(duce)e (translation)g(in)m(v)-5 b(arian)m(t)36 b(cluster)50 4314 y(w)m(eigh)m(ts)42 b Fr(w)452 4281 y Fq(q)492 4314 y Fz(\()p Fr(C)7 b Fz(\))42 b(whic)m(h)e(are)i(simply)d(equal)i(to)h (the)f(w)m(eigh)m(t)h(of)g(a)g(translate)f(of)h(the)g(cluster)e Fr(C)48 b Fz(that)50 4434 y(has)32 b(supp)s(ort)f(in)668 4412 y Fk(e)662 4434 y Fz(\003)725 4448 y Fq(q)798 4434 y Fz(and)h(do)s(es)g(not)h(touc)m(h)g(an)m(y)g(con)m(tour.)49 b(W)-8 b(e)34 b(ma)m(y)f(also)g(consider)e(the)i(ensem)m(ble)f Fp(F)3669 4402 y Fq(q)50 4544 y Fz(of)k(all)e(p)s(olymers)g(that)i(ma)m (y)g(exist)f(in)g(an)g(in\014nite)e(v)m(olume)j(restricted)f(ensem)m (ble)g(of)h(color)f Fr(q)k Fz(and)c(the)50 4653 y(corresp)s(onding)28 b(set)j(of)g(all)e(cluster)h Fp(C)1353 4620 y Fq(q)1393 4653 y Fz(.)41 b(Let)30 b(us)g(also)g(de\014ne)1305 4945 y Fr(f)1350 4959 y Fq(q)1414 4945 y Fp(\021)25 b Fr(U)10 b Fz(\()p Fr(q)s Fz(\))21 b Fp(\000)1922 4859 y Fk(X)1808 5054 y Fq(C)t Fl(2C)1955 5035 y Fs(q)1992 5054 y Fq(;)12 b(C)p 2026 5069 57 4 v 5 w Fl(3)p Fm(0)2210 4884 y Fk(b)-67 b Fr(w)2261 4851 y Fq(q)2302 4884 y Fz(\()p Fr(C)7 b Fz(\))p 2194 4924 250 4 v 2258 5007 a Fp(j)p Fr(C)p 2283 5022 72 4 v 7 w Fp(j)3477 4945 y Fz(\(5)p Fr(:)p Fz(54\))50 5289 y(Then)29 b(Theorem)h(5.6)i(allo)m(ws)d(us)h(to)h(express)f(the)h (partition)e(function)g(as)h(follo)m(ws.)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)i(13:49)1146 b Fx(28)p eop %%Page: 29 29 29 28 bop 50 340 a Fh(Theorem)31 b(5.7.)p Fw(Ther)-5 b(e)28 b(exist)f Fr(\034)1205 354 y Fm(0)1271 340 y Fr(<)e Fp(1)p Fw(,)j(indep)-5 b(endent)29 b(of)f Fr(R)g Fw(such)f(that)i(the)f (fol)5 b(lowing)28 b(holds)h(if)e Fr(\034)35 b(>)25 b(\034)3641 354 y Fm(0)3682 340 y Fw(:)400 660 y Fr(Z)462 674 y Fm(\003)542 660 y Fz(=3D)740 573 y Fk(X)628 773 y Fl(f)p Fd(\000)714 782 y Ft(1)750 773 y Fq(;:::)n Fd(\000)885 782 y Fs(m)946 773 y Fl(g)1057 546 y Fq(m)1031 573 y Fk(Y)1030 767 y Fq(i)p Fm(=3D1)1163 660 y Fr(e)1205 622 y Fl(\000)p Fq(E)s Fm(\()p Fd(\000)1395 631 y Fs(i)1426 622 y Fm(\))1474 660 y Fz(exp)1628 477 y Fk(0)1628 641 y(@)1708 660 y Fp(\000)1799 573 y Fk(X)1794 768 y Fq(q)r Fl(2)p Fq(Q)1950 660 y Fr(U)10 b Fz(\()p Fr(q)s Fz(\))p Fp(j)p Fz(\003)2224 674 y Fq(q)2265 660 y Fp(j)21 b Fz(+)2463 573 y Fk(X)2402 773 y Fq(C)t Fl(2C)s Fm(\()p Fd(\000)p Fm(\))2687 660 y Fk(b)-67 b Fr(w)2736 674 y Fd(\000)2790 660 y Fz(\()p Fr(C)7 b Fz(\)\))2967 477 y Fk(1)2967 641 y(A)638 1051 y Fz(=3D)846 964 y Fk(X)734 1164 y Fl(f)p Fd(\000)820 1173 y Ft(1)856 1164 y Fq(;:::)n Fd(\000)991 1173 y Fs(m)1052 1164 y Fl(g)1163 937 y Fq(m)1137 964 y Fk(Y)1137 1158 y Fq(i)p Fm(=3D1)1269 1051 y Fr(e)1311 1013 y Fl(\000)p Fq(E)s Fm(\()p Fd(\000)1501 1022 y Fs(i)1532 1013 y Fm(\))1580 1051 y Fz(exp)1734 841 y Fk(0)1734 1001 y(B)1734 1059 y(@)1814 1051 y Fp(\000)1905 964 y Fk(X)1900 1159 y Fq(q)r Fl(2)p Fq(Q)2056 1051 y Fr(f)2101 1065 y Fq(q)2141 1051 y Fp(j)p Fz(\003)2229 1065 y Fq(q)2270 1051 y Fp(j)20 b Fz(+)2549 964 y Fk(X)2406 1186 y Fq(C)t Fl(2)2517 1182 y Fk(e)2511 1186 y Fl(C)t Fm(:)p Fq(C)p 2575 1201 57 4 v 4 w Fl(\\)p Fd(\000)p 2680 1201 50 4 v 1 w Fl(6)p Fm(=3D)p Fl(;)2854 1051 y Fk(e)-67 b Fr(w)r Fz(\()p Fr(C)7 b Fz(\))3047 841 y Fk(1)3047 1001 y(C)3047 1059 y(A)3477 869 y Fz(\(5)p Fr(:)p Fz(55\))50 1403 y Fw(wher)-5 b(e)34 b FB(\000)25 b Fp(\021)g(f)p Fr(g)s(b)621 1417 y Fm(1)663 1403 y Fr(;)15 b(:)g(:)g(:)i(;)e FB(\000)p Fz(?)p Fr(n)p Fp(g)p Fw(,)33 b Fp(C)d(\021)c(C)5 b Fz(\()p FB(\000)p Fz(\))33 b Fw(is)g(the)g(c)-5 b(ol)5 b(le)-5 b(ction)34 b(of)f(clusters)g(c)-5 b(onstructe)g(d)35 b(by)d(The)-5 b(or)g(em)35 b(5.6)50 1512 y(and)28 b(in)f(the)h(se)-5 b(c)g(ond)28 b(formula)h(we)e(use)g(the)h(r)-5 b(esummation)29 b(\(5.54\).)42 b(Then)27 b(the)h(sum)f(is)h(only)g(over)f(clusters)50 1632 y(interse)-5 b(cting)33 b FB(\000)p 539 1647 63 4 v Fw(.)41 b(The)33 b(values)49 b Fk(e)-67 b Fr(w)s Fz(\()p Fr(C)7 b Fz(\))32 b Fw(ar)-5 b(e)34 b(then)f(de\014ne)-5 b(d,)33 b(for)h(any)f Fr(C)f Fp(2)2578 1609 y Fk(e)2565 1632 y Fp(C)37 b Fw(such)c(that)h Fr(C)p 3043 1647 72 4 v 27 w Fp(\\)19 b FB(\000)p 3215 1647 63 4 v 25 w Fp(6)p Fz(=3D)25 b Fp(;)33 b Fw(as)1200 1900 y Fk(e)-67 b Fr(w)r Fz(\()p Fr(C)7 b Fz(\))26 b(=3D)41 b Fk(b)-67 b Fr(w)1580 1914 y Fd(\000)1634 1900 y Fz(\()p Fr(C)7 b Fz(\))20 b Fp(\000)1887 1814 y Fk(X)1935 2005 y Fq(q)2049 1900 y Fk(b)-67 b Fr(w)2100 1863 y Fq(q)2141 1900 y Fz(\()p Fr(C)7 b Fz(\))2294 1839 y Fp(j)p Fr(C)p 2319 1854 72 4 v 27 w Fp(n)21 b FB(\000)p 2477 1854 63 4 v Fp(j)p 2294 1879 271 4 v 2369 1963 a(j)p Fr(C)p 2394 1978 72 4 v 6 w Fp(j)3477 1900 y Fz(\(5)p Fr(:)p Fz(56\))50 2210 y Fw(wher)-5 b(e)34 b(we)e(set)49 b Fk(b)-67 b Fr(w)648 2224 y Fd(\000)727 2210 y Fz(=3D)25 b(0)33 b Fw(if)f Fr(C)g Fp(62)25 b(C)5 b Fz(\()p FB(\000)p Fz(\))p Fw(.)159 2392 y(The)41 b(quantities)g Fr(f)822 2406 y Fq(q)903 2392 y Fw(ar)-5 b(e)42 b(analytic)g(functions)f(of)g(the)g(p)-5 b(ar)g(ameters)44 b(of)e(the)f(Hamiltonian,)j(and)e(the)50 2501 y(cluster)33 b(weights)49 b Fk(e)-67 b Fr(w)s Fz(\()p Fr(C)7 b Fz(\))32 b Fw(satisfy)i(an)f(estimate)1405 2649 y Fk(X)1366 2844 y Fq(C)t Fm(:)p Fq(C)p 1443 2859 57 4 v 5 w Fl(3)p Fq(i)1590 2735 y Fp(j)16 b Fk(e)-67 b Fr(w)s Fz(\()p Fr(C)7 b Fz(\))p Fp(j)p Fr(e)1892 2697 y Fq(c\034)g Fl(j)p Fq(C)p 1988 2712 V 4 w Fl(j)2096 2735 y Fp(\024)25 b Fr(e)2234 2697 y Fl(\000)p Fq(c)2324 2670 y Fj(0)2348 2697 y Fq(\034)3477 2735 y Fz(\(5)p Fr(:)p Fz(57\))50 3055 y Fw(for)33 b Fr(c;)15 b(c)313 3022 y Fl(0)365 3055 y Fr(>)25 b Fz(0)33 b Fw(indep)-5 b(endent)34 b(of)f Fr(R)q Fw(.)50 3291 y(Pr)-5 b(o)g(of.)48 b Fz(This)30 b(theorem)j(is)f(a)h(straigh)m(tforw)m(ard)f(consequence)h (of)g(Corollary)e(5.5)j(and)e(Theorem)g(5.6,)i(as)50 3400 y(explained)28 b(ab)s(o)m(v)m(e.)42 b Fp(})159 3636 y Fz(With)j(Theorem)g(5.7)i(w)m(e)f(ha)m(v)m(e)g(ac)m(hiev)m(ed)g(the)g (goal)g(of)g(this)e(pap)s(er:)70 b(w)m(e)46 b(ha)m(v)m(e)h(form)m (ulated)e(the)50 3745 y(partition)27 b(function)g(to)i(our)f(mo)s(dels) f(in)g(the)i(form)f(of)h(an)f(abstract)h(Pirogo)m(v{Sinai)f(mo)s(del)f (in)g(the)i(sense)50 3854 y(of)k([Z3],)h(with)d(b)s(ounds)f(on)j(the)g (P)m(eierls)e(constan)m(t)j(and)e(the)h(cluster)f(w)m(eigh)m(ts)h(that) g(do)g(not)g(dep)s(end)e(on)50 3963 y(the)f(range)g(of)g(the)f(in)m (teraction)h Fr(R)g Fz(but)f(only)g(on)h(the)g(o)m(v)m(erall)f (strength)h(expressed)f(b)m(y)h Fr(\034)10 b Fz(.)40 b(Theorem)30 b(2.2)50 4072 y(no)m(w)g(follo)m(ws)g(from)g(the)g (general)h(results)e(of)h([Z3].)1615 4362 y FA(References)-47 4598 y Fz([B1])47 b(A.)26 b(Bo)m(vier,)i(\\The)d(Kac)i(v)m(ersion)e(of) h(the)g(Sherrington{Kirkpatric)m(k)d(mo)s(del)h(at)j(high)d(temp)s (eratures",)159 4707 y(J.)30 b(Stat.)42 b(Ph)m(ys.)e FB(91)p Fz(,)31 b(459{474)i(\(1998\).)-180 4889 y([BGP1])47 b(A.)34 b(Bo)m(vier,)i(V.)e(Ga)m(yrard,)h(and)f(P)-8 b(.)34 b(Picco,)h(\\Large)g(deviation)e(principles)e(for)i(the)h (Hop\014eld)f(mo)s(del)159 4998 y(and)d(the)g(Kac{Hop\014eld)g(mo)s (del",)g(Prob.)40 b(Theor.)h(Rel.)f(Fields)29 b FB(101)p Fz(,)i(511{546)j(\(1995\).)-180 5180 y([BGP2])47 b(A.)34 b(Bo)m(vier,)i(V.)f(Ga)m(yrard,)g(and)f(P)-8 b(.)34 b(Picco,)i (\\Distribution)c(of)i(o)m(v)m(erlap)h(pro\014les)e(in)f(the)j(one)f (dimen-)159 5289 y(sional)29 b(Kac{Hop\014eld)h(mo)s(del",)g(Comm)m (un.)40 b(Math.)h(Ph)m(ys.)f FB(186)p Fz(,)32 b(323{379)h(\(1997\).)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)f(13:49)1146 b Fx(29)p eop %%Page: 30 30 30 29 bop -180 340 a Fz([BGP3])47 b(A.)39 b(Bo)m(vier,)k(V.)c(Ga)m (yrard,)j(and)d(P)-8 b(.)39 b(Picco,)j(\\T)m(ypical)d(pro\014les)e(of)j (the)f(Kac{Hop\014eld)g(mo)s(del",)i(in)159 449 y(\\Mathematical)26 b(asp)s(ects)e(of)h(spin)d(glasses)j(and)f(neural)f(net)m(w)m(orks",)k (A.)e(Bo)m(vier)f(and)g(P)-8 b(.)25 b(Picco)g(\(eds.\),)159 558 y(Progress)30 b(in)f(Probabilit)m(y)g FB(41)p Fz(,)i(187{242)i (\(Birkh\177)-45 b(auser,)30 b(Boston)h(1998\).)-174 740 y([BKL1])46 b(J.)28 b(Bricmon)m(t,)g(K.)g(Kuro)s(da,)f(J.)g(L.)h (Leb)s(o)m(witz,)g(\\The)g(structure)f(of)h(Gibbs)e(states)j(and)e (phase)h(co)s(exis-)159 849 y(tence)21 b(for)f(non-symmetric)g(con)m (tin)m(uum)f(Widom{Ro)m(wlinson)f(mo)s(del)h(",)k(Z.W)-8 b(ahrsc)m(heinlic)m(hk)m(eitstheorie)159 958 y(V)g(erw.)41 b(Geb.)g FB(67)p Fz(,)31 b(121{138)i(\(1984\).)-205 1140 y([BKL2])77 b(J.)43 b(Bricmon)m(t,)j(K.)d(Kuro)s(da,)i(J.)d(L.)h(Leb)s (o)m(witz,)j(\\First)d(order)f(phase)g(transitions)g(in)f(lattice)i (and)159 1249 y(con)m(tin)m(uous)37 b(systems:)55 b(Extension)36 b(of)i(Pirogo)m(v{Sinai)e(theory",)k(Comm)m(un.)61 b(Math.)h(Ph)m(ys.)g FB(101)p Fz(,)159 1358 y(501{538)33 b(\(1985\).)-52 1540 y([BS])46 b(J.)35 b(Bricmon)m(t,)h(J.)f(Sla)m(wn)m(y)-8 b(,)36 b(\\Phase)g(transitions)d(in)h(systems)h(with)e(a)j(\014nite)e (n)m(um)m(b)s(er)f(of)j(dominan)m(t)159 1649 y(ground)29 b(states",)k(J.)d(Statist.)41 b(Ph)m(ys.)f FB(54)31 b Fz(\(1989\),)i(89{161)g(\(1989\).)-102 1831 y([BZ1])46 b(A.)30 b(Bo)m(vier)h(and)f(M.)g(Zahradn)-10 b(\023)-35 b(\020k,)29 b(\\The)h(lo)m(w)g(temp)s(erature)g(phase)g(of)g(Kac{Ising) g(mo)s(dels",)g(J.)g(Stat.)159 1940 y(Ph)m(ys.)40 b FB(87)p Fz(,)31 b(311{332)j(\(1997\).)-102 2122 y([BZ2])46 b(A.)27 b(Bo)m(vier)g(and)f(M.)h(Zahradn)-10 b(\023)-35 b(\020k,)26 b(\\A)h(simple)d(inductiv)m(e)h(approac)m(h)i(to)h(the)f(problem)e(of)h (con)m(v)m(ergence)159 2231 y(of)k(cluster)g(expansions)f(of)i(p)s (olymer)e(mo)s(dels",)g(J.)i(Stat.)41 b(Ph)m(ys.)p FB(100)p Fz(,)31 b(765{778)j(\(2000\).)-102 2413 y([BZ3])46 b(A.)31 b(Bo)m(vier)g(and)e(M.)i(Zahradn)-10 b(\023)-35 b(\020k,)29 b(in)g(preparation.)-92 2595 y([Bo1])47 b(Th.)75 b(Bo)s(dineau,)45 b(\\In)m(terface)f(for)e(one-dimensional)e(random)h(Kac)i(p)s(oten)m (tials,)i(Ann.)75 b(Inst.)h(H.)159 2704 y(P)m(oincar)m(\023)-43 b(e.)42 b(Probab.)e(Statist.)h FB(33)p Fz(,)31 b(559{590)i(\(1997\).) -148 2885 y([CMP])46 b(M.)33 b(Cassandro,)g(R.)f(Marra,)i(and)e(E.)h (Presutti,)f(\\Corrections)h(to)g(the)g(critical)e(temp)s(erature)i(in) e(2d)159 2995 y(Ising)e(systems)h(with)g(Kac)g(p)s(oten)m(tials",)h(J.) f(Stat.)41 b(Ph)m(ys.)g FB(78)p Fz(,)31 b(1131{1138)j(\(1995\).)-135 3176 y([COP])45 b(M.)35 b(Cassandro,)g(E.)f(Orlandi,)f(and)h(E.)g (Presutti,)h(\\In)m(terfaces)h(and)e(t)m(ypical)g(Gibbs)e (con\014gurations)159 3285 y(for)e(one-dimensional)e(Kac)j(p)s(oten)m (tials",)f(Prob.)41 b(Theor.)f(Rel.)g(Fields)29 b FB(96)p Fz(,)i(57{96)i(\(1993\).)-65 3467 y([CP])46 b(M.)39 b(Cassandro)e(and)h (E.)g(Presutti,)i(\\Phase)f(transitions)d(in)h(Ising)g(systems)h(with)f (long)h(but)g(\014nite)159 3576 y(range",)31 b(Mark)m(o)m(v)h(Pro)s(c.) 41 b(Rel.)f(Fields)29 b(\(1996\).)-205 3758 y([DOPT])46 b(A.)22 b(De)h(Masi,)h(E.)f(Orlandi,)e(E.)h(Presutti,)i(and)d(L.)h(T)-8 b(riolo,)24 b(\\Glaub)s(er)d(ev)m(olution)h(with)e(Kac)j(p)s(oten)m (tials,)159 3867 y(I.)f(Mesoscopic)h(and)f(macroscopic)h(limits,)f(in)m (terface)h(dynamics",)g(Nonlinearit)m(y)e FB(7)p Fz(,)k(633{696)g (\(1994\);)159 3976 y(\\I)s(I.)i(Fluctuations",)h(Nonlinearit)m(y)f FB(9)p Fz(,)h(27{51)i(\(1996\).)42 b(\\I)s(I)s(I.)27 b(Spino)s(dal)d(decomp)s(osition",)k(Spino)s(dal)159 4085 y(decomp)s(osition.)39 b(Nonlinearit)m(y)29 b FB(9)i Fz(53{114)i(\(1996\).)-52 4267 y([Do])47 b(R.)26 b(L.)g(Dobrushin,)f (\\Estimates)i(of)f(Semiin)m(v)-5 b(arian)m(ts)24 b(for)i(the)g(Ising)f (Mo)s(del)g(at)i(Lo)m(w)g(T)-8 b(emp)s(eratures",)159 4376 y(59{81,)40 b(T)-8 b(opics)35 b(in)f(Statistical)i(Ph)m(ysics,)g (AMS)g(T)-8 b(ranslation)34 b(Series)h(2,)j FB(177)p Fz(,)g(AMS,)e(Adv)-5 b(ances)36 b(in)159 4485 y(the)30 b(Mathematical)i(Sciences)e({)h(32,)g(\(1995\).)-62 4667 y([DZ])46 b(R.L.)23 b(Dobrushin)e(and)i(M.)g(Zahradn)-10 b(\023)-35 b(\020k,)23 b(\\Phase)g(diagrams)g(for)g(con)m(tin)m (uous-spin)e(mo)s(dels:)36 b(an)23 b(exten-)159 4776 y(sion)32 b(of)h(the)g(Pirogo)m(v{Sinai)f(theory",)i(in)e (\\Mathematical)i(Problems)d(of)i(Statistical)g(Mec)m(hanics",)159 4885 y(R.L.)d(Dobrushin,)f(Ed.)40 b(\(D.)31 b(Reidel)e(Publishing)d (Compan)m(y)-8 b(,)31 b(Dordrec)m(h)m(t,)h(1986\).)-61 5067 y([HZ])46 b(P)-8 b(.Holic)m(k)q(\023)-47 b(y)33 b(,)g(M.)f(Zahradn)-10 b(\023)-35 b(\020k,\\Strati\014ed)30 b(Lo)m(w)i(T)-8 b(emp)s(erature)32 b(Phases)g(of)g(Strati\014ed)f(Spin) e(Mo)s(dels.)159 5176 y(A)h(general)h(P)-8 b(.S.)30 b(Approac)m(h)h(",) g(submitted)e(to)i(JSP)-8 b(.)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(30)p eop %%Page: 31 31 31 30 bop -70 340 a Fz([KP])46 b(R.)32 b(Kotec)m(k)q(\023)-47 b(y,)34 b(D.)f(Preiss,)e(\\Cluster)f(expansions)h(for)g(abstract)i(p)s (olymer)d(mo)s(dels",)h(Comm.)44 b(Math.)159 449 y(Ph)m(ys.)c FB(103)p Fz(,)32 b(491{498)h(\(1986\).)-144 631 y([KUH])46 b(M.)29 b(Kac,)g(G.)g(Uhlen)m(b)s(ec)m(k,)f(and)g(P)-8 b(.C.)29 b(Hemmer,)g(\\On)f(the)g(v)-5 b(an)28 b(der)g(W)-8 b(aals)29 b(theory)f(of)h(v)-5 b(ap)s(our-liquid)159 740 y(equilibrium.)35 b(I.)26 b(Discussion)f(of)h(a)h(one-dimensional)e (mo)s(del")g(J.)i(Math.)40 b(Ph)m(ys.)f FB(4)p Fz(,)27 b(216{228)j(\(1963\);)159 849 y(\\I)s(I.)40 b(Discussion)e(of)j(the)f (distribution)c(functions")j(J.)h(Math.)71 b(Ph)m(ys.)f FB(4)p Fz(,)43 b(229{247)g(\(1963\);)48 b(\\I)s(I)s(I.)159 958 y(Discussion)29 b(of)h(the)h(critical)e(region",)i(J.)f(Math.)41 b(Ph)m(ys.)g FB(5)p Fz(,)30 b(60{74)j(\(1964\).)-139 1140 y([LMP])46 b(J.L.)24 b(Leb)s(o)m(witz,)h(A.)f(Mazel,)i(and)d(E.)h (Presutti,)h(\\Liquid-v)-5 b(ap)s(or)21 b(phase)i(transitions)g(for)g (systems)h(with)159 1249 y(\014nite-range)30 b(in)m(teractions",)h(J.)f (Statist.)41 b(Ph)m(ys.)f FB(94)p Fz(,)31 b(955{1025)j(\(1999\).)-162 1431 y([NOZ])76 b(F.)24 b(Nardi,)h(E.)f(Olivieri,)e(M.)j(Zahradn)-10 b(\023)-35 b(\020k,\\On)22 b(the)i(Ising)f(mo)s(del)g(with)f(strongly)i (anisotropic)f(external)159 1540 y(\014eld",)30 b(Jour.)40 b(Stat.)h(Ph)m(ys.)f FB(97)p Fz(,87{144)34 b(\(1999\).)1 1722 y([P])46 b(R.)40 b(P)m(eierls,)h(\\On)e(the)h(Ising)e(mo)s(del)h (of)g(ferromagnetism",)k(Pro)s(c.)68 b(Cam)m(bridge)39 b(Phil.)66 b(So)s(c.)i FB(32)p Fz(,)159 1831 y(477{481)33 b(\(1936\).)-68 2013 y([Z1])76 b(M.)31 b(Zahradn)-10 b(\023)-35 b(\020k)28 b(\\An)j(alternativ)m(e)g(v)m(ersion)f(of)h (Pirogo)m(v{Sinai)e(theory",)i(Comm.)g(Math.)42 b(Ph)m(ys.)30 b FB(93)p Fz(,)159 2122 y(559{581)j(\(1984\).)-68 2304 y([Z2])76 b(M.)32 b(Zahradn)-10 b(\023)-35 b(\020k,)30 b(\\A)i(short)f(course)h(in)e(the)i(Pirogo)m(v{Sinai)f(theory",)h (Rendicon)m(ti)f(Mat.)45 b(Appl.)e FB(18)p Fz(,)159 2413 y(411{486)33 b(\(1998\).)-38 2595 y([Z3])46 b(M.)34 b(Zahradn)-10 b(\023)-35 b(\020k,)33 b(\\Cluster)g(expansions)f(of)i(small)e(con)m (tours)j(in)d(abstract)j(Pirogo)m(v{Sinai)d(mo)s(dels",)159 2704 y(preprin)m(t)c(2001.)50 2922 y(An)m(ton)j(Bo)m(vier)2474 b(Milo)-5 b(\024)-41 b(s)30 b(Zahradn)-10 b(\023)-35 b(\020k)50 3031 y(W)-8 b(eierstrass)31 b(Institute)1392 b(F)-8 b(acult)m(y)32 b(of)e(Mathematics)i(and)e(Ph)m(ysics)50 3140 y(Mohrenstrasse)g(39)2233 b(Charles)29 b(Univ)m(ersit)m(y)50 3249 y(10117)j(Berlin)2606 b(Sok)m(olo)m(vsk\023)-45 b(a)32 b(83)50 3358 y(German)m(y)2724 b(18600)32 b(Praha)e(8)50 3467 y(e-mail:)40 b(b)s(o)m(vier@wias-b)s(erlin.de)1868 b(Czec)m(h)31 b(Republic)2310 3576 y(e-mail:)40 b(mzahrad@k)-5 b(arlin.m\013.cuni.cz)50 5398 y Ft(29)p Fs(=3Daug)r(ust=3D)p Ft(2001;)32 b(13:49)1146 b Fx(31)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF --==_Exmh_3081196201500-- ---------------0109051105251--