Content-Type: multipart/mixed; boundary="-------------0109051146907" This is a multi-part message in MIME format. ---------------0109051146907 Content-Type: text/plain; name="01-321.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-321.keywords" Low temperature Gibbs states, contours, cluster expansion, contour functional, abstract Pirogov -- Sinai model with cluster field, Pirogov -- Sinai theory ---------------0109051146907 Content-Type: application/postscript; name="psaugust.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="psaugust.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: psaugust.dvi %%Pages: 48 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips psaugust.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.09.05:1834 %%BeginProcSet: texc.pro %! /TeXDict 250 dict def 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b(addition)f(to)j(the)f(P)m(eierls)e(condition)h(imp)s(osed)f (on)h(energies)h(of)f Fu(contours)50 1976 y FB(of)35 b(our)g(abstract)h(mo)s(del)e(w)m(e)i(assume)f(also)g(a)h(v)m(alidit)m (y)e(of)h(suitably)f(form)m(ulated)g(condition)g(\(1.13\))k(on)50 2085 y(quic)m(k)32 b(deca)m(y)h(of)f(the)h(\\cluster)f(\014eld";)g(the) h(latter)f(b)s(eing)f(a)i(concept)g(greatly)g(generalizing)e(the)h (notion)50 2195 y(of)e(an)h(external)f(\\magnetic")i(\014eld)d(acting)h (on)h(single)e(spins.)159 2376 y(It)h(is)f(instructiv)m(e)f(to)j (imagine)d(that)j(our)e(\\cluster)h(\014eld")e(is)h(actually)h(a)g (cluster)f(expansion)f(of)i(some)50 2485 y Fu(r)-5 b(estricte)g(d)37 b(ensemble)j FB(of)34 b(some)g(previous)e(mo)s(del,)h(in)g(the)g (regions)g(outside)g(con)m(tours.)51 b(Dep)s(ending)32 b(on)50 2595 y(the)f(concrete)i(mo)s(del)d(this)g(can)i(b)s(e)e(either) h(a)h(simple)d(expansion)h(of)h(high)f(temp)s(erature)h(t)m(yp)s(e,)h (of)f(some)50 2704 y(restricted)25 b(ensem)m(ble)f(of)i(\\almost)f (ground)f(con\014gurations")h(\(e.g.)40 b(in)24 b(the)i(con)m(tin)m (uous)e(spin)g(mo)s(dels\))g(or)50 2813 y(something)j(more)i (complicated)e(lik)m(e)h(the)g(situation)f(analyzed)h(in)e([BZK].)j (Or,)f(alternativ)m(ely)f(it)h(can)g(b)s(e)50 2922 y(also)g(that)i(the) f(considered)e(cluster)h(\014eld)f(is)h(actually)g(a)h(cluster)f (expansion)g(of)g(some)i(lo)m(w)e(temp)s(erature)50 3031 y(p)s(olymer)h(mo)s(del)g(of)h(\\small")g(con)m(tours.)159 3213 y(By)e(small)f(con)m(tours)h(w)m(e)g(will)e(mean)i(the)g(\\w)m (ell)f(b)s(eha)m(v)m(ed")i(ones,)f(not)h(exhibiting)c(the)j(c)m (haracteristic)50 3322 y(di\016culties)23 b(whic)m(h)i(large)h(con)m (tours)h(cause)f(in)f(nonsymmetric)g(situations)g(of)h(P)m(.S.)g (theory)-8 b(.)40 b(Finally)-8 b(,)25 b(one)50 3431 y(should)k(k)m(eep) j(in)e(mind)f(the)j(most)f(general)h(p)s(ossibilit)m(y)27 b(that)32 b(the)g(considered)e(\\starting")i(cluster)e(\014eld)50 3540 y(actually)g(app)s(eared)f(as)i(a)g(result)e(of)i(a)f Fu(combination)39 b FB(of)31 b(b)s(oth)e(suc)m(h)h(expansion)g(pro)s (cedures.)159 3722 y(So,)37 b(the)g(\\preparation")f(of)g(some)g (original)e(\(spin\))h(mo)s(del)g(to)i(an)f(abstract)h(Pirogo)m(v)f (Sinai)e(mo)s(del)50 3831 y(of)d(the)g(t)m(yp)s(e)g(\(1.11\))i(should)c (b)s(e)i(considered)f(as)h(a)g(task)h(whic)m(h)d(is)h Fu(complementary)42 b FB(to)32 b(the)f(task)g(of)h(the)50 3940 y(presen)m(ted)g(pap)s(er.)44 b(The)31 b(presen)m(ted)h(pap)s(er)f (could)g(b)s(e)g(read)h(indep)s(enden)m(tly)d(of)j(the)g(pap)s(er)e ([BZK])i(and)50 4049 y(its)e(sub)5 b(ject)30 b(is)f(in)g(principle)e (considerably)i(more)h(general.)41 b(Ho)m(w)m(ev)m(er,)32 b(the)f(result)e(of)i([BZK])f(still)e(giv)m(es)50 4158 y(a)g(most)g(relev)m(an)m(t)h(\\starting)f(mo)s(del")g(for)f(whic)m(h)g (the)h(tec)m(hnology)h(of)f(this)f(new)h(pap)s(er)e(is)h(dev)m(elop)s (ed.)40 b(It)50 4267 y(is,)28 b(at)g(the)h(momen)m(t,)g(the)g(main)e (example)g(whic)m(h)g(can)i(b)s(e)e(treated)i(b)m(y)f(our)g(new)f(v)m (ersion)h(of)g(P)m(.S.)h(theory)50 4376 y(but)g(whic)m(h)h(can)p Fu(not)40 b FB(b)m(y)30 b(treated)h(b)m(y)g(other)f(v)m(ersions)g(of)g (that)h(theory)g(app)s(earing)e(in)g(the)i(literature.)159 4558 y(While)g(w)m(e)h(exp)s(ect)h(that)f(in)f(future,)h(there)g(will)e (b)s(e)h(other)h(examples)g(\(than)g(the)g(output)g(of)g([BZK]\))50 4667 y(where)26 b(our)g(metho)s(d)g(turns)g(out)g(to)i(b)s(e)e(useful,) g(the)g(situation)g(of)h([BZK])f(still)f(giv)m(es)i(a)g(main)f(motiv)m (ation)50 4776 y(for)e(the)h(presen)m(ted)f(pap)s(er.)38 b(In)24 b(fact,)j(man)m(y)d(of)h(the)g(t)m(ypical)f(applications)f(of)h (the)h(P)m(.S.)g(theory)g(treated)g(in)50 4885 y(the)j(existing)g (literature)f(already)h(\014t)g(the)h(more)f(sp)s(ecial)f(sc)m(heme)i (\(1.1\))h(presen)m(ted)e(b)s(elo)m(w.)40 b(W)-8 b(e)29 b(con)m(v)m(ert)50 4995 y(it)k(\014rst)f(\(and)h(this)f(w)m(as)i (already)f(done)g(in)f([BZK]\))h(in)m(to)h(a)f(more)h(general)f(sc)m (heme)h(\(1.11\).)52 b(The)32 b(latter)50 5104 y(situation)i Fu(is)43 b FB(more)35 b(general)h(and)f(w)m(e)g(exp)s(ect)h(it)f(to)h (b)s(e)f(useful)f(in)g(further)g(applications)f(that)j(do)f(not)50 5213 y(allo)m(w)26 b(form)m(ulation)f(of)i(a)g(t)m(yp)s(e)f(\(1.1\).)42 b(F)-8 b(or)27 b(example,)g(man)m(y)g(mo)s(dels)e(of)h(con)m(tin)m (uous)h(spins,)e(also)i(of)f(Kac)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)j(18:34)1099 b Fz(2)p eop %%Page: 3 3 3 2 bop 50 340 a FB(t)m(yp)s(e,)31 b(seem)f(to)h(o\013er)g(nice)f (reform)m(ulations)f(of)i(the)f(t)m(yp)s(e)h(\(1.11\),)i(but)c(not)i (of)f(the)h(t)m(yp)s(e)g(\(1.1\).)159 522 y(Also)i(conceptionally)-8 b(,)34 b(the)g(sc)m(heme)g(\(1.11\))i(is)c(more)i(adequate)g(than)f (\(1.1\))j(b)s(ecause)d(the)h(presence)50 631 y(of)e(the)g(\\cluster)g (\014eld")f(in)g(\(1.23\))j(already)e(suggests)g(also)g(the)h(basic)e Fu(method)9 b FB(,)35 b(of)d(successiv)m(e)g(\\partial)50 740 y(expansions",)22 b(ho)m(w)g(to)g(treat)g(suc)m(h)f(a)h(mo)s(del.) 36 b(Th)m(us,)22 b(our)f(basic)g(metho)s(d)g(is)f(an)h(analytical)g (one,)i(and)e(with)50 849 y(suc)m(h)36 b(an)g(approac)m(h)g(the)h(P)m (.S.)f(theory)h(b)s(ecomes)g(part)f(of)g(analysis)f(rather)h(than)g (probabilit)m(y)e(theory)-8 b(.)50 958 y(In)39 b(fact)i(the)f(main)f(b) s(o)s(dy)f(of)i(the)g(theory)g(do)s(es)f(not)i(c)m(hange)g(at)f(all)f (if)g(also)g Fu(complex)53 b FB(Hamiltonians)50 1067 y(are)31 b(considered.)42 b(One)31 b(has)g(to)h(b)s(e)e(only)h(careful) f(to)i(express)f(things)f(in)g(terms)h(of)g(partition)f(functions,)50 1176 y(rather)g(than)g(with)f(probabilities.)159 1358 y(A)39 b(probabilistically)c(inclined)h(reader)j(w)m(ould)f(exp)s(ect,) k(in)c(the)h(pap)s(er)f(on)h(Pirogo)m(v{Sinai)f(theory)-8 b(,)50 1467 y(some)32 b(more)g(explicit)e(statemen)m(ts)k(concerning)d (the)h(probabilities)d(of)j(ev)m(en)m(ts)h(lik)m(e)e(that)i(a)f(spin)e (ha)m(ving)50 1576 y(in)25 b(a)i(giv)m(en)f(site)h(a)f(sp)s(eci\014ed)f (v)m(alue,)j(inside)c(a)j(\014nite)e(v)m(olume)h(and)g(under)f(a)i(giv) m(en)f(b)s(oundary)f(condition.)50 1685 y(Strictly)k(sp)s(eaking,)g (answ)m(ers)h(to)h(suc)m(h)e(problems)g(are)h(going)g(b)s(ey)m(ond)g (the)g(basic)g(tec)m(hnique)f(dev)m(elop)s(ed)50 1795 y(here)h({)h(whic)m(h)e(is)g(form)m(ulated)h(solely)g Fu(in)i(the)h(language)g(of)g(contours)39 b FB(\(of)31 b(some)g(original)d(spin)h(mo)s(del\).)159 1976 y(The)37 b(translation)f(from)g(a)i(spin)d(mo)s(del)h(to)i(a)g(con)m(tour)g(one) f(is)f(a)i(mo)s(del)e(dep)s(enden)m(t)g(task,)k(and)c(so)50 2085 y(some)e(questions)f(concerning)g(the)h(original)e(mo)s(del)h(in)f (principle)f(cannot)j(b)s(e)f(\(at)i(least)f(immediately\))50 2195 y(form)m(ulated)41 b(in)g(terms)g(of)h(abstract)h(con)m(tours.)76 b(This)40 b(is)h(not)h(to)g(sa)m(y)h(that)f(suc)m(h)g(questions)f(do)g (not)50 2304 y(deserv)m(e)29 b(a)h(more)f(detailed)f(treatmen)m(t,)j (whic)m(h)d(is)g(unfortunately)g(mostly)g(omitted)h(here)g({)h(though)f (the)50 2413 y(imp)s(ortan)m(t)g(preparatory)i(w)m(ork)f(enabling)f(to) i(solv)m(e)g(these)f(questions)g(is)f(dev)m(elop)s(ed)h(in)f(detail.) 159 2595 y(Concerning)34 b(the)h(probabilistically)c(\015a)m(v)m(ored) 36 b(questions)e(in)g(general,)i(w)m(e)g(minimize)c Fu(intentional)5 b(ly)50 2704 y FB(their)39 b(discussion)f(here.)71 b(It)41 b(w)m(ould)e(not)i(b)s(e)f(\\st)m(ylish")g(to)h(mak)m(e)h(a)f (\(false!\))71 b(impression)38 b(that)j(truly)50 2813 y(probabilistic)23 b(argumen)m(ts)j(should)f(pla)m(y)h(an)m(y)g(imp)s (ortan)m(t)g(role)g(in)f(the)h(dev)m(elopmen)m(t)h(of)g(a)f(theory)h (whic)m(h)50 2922 y(should)40 b(deal,)46 b(in)41 b(mostly)h(unc)m (hanged)g(form,)j(also)e(with)e(complex)h(Hamiltonians.)75 b(The)42 b(idea)g(that)50 3031 y(con)m(tours)d(should)f(b)s(e)g(view)m (ed)h(as)g(an)g(energetical)h(barriers)d(of)i(course)g(suggests)h(that) g(con)m(tours)f(will)50 3140 y(ha)m(v)m(e)h(a)g(small)e(probabilit)m(y) e(\(in)j(ensem)m(bles)f(with)g(\\stable")i(b)s(oundary)d(condition\),)k (and)e(surely)f(the)50 3249 y(latter)33 b(statemen)m(t)h(helps)d(to)i (visualize)e(the)h(structure)g(of)h(the)f(in\014nite)f(v)m(olume)h (Gibbs)f(state\(s\))j(of)f(the)50 3358 y(mo)s(del.)43 b(Ho)m(w)m(ev)m(er,)34 b(the)e(reader)f(will)e(realize)i(that)h(there)g (is)e(surprisingly)d(little)k(of)g(probabilit)m(y)f(theory)50 3467 y(\(and)21 b(classical)f(theory)h(of)h(Gibbs)d(states)j(in)e (particular\))g(in)g(the)h(Pirogo)m(v)h(Sinai)d(theory)-8 b(.)38 b(The)21 b(tec)m(hnology)50 3576 y(is)29 b(mostly)h(dev)m(elop)s (ed)g(in)f(\014nite)g(v)m(olumes,)i(and)e(for)h(complex)h(Hamiltonians) d(in)h(principle)e(.)1264 3951 y FC(1.)51 b(Setting)33 b(of)f(the)h(pr)n(oblem)50 4187 y FB(As)39 b(w)m(e)i(already)e(noticed) g(ab)s(o)m(v)m(e,)44 b(w)m(e)c(giv)m(e)g(t)m(w)m(o)h(p)s(ossible)d (settings.)68 b(While)39 b(the)h(second)f(one)h(\(1.11\))50 4296 y(will)35 b(b)s(e)i(the)h(actual)g(reference)g(p)s(oin)m(t)f(for)g (our)g(further)f(w)m(ork,)k(the)e(\014rst)f(one)h(comes)h(as)e(an)h (example)50 4405 y(reminding)29 b(from)j(where)g(w)m(e)h(actually)f (reac)m(hed,)i(in)d(the)i(last)f(section)h(of)f([BZK],)h(our)f(second)g (setting.)50 4515 y(T)-8 b(o)28 b(get)i(\(1.11\))g(from)e(\(1.1\),)i (one)f(has)f(to)g(tak)m(e)i(cluster)e(expansion)f(of)h(all)f(the)h(p)s (olymers)f Ft(P)41 b FB(from)27 b(\(1.1\)\):)50 4793 y FD(A)35 b(starting)f(example.)159 5071 y FB(Let)e(\003)f(b)s(e)g(a)h Ft(\027)6 b FB({dimensional)29 b(\014nite)h(torus,)i Ft(\027)g Fs(\025)27 b FB(2.)44 b(Assume)31 b(that)h(w)m(e)g(ha)m(v)m (e)h(some)f(family)e Fs(f)p Ft(q)g Fs(2)c Ft(Q)p Fs(g)50 5180 y FB(of)45 b(\\reference)g(colors")g(whose)f(elemen)m(ts)h(are)g (t)m(ypically)e(in)m(terpreted)h(as)h(spins)d Ft(q)48 b FB(for)c(whic)m(h)f(some)50 5289 y(\\original)33 b(Hamiltonian")f (\(whic)m(h)i(w)m(as)g(later)g(con)m(v)m(erted)i(to)f(a)f(form)g (\(1.1\)\))i(had)e(its)f(\\almost)i(ground)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1099 b Fz(3)p eop %%Page: 4 4 4 3 bop 50 340 a FB(v)m(alue")302 307 y Fr(1)370 340 y FB(at)28 b(the)f(constan)m(t)h(con\014guration)f Fs(f)p Ft(x)1639 354 y Fq(i)1695 340 y FB(=)e Ft(q)s(;)15 b(i)26 b Fs(2)e FB(\003)p Fs(g)p FB(.)41 b(Assume)26 b(that)i(some)f(\\con)m (tours")i(denoted)50 449 y(b)m(y)36 b(sym)m(b)s(ols)f FD(\000)h FB(and)g(also)g(some)h(\\p)s(olymers")f(denoted)g(b)m(y)g (sym)m(b)s(ols)f Ft(P)50 b FB(are)36 b(already)g(de\014ned)f(in)h(the) 50 558 y(considered)29 b(mo)s(del.)40 b(See)30 b(b)s(elo)m(w)g(for)g (more)g(details)g(on)g(these)h(ob)5 b(jects.)159 740 y(Normally)-8 b(,)40 b(if)e(w)m(e)h(relate)g(the)g(general)g(picture)f (\(1.1\))j(to)e(a)g(corresp)s(onding)e(original)g(spin)g(mo)s(del,)50 849 y(then)d(con)m(tour)h FD(\000)f FB(is)g(de\014ned)f(as)i(a)g (triple)e FD(\000)f FB(=)f(\()p Ft(G)h Fs(\021)g FD(\000)p 1988 864 63 4 v Ft(;)15 b(x)2143 863 y Fq(G)2200 845 y Fp(0)2229 849 y Ft(;)g(I)7 b FB(\))35 b(where)f FD(\000)p 2653 864 V 32 w Fs(\032)e FB(\003)j(is)e(the)i(\\supp)s(ort")f(of)50 958 y(a)g(con)m(tour)h FD(\000)p FB(,)g Ft(x)639 972 y Fq(G)696 954 y Fp(0)759 958 y FB(a)g(spin)d(con\014guration)i(on)g (some)g Ft(G)2023 925 y Fo(0)2080 958 y Fs(\032)d Ft(G)j FB(\(often)h(w)m(e)g(ha)m(v)m(e)g(simply)d Ft(G)3275 925 y Fo(0)3366 958 y Fs(\021)f Ft(G)i FB(but)50 1067 y(imagine)27 b(also)g(the)h(p)s(ossibilit)m(y)c(when)j(w)m(e)h (\\factorize)h(o)m(v)m(er)g(spins)d(inside)f FD(\000)p FB(";)k(then)f Ft(G)3036 1034 y Fo(0)3088 1067 y FB(is)f(the)h(b)s (oundary)50 1176 y(of)41 b Ft(G)p FB(\))g(and)g Ft(I)48 b FB(is)41 b(some)g(additional)f(information)f(\(often)j(empt)m(y)g (one\))g(form)m(ulating)e(p)s(ossibly)e(some)50 1285 y(other)i(restrictions)e(on)i(allo)m(w)m(ed)f(ob)5 b(jects)41 b FD(\000)e FB(resp.)68 b(some)41 b(conditions)d(on)h(their)g (compatibilit)m(y)f(with)50 1395 y(other)31 b(ob)5 b(jects.)44 b(While)29 b(p)s(olymers)h(used)g(in)g(suc)m(h)g(a)i(sc)m(heme)f (\(1.1\))i(will)c(b)s(e)h(alw)m(a)m(ys)i(\\mono)s(c)m(hromatic")50 1504 y({)i(in)e(the)h(sense)g(that)h(a)g(\014xed)f(color)g Ft(q)g Fs(2)d Ft(Q)j FB(will)d(b)s(e)j(alw)m(a)m(ys)h(assigned)f(to)h (them,)g(con)m(tours)g(will)c(b)s(e)j(in)50 1613 y(general)26 b(\\p)s(olyc)m(hromatic")f(ob)5 b(jects)27 b(assigning)d(p)s(ossibly)f Fu(di\013er)-5 b(ent)35 b FB(colors)25 b(\(usually)f(these)i(are)g (just)g(the)50 1722 y(b)s(oundary)i(v)m(alues)i(of)h(spins)d Ft(x)1116 1736 y Fq(G)1177 1722 y FB(\))j(to)g(the)g(connected)g(comp)s (onen)m(ts)f(of)h(their)e(supp)s(orts.)159 1904 y(Let)21 b(some)g(\(in\)compatibilit)m(y)e(relation)h(denoted)h(b)m(y)f Ft(c)h FB(\(resp.)38 b Ft(\023)p FB(\))21 b(b)s(et)m(w)m(een)g(all)f (these)h(ob)5 b(jects)22 b(b)s(e)e(giv)m(en)50 2013 y(and)31 b(also)g(some)h(translation)e(in)m(v)m(arian)m(t)h(\(incompatibilit)m (y)e(ma)m(y)j(cause)g(a)g(violation)e(of)i(this)e(prop)s(ert)m(y\))50 2122 y(w)m(eigh)m(ts)e Ft(w)436 2136 y Fn(\000)517 2122 y FB(and)f Ft(w)756 2136 y Fq(P)817 2122 y FB(.)40 b(The)27 b(relation)g Ft(\023)g FB(will)e(b)s(e)i(considered)g(as)h(some)g (abstract)g(relation)f(but)g(t)m(ypically)50 2231 y(the)44 b(compatibilit)m(y)d(of)j(con)m(tours)g(and)f(p)s(olymers)f(is)g (de\014ned)g(in)h(terms)g(of)h(disconnectedness)e(resp.)50 2340 y(disjoin)m(tness)f(of)j(the)g(\\supp)s(orts")e FD(\000)p 1312 2355 V 43 w FB(resp.)80 b Ft(P)p 1686 2355 71 4 v 56 w FB(of)44 b FD(\000)f FB(resp.)80 b Ft(P)13 b FB(.)80 b(F)-8 b(or)44 b(con)m(tours)g(one)g(also)f(requires)50 2449 y(\\matc)m(hing)c(compatibilit)m(y")e(of)i(the)g(colours)f (determined)f(b)m(y)h(them.)66 b(In)37 b(particular,)j(in)d([BZK])i (the)50 2558 y(supp)s(orts)24 b(of)h(con)m(tours)i(are)f(required)d(to) k(b)s(e)e(disconnected)g(while)f(compatibilit)m(y)g(of)h(a)h(p)s (olymer)e Ft(P)39 b FB(with)50 2667 y(an)30 b(other)h(ob)5 b(ject)31 b Ft(P)760 2634 y Fo(0)816 2667 y FB(or)f FD(\000)g FB(means)h(that)g(the)f(supp)s(ort)f(of)h Ft(P)44 b FB(do)s(es)30 b(not)g(in)m(tersect)h(its)f(supp)s(ort.)159 2849 y(In)38 b(man)m(y)g(previous)f(situations)h(of)g(Pirogo)m(v)h(-)g(Sinai)d (theory)j(there)g(w)m(as)g(no)f(need)g(to)i(distinguish)50 2958 y(mono)s(c)m(hromatic)26 b(con)m(tours)g(and)e(p)s(olymers;)i (then)f(p)s(olymers)f(disapp)s(ear)f(from)i(the)h(setting)g(\(1.1\).)41 b(This)50 3067 y(is)32 b(actually)g(the)i(situation)e(whic)m(h)f(w)m (as)j(called)e(an)h(\\abstract)h(P)m(.S.)g(mo)s(del")e(b)s(efore)h(b)m (y)g(the)g(author)g(of)50 3176 y(this)g(article.)53 b(Ho)m(w)m(ev)m (er,)38 b(in)33 b([BZK])i(and)e(in)h(other)g(applications)f(to)s(o,)k (mono)s(c)m(hromatic)d(con)m(tours)h(are)50 3285 y(ob)5 b(jects)27 b(of)g(rather)f(a)h(di\013eren)m(t)f(nature)g(\(\\connected) i(sets)f(&)f(con\014guration)g(on)g(them\))h(than)f(p)s(olymers)50 3395 y(\(whic)m(h)20 b(are)i(considered)e(as)h(graphs)g(there;)k(their) 20 b(\\gas")j(represen)m(ting)d(the)i(restricted)f(ensem)m(ble)f (outside)50 3504 y(con)m(tours\).)159 3685 y(The)30 b(partition)f (function)g(on)h(a)h(torus)f(\003)g(will)e(b)s(e)i(then)g(de\014ned)f (as)696 3975 y Ft(Z)7 b FB(\(\003\))26 b(=)1280 3889 y Fm(X)1020 4088 y Fo(f)p Fn(\000)1106 4097 y Fw(1)1142 4088 y Fq(;:::)n Fn(\000)1277 4097 y Fv(m)1338 4088 y Fq(;P)1406 4097 y Fw(1)1441 4088 y Fq(;:::)n(;P)1595 4098 y Fv(k)1635 4088 y Fo(g)1744 3861 y Fq(m)1718 3889 y Fm(Y)1718 4082 y Fq(i)p Fr(=1)1851 3975 y Ft(w)1916 3989 y Fn(\000)1965 3998 y Fv(i)2026 3889 y Fm(Y)2013 4084 y Fq(q)r Fo(2)p Fq(Q)2170 3975 y FB(exp\()p Fs(\000)p Ft(e)2457 3989 y Fq(q)2513 3975 y Fs(j)p FB(\003)2601 3989 y Fq(q)2641 3975 y Fs(j)p FB(\))2790 3861 y Fq(k)2752 3889 y Fm(Y)2748 4082 y Fq(j)t Fr(=1)2888 3975 y Ft(w)2953 3989 y Fq(P)2999 3998 y Fv(i)3523 3975 y FB(\(1)p Ft(:)p FB(1\))50 4281 y(where)32 b(the)g(sum)g(is)f(o)m(v)m(er)j(all)d (compatible)h(collections)f(of)i(con)m(tours)g(and)f(p)s(olymers.)45 b(Suc)m(h)31 b(collections)50 4390 y(will)d(b)s(e)h(simply)f(called)i Fu(con\014gur)-5 b(ations)40 b FB(of)30 b(our)g(mo)s(del,)f(sometimes)i (denoted)f(also)h(b)m(y)f(sym)m(b)s(ol)f Ft(x)3502 4404 y Fr(\003)3556 4390 y FB(.)159 4572 y(The)38 b(set)g(\003)566 4586 y Fq(q)645 4572 y FB(in)e(\(1.1\))k(denotes)f(the)f(collection)g (of)g(all)f(\\)p Ft(q)s FB({correct")k(p)s(oin)m(ts)c(of)h(the)g (con\014guration)50 4681 y Fs(f)p FD(\000)158 4695 y Fq(i)188 4681 y Fs(g)29 b([)f(f)p Ft(P)454 4695 y Fq(j)493 4681 y Fs(g)43 b FB(i.e.)77 b(of)42 b(all)g(p)s(oin)m(ts)f(outside)h (of)h(con)m(tours)g(ha)m(ving)f(a)g(prescrib)s(ed)e(color)j Ft(q)s FB(.)77 b(Then)41 b(w)m(e)50 4790 y(ha)m(v)m(e)h(the)g(decomp)s (osition)e(\003)k(=)f Fs([)1330 4804 y Fq(q)r Fo(2)p Fq(Q)1476 4790 y FB(\003)1539 4804 y Fq(q)1607 4790 y Fs([)27 b([)1756 4804 y Fq(i)1786 4790 y FD(\000)p 1786 4805 63 4 v 23 x Fq(i)1921 4790 y FB(where)41 b FD(\000)p 2195 4805 V 41 w FB(denotes)h(the)f(supp)s(ort)f(of)h FD(\000)p FB(.)74 b(This)40 b(is)50 4899 y(nonam)m(biguous)34 b(for)i(nonempt)m(y)g(systems)g Fs(f)p FD(\000)1680 4913 y Fq(i)1710 4899 y Fs(g)p FB(only;)j(an)c(imp)s(ortan)m(t)g(additional) f(remark)i(is)f(that)h(w)m(e)50 5009 y(consider)f(no)h(single)e(\\empt) m(y")j(con\014guration)f Fs(f)p FD(\000)1839 5023 y Fq(i)1869 5009 y Fs(g)h FB(but,)g(instead,)f(a)h(whole)e(collection)g(\(mark)m (ed)i(b)m(y)p 50 5111 546 4 v 159 5172 a Fl(1)197 5198 y Fz(\\Almost)24 b(ground)f(state")h(should)h(b)r(e)f(understo)r(o)r(d) h(as)f(a)g(\\ground)f(state)i(for)e(a)h(suitably)h(\(sligh)n(tly\))g(c) n(hanged)f(Hamil-)50 5289 y(tonian".)34 b(Ho)n(w)n(ev)n(er,)26 b(this)g(is)h(mean)n(t)e(just)h(for)f(an)g(in)n(tuition,)i(w)n(e)f (will)h(not)e(ha)n(v)n(e)g(to)h(mak)n(e)f(precise)i(suc)n(h)g(a)e (notion.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)k(18:34)1099 b Fz(4)p eop %%Page: 5 5 5 4 bop 50 340 a Ft(q)28 b Fs(2)d Ft(Q)p FB(\))i(of)g(con\014gurations) f(in)m(terpreted)h(as)g(\\colored)h(emptiness")e Ft(x)2465 296 y Fq(q)2465 366 y Fr(\003)2544 340 y FB(=)f Fs(f)p Ft(P)2756 296 y Fq(q)2743 365 y(j)2797 340 y Fs(g)i FB(\(emptiness)g (\014lled)e(only)50 472 y(b)m(y)30 b(p)s(olymers\))f(i.e.)41 b(con\014gurations)29 b(with)g Fs(f)p FD(\000)1659 486 y Fq(i)1690 472 y Fs(g)d FB(=)f Fs(;)30 b FB(and)g(with)f(\003)2379 439 y Fq(q)2445 472 y Fs(\021)c FB(\003,)30 b(\003)2722 439 y Fq(q)2757 412 y Fp(0)2812 472 y FB(=)25 b Fs(;)31 b FB(for)f(other)h Ft(q)3405 439 y Fo(0)3456 472 y Fs(6)p FB(=)25 b Ft(q)s FB(.)50 726 y Fu(Note.)46 b FB(It)32 b(is)f(instructiv)m(e)g(to)i(write)e(\014rst)g(the)h(\\ground")g(terms) g Ft(e)2347 693 y Fo(\000)p Fq(e)2439 702 y Fv(q)2475 693 y Fo(j)p Fr(\003)2547 702 y Fv(q)2582 693 y Fo(j)2640 726 y FB(and)f(only)h(then)f(the)h(p)s(olymer)50 835 y(activities)g Ft(w)504 849 y Fq(P)598 835 y FB(to)h(emphasize)g(that)g (while)e(con)m(tours)i FD(\000)g FB(ma)m(y)g(c)m(hange)h(their)e (external)h(color)g({)g(w)m(e)g(will)50 944 y(often)44 b(write)f FD(\000)48 b FB(=)g FD(\000)832 911 y Fq(q)916 944 y FB(if)43 b(the)h(latter)h(is)e(equal)g(to)i Ft(q)i FB({)d(to)h(another)f(color,)k(sa)m(y)j(~)-51 b Ft(q)r FB(,)48 b(in)43 b(the)h(in)m(terior)50 1053 y(comp)s(onen)m(ts)c (\(usually)e(denoted)i(b)m(y)g(in)m(t)15 b FD(\000)41 b FB(=)g(in)m(t)1859 1067 y Fr(~)-42 b Fq(q)1909 1053 y FD(\000)p FB(\))40 b(of)g(the)g(complemen)m(t)g(of)h(the)f(supp)s (ort)e FD(\000)p 3469 1068 63 4 v FB(.)69 b(In)50 1162 y(con)m(trary)38 b(to)g(con)m(tours,)i(p)s(olymers)c(will)e(b)s(e)j (alw)m(a)m(ys)h(\\mono)s(c)m(hromatic")h(ob)5 b(jects)38 b(living)d(in)h(some)i(\003)3670 1176 y Fq(q)50 1271 y FB(and)g(ha)m(ving)g(a)h(constan)m(t)g(color)g Ft(q)s FB(.)65 b(Their)36 b(in)m(terpretation)i(is)g(is)f(fact)j(less)d(\\ph)m (ysical")h(than)h(that)g(of)50 1380 y(con)m(tours,)31 b(and)f(it)g(will)e(b)s(e)h(also)i(necessary)g(to)g(assume)f(that)h (their)e(activities)h Ft(w)2882 1394 y Fq(P)2973 1380 y FB(are)h(not)g(necessarily)50 1489 y(p)s(ositiv)m(e,)36 b(while)e(for)h(con)m(tours)i(their)e(activities)g(will)e(b)s(e)i(alw)m (a)m(ys)2359 1456 y Fr(2)2436 1489 y FB(of)h(the)g(t)m(yp)s(e)g Ft(w)2982 1503 y Fn(\000)3070 1489 y FB(=)e Ft(e)3217 1456 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))3477 1489 y FB(where)50 1598 y Ft(E)5 b FB(\()p FD(\000)p FB(\))31 b(is)e(a)i(p)s(ositiv)m(e)f(\(and)g(large\))h Fu(ener)-5 b(gy)38 b FB(of)31 b(the)f(con)m(tour)h FD(\000)p FB(.)159 1780 y(The)j(notion)h(of)g(an)g(in)m(terior)f(of)h(a)g(con)m (tour)h(on)f(a)g(\(usually)e Fu(\014nite)7 b FB(\))36 b(torus)e(clearly)h(has)f(to)i(b)s(e)f(more)50 1889 y(sp)s(eci\014ed.) 58 b(Let)37 b(us)f(call)f(a)i(con)m(tour)h FD(\000)e FB(a)h(nonastronomical)f(one)h(if)e(its)h(size)h(is)f(considerably)e (smaller)50 1998 y(than)c(the)h(size)f(of)g(our)g(\\univ)m(erse")g (\003)h(sa)m(y)1410 2236 y(diam)14 b FD(\000)p 1622 2251 V 56 w Ft(<)55 b FB(1)p Ft(=)p FB(2)47 b(diam)14 b(\003)p Ft(:)1174 b FB(\(1)p Ft(:)p FB(2\))50 2474 y(Let)28 b(us)g(mak)m(e)h(a) g(tec)m(hnically)e(con)m(v)m(enien)m(t)i(agreemen)m(t)h(that)f Fu(only)37 b FB(con\014gurations)28 b(with)e Fu(no)35 b FB(astronomi-)50 2584 y(cally)26 b(large)g(con)m(tours)h(will)d(b)s (e)i(considered)f(in)g(what)h(follo)m(ws.)39 b(In)25 b(particular,)h(partition)f(function)g Ft(Z)7 b FB(\(\003\))50 2693 y(will)28 b(b)s(e)j(alw)m(a)m(ys)h(tak)m(en)g(only)f(o)m(v)m(er)h (them.)44 b(F)-8 b(ew)32 b(simple)d(commen)m(ts)j(on)g(ho)m(w)f(to)h (adapt)g(our)e(study)h(also)50 2802 y(to)41 b(the)f(\\full")f (partition)g(function)g(tak)m(en)i(also)g(o)m(v)m(er)g (con\014gurations)f(\(of)h(con)m(tours)f(and)g(p)s(olymers\))50 2911 y(with)30 b(no)i(\\nonastronomical")g(constrain)m(t)f(\(suc)m(h)h (a)g(constrain)m(t)g(turns)e(out)i(to)h(b)s(e)e(of)g(extremely)h(small) 50 3020 y(relev)m(ance)f(in)e(large)i(tori\))f(will)e(b)s(e)h(giv)m(en) i(at)g(the)f(end)g(of)h(the)f(pap)s(er.)159 3202 y(More)35 b(generally)-8 b(,)35 b(w)m(e)f(will)e(consider)h(also)h(partitition)e (function)h(with)g(some)i(limitation)d Fs(C)39 b FB(,)c(whic)m(h)50 3311 y(has)d(to)g(b)s(e)g(ful\014lled)c(for)k(all)f(considered)g (con\014gurations)g(for)h(whic)m(h)f(w)m(e)h(sum)f(in)g(\(1.1\).)47 b(T)m(ypically)31 b(the)50 3420 y(constrain)m(t)42 b Fs(C)k FB(will)39 b(b)s(e)i(just)h(a)g(\014xed)f(collection)g(of)h(con) m(tours)g(or)g(\(less)f(naturally\))f(p)s(olymers.)73 b(The)50 3529 y(partition)29 b(function)g Ft(Z)847 3543 y Fo(C)924 3529 y FB(will)f(b)s(e)i(then)g(de\014ned)f(as)706 3852 y Ft(Z)768 3866 y Fo(C)815 3852 y FB(\(\003\))d(=)1331 3765 y Fm(X)1070 3965 y Fo(f)p Fn(\000)1156 3974 y Fw(1)1192 3965 y Fq(;:::)o Fn(\000)1328 3974 y Fv(m)1388 3965 y Fq(;P)1456 3974 y Fw(1)1492 3965 y Fq(;:::)n(;P)1646 3975 y Fv(k)1686 3965 y Fo(g)1765 3738 y Fq(m)1739 3765 y Fm(Y)1738 3959 y Fq(i)p Fr(=1)1871 3852 y Ft(w)1936 3866 y Fn(\000)1985 3875 y Fv(i)2046 3765 y Fm(Y)2034 3960 y Fq(q)r Fo(2)p Fq(Q)2191 3852 y FB(exp\()p Fs(\000)p Ft(e)2478 3866 y Fq(q)2533 3852 y Fs(j)p FB(\003)2621 3866 y Fq(q)2662 3852 y Fs(j)p FB(\))2780 3738 y Fq(k)2742 3765 y Fm(Y)2737 3959 y Fq(j)t Fr(=1)2878 3852 y Ft(w)2943 3866 y Fq(P)2989 3875 y Fv(j)3028 3852 y Ft(:)470 b FB(\(1)p Ft(:)p FB(3\))50 4191 y(Here,)30 b(the)g(sum)e(is)h(tak)m(en)h(o)m(v)m 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(ould)e(b)s(e)f(incon)m(v)m(enien)m(t.)p 50 5202 546 4 v 159 5263 a Fl(2)197 5289 y Fz(A)n(t)c(least)i(in)f(this)h(pap)r (er,)f(where)h(w)n(e)e(do)h(not)f(dev)n(elop)h(a)f(systematic)i (treatmen)n(t)f(of)f Fy(complex)34 b Fz(Hamiltonians.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1099 b Fz(5)p eop %%Page: 6 6 6 5 bop 50 340 a FD(The)34 b(P)m(eierls)i(condition)g(for)f(con)m (tours)h(and)f(p)s(olymers.)159 574 y FB(The)d(w)m(eigh)m(ts)i Ft(w)740 588 y Fn(\000)827 574 y FB(and)e Ft(w)1071 588 y Fq(P)1165 574 y FB(will)e(b)s(e)j(assumed)f(to)i(satisfy)e(the)h (follo)m(wing)f(b)s(ounds:)44 b(W)-8 b(eigh)m(ts)34 b Ft(w)3550 588 y Fn(\000)3637 574 y FB(of)50 683 y(con)m(tours)27 b(are)g(giv)m(en)g(as)g Ft(e)945 650 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))1197 683 y FB(where)f Ft(E)5 b FB(\()p FD(\000)p FB(\))28 b(is)e(the)h(\\energy")h(of)f FD(\000)g FB(satisfying)e(the)i(P)m(eierls)f(condition)1627 893 y Ft(E)5 b FB(\()p FD(\000)p FB(\))26 b Fs(\025)f Ft(\034)g Fs(j)p FD(\000)p 2044 908 63 4 v Fs(j)1391 b FB(\(1)p Ft(:)p FB(5\))50 1102 y(where)34 b Ft(\034)42 b Fs(\035)32 b FB(1.)54 b(The)34 b(constan)m(t)i Ft(\034)44 b FB(has)35 b(the)f(order)g(of)h(in)m(v)m(erse)g(temp)s(erature)f(in)f (t)m(ypical)h(applications.)50 1211 y(Belo)m(w)d(w)m(e)g(will)c (rewrite)j(this)f(b)s(ound)f(in)i(another,)g(more)h(suitable)e(\\exp)s (onen)m(tial")h(form)g(\(1.14\).)159 1393 y(Analogous)i(condition)f (will)f(b)s(e)h(assumed)h(for)g(the)g(p)s(olymers,)f(but)h(in)e(suc)m (h)i(a)h(case)g(it)f(will)d(b)s(e)j(ev)m(en)50 1502 y(more)41 b(advisable)e(to)i(write)f(it)g(in)f(the)i(exp)s(onen)m(tial)f(form)g (b)s(ecause)h Ft(w)2578 1516 y Fq(P)2679 1502 y FB(ma)m(y)g(b)s(e)f (negativ)m(e,)45 b(ev)m(en)c(in)50 1611 y(mo)s(dels)28 b(with)g(real)h(Hamiltonian)e Ft(H)36 b FB(\(in)29 b(con)m(trary)h(to)g (the)f(w)m(eigh)m(ts)h Ft(w)2527 1625 y Fn(\000)2580 1611 y FB(\).)41 b(In)29 b(fact,)h(our)f(main)f(example)50 1720 y([BZK])d(requires)f(p)s(olymers)f(to)j(b)s(e)e(geometrical)i(ob)5 b(jects)26 b(of)f(either)g(a)g(v)m(ery)h(simple)d(geometric)j (structure)50 1829 y(\(\\thin")35 b(graphs,)g(t)m(ypically)f(with)g (only)g(a)i(few)e(b)s(onds,)h(called)f(\\constellations")h(in)f ([BZK]\))i(or)f(m)m(uc)m(h)50 1939 y(more)26 b(\\dense")g(ob)5 b(jects,)28 b(with)d(someho)m(w)h(more)g(complicated)g(geometric)h (nature)f(\(\\galaxies"\))h(similar)50 2048 y(to)k(con)m(tours.)41 b(It)31 b(is)e(b)s(est)h(to)h(treat)h(them)e(in)f(a)i(uni\014ed)d(w)m (a)m(y)j(as)g(graphs)e(with)g(w)m(eigh)m(ts)i(giv)m(en)g(as)976 2270 y Fs(j)p Ft(w)1066 2284 y Fq(P)1127 2270 y Fs(j)25 b(\024)1284 2184 y Fm(Y)1273 2379 y Fq(b)p Fo(2)p Fq(P)1426 2270 y Fs(j)p Ft(w)1516 2284 y Fq(b)1552 2270 y Fs(j)213 b FB(where)2219 2184 y Fm(X)2232 2379 y Fq(b)p Fo(3)p Fq(i)2366 2270 y Fs(j)p Ft(w)2456 2284 y Fq(b)2493 2270 y Fs(j)25 b(\024)g Ft(e)2681 2233 y Fo(\000)t Fr(~)-40 b Fq(\034)3523 2270 y FB(\(1)p Ft(:)p FB(6\))50 2566 y(with)33 b(another)39 b(~)-50 b Ft(\034)41 b Fs(\035)31 b FB(1.)52 b(The)34 b(constan)m(ts)h Ft(\034)44 b FB(and)38 b(~)-50 b Ft(\034)44 b FB(in)m(tro)s(duced)33 b(here)g(do)h(not)h (corresp)s(ond)d(literally)g(to)50 2675 y(the)d(constan)m(ts)h(denoted) f(b)m(y)g(the)g(same)h(letters)f(in)e([BZK].)j(Later)f(w)m(e)h(will)c (rewrite)i(the)h(b)s(ound)e(\(1.6\))k(in)50 2784 y(another)f(w)m(a)m(y) -8 b(,)32 b(based)e(on)h(the)f(follo)m(wing)f(standard)h(tree)h (summation)e(b)s(ound)f(\(see)k(e.g.)41 b([BZK]\):)50 3018 y FD(Lemma.)50 b Fu(L)-5 b(et)33 b(some)g(tr)-5 b(e)g(e)34 b(weights)f Ft(w)1415 3032 y Fq(t)1480 3018 y Fu(be)f(given)g(satisfying)h(the)g(fol)5 b(lowing)34 b(estimates:)802 3241 y Fs(j)p Ft(w)892 3255 y Fq(t)924 3241 y Fs(j)58 b(\024)g Ft(e)1178 3203 y Fo(\000)p Fq(\020)t Fo(j)p Fq(t)p 1294 3215 28 4 v Fo(j)1395 3154 y Fm(Y)1399 3350 y Fq(b)p Fo(2)p Fq(t)1544 3241 y FB(\()p Ft(e)1621 3203 y Fq(w)1674 3213 y Fv(b)1732 3241 y Fs(\000)20 b FB(1\))159 b Fu(wher)-5 b(e)2492 3154 y Fm(X)2504 3350 y Fq(b)p Fo(3)p Fq(i)2638 3241 y Fs(j)p Ft(w)2728 3255 y Fq(b)2765 3241 y Fs(j)25 b(\024)g Ft(\020)572 b FB(\(1)p Ft(:)p FB(7\))50 3542 y Fu(and)31 b(wher)-5 b(e)30 b(the)h(pr)-5 b(oduct)31 b(is)f(over)g(al)5 b(l)30 b(bonds)h Ft(b)f Fu(of)g(the)g(tr)-5 b(e)g(e)30 b Ft(t)p Fu(.)41 b(Then)30 b(the)g(sum)2789 3473 y Fm(P)2885 3569 y Fq(t)p Fr(:)p Fq(r)r Fr(\()p Fq(t)p Fr(\)=)p Fq(i)3154 3542 y Fs(j)p Ft(w)3244 3556 y Fq(t)3277 3542 y Fs(j)g Fu(over)g(tr)-5 b(e)g(es)50 3662 y(having)33 b(the)g(\\r)-5 b(oot")34 b Ft(i)f Fu(\(and)g(at)h(le)-5 b(ast)33 b(one)g(bond\))h(can)f(be)f (bounde)-5 b(d)34 b(as)f(fol)5 b(lows)1411 3803 y Fm(X)1352 4002 y Fq(t)p Fr(:)p Fq(r)r Fr(\()p Fq(t)p Fr(\)=)p Fq(i)1616 3890 y Fs(j)p Ft(w)1706 3904 y Fq(t)1739 3890 y Fs(j)25 b(\024)g FB(1)c Fs(\000)f Ft(e)2084 3852 y Fo(\000)p Fq(\020)2240 3890 y Fs(\024)25 b Ft(\020)7 b(:)1115 b FB(\(1)p Ft(:)p FB(8\))159 4278 y(No)m(w,)38 b(with)c(the)h(help)f(of)i (suc)m(h)f(a)h(lemma)f(one)h(can)f(rewrite)g(\(see)i(again)e([BZK])h (for)f(more)g(details\))50 4387 y(the)30 b(b)s(ound)f(\(1.6\))j(in)d (the)h(follo)m(wing)f(w)m(a)m(y)-8 b(,)32 b(and)e(this)f(will)f(b)s(e)i (our)g(actual)g(starting)h(assumption)d(later:)50 4621 y FD(Reform)m(ulated)34 b(assumption)h(\(1.6\))f(on)h(p)s(olymer)g(w)m (eigh)m(ts)g Ft(w)2503 4635 y Fq(P)2563 4621 y FD(.)159 4856 y FB(With)e(the)g(help)f(of)i(the)g(ab)s(o)m(v)m(e)g(lemma,)g (taking)f Ft(\020)k FB(=)30 b Ft(e)2108 4823 y Fo(\000)t Fr(~)-40 b Fq(\034)2244 4856 y FB(one)34 b(can)g(ob)m(viously)e (rewrite)g(the)i(b)s(ound)50 4965 y(\(1.6\))e(in)d(the)i(follo)m(wing)d (w)m(a)m(y)-8 b(,)32 b(with)e(suitable)e(small)h(\(see)j(b)s(elo)m(w)d (for)h(more)h(details\))37 b(~)-52 b Ft(p)688 5106 y Fm(X)650 5301 y Fq(P)10 b Fr(:)p Fq(P)p 727 5313 57 4 v 10 w Fo(3)p Fq(i)903 5192 y Fs(j)p Ft(w)993 5206 y Fq(P)1054 5192 y Fs(j)30 b Ft(e)1151 5155 y Fq(p)p Fo(j)p Fq(P)p 1210 5167 V 11 w Fo(j)1348 5192 y Fs(\024)63 b FB(~)-52 b Ft(p)55 b Fs(\034)g FB(1)h Fs(\034)g Ft(p)181 b FB(where)h Ft(p)25 b FB(=)30 b(~)-50 b Ft(\034)30 b Fs(\000)20 b Ft(e)2982 5155 y Fo(\000)t Fr(~)-40 b Fq(\034)3085 5192 y Ft(:)413 b FB(\(1)p Ft(:)p FB(9\))50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1099 b Fz(6)p eop %%Page: 7 7 7 6 bop 50 340 a Fu(Note.)46 b FB(The)39 b(constan)m(t)i Ft(p)f FB(in)e(b)s(ounds)g(of)h(the)h(t)m(yp)s(e)g(\(1.9\))i(is)c (often)j(in)d(the)i(literature)f([ES],[KP])g(etc.)50 449 y(denoted)j(b)m(y)g(sym)m(b)s(ol)f Ft(a)p FB(.)75 b(Ho)m(w)m(ev)m(er,)47 b(w)m(e)c(c)m(ho)s(ose)g(the)f(letter)g Ft(p)g FB(for)g(p)s(olymers)e Ft(P)55 b FB(and)42 b(analogously)-8 b(,)50 558 y(b)s(elo)m(w,)28 b Ft(g)j FB(for)d(con)m(tours)g FD(\000)p FB(.)40 b(Later)28 b(w)m(e)g(rewrite)g(also)f(\(1.5\))j(in)d (a)h(similar)d(w)m(a)m(y)k(as)f(\(1.9\))i(but)d(in)f(that)j(case)50 667 y(w)m(e)i(will)c(ha)m(v)m(e)32 b(to)f(strengthen)f(the)h (requiremen)m(ts)e(on)i(the)f(corresp)s(onding)e(constan)m(t)k Ft(g)s FB(.)159 849 y(So,)47 b(this)c(is)f(our)h(basic)h(example.)80 b(The)43 b(imp)s(ortan)m(t)g(idea)g(no)m(w)h(is)e(to)j(apply)d Fu(cluster)j(expansion)50 958 y FB(b)s(oth)32 b(to)h(p)s(olymers)e(and) h(\(later\))i(also)e(to)i(con)m(tours)f(of)g(suc)m(h)f(a)h(mo)s(del.)46 b(There)33 b(is,)f(ho)m(w)m(ev)m(er,)j(a)e(crucial)50 1067 y(di\013erence)41 b(b)s(et)m(w)m(een)h(the)g(\\mono)s(c)m (hromatic")h(p)s(olymers)d(on)h(one)h(side)f({)h(whose)f(expansion)g (should)50 1176 y(cause)31 b(no)f(problems)e(under)h(assumptions)f(of)i (the)h(t)m(yp)s(e)f(\(1.9\))i({)e(and)g(on)g(the)g(other)h(side)e(the)h (con)m(tours)50 1285 y(ha)m(ving)35 b(in)g(general)g(\\p)s(olyc)m (hromatic")h(b)s(oundaries)e(inducing)f(di\013eren)m(t)i(colors)h Ft(q)i FB(in)d(the)h(rest)g(of)g(the)50 1395 y(space.)k(The)25 b(latter)h(is)e(the)i(feature)g(whic)m(h)f(is)f(v)m(ery)i(c)m (haracteristic)h(for)e(the)h(applications)e(of)i(P)m(.S.)g(theory)50 1504 y(and)k(distinguishes)c(them)31 b(from)f(the)g(standard)g(p)s (olymer)f(lo)m(w)h(temp)s(erature)g(expansions.)159 1685 y(Nev)m(ertheless,)43 b(the)c(\\zeroth")j(step)d(of)h(our)f(expansion)f (namely)h(the)h(expansion)e(of)i(all)e(p)s(olymers)50 1795 y(causes)31 b(no)f(problems)f(and)h(it)g(is)f(really)h(advisable)f (to)i(do)g(it.)40 b(It)31 b(en)m(visages)g(our)f(basic)g(future)g (strategy)-8 b(,)50 1904 y(whic)m(h)26 b(can)i(b)s(e)e(c)m (haracterized)j(as)e(an)h(attempt)g(to)g(expand,)g(in)e(a)i(sequence)f (of)h(steps,)g(also)f(all)f Fu(contours)50 2013 y FB(of)37 b(the)f(mo)s(del.)58 b(The)36 b(p)s(olyc)m(hromatic)g(c)m(haracter)i (of)f(con)m(tours)g(will)d(then)i(require)f(an)i(application)e(of)50 2122 y(additional)29 b(to)s(ols)j(dev)m(elop)s(ed)e(to)j(deal)e(with)f (suc)m(h)h(a)h(situation,)f(ha)m(ving)g(no)g(analogy)h(in)e(the)i (theory)g(of)50 2231 y(p)s(olymer)k(mo)s(dels.)62 b(These)38 b(to)s(ols,)h(namely)f(the)g(basic)f(notion)g(of)h(a)h(con)m(tour)f (functional,)h(are)f(rather)50 2340 y(c)m(haracteristic)31 b(for)f(the)h(metho)s(d)e(called)h(the)h(P)m(.S.)f(theory)-8 b(.)159 2522 y(So,)22 b(imagine)e(that)h(while)d(\014xing)h(a)h(family) f Fs(f)p FD(\000)1726 2536 y Fq(i)1757 2522 y Fs(g)h FB(of)h(con)m(tours)g(of)f(a)h(con\014guration)e(in)g(\(1.1\))j(w)m(e)f (expand,)50 2631 y(b)m(y)k(standard)g(cluster)g(expansion)g(\(Ma)m(y)m (er)j(expansion\))d(metho)s(d)g(\(discussed)f(later)i(in)e(some)i (detail,)g(but)50 2740 y(see)43 b(also)f([BZE])h(and)f([NOZ]\))h(the)f (abstract)i(p)s(olymer)d(mo)s(del)g(\\living")g(outside)h(of)g(the)h (collection)50 2849 y Fs(f)p FD(\000)158 2863 y Fq(i)188 2849 y Fs(g)p FB(.)e(Then)27 b(w)m(e)h(can)g(reform)m(ulate)g(\(1.1\))i (in)d(the)h(follo)m(wing)f(w)m(a)m(y)-8 b(,)30 b(and)d(this)g(will)e(b) s(e)j(an)g(actual)g(starting)50 2958 y(p)s(oin)m(t)j(of)i(our)f(pap)s (er;)h(the)g(situation)f(\(1.1\))i(b)s(eing)e(no)m(w)g(considered)g (just)g(as)h(an)f(example)h(leading,)f(b)m(y)50 3067 y(expansion,)d(to)i(the)g(follo)m(wing)e(more)h(general)h(setting:)50 3300 y FD(Our)j(new,)h(reform)m(ulated)f(\\zeroth)h(lev)m(el")g(mo)s (del.)159 3533 y FB(Assume)f(that)g(some)h(ob)5 b(jects)35 b(called)f(con)m(tours)g FD(\000)g FB(are)h(de\014ned)e(ha)m(ving)h (the)g(prop)s(erties)f(describ)s(ed)50 3642 y(in)28 b(\(1.1\).)42 b(Assume)29 b(that)h(their)e(w)m(eigh)m(ts)i Ft(w)1543 3656 y Fn(\000)1626 3642 y FB(satisfy)f(the)h(b)s(ound)d(\(1.5\).)42 b(Usually)-8 b(,)29 b(the)g(w)m(eigh)m(ts)h Ft(w)3485 3656 y Fn(\000)3568 3642 y FB(will)50 3755 y(b)s(e)g(translation)f(in)m (v)m(arian)m(t)h(quan)m(tities)g(giv)m(en)g(as)h Ft(w)1852 3769 y Fn(\000)1931 3755 y FB(=)25 b Ft(e)2069 3722 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))2293 3755 y FB(;)31 b(then)f(the)h(quan)m(tit)m(y)1212 3971 y Ft(H)7 b FB(\()p Fs(f)p FD(\000)1438 3985 y Fq(i)1469 3971 y Fs(g)p FB(\))26 b Fs(\021)1671 3884 y Fm(X)1724 4078 y Fq(i)1817 3971 y Ft(E)5 b FB(\()p FD(\000)1987 3985 y Fq(i)2018 3971 y FB(\))21 b(+)2165 3884 y Fm(X)2213 4075 y Fq(q)2311 3971 y Ft(e)2353 3985 y Fq(q)2394 3971 y Fs(j)p FB(\003)2482 3933 y Fq(q)2522 3971 y Fs(j)930 b FB(\(1)p Ft(:)p FB(10\))50 4268 y(could)34 b(b)s(e)g(called)g(the)h (\\energy")h(of)f(the)g(\(factorized\))h(\\con\014guration")f Fs(f)p FD(\000)2749 4282 y Fq(i)2780 4268 y Fs(g)p FB(.)54 b(No)m(w)35 b(assume)g(further)50 4377 y(that)30 b(some)h(\\cluster)f (\014eld")f(\(see)i(b)s(elo)m(w)e(for)h(more)g(details\))f Ft(w)2248 4391 y Fq(T)2336 4377 y FB(is)g(giv)m(en,)h(with)f (translation)g(in)m(v)m(arian)m(t)50 4486 y(\(for)h Ft(T)43 b FB(not)31 b(touc)m(hing)f(\003)916 4453 y Fq(c)953 4486 y FB(\))h(v)m(alues.)41 b(Consider)28 b(the)j(partition)e (function)g(on)h(a)h(torus)f(\003)736 4762 y Ft(Z)7 b FB(\(\003\))25 b(=)1171 4676 y Fm(X)1059 4875 y Fo(f)p Fn(\000)1145 4884 y Fw(1)1181 4875 y Fq(;:::)o Fn(\000)1317 4884 y Fv(m)1377 4875 y Fo(g)1456 4648 y Fq(m)1431 4676 y Fm(Y)1430 4869 y Fq(i)p Fr(=1)1563 4762 y Ft(w)1628 4776 y Fn(\000)1677 4785 y Fv(i)1725 4676 y Fm(Y)1766 4866 y Fq(q)1857 4762 y FB(exp\()p Fs(\000)p 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(v)m(e)g(only)d(the)i(expansion)f(of)h(the)g(full)d(partition)h (function)h Ft(Z)2999 3667 y Fr(\003)3082 3653 y FB(on)g(a)h(torus.)40 b(One)50 3762 y(w)m(ould)24 b(lik)m(e)i(also)g(to)g(expand)f(some)i(sp) s(ecial,)e Fs(C)31 b FB({)c(conditioned)d(partition)h(functions)f(in)m (tro)s(duced)g(ab)s(o)m(v)m(e.)159 3944 y(T)-8 b(ak)m(e)28 b(a)f(simplest)d(and)i(most)h(basic)f(probabilistic)d(question)i(\(in)h (the)g(setting)h(w)m(e)g(ha)m(v)m(e\):)40 b(\\What)28 b(is)d(a)50 4053 y(probabilit)m(y)c(that)j(a)f(con)m(tour)h FD(\000)g FB(app)s(ears)e(in)g(the)i(Gibbs)d(ensem)m(ble)i(with)f(the)i (partition)e(function)g Ft(Z)7 b FB(\(\003\)")50 4162 y(?)38 b(Or)22 b(more)i(generally:)36 b(\\What)25 b(is)d(a)i (probabilit)m(y)c(of)k(the)f(same)h(ev)m(en)m(t)h(in)d(the)h(ensem)m (ble)g(conditioned)f(b)m(y)50 4272 y(the)32 b(assumption)d(of)j(a)g (required)e(presence)h(of)h(some)g(con)m(tour)g(subsystem)f Fs(C)5 b FB("?)44 b(Then,)31 b(b)m(y)g(c)m(ho)s(osing)h(\003)50 4381 y(and)27 b Fs(C)32 b FB(prop)s(erly)25 b(w)m(e)j(can)g(mo)s(del)e (an)m(y)i(b)s(oundary)d(condition)h(on)h(an)m(y)h(\014nite)e(subset)h (of)g Fi(Z)3160 4348 y Fq(\027)3228 4381 y FB(and)g(then)g(w)m(e)50 4490 y(can)c(go)h(to)g(the)g(thermo)s(dynamic)d(limit.)36 b(T)-8 b(o)24 b(conclude,)g(our)f(restriction)f(to)i(tori,)h(with)d(a)h (suitable)f(c)m(hoice)50 4599 y(of)33 b(a)h(conditioning)d(form)m (ulated)h(in)g(terms)h(of)h(a)f(presence)g(\(absence\))i(of)e(some)h (con)m(tour)f(subsystem)g Fs(C)50 4708 y FB(is)28 b(not)i(at)g(all)e (restrictiv)m(e,)i(and)e(all)g(limit)g(Gibbs)f(states)k(on)e Fi(Z)2213 4675 y Fq(\027)2283 4708 y FB(can)h(b)s(e)e(attained)i(as)f (limits)e(of)j(suitably)50 4817 y(conditioned)f(ensem)m(bles)g(on)i (tori.)159 4999 y(Fixing)d(a)j(restrictiv)m(e)e(condition)g Fs(C)34 b FB(has)c(precisely)e(the)i(aim)f(to)i(allo)m(w)e(a)h(prop)s (er)f(form)m(ulation)f(of)i(ex-)50 5108 y(pansion)e(results)g(also)h (for)g(the)h(probabilities)c(of)k(corresp)s(ondingly)c(conditioned)i (ev)m(en)m(ts.)42 b(Conditioned)p 50 5202 546 4 v 159 5263 a Fl(4)197 5289 y Fz(The)26 b(quan)n(tit)n(y)f Fh(f)710 5263 y Fd(1)781 5289 y Fz(,)h(our)g(main)f(output)g(of)g(the)h (expansions)h(constructed)g(b)r(elo)n(w,)f(will)h(b)r(e)f(equal)g(to)f Fh(h)p Fz(.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)k(18:34)1099 b Fz(9)p eop %%Page: 10 10 10 9 bop 50 340 a FB(ev)m(en)m(ts)31 b(are)g(then)f(in)m(terp)s(ereted) f(as)i(ev)m(en)m(ts)g(in)e(the)i(ensem)m(ble)e(with)g(the)i(partition)e (function)g(\(1.12\).)42 b(In)50 449 y(practise,)26 b(the)g(limitation) e Fs(C)31 b FB(will)23 b(often)j(include)d(a)j(requiremen)m(t)f(of)h(a) g(\\\014xed)g(color)f Ft(q)k FB(at)d(in\014nit)m(y";)g(then)50 558 y(the)k(most)h(in)m(teresting)f(case)h(is)f(that)h(of)f Ft(q)j FB(b)s(eing)c(the)i(external)f(color)h(of)f FD(\000)p FB(.)159 740 y(Conditional)e(probabilit)m(y)h(in)g(the)i(ensem)m(ble)f (with)g(partition)f(function)h(\(1.12\))j(\(with)d(conditioning)50 849 y(b)s(eing)42 b(form)m(ulated)i(in)f(terms)h(of)g(a)g(limitation)e Fs(C)49 b FB(whic)m(h)43 b(has)h(to)h(b)s(e)e(ful\014lled)e(for)j(an)m (y)g(considered)50 958 y(con\014guration\))28 b(of)g(the)g(ev)m(en)m(t) i(\\con)m(tour)f FD(\000)f FB(o)s(ccurs,)h(as)f(a)g(subsystem)g(of)g (the)g(family)f(of)h(all)f(con)m(tours)i(of)50 1067 y(a)i(giv)m(en)f (con\014guration")g(is)g(then)g(giv)m(en)g(b)m(y)g(form)m(ulas)g(lik)m (e)f(\(see)j(more)e(precisely)f(later)i(in)e(\(6.6\)\))641 1316 y Ft(P)712 1278 y Fo(C)699 1338 y Fr(\003)759 1316 y FB(\()p FD(\000)p FB(\))d(=)1025 1254 y Ft(Z)1087 1268 y Fo(C)s([)p Fn(\000)1232 1254 y FB(\(\003\))p 1025 1295 342 4 v 1074 1378 a Ft(Z)1136 1392 y Fo(C)1183 1378 y FB(\(\003\))1463 1316 y(=)86 b(exp)o(\()p Fs(\000)p Ft(E)5 b FB(\()p FD(\000)p FB(\))22 b(+)2353 1229 y Fm(X)2182 1429 y Fq(T)10 b(\023)p Fr(\()p Fn(\000)p Fo([)p Fr(in)n(t)i Fn(\000)p Fo([C)s Fr(\))2657 1316 y FB(\()p Ft(w)2757 1330 y Fq(T)2835 1316 y Fs(\000)38 b FB(~)-64 b Ft(w)2990 1330 y Fq(T)3048 1316 y FB(\)\))359 b(\(1)p Ft(:)p FB(17\))50 1613 y(where)28 b(in)f(the)h(n)m(umerator)g(w)m(e)h(add)f(the)g (limitation)e(\\)p FD(\000)j FB(m)m(ust)f(app)s(ear")g(to)h(the)g (previous)d(limitation)g Fs(C)5 b FB(.)50 1722 y(The)23 b(second)h(equation)f(suggests)h(that)h(later)e(w)m(e)h(will)d(tak)m(e) 26 b(the)e(expansion)e(of)i(b)s(oth)f(the)h(n)m(umerator)f(and)50 1831 y(the)36 b(denominator)g(of)g(the)h(fraction.)58 b(Then)35 b(the)i(\(small)e(and)g(quic)m(kly)h(deca)m(ying!\))59 b(expansion)35 b(terms)50 1940 y Ft(w)115 1954 y Fq(T)204 1940 y FB(resp.)64 b(~)-64 b Ft(w)502 1954 y Fq(T)591 1940 y FB(\(in)31 b(the)h(expressions)f(of)h(partition)f(functions)f (corresp)s(o)s(ding)g(to)j(situations)e(when)g FD(\000)h FB(is)50 2049 y(presen)m(t)38 b(resp.)62 b(absen)m(t\))38 b(in)f(eac)m(h)i(of)f(the)g(t)m(w)m(o)h(\(complicated!\))63 b(series)37 b(are)h(the)g(same)g(and)f(therefore)50 2158 y(cancel)31 b(if)e(the)h(cluster)g Ft(T)43 b FB(is)30 b(outside)f FD(\000)p 1357 2173 63 4 v 20 w Fs([)20 b FB(in)m(t)15 b FD(\000)p FB(.)159 2340 y(F)-8 b(or)43 b Ft(T)58 b Fs(\032)45 b FB(in)m(t)15 b FD(\000)42 b FB(the)h(situation)e(is)h Fu(mor)-5 b(e)45 b(delicate)7 b FB(,)46 b(a)d(cancellation)f(\(roughly\))f(or)i(ev)m(en)g(v)m(olume) 50 2449 y(energy)28 b(increase)g(tak)m(es)i(place)e(only)f(for)h(sp)s (ecial,)f(\\stable")i(external)f(colors)g Ft(q)j FB(of)d FD(\000)p FB(.)40 b(\(This)27 b(will)e(b)s(e)j(the)50 2558 y(situation)j(where)g(extremal)h(Gibbs)f(state)i(of)f(a)g(t)m(yp)s (e)g Ft(q)j FB(exists.\))45 b(The)32 b(w)m(eigh)m(ts)g Ft(w)2932 2572 y Fq(T)3021 2558 y FB(resp.)64 b(~)-64 b Ft(w)3319 2572 y Fq(T)3408 2558 y FB(for)32 b Ft(T)13 b(\023)p FD(\000)50 2667 y FB(do)32 b(not)g(cancel)g(but)f(their)g(sum) g(is)g(unimp)s(ortan)m(t,)g(m)m(uc)m(h)h(smaller)f(than)g(the)i (leading)d(term)i Ft(E)5 b FB(\()p FD(\000)p FB(\).)47 b(So,)50 2777 y(only)23 b(for)i(a)f(stable)g Ft(q)k FB(w)m(e)d(really)e (get)i(a)g(useful)e(form)m(ula,)i(with)e(only)h(those)g(\(quic)m(kly)g 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b(the)f(answ)m(ers)g(ha)m (v)m(e)i(to)f(b)s(e)f(translated)g(from)50 3663 y(the)f(language)g(of)g (con)m(tours)g(bac)m(k)g(to)h(the)e(spin)f(mo)s(del.)39 b(Often,)29 b(this)f(is)f(a)i(straigh)m(tforw)m(ard)g(task)g(lik)m(e)f (in)50 3772 y(the)33 b(ab)s(o)m(v)m(e)h(example)e(but)g(it)g(can)h(b)s (e)f(a)h(more)g(tedious)f(one)h(for)g(more)f(complicated)h(ev)m(en)m (ts.)49 b(An)m(yw)m(a)m(y)-8 b(,)50 3881 y(this)29 b(is)h(\\\(spin\))f (mo)s(del)g(dep)s(enden)m(t")h(w)m(ork,)h(not)f(directly)g(related)g (to)h(the)g(aim)e(of)i(this)e(article.)159 4063 y(F)-8 b(or)37 b(example,)g(the)f(b)s(eha)m(viour)f(of)h(blo)s(c)m(kspins)d (can)j(b)s(e)g(ev)m(en)m(tually)-8 b(,)38 b(of)e(course,)h(also)f (describ)s(ed)e(in)50 4172 y(the)j(language)g(of)g(con)m(tours)g(of)g (the)g(spin)e(mo)s(del)g(but)i(the)f(expansion)g(form)m(ulas)g(for)g (suc)m(h)h(a)g(con)m(tour)50 4281 y(description)e(will)g(lo)s(ok)i (more)h(cum)m(b)s(ersome)f(than)g(for)g(the)h(ev)m(en)m(ts)h(describ)s 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m(w)m(o)h(suc)m(h)f(mo)s(dels,)f(and)g(let)g Fs(A)3004 5303 y Fr(1)3077 5289 y FB(resp.)47 b Fs(A)3385 5303 y Fr(2)3458 5289 y FB(b)s(e)32 b(the)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)d(18:34)1080 b Fz(10)p eop %%Page: 11 11 11 10 bop 50 340 a FB(families)32 b(of)j(a)m(v)m(ailable)f(con)m(tours) h(used)e(in)g(the)i(\014rst)e(resp.)52 b(second)34 b(mo)s(del.)52 b(In)33 b(fact)i(w)m(e)g(will)d(consider)50 449 y(only)25 b(the)h(case)h(when)e Fs(A)891 463 y Fr(1)957 449 y Fs(\032)g(A)1126 463 y Fr(2)1166 449 y FB(.)39 b(Let)27 b Fs(G)k(\032)25 b(A)1643 463 y Fr(1)1694 449 y Fs(\\)11 b(A)1839 463 y Fr(2)1880 449 y FB(.)39 b(Sa)m(y)26 b(that)h(these)g(t)m(w)m(o)g(mo)s (dels)e(are)h Fs(G)31 b FB({)c(equiv)m(alen)m(t)50 558 y(if)33 b(for)h(all)g(limitations)e Fs(C)39 b FB(expressed)34 b(in)f(terms)h(of)h(con)m(tours)g(from)e Fs(G)40 b FB(the)35 b(partition)e(functions)g Ft(Z)3529 572 y Fo(C)3576 558 y FB(\(\003\))50 667 y(tak)m(en)e(o)m(v)m(er)g(all)d(con\014gurations)h (\(=)h(compatible)f(systems)g(of)h(con)m(tours)g(from)f Fs(A)2894 681 y Fr(1)2964 667 y FB(resp.)40 b Fs(A)3265 681 y Fr(2)3335 667 y FB(satisfying)50 776 y(the)30 b(limitation)f Fs(C)5 b FB(\))30 b(are)h(the)g(same)g(for)f Fu(al)5 b(l)33 b(tori)40 b FB(\003,)31 b(in)e(b)s(oth)h(mo)s(dels.)159 958 y(W)-8 b(e)24 b(alw)m(a)m(ys)g(consider)f(here,)i(at)f(the)f (\014rst)g(instance,)i(the)e(case)i Fs(G)30 b FB(=)25 b Fs(;)p FB(.)39 b(Ho)m(w)m(ev)m(er,)27 b(for)c(a)h(more)g(detailed)50 1067 y(analysis)g(of)i(Gibbs)e(probabilities)e(one)j(has)h(to)g(c)m(ho) s(ose)h(a)f(suitably)d(ric)m(her)i(family)f Fs(G)5 b FB(.)39 b(If)25 b(the)h(t)m(w)m(o)h(mo)s(dels)50 1176 y(are)k Fs(G)k FB(equiv)m(alen)m(t)c(for)f Fs(G)h FB(=)25 b Fs(A)1119 1190 y Fr(1)1179 1176 y Fs(\\)20 b(A)1333 1190 y Fr(2)1404 1176 y FB(then)30 b(w)m(e)g(will)e(simply)g(call)i (them)g Fu(e)-5 b(quivalent)9 b FB(.)159 1358 y(Needless)32 b(to)h(sa)m(y)-8 b(,)33 b(t)m(w)m(o)h(equiv)m(alen)m(t)e(\(ev)m(en)h Fs(;)g FB({)f(equiv)m(alen)m(t!\))46 b(mo)s(dels)31 b(ha)m(v)m(e)i(the) g Fu(same)h(fr)-5 b(e)g(e)35 b(ener)-5 b(gy)8 b FB(.)50 1467 y(Tw)m(o)29 b(equiv)m(alen)m(t)h(mo)s(dels)e(moreo)m(v)m(er)j(ha)m (v)m(e)g(the)e(same)h(probabilities)c(of)k(ev)m(en)m(ts)h(whic)m(h)d (are)i(expressible)50 1576 y(in)f(the)h(language)h(of)g(con)m(tours)g (from)f Fs(A)1450 1590 y Fr(1)1510 1576 y Fs(\\)20 b(A)1664 1590 y Fr(2)1704 1576 y FB(.)50 1796 y FD(A)35 b(generalized)g(approac) m(h)h(to)f(the)f(mo)s(del)g(\(1.11\))h(with)f(\(1.13\))h(and)g (\(1.14\).)159 2016 y FB(It)j(is)f(in)g(the)h(v)m(ery)h(nature)e(of)i (the)f(con)m(tour)h(approac)m(h)f(\(=)g(the)g(P)m(.S.)h(theory\))f (that)h(an)m(y)f(question)50 2125 y(in)c(the)h(mo)s(del)g(\(1.11\))i({) f(or)f(in)f(some)i(original)e(spin)f(mo)s(del,)j(whic)m(h)e(w)m(as)h (later)h(rephrased)e(b)m(y)h(\(1.11\),)50 2235 y(p)s(ossibly)40 b(through)j(\(1.1\){)j(is)c(form)m(ulated)h(in)g(the)g(language)h(of)g (con)m(tours.)80 b(T)-8 b(ranslation,)46 b(in)m(to)e(the)50 2344 y(language)28 b(of)f(con)m(tours,)i(is)d(then)i(needed)f(for)g(an) m(y)h(question)e(form)m(ulated)h(directly)f(in)g(terms)i(of)f(spins)f (of)50 2453 y(the)k(original)d(mo)s(del.)40 b(P)m(ossible)28 b(complexit)m(y)i(of)f(a)h(translation)f(of)h(suc)m(h)f(a)h(question)f (a\013ects,)j(of)d(course,)50 2562 y(also)j(the)h(complexit)m(y)f(of)g (the)h(translation,)f(bac)m(k)h(to)g(the)f(language)h(of)f(spins,)f(of) i(the)f(answ)m(er.)47 b(Ev)m(en)m(ts)50 2671 y(in)37 b(the)j(original)d(spin)g(mo)s(del)h(whic)m(h)f(are)j(not)f(straigh)m (tforw)m(ardly)f(expressed)g(in)g(terms)h(of)g(con)m(tours)50 2780 y(m)m(ust)f(b)s(e)g(then)h(dealt)f(with)f(the)i(help)e(of)i(the)g (conditional)e(exp)s(ectations)i(w.r.)65 b(to)39 b(v)m(arious)f(p)s (ossible)50 2889 y(con\014gurations)29 b(of)i(con)m(tours)g(allo)m (wing)e(a)i(presence)f(of)h(the)f(giv)m(en)h(ev)m(en)m(t.)159 3071 y(W)-8 b(e)33 b(already)f(argued)g(that)h(it)e(is)h(advisable)e (to)j(minimize)d(the)i(usage)h(of)f(probabilistic)d(concepts)k(in)50 3180 y(all)c(these)i(in)m(v)m(estigations,)f(giving)f(emphasis)g(to)i (the)f(expansion)f(of)i(partition)e(functions.)39 b(This)28 b(allo)m(ws)50 3289 y(also)i(a)h(later)f(generalization)g(to)h(complex) g(Hamiltonians.)159 3471 y(When)f(computing)f(the)h(partition)e (function,)h(the)i(k)m(ey)f(p)s(oin)m(t)f(is)g(to)i(k)m(eep)f(in)f (mind)f(the)i(structure)g(of)50 3580 y(considered)25 b(con\014gurations)h Fs(f)p FD(\000)1182 3594 y Fq(i)1212 3580 y Fs(g)p FB(.)40 b(An)m(y)27 b(suc)m(h)f(con\014guration)g (induces)f(a)h(coloring)g(of)h(the)f(rest)h(\()p Fs([)3514 3594 y Fq(i)3544 3580 y FD(\000)3607 3594 y Fq(i)p 3544 3608 94 4 v 3637 3580 a FB(\))3672 3547 y Fq(c)50 3689 y FB(of)38 b(the)f(torus)h(\003,)i(and)d(the)g(in)m(terior)g(comp)s (onen)m(ts)h(of)f(\()p FD(\000)p 2022 3704 63 4 v 1 w FB(\))2121 3656 y Fq(c)2196 3689 y FB(ha)m(v)m(e)h(in)f(general)g(a)h (color)g(di\013eren)m(t)f(from)50 3798 y(that)j(of)g(the)g(exterior)g (of)g FD(\000)p FB(.)69 b(This)37 b(creates,)44 b(together)d(with)e (the)h(fact)h(that)f(generally)f(w)m(e)h(ha)m(v)m(e)h Fu(no)50 3907 y(symmetry)g FB(in)30 b(the)i(mo)s(del,)f(all)f(the)i (di\016culties)d(\(and)i(the)h(in)m(terest\))g(of)f(Pirogo)m(v)h({)g (Sinai)d(theory)-8 b(.)45 b(The)50 4016 y(compatibilit)m(y)36 b(of)i(con)m(tours)g(and)f(clusters)g(should)e(b)s(e)i(view)m(ed)h(as)g (rather)f(an)h(abstract)g(relation,)h(its)50 4125 y(details)29 b(b)s(eing)g(mostly)h(irrelev)m(an)m(t)g(b)s(elo)m(w.)159 4307 y(It)45 b(will)e(b)s(e)h(imp)s(ortan)m(t)h(to)h(sligh)m(tly)d (generalize)i(the)h(mo)s(del)e(\(1.11\),)51 b(k)m(eeping)45 b(a)h(careful)e(ey)m(e)j(on)50 4416 y(the)40 b(v)m(alidit)m(y)f(of)i (the)f(P)m(eierls)f(conditions)g(\(1.14\))j(and)e(\(1.13\),)45 b(with)39 b(suitably)f(large)j Ft(g)i FB(and)d Ft(c)p FB(.)70 b(The)50 4525 y(generalization)30 b(will)e(ha)m(v)m(e)j(t)m(w)m (o)h(main)d(asp)s(ects:)159 4707 y(1\))i(a)g(broader)f(approac)m(h)g (to)h(considered)f(collections)f(of)i(allo)m(w)m(ed)f(con)m(tours)h FD(\000)p FB(,)g(and)e(their)h(energies)159 4889 y(2\))h(a)g(broader)f (approac)m(h)g(to)h(considered)f(collections)f(of)i(clusters)f Ft(T)13 b FB(,)30 b(and)g(their)f(w)m(eigh)m(ts)159 5071 y(While)45 b(1\))j(could)e(ha)m(v)m(e)i(b)s(een)e(tric)m(ky)g(in)g (some)h(other)g(situations)f(\(for)h(other)g(mo)s(dels,)i(notably)50 5180 y(for)37 b(those)i(with)d(con)m(tin)m(uous)h(spins\),)h(here)g(it) f(will)e(ha)m(v)m(e)k(alw)m(a)m(ys)g(a)f(v)m(ery)g(straigh)m(tforw)m (ard)f(meaning.)50 5289 y(Namely)-8 b(,)28 b(w)m(e)g(ha)m(v)m(e)h(to)f (acquire)f(only)f(the)i(idea)e(that)i(also)g(appropriately)d(selected)j Fu(subsets)35 b FB(of)27 b(the)h(orig-)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)h(18:34)1080 b Fz(11)p eop %%Page: 12 12 12 11 bop 50 340 a FB(inal)28 b(family)h Fs(A)g FB(of)h(all)f(con)m (tours)i(should)d(b)s(e)i(considered,)f(and)g(also)h(that)h(some)f (collections)g(of)g(con)m(tour)50 449 y(should)e(b)s(e)i(\\glued")g (together)i(later)e(i.e.)41 b(considered)29 b(as)i(a)g(single)e(en)m (tit)m(y)-8 b(.)159 631 y(On)33 b(the)h(other)g(hand,)g(a)g(careful)f (generalization)h(of)g(a)g(notion)f(2\))i(of)f(a)g(cluster)f(will)f(b)s (e)h(absolutely)50 740 y(necessary)j(for)g(the)g(approac)m(h)g(to)h (Pirogo)m(v)f({)h(Sinai)d(theory)i(w)m(e)g(are)h(dev)m(eloping)e(here.) 57 b(This)34 b(will)f(b)s(e)50 849 y(c)m(haracterized)c(b)m(y)g(the)g (desire)e(to)i Fu(iter)-5 b(atively)32 b(expand)h(the)e(smal)5 b(lest)32 b(contours)37 b FB(resp.)j(con)m(tour)29 b(systems)50 958 y(of)34 b(the)f(mo)s(del)g(and)g(to)h(con)m(v)m(ert)h(them)f(in)m (to)f(some)h Fu(new)5 b(ly)36 b(forme)-5 b(d)45 b FB(\(more)34 b(complicated\))f(cluster)g(\014eld)50 1067 y Ft(w)115 1081 y Fq(T)212 1067 y FB({)41 b(suc)m(h)f(that)h(a)f(con)m(trol)h(o)m (v)m(er)h(the)e(deca)m(y)h(of)g(\(b)s(oth)f(the)g(old)g(and)f(new\))h (cluster)g(w)m(eigh)m(ts)g Ft(w)3576 1081 y Fq(t)3649 1067 y FB(is)50 1176 y(retained,)30 b(ev)m(en)h(when)e(using)g(an)h (in\014nite)f(n)m(um)m(b)s(er)g(of)h(suc)m(h)g(expansion)f(steps.)159 1358 y(What)23 b(it)g(means)f(\\to)j(expand")d(will)e(b)s(e)i (discussed)f(b)s(elo)m(w)h(in)g(detail,)i(in)d(particular)g(see)j(our)e (Theorem)50 1467 y(4)37 b(and)f(its)g(equation)h(\(3.17\))i(whic)m(h)d (is)g(really)f(the)i(basic)g(tec)m(hnical)f(step)h(of)g(our)g(approac)m (h.)60 b(A)m(t)38 b(this)50 1576 y(p)s(oin)m(t)d(it)h(su\016ces)g(to)h (men)m(tion)f(a)g(\\principle)e(of)i(corresp)s(ondence")g(stating)g (that)h(our)f(newly)f(de\014ned)50 1685 y(op)s(eration)g(of)i (expansion)d(of)j(con)m(tours)f(should)e(b)s(e)i Fu(identical)46 b FB(with)35 b(the)h(w)m(ell)f(understo)s(o)s(d)f(op)s(eration)50 1795 y(of)i(an)f(expansion)g(of)g Fu(polymers)45 b FB(in)35 b(the)g(cases)i(when)d(i\))i(expanded)e(con)m(tours)j(are)f(\\mono)s(c) m(hromatic")50 1904 y(\(suc)m(h)c(that)h(neigh)m(b)s(oring)d(v)m(alues) i(of)h(spins)d Ft(q)35 b FB(are)e(ev)m(erywhere)g(the)f(same)h(along)f (the)h(b)s(oundary)d(of)j FD(\000)p 3612 1919 63 4 v -1 w FB(\))50 2013 y(resp.)38 b(more)27 b(generally)e(in)g(the)h(cases) i(where)d(ii\))g(there)i(is)e(a)h Fu(symmetry)36 b FB(b)s(et)m(w)m(een) 27 b(the)f(di\013eren)m(t)g(\\colors")50 2122 y(of)k(the)h(mo)s(del.)39 b(The)30 b(main)f(aim)h(of)g(the)g(whole)g(construction)g(is)f(to)i (construct)f(a)h(cluster)f(expansion)f(of)50 2231 y(the)h(partition)f (function)g Ft(Z)7 b FB(\(\003\))31 b(and)f(to)h(deriv)m(e)f(an)g (expansion)f(form)m(ula)h(for)g(free)h(energy)f(from)g(it.)50 2460 y Fu(Note)36 b(on)h(an)f(abstr)-5 b(act)38 b(P)n(.S.)d(model)j (with)f(tr)-5 b(e)g(e)36 b(like)g(contours.)48 b FB(The)34 b(clusters)f Ft(T)47 b FB(used)33 b(in)g(\(1.23\))k(ha)m(v)m(e)50 2569 y(essen)m(tially)31 b(a)h(tree)g(structure.)44 b(In)31 b(fact)i(one)f(could)f(treat)h(not)g(only)f(these)h(ob)5 b(jects)33 b(but)e(also)g Fu(contours)50 2678 y FB(more)c (systematically)f(as)h(graphs)f(resp.)38 b(trees)28 b(carrying)e(some)h (additional)d(information,)i(in)f(particular)50 2787 y(ab)s(out)40 b(the)g(\\color")h(from)f Ft(Q)g FB(attac)m(hed)i(to)f (the)f Fu(conne)-5 b(cte)g(d)43 b(components)50 b FB(of)40 b(the)h(complemen)m(t)f(of)g(a)50 2896 y(supp)s(ort)32 b(of)j(a)f(con)m(tour.)53 b(The)34 b(prescriptions)e(on)i(the)g(colors) h(of)f(the)h(con)m(tours)f(should)f(b)s(e)g(\\matc)m(hing)50 3005 y(compatible",)25 b(in)d(the)i(complemen)m(t)f(of)h(the)g(union)d (of)j(supp)s(orts)e(of)h(these)h(ob)5 b(jects.)39 b(One)23 b(could)g(reform)m(u-)50 3114 y(late)h(the)h(abstract)g(Pirogo)m(v)f({) h(Sinai)d(theory)i(for)g(suc)m(h)g(a)g(more)g(general)h(setting,)g (with)e(con)m(tours)i(treated)50 3223 y(as)33 b(graphs)g(\(trees\))i (inducing)30 b(a)k(\\`connected)h(coloring")e(of)h(the)f(rest)h(of)f (the)h(space.)50 b(W)-8 b(e)34 b(exp)s(ect)g(that)50 3332 y(for)c(some)h(more)g(complicated)g(situations,)f(lik)m(e)g(for)g (mo)s(dels)g(with)f(in\014nite)g(range)i(\\tail")g(in)m(teractions,)50 3441 y(this)e(could)h(actually)f(the)i(most)g(adequate)g(approac)m(h.) 50 3670 y FD(Auxiliary)k(ordering)h(on)f(the)f(collection)j Fs(A)d FD(of)h(all)g(con)m(tours)h(of)f(the)f(mo)s(del.)159 3899 y FB(The)22 b(idea)h(of)g(the)h(forthcoming)e(expansions)g(will)e (b)s(e)j(to)g(expand)g(the)g(mo)s(del)f(successiv)m(ely)-8 b(,)25 b(according)50 4008 y(to)c(the)g(size)g(of)g(con)m(tours)g(or)g (con)m(tours)h(systems)e(whic)m(h)g(ha)m(v)m(e)i(to)g(b)s(e)e (expanded.)37 b(It)20 b(is)g(natural)g(to)i(consider)50 4117 y(the)30 b(follo)m(wing)f(partial)g(ordering)g Fs(\036)i FB(on)f(the)g(family)f(of)i(all)e(con)m(tours)i Fs(A)p FB(,)f(giv)m(en)g(as)1364 4301 y FD(\000)25 b Fs(\036)g FD(\000)1611 4263 y Fo(0)1722 4301 y Fs(\()-15 b(\))86 b FD(\000)p 1975 4316 V 25 w Fs(\032)25 b FB(in)m(t)14 b FD(\000)2345 4263 y Fo(0)2371 4301 y Ft(:)1081 b FB(\(1)p Ft(:)p FB(18\))50 4485 y(Recall)26 b(that)h(our)f(\\univ)m(erse")g(is)g (a)h(\(p)s(ossibly)d(v)m(ery)i(large\))h(torus)f(\003,)i(and)e(w)m(e)h (restrict)f(our)g(in)m(terest)h(only)50 4594 y(to)g(ensem)m(bles)e(of)h (con\014gurations)f(\(and)h(w)m(e)h(tak)m(e)g(the)g(partition)d (function)h Ft(Z)7 b FB(\(\003\))26 b(o)m(v)m(er)i(those)e(ensem)m (bles)50 4703 y(only\))k(ha)m(ving)g(con)m(tours)h(of)f (\\nonastronomical")h(size)f(only)-8 b(,)30 b(suc)m(h)g(that)1188 4887 y FD(\000)c Fs(2)e(A)h FB(=)-15 b Fs(\))25 b FB(diam)14 b FD(\000)p 1844 4902 V 56 w Ft(<)55 b FB(1)p Ft(=)p FB(2)47 b(diam)14 b(\003)p Ft(:)906 b FB(\(1)p Ft(:)p FB(19\))50 5071 y(This)32 b(assumption)h(on)h(the)h(the)f(diameter)g (of)h FD(\000)f FB(clari\014es)f(the)i(notion)e(of)i(in)m(terior)e (comp)s(onen)m(ts)h(of)h FD(\000)p 3610 5086 V -40 x Fq(c)50 5180 y FB(on)30 b(\003.)42 b(It)31 b(is)e(rather)i(elemen)m (tary)g(to)g(sho)m(w)g(that)g(with)e(P)m(eierls)h(condition,)g(con)m (tribution)f(of)i(con\014gura-)50 5289 y(tions)f(con)m(taining)g(some)h (\\astronomical")g(con)m(tour)g(is)f(extremely)g(small,)g(when)f(diam) 14 b(\003)26 b Fs(!)g(1)p FB(,)k(in)g(the)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)f(18:34)1080 b Fz(12)p eop %%Page: 13 13 13 12 bop 50 340 a FB(ensem)m(ble)31 b(with)f(the)i(full)d(partition)i (function)f Ft(Z)1764 354 y Fq(f)7 b(ull)1901 340 y FB(\(\003\).)45 b(Their)30 b(relativ)m(e)i(con)m(tribution)e(is)g(actually)i(of)50 452 y(of)e(order)g Ft(e)433 419 y Fo(\000)p Fq(C)t(\034)20 b Fr(diam)11 b(\003)824 452 y FB(.)40 b(W)-8 b(e)32 b(omit)e(a)h(more)f (detailed)g(discussion)d(here.)159 634 y(Notice)k(that)g(the)g (ordering)e(ab)s(o)m(v)m(e)i(is)f(in)m(v)m(arian)m(t)g(w.r.)40 b(to)31 b(all)f(translations)f(in)g(\003.)50 864 y FD(\\Expandable")34 b(con)m(tours)i(in)f(a)g(mo)s(del)f(\(1.23\).)47 b(Extremal)33 b(con)m(tour)j(systems.)159 1094 y FB(Denote)f(b)m(y)e Fs(A)e FB(=)f Fs([)869 1108 y Fq(q)909 1094 y Fs(A)982 1061 y Fq(q)1021 1094 y FB(,)35 b(more)f(precisely)e Fs(A)p FB(\(\003\))f(=)f Fs([)2086 1108 y Fq(q)2126 1094 y Fs(A)2199 1061 y Fq(q)2239 1094 y FB(\(\003\))k(the)g(system)f(of)h (all)f(con)m(tours)h(of)g(the)50 1204 y(mo)s(del)h(\(1.23\))k(in)c(a)h (giv)m(en)h(torus)f(\003.)58 b(In)36 b(the)g(follo)m(wing)f(w)m(e)i (will)d(select,)k(for)e(an)m(y)h Ft(q)h Fs(2)d Ft(Q)p FB(,)j(a)e(suitable)50 1313 y(subsystem)26 b Fs(H)563 1280 y Fq(q)628 1313 y Fs(\032)f(A)797 1280 y Fq(q)863 1313 y FB(\(of)j(\\expandable,)f(small")f(con)m(tours;)j(see)f(b)s(elo) m(w\).)39 b(The)26 b(subsystem)g Fs(H)3381 1280 y Fq(q)3448 1313 y FB(will)e(b)s(e)50 1422 y(nonempt)m(y)30 b(for)g Fu(at)j(le)-5 b(ast)34 b(one)k Ft(q)28 b Fs(2)d Ft(Q)p FB(.)40 b(W)-8 b(e)32 b(will)27 b(assume)k(that)g(its)e(c)m(hoice)j(is) d(alw)m(a)m(ys)i(made)f(suc)m(h)g(that)1086 1620 y FD(\000)1149 1582 y Fo(0)1200 1620 y Fs(2)25 b(H)1364 1582 y Fq(q)1403 1620 y Ft(;)46 b FD(\000)25 b Fs(\036)g FD(\000)1721 1582 y Fo(0)1863 1620 y FB(=)-15 b Fs(\))25 b FD(\000)35 b Ft(=)-55 b Fs(2)55 b([)2300 1634 y Fq(q)2335 1615 y Fp(0)2360 1634 y Fo(2)p Fq(Q)2531 1620 y Fs(H)2609 1582 y Fq(q)2644 1555 y Fp(0)3477 1620 y 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Fq(q)2426 2401 y Fs(n)c(H)2570 2363 y Fq(q)3477 2401 y FB(\(1)p Ft(:)p FB(21\))50 2591 y(could)32 b(b)s(e)g(p)s(ossibly)f(placed)h(compatibly)g(in)m(to)h(the) h(in)m(terior)e(of)h Fs(D)2381 2605 y Fq(ext)2489 2591 y FB(.)49 b(Extremal)32 b(con)m(tour)i(systems)f Fs(D)50 2700 y FB(will)28 b(b)s(e)h(later)i(treated)g(as)g(smallest)e Fu(expandable)39 b FB(ob)5 b(jects)31 b(of)g(the)f(mo)s(del.)159 2882 y(T)-8 b(o)27 b(ha)m(v)m(e)g(a)g(concrete)h(idea)e(of)g(the)h(c)m (hoice)g(of)g Fs(H)1805 2849 y Fq(q)1871 2882 y FB(imagine)e(the)i (collection)f(of)g(all)g(con)m(tours)h FD(\000)3406 2849 y Fq(q)3472 2882 y FB(whose)50 2991 y(diameter)c(is)g(the)i(smallest)e (p)s(ossible,)g(in)g(the)h(family)e Fs(A)1957 2958 y Fq(q)1997 2991 y FB(.)38 b(Assume)24 b(for)g(simplicit)m(y)d(that)j Ft(w)3187 3005 y Fq(T)3270 2991 y Fs(\021)h FB(0)f(in)f(suc)m(h)50 3100 y(a)31 b(starting)f(example.)42 b(Ho)m(w)m(ev)m(er,)33 b(ev)m(en)e(then)g(w)m(e)g(ha)m(v)m(e)h(to)f(assume)g(that)g(all)f Ft(e)2801 3114 y Fq(q)2872 3100 y FB(are)h(roughly)e(the)i(same)50 3209 y(to)f(mak)m(e)h(suc)m(h)f(a)g(c)m(hoice)h(of)f(a)g(nonempt)m(y)g Fs(H)1627 3176 y Fq(q)1697 3209 y FB(a)g(reasonable)g(one)g(for)g (future)f(expansions.)39 b(Otherwise,)50 3318 y(one)30 b(has)h(to)g(tak)m(e)h Fs(H)765 3285 y Fq(q)830 3318 y FB(=)25 b Fs(;)30 b FB(if)g Ft(e)1127 3332 y Fq(q)1198 3318 y FB(deviates)g(from)g(min)1915 3332 y Fq(q)1950 3314 y Fp(0)1976 3332 y Fo(2)p Fq(Q)2102 3318 y Ft(U)10 b FB(\()p Ft(q)2253 3285 y Fo(0)2278 3318 y FB(\))31 b(to)s(o)g(m)m(uc)m(h.)50 3548 y Fu(Note.)46 b FB(So,)27 b(for)e(an)m(y)g(con)m(tour)h(from)f Fs(H)1372 3515 y Fq(q)1437 3548 y FB(w)m(e)h(assume)f(that)g(inside)e(of)j(in)m(t)14 b FD(\000)2604 3515 y Fq(q)2644 3548 y FB(,)27 b(no)e(other)h(\\small") e(con)m(tours)50 3661 y(from)38 b(some)h Fs(H)587 3628 y Fq(q)622 3601 y Fp(0)690 3661 y FB(can)g(reside,)h(whatev)m(er)f(the) g(whole)f(collection)g(of)h(in)m(terior)e(con)m(tours)j(inside)c FD(\000)3545 3628 y Fq(q)3623 3661 y FB(is.)50 3770 y(There)42 b(ma)m(y)i(b)s(e)f(a)g(ro)s(om,)k(inside)41 b(of)i(in)m(t)15 b FD(\000)1611 3737 y Fq(q)1651 3770 y FB(,)46 b(for)d(\\nonsmall")f (\(sligh)m(tly)g(parado)m(xical)h(name)g(here!\))50 3888 y(con)m(tours)30 b(from)f(some)g Fs(R)933 3855 y Fq(q)968 3827 y Fp(0)998 3888 y FB(.)40 b(This)28 b(will)f(b)s(e)h(quite)h (normal)f(and)h(c)m(haracteristic,)i(esp)s(ecially)c(in)h(the)i(later) 50 3997 y(expansion)h(stages)j(when)d(the)i(size)f(of)h(\\small")f(con) m(tours)h(from)f Fs(H)2418 3964 y Fq(q)2490 3997 y FB(will)e(b)s(e)i (already)g(quite)g(large.)47 b(A)m(t)50 4109 y(the)35 b(same)g(time,)h(some)g(con)m(tours)f(geometrically)g(m)m(uc)m(h)g (smaller)e(than)i FD(\000)g FB(but)f(from)h(some)g Fs(R)3428 4076 y Fq(q)3463 4049 y Fp(0)3527 4109 y FB(\(this)50 4227 y(can)30 b(happ)s(en)f(only)h(for)g(those)h Ft(q)1158 4194 y Fo(0)1213 4227 y FB(for)f(whic)m(h)g Fs(H)1691 4194 y Fq(q)1726 4166 y Fp(0)1785 4227 y FB(is)g(empt)m(y!\))41 b(will)28 b(p)s(ersist)h(in)g(the)h(mo)s(del.)159 4408 y(T)m(ypically)d(w)m(e)i(will)d(b)s(e)j(in)e(a)i(follo)m(wing)f (situation.)39 b(Assume)28 b(that)i(there)f(is,)f(for)h(an)m(y)g (nonempt)m(y)g Fs(H)3645 4375 y Fq(q)3685 4408 y FB(,)50 4517 y(a)35 b(common)h(diametr)f Ft(N)916 4531 y Fq(q)991 4517 y FB(of)g(all)g(con)m(tours)g(from)g Fs(H)1900 4484 y Fq(q)1940 4517 y FB(.)55 b(Assume)35 b(that)h(con)m(tours)g(from)f (corresp)s(onding)50 4630 y(families)d Fs(R)462 4597 y Fq(q)536 4630 y FB(ha)m(v)m(e)j(a)g(size)f(at)g(least)h Ft(R)1405 4644 y Fq(q)1476 4630 y Fs(\025)c Ft(N)1651 4644 y Fq(q)1691 4630 y FB(.)52 b(W)-8 b(rite)35 b Ft(N)2098 4644 y Fq(k)2174 4630 y Fs(\021)c Ft(N)2349 4644 y Fq(q)2382 4654 y Fv(k)2427 4630 y Ft(;)49 b(R)2570 4644 y Fq(k)2646 4630 y Fs(\021)31 b Ft(R)2817 4644 y Fq(q)2850 4654 y Fv(k)2895 4630 y Ft(;)49 b Fs(H)3047 4597 y Fq(k)3123 4630 y Fs(\021)31 b(H)3303 4597 y Fq(q)3336 4607 y Fv(k)3380 4630 y FB(.)52 b(Let)34 b(us)50 4739 y(arrange)d(the)f(elemen)m(ts)h 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(rather)g(misleading)f(if)g(w)m(e)i(lo)s(ok)f(at)i(the)f(dynamics)e(of) i(con)m(tours.)48 b(Then)50 2478 y(the)31 b(large)g(\()p Fs(\021)g FB(ha)m(ving)g(an)g(instable)e(external)i(color)g(in)f(our)g (terminology\))h(con)m(tours)h(are)f(dynamically)50 2588 y(more)f(stable)g(than)h(t)m(ypical)f(small)e(ones)j(\(whic)m(h)e(ev)m (olv)m(e)j(in)d(time)h(m)m(uc)m(h)h(more)f(rapidly\).)50 2819 y FD(Notation.)52 b FB(Denote)31 b(b)m(y)f(sym)m(b)s(ols)f FD(\000)1384 2833 y Fo(\014)1445 2819 y FB(,)i(more)f(precisely)f FD(\000)2163 2775 y Fq(q)2163 2841 y Fo(\014)2224 2819 y FB(,)i(the)f(extremal)g(con)m(tour)h(systems)f(of)h(the)50 2928 y(t)m(yp)s(e)36 b(\()p FD(\000)p Ft(;)15 b Fs(f)p FD(\000)506 2942 y Fq(i)538 2928 y Fs(g)p FB(\))37 b(where)f FD(\000)p FB(,)i(the)e(external)g(con)m(tour)h(of)g(the)g(system)f FD(\000)2549 2942 y Fo(\014)2610 2928 y FB(,)i(b)s(elongs)e(to)h Fs(H)3200 2942 y Fq(q)3276 2928 y FB(and)f FD(\000)3522 2942 y Fq(i)3588 2928 y FB(are)50 3037 y(inside)31 b(of)j FD(\000)p FB(,)g(b)s(elonging)e(to)j(some)f Fs(R)1378 3051 y Fq(q)1413 3032 y Fp(0)1442 3037 y FB(.)51 b(W)-8 b(e)35 b(will)c(use)i(the)h(sym)m(b)s(ol)e Fs(H)2565 2993 y Fq(q)2564 3059 y Fo(\014)2659 3037 y FB(to)i(denote)g(the)g (collection)g(of)50 3146 y(suc)m(h)c(extremal)g(con)m(tour)h(systems)g (con)m(taining)f(precisely)f(one)h(external)h(con)m(tour)g FD(\000)3026 3113 y Fq(q)3091 3146 y Fs(2)25 b(H)3255 3113 y Fq(q)3295 3146 y FB(.)50 3377 y FD(On)34 b(the)h(c)m(hoice)h(of) f(\(recursiv)m(ely)h(de\014ned\))e(clusters.)159 3608 y FB(No)m(w)f(w)m(e)g(will)c(discuss)i(our)h(general)g(c)m(hoice)h(of)g (the)f(\\clusters")g(i.e.)47 b(indices)30 b(marking)h(the)i(\\cluster) 50 3717 y(\014eld")g Fs(f)p Ft(w)406 3731 y Fq(T)463 3717 y Fs(g)h FB(in)f(our)g(abstract)i(Hamiltonian)d(\(1.23\).)53 b(W)-8 b(e)35 b(already)e(argued)h(that)g(w)m(e)g(need)g(a)g(general)50 3826 y(enough)c(notion,)h(suc)m(h)g(that)g(\\clusters)g(of)g(con)m (tours)g(and)g(other)g(clusters)f(are)h(again)g(clusters".)42 b(While)50 3935 y(the)37 b(set)h(of)f(considered)f(con)m(tours)h(in)f (partially)f(expanded)h(mo)s(dels)g(constructed)i(b)s(elo)m(w)e(will)e (alw)m(a)m(ys)50 4044 y(b)s(e)h(a)h(suitable)e Fu(subset)44 b FB(of)36 b(the)g(family)e Fs([)1500 4058 y Fq(q)1540 4044 y Fs(A)1613 4011 y Fq(q)1688 4044 y FB(\(of)j(all)d(con)m(tours)i (in)f(the)h(starting)f(lev)m(el)h(mo)s(del)e(\(1.11\)\))50 4153 y(the)c(concept)h(of)f(a)h(cluster)e(has)h(to)g(b)s(e)g (generalized,)g(to)h(b)s(ecome)f(useful)e(in)h(situations)g(where)g (iterativ)m(e)50 4262 y(partial)h(expansions)g(of)h(the)g(mo)s(del)f (are)i(considered.)42 b(Then,)30 b(a)i(recursiv)m(e)e(approac)m(h)i(to) g(the)f(notion)g(of)50 4372 y(a)g(cluster)e(is)h(natural,)f(also)i(in)e (the)h(inductiv)m(e)f(study)h(of)g(the)h(b)s(ounds)d(for)i(cluster)g(w) m(eigh)m(ts.)159 4553 y(Ha)m(ving)23 b(established)f(all)g(these)h(b)s (ounds)e(one)j(could)e(simplify)e(the)j(structure)f(of)i(clusters,)g (\\factorize")50 4662 y(them)29 b(w.r.)40 b(to)30 b(their)e(supp)s (ort,)g(and)h(treat)h(them)f(as)g(connected)i(sets)e(or)g(graphs.)40 b(Ho)m(w)m(ev)m(er,)32 b(to)e(do)f(this)50 4772 y(ia)h(adequate)h(only) f(when)f(a)i(summary)e(information)g(on)h(these)h(ob)5 b(jects)31 b(is)f(required,)e(when)i(a)h(detailed)50 4881 y(discussion)d(of)i(the)h(con)m(v)m(ergence)h(of)f(expansion)e (terms)h(w)m(as)h(already)f(done.)159 5062 y(F)-8 b(or)34 b(an)m(y)g Fs(H)578 5029 y Fq(q)618 5062 y Ft(;)15 b(q)33 b Fs(2)d Ft(Q)j FB(w)m(e)h(will)d(consider)h(the)i(family)e Fs(H)2116 5019 y Fq(q)2115 5085 y Fo(\014)2206 5062 y Fs(\033)e(H)2385 5029 y Fq(q)2458 5062 y FB(of)k(all)e(extremal)i(con)m (tour)g Fu(systems)50 5172 y FD(\000)113 5186 y Fo(\014)199 5172 y FB(=)25 b(\()p FD(\000)p Ft(;)15 b Fs(f)p FD(\000)541 5186 y Fq(k)587 5172 y Fs(g)p FB(\))29 b(ha)m(ving)e(one)i(external)e (con)m(tour)i FD(\000)f FB(from)f Fs(H)2209 5139 y Fq(q)2277 5172 y FB(and)h(p)s(ossibly)d(some)j(in)m(terior,)g(\\big)f(and)50 5289 y(nonexpandable)37 b(con)m(tours)j FD(\000)1121 5303 y Fq(k)1205 5289 y FB(tak)m(en)g(from)f Fs(R)1762 5256 y Fq(q)1797 5229 y Fp(0)1827 5289 y FB(,)j Ft(q)1938 5256 y Fo(0)2003 5289 y Fs(6)p FB(=)e Ft(q)i FB(\(and)d(b)s(ordering)e (\\holes")j(with)d(a)j(\\more)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(14)p eop %%Page: 15 15 15 14 bop 50 340 a FB(stable")37 b(color\).)60 b(Let)37 b(further)f Fs(T)1209 354 y Fq(q)1248 340 y Ft(;)15 b(q)39 b Fs(2)d Ft(Q)g FB(b)s(e)h(a)g(family)e(of)i(\\clusters")g(i.e.)59 b(ob)5 b(jects)38 b(already)e(de\014ned)50 449 y(b)s(efore,)i(denoted)f (b)m(y)g(sym)m(b)s(ols)e Ft(T)1245 463 y Fq(q)1323 449 y FB(or)h(simply)f Ft(T)13 b FB(.)60 b(W)-8 b(e)38 b(assume)f(that)g (for)g(these)g(ob)5 b(jects)38 b(w)m(e)f(already)50 558 y(ha)m(v)m(e)29 b(the)e(notion)h(of)f(a)h(supp)s(ort,)f(denoted)h(b)m (y)f Ft(T)p 1688 573 66 4 v 13 w FB(,)h(whic)m(h)f(is)f(de\014ned)h(as) h(a)g(connected)g(graph)f(in)f(\003.)40 b(The)50 667 y(forthcoming)30 b(de\014nition)e(is)h(a)i(preparation)e(for)h(our)g (main)g(result)f(\(Theorem)h(4\))h(stated)h(b)s(elo)m(w:)50 888 y FD(Newly)i(formed)g FB(\()p Fs(H)842 844 y Fq(q)841 910 y Fo(\014)903 888 y Ft(;)15 b Fs(T)1016 855 y Fq(q)1056 888 y FB(\))35 b FD(clusters.)159 1120 y FB(Sa)m(y)24 b(that)g(a)g(pair)e Fi(C)49 b FB(=)25 b(\()p Fs(f)p FD(\000)1093 1134 y Fo(\014)1155 1080 y Fq(i)1186 1120 y Fs(g)p Ft(;)15 b Fs(f)p Ft(T)1369 1134 y Fq(j)1408 1120 y Fs(g)p FB(\))25 b(where)e Fs(f)p FD(\000)1877 1134 y Fo(\014)1939 1080 y Fq(i)1969 1120 y Fs(g)j(\032)f(H)2214 1076 y Fq(q)2213 1143 y Fo(\014)2297 1120 y FB(and)e Fs(f)p Ft(T)2565 1134 y Fq(j)2604 1120 y Fs(g)j(\032)f(T)2843 1087 y Fq(q)2907 1120 y FB(is)e(a)h(\()p Fs(H)3174 1076 y Fq(q)3173 1143 y Fo(\014)3234 1120 y Ft(;)15 b Fs(T)3347 1087 y Fq(q)3388 1120 y FB(\))23 b(cluster)50 1251 y(if)i(it)g(is)g Fu(conne)-5 b(cte)g(d)36 b 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b(that)f(with)e(suc) m(h)h(a)h(de\014nition)e(all)g(the)i(pairs)e Fs(f)p Ft(T)8 b(;)3616 1583 y FB(~)3598 1606 y Ft(T)13 b Fs(g)50 1736 y FB(of)34 b(clusters)f(are)h(m)m(utually)f(compatible.)51 b(Con)m(tour)34 b(systems)g FD(\000)2308 1750 y Fo(\014)2369 1696 y Fq(i)2433 1736 y FB(will)d(b)s(e)j(called)f(the)h(\\n)m(uclei")g (of)g Fi(C)18 b FB(.)50 1845 y(A)30 b(t)m(ypical)g(non)m(trivial)f (cluster)g(has)h(one)h(n)m(ucleus,)f(and)f(empt)m(y)i(or)g(nonempt)m(y) f(collection)g(of)g(\\legs")i Ft(T)3649 1859 y Fq(i)3679 1845 y FB(.)159 2027 y(W)-8 b(e)26 b(will)d(no)m(w)i(generalize)g(the)h (ab)s(o)m(v)m(e)g(notion)f(also)g(to)h(all)e Fu(multiindices)33 b Fi(T)27 b FB(=)e Fs(f)p Ft(n)2954 2041 y Fn(\000)3003 2050 y Fp(\014)3062 2027 y Ft(;)15 b(n)3157 2041 y Fq(T)3214 2027 y Fs(g)26 b FB(i.e.)38 b(in)m(teger)50 2138 y(v)m(alued)23 b(functions)g(on)h(the)g(set)h(of)f(indices)e Fs(H)1590 2094 y Fq(q)1589 2160 y Fo(\014)1658 2138 y Fs([)8 b(T)1798 2105 y Fq(q)1863 2138 y FB(whic)m(h)22 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FB(,)50 449 y(but)29 b(for)i Ft(T)38 b Fs(\032)25 b Ft(M)40 b FB(w)m(e)31 b(will)d(alw)m(a)m(ys)i(assume)h(that)g(it)e (is)h(translation)f(in)m(v)m(arian)m(t.)159 631 y(No)m(w)i(w)m(e)g(are) g(ready)f(to)h(generalize)f(our)g(basic)g(mo)s(del)f(\(1.11\):)50 865 y FD(An)35 b(abstract)f(Pirogo)m(v)j(Sinai)e(mo)s(del)f(with)g (recursiv)m(ely)i(de\014ned)f(\\cluster)g(\014eld".)159 1099 y FB(F)-8 b(rom)26 b(no)m(w)g(on,)h(w)m(e)f(will)d(consider)i (again)h(a)g(mo)s(del)e(of)i(the)g(t)m(yp)s(e)g(\(1.11\))i(with)d(the)h (partition)e(function)541 1381 y Ft(Z)7 b FB(\(\003\))26 b(=)977 1295 y Fm(X)865 1494 y Fo(f)p Fn(\000)951 1503 y Fw(1)987 1494 y Fq(;:::)n Fn(\000)1122 1503 y Fv(m)1183 1494 y Fo(g)1262 1267 y Fq(m)1236 1295 y Fm(Y)1235 1488 y Fq(i)p Fr(=1)1398 1381 y FB(exp\()p Fs(\000)p Ft(E)5 b FB(\()p FD(\000)1813 1395 y Fq(i)1844 1381 y FB(\)\))1929 1295 y Fm(Y)1970 1485 y Fq(q)2061 1381 y FB(exp\()p Fs(\000)p Ft(e)2348 1395 y Fq(q)2389 1381 y Fs(j)p FB(\003)2477 1395 y Fq(q)2517 1381 y Fs(j)p FB(\))46 b(exp\()2843 1295 y Fm(X)2797 1490 y Fq(T)2840 1499 y Fv(q)2877 1490 y Fk(@)p Fr(\003)2984 1499 y Fv(q)3034 1381 y Ft(w)3099 1395 y Fq(T)3142 1404 y Fv(q)3183 1381 y FB(\))259 b(\(1)p Ft(:)p FB(23\))50 1691 y(but)34 b(with)g(a)h(more)g(general)g(notion)f (of)h(a)h(cluster)e(in)m(tro)s(duced)f(ab)s(o)m(v)m(e.)56 b(Instead)34 b(of)h(sa)m(ying)g(that)h Ft(T)3574 1705 y Fq(q)3649 1691 y FB(is)50 1800 y(incompatible)h(with)g(\003)877 1814 y Fq(q)956 1800 y FB(w)m(e)i(are)g(using)f(also)g(the)h(ab)s(o)m (v)m(e)h(notation)f Ft(T)2551 1814 y Fq(q)2631 1800 y Fj(@)g FB(\003)2804 1814 y Fq(q)2844 1800 y FB(.)66 b(W)-8 b(e)40 b(emphasize)e(that)50 1909 y(w)m(e)32 b(replace)g(here)f(the)h (usual)e(condition)g Ft(T)40 b Fs(\032)27 b FB(\003)1744 1923 y Fq(q)1816 1909 y FB(used)k(in)f(the)i(theory)g(of)g(p)s(olymer)e (mo)s(dels)g(\(and)i(also)50 2018 y(in)j(the)i(theory)g(of)f (\\partially)f(expanded")h(p)s(olymer)f(mo)s(dels,)i(see)h([BZK]\))e(b) m(y)h(suc)m(h)f(a)h(more)g(general)50 2127 y(condition.)42 b(Then)30 b(the)i(translation)e(in)m(v)m(ariancy)g(of)i Ft(w)1954 2141 y Fq(T)2043 2127 y FB(is)e(assumed)g(to)i(hold)e(only)h (if)f Ft(T)44 b FB(is)30 b Fu(compatible)50 2236 y FB(with)h(\003)322 2203 y Fq(q)363 2236 y FB(,)i Ft(T)42 b Fs(\032)29 b FB(\003)679 2203 y Fq(q)752 2236 y FB(while)i(for)i(clusters)f Ft(T)1532 2203 y Fq(q)1605 2236 y FB(touc)m(hing)g(\(\003)2076 2203 y Fq(q)2117 2236 y FB(\))2152 2203 y Fq(c)2189 2236 y FB(,)i(no)f(translation)e(in)m(v)m(ariancy)i(is)e(assumed)50 2345 y(and)26 b Ft(w)288 2359 y Fq(T)372 2345 y FB(ma)m(y)h(dep)s(end)d (on)j(the)g(\\shap)s(e")f(of)h(\003)1657 2312 y Fq(q)1724 2345 y FB(around)e Ft(T)13 b FB(,)27 b(to)s(o.)41 b(\(W)-8 b(e)28 b(could)d(assume)i(some)f(in)m(v)m(ariancy)50 2454 y(w.r.)40 b(to)31 b(the)g(shifts)e(of)h(the)h(pair)e Ft(T)8 b(;)15 b(T)33 b Fs(\\)20 b FB(\003)1529 2421 y Fq(q)1599 2454 y FB(but)30 b(w)m(e)h(will)c(not)k(do)f(it.\))159 2636 y(Consider)e(also)j(the)f(more)h(general,)g Fs(C)k FB({)c(conditioned)e(partition)g(function)521 2913 y Ft(Z)583 2927 y Fo(C)630 2913 y FB(\(\003\))d(=)997 2826 y Fm(X)885 3025 y Fo(f)p Fn(\000)971 3034 y Fw(1)1007 3025 y Fq(;:::)n Fn(\000)1142 3034 y Fv(m)1203 3025 y Fo(g)1282 2799 y Fq(m)1256 2826 y Fm(Y)1255 3020 y Fq(i)p Fr(=1)1419 2913 y FB(exp\()p Fs(\000)p Ft(E)5 b FB(\()p FD(\000)1834 2927 y Fq(i)1864 2913 y FB(\)\))1949 2826 y Fm(Y)1991 3017 y Fq(q)2082 2913 y FB(exp)o(\()p Fs(\000)p Ft(e)2368 2927 y Fq(q)2409 2913 y Fs(j)p FB(\003)2497 2927 y Fq(q)2538 2913 y Fs(j)p FB(\))46 b(exp)o(\()2863 2826 y Fm(X)2817 3021 y Fq(T)2860 3030 y Fv(q)2897 3021 y Fk(@)p Fr(\003)3004 3030 y Fv(q)3055 2913 y Ft(w)3120 2927 y Fq(T)3163 2936 y Fv(q)3203 2913 y FB(\))239 b(\(1)p Ft(:)p FB(24\))50 3227 y(where)34 b Fs(C)40 b FB(is)33 b(some)i(limitation)d(on)j(allo)m(w)m(ed)f(con\014gurations)g(\(in)g (the)g(sum)g(\(1.24\)\))j(form)m(ulated)d(in)g(the)50 3336 y(language)d(of)f(con)m(tours)h(\(tak)m(en)h(from)e(some)h(giv)m (en)f(subsystem)f Fs(S)37 b FB(of)31 b Fs(A)p FB(\).)50 3571 y Fu(Note.)59 b(Boundary)41 b(condition)e(\\at)g(in\014nity")g(on) g(a)g(torus)g FB(\003)p Fu(.)47 b FB(Imagine)36 b(e.g.)61 b(the)37 b(limitation)e Fs(C)41 b(\021)35 b(C)3669 3538 y Fq(q)50 3680 y FB(sa)m(ying)22 b(that)g(external)g(con)m(tours)h(of)f (an)m(y)g(con\014guration)f Ft(x)2062 3694 y Fr(\003)2138 3680 y FB(in)g(the)h(sum)f(\(1.24\))j(ha)m(v)m(e)g(a)e(\014xed)f (prescrib)s(ed)50 3789 y(color)40 b Ft(q)s FB(.)71 b(Denote)42 b(further)d(b)m(y)i Fs(S)1252 3803 y Fq(q)1332 3789 y FB(the)g(collection)f(of)h(all)e(con)m(tours)i(con)m(taining)f (\\in\014nit)m(y")g(in)f(their)50 3898 y(in)m(terior.)f(Then)26 b(the)h(limitations)d Fs(C)1291 3912 y Fq(q)1358 3898 y FB(&)i(p)s(ossibly)e(also)j(additional)e(condition)g(that)j(\\con)m (tours)f(from)g Fs(S)3670 3912 y Fq(q)50 4007 y FB(are)34 b(all)f(forbidden)e(in)i Ft(x)907 4021 y Fr(\003)961 4007 y FB(")i(giv)m(e)f(reasonable)g(c)m(hoices)g(of)g(what)g(could)f (b)s(e)g(called)h(the)g Ft(q)s FB({th)g(b)s(oundary)50 4116 y(condition)29 b(on)h(\003.)41 b(The)29 b(summation)g(in)g (\(1.23\))k(resp.)40 b(in)29 b(\(1.24\))j(is)e(then)g(o)m(v)m(er)h(all) f(compatible)f(con)m(tour)50 4225 y(systems)h(resp.)40 b(o)m(v)m(er)32 b(all)d(compatible)h(con)m(tour)h(systems)f(satisfying) f(suc)m(h)h(a)h(limitation.)159 4407 y(Belo)m(w)36 b(w)m(e)h(shall)d (assume)i(that)g(the)g(b)s(ounds)e(\(1.13\))k(and)d(\(1.14\),)40 b(\(1.15\))e(for)e(the)g(cluster)f(w)m(eigh)m(ts)50 4516 y(and)28 b(for)g(the)h(con)m(tour)h(w)m(eigh)m(ts)f(are)g(v)m(alid.)39 b(This)27 b(is)h(a)h(con)m(v)m(enien)m(t)h(w)m(a)m(y)f(ho)m(w)g(the)g (p)s(olymer)e(and)h(cluster)50 4625 y(b)s(ounds)f(are)i(form)m(ulated,) g(and)g(w)m(e)g(refer)g(to)h([KP],)g([D)m(V],)g(and)f(esp)s(ecially)e (to)j([BZK])f(\(see)h(also)g([NOZ]\))50 4734 y(for)j(an)h(additional)e (information)g(\(and)i(more)g(reference)g(to)h(previous)e(w)m(ork\))h (on)g(these)g(b)s(ounds.)49 b(The)50 4843 y(constan)m(t)41 b Ft(g)472 4810 y Fo(\000)p Fr(1)610 4843 y FB(can)g(b)s(e)e(in)m (terpreted)h(as)g(the)g(correlation)g(length)g(of)g(the)g(con)m(tour)h (gas)g(\(of)f(con)m(tours)50 4953 y FD(\000)p FB(,)j(without)c(the)i (cluster)f(\014eld\))f(and)h(similarly)d(the)k(quan)m(tit)m(y)g Ft(c)2391 4920 y Fo(\000)p Fr(1)2529 4953 y FB(is)f(a)h(correlation)f (length)g(of)h(the)50 5062 y(\\expanded)30 b(gas")h(describ)s(ed)e(b)m (y)h(the)g(cluster)g(quan)m(tities)g Ft(w)2168 5076 y Fq(T)2225 5062 y FB(.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)f(18:34)1080 b Fz(16)p eop %%Page: 17 17 17 16 bop 50 340 a FD(Rephrasemen)m(t)34 b(of)h(the)g(con)m(tour)g(and) g(cluster)h(estimates)d(\(1.14\))i(and)g(\(1.13\).)159 574 y FB(Later,)29 b(w)m(e)g(will)c(need)i(to)i(rein)m(terpret)e(the)i (b)s(ounds)c(\(1.14\))30 b(in)d(view)g(of)h(the)g(construction)g(of)g (con)m(tour)50 683 y(functional)h Ft(F)13 b FB(\()p FD(\000)p FB(\),)31 b(more)f(sp)s(eci\014cally)e Ft(F)13 b FB(\()p FD(\000)1593 697 y Fo(\014)1655 683 y FB(\).)41 b(But)31 b(\014rst)e(recall)h(the)h(b)s(ound)d(\(1.13\))1279 824 y Fm(X)1259 1019 y Fq(T)10 b Fo(3)p Fq(x)1402 1028 y Fv(i)1445 910 y Fs(j)p Ft(w)1535 924 y Fq(T)1593 910 y Fs(j)31 b Ft(e)1691 873 y Fq(c)p Fo(j)p Fq(T)p 1746 885 53 4 v 9 w Fo(j)1879 910 y Fs(\024)86 b Ft(\017)25 b Fs(\034)56 b FB(1)25 b Fs(\034)56 b Ft(c)976 b FB(\(1)p Ft(:)p FB(25\))50 1209 y(whic)m(h)29 b(will)f(determine)h(our)h(actual) h(c)m(hoice)g(of)f(the)h(small)e(constan)m(t)j Ft(\017)e FB(in)f(all)g(the)i(b)s(ounds)d(b)s(elo)m(w.)159 1391 y(More)f(in)m(teresting)e(will)f(b)s(e)i(a)g(reform)m(ulation)f(of)i (\(1.14\),)i(motiv)m(ated)f(b)m(y)e(the)g(de\014nition)e(of)i(a)h(con)m (tour)50 1500 y(functional)i(b)s(elo)m(w.)40 b(Rewrite)29 b(it)h(as)h(follo)m(ws,)f(in)m(tro)s(ducing)e(a)j(new)e(auxiliary)f (constan)m(t)k Ft(a)e FB(\()3266 1448 y Ft(:)3242 1500 y FB(=)c Ft(g)s FB(\))1612 1709 y Ft(a)g FB(=)f Ft(g)e Fs(\000)d Ft(\017)g Fs(\000)g Ft(!)1333 b FB(\(1)p Ft(:)p FB(26\))50 1917 y(where)30 b Ft(\017)g FB(is)f(from)h(\(1.13\))j(and)d Ft(!)j FB(is)c(the)i(suprem)m(um)d(of)j(the)g(l.h.s.)39 b(of)31 b(\(1.14\),)i(written)c(no)m(w)i(as)1297 2058 y Fm(X)1300 2253 y Fn(\000)p 1300 2265 50 4 v 1 w Fo(3)p Fq(i)1474 2144 y Ft(e)1516 2107 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))1770 2144 y Ft(e)1812 2107 y Fr(\()p Fq(a)p Fr(+)p Fq(\017)p Fr(+)p Fq(!)r Fr(\))p Fo(j)p Fn(\000)p 2119 2119 V 2 w Fo(j)2252 2144 y Fs(\024)55 b Ft(!)s(:)1014 b FB(\(1)p Ft(:)p FB(27\))50 2455 y(More)31 b(sp)s(eci\014cally)-8 b(,)29 b(also)h(the)h(b)s(ound)d(\(1.15\))33 b(can)d(b)s(e)g(rewritten)f(as)954 2595 y Fm(X)771 2791 y Fn(\000)p 771 2803 V 1 w Fo(3)p Fq(i;)11 b Fr(diam)h Fn(\000)p 1099 2803 V Fr(=)p Fq(N)1313 2682 y Ft(e)1355 2644 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))1610 2682 y Ft(e)1652 2644 y Fr(\()p Fq(a)p Fr(+)p Fq(\017)p Fr(+)p Fq(!)r Fr(\))p Fo(j)p Fn(\000)p 1959 2656 V 2 w Fo(j)2091 2682 y Fs(\024)56 b Ft(!)2275 2696 y Fq(N)2373 2682 y FB(;)2535 2595 y Fm(X)2569 2790 y Fq(N)2682 2682 y Ft(!)2739 2696 y Fq(N)2832 2682 y FB(=)25 b Ft(!)492 b FB(\(1)p Ft(:)p FB(28\))50 3001 y(where)30 b Ft(!)370 3015 y Fq(N)469 3001 y FB(deca)m(y)h(lik)m(e)f Ft(!)945 3015 y Fq(N)1069 3001 y Fs(\024)55 b Ft(K)37 b(\024)1361 2968 y Fq(N)1430 3001 y Ft(;)46 b(\024)26 b(<)f FB(1)30 b(and)g Ft(K)37 b FB(is)30 b(a)g(suitable)f(constan)m(t)j(indep)s (enden)m(t)c(on)i Ft(a)p FB(.)159 3183 y(Recall)i(that)i Ft(a)f FB(is)f(still)f(of)i(the)g(order)g Ft(g)j FB(\(and)d Ft(\034)10 b FB(\))33 b(while)e Ft(!)36 b FB(is)c(of)h(the)g(order)f Ft(e)2882 3150 y Fo(\000)p Fq(C)t(\034)3042 3183 y FB(,)h(so)h(for)e(a) i(large)f Ft(\034)50 3292 y FB(w)m(e)e(surely)e(ha)m(v)m(e)i(a)g (trivial)d(inequalit)m(y)-8 b(,)30 b(see)h(more)f(precisely)f(b)s(elo)m (w)h(in)f(\(2.10\))1332 3500 y Ft(!)g(<)c(a)151 b FB(i.e.)h(2)p Ft(!)23 b FB(+)d Ft(\017)25 b(<)g(g)s(:)1051 b FB(\(1)p Ft(:)p FB(29\))50 3782 y Fu(Note.)46 b FB(While)32 b(the)i(cluster)f(b) s(ound)e(\(1.25\))k(is)e(still)e(just)i(\(1.13\),)j(the)e(presence)f (of)h(another)f(small)f(con-)50 3891 y(stan)m(ts)d Ft(!)j FB(and)c Ft(\017)h FB(newly)e(incorp)s(orated)h(in)m(to)g(\(1.27\))j (and)d(later)h(estimates)h(\(compare)f(the)g(form)m(ulations)50 4000 y(of)j(Theorem)f(1)h(and)f(Theorem)h(2)g(b)s(elo)m(w,)f(where)g (another)h(auxiliary)-8 b(,)30 b(large)i(enough,)g(constan)m(t)h(\012)e (will)50 4109 y(b)s(e)j(also)g(in)m(tro)s(duced\))f(will)f(b)s(e)i(imp) s(ortan)m(t)f(and)h(not)h(trivial.)50 b(While)34 b(there)g(is)g(no)g (necessit)m(y)h(to)g(in)m(tro-)50 4218 y(duce)28 b(suc)m(h)g(new)g (constan)m(ts)i(in)d(p)s(olymer)g(expansion)g(problems)g(\(lik)m(e)h (in)f(partially)g(expanded)g(p)s(olymer)50 4327 y(mo)s(dels)37 b(of)h([BZE]\))h(these)g(constan)m(ts)h(will)c(b)s(e)h(essen)m(tial)i (later,)h(in)d(the)i(con)m(text)h(of)f(general)f(Pirogo)m(v)50 4436 y(Sinai)28 b(theory)j(dealing)e(with)g(matc)m(hing)h(compatible)g (collections)g(of)g(\\p)s(olyc)m(hromatic")h(con)m(tours.)159 4618 y(This)24 b(is)i(related)g(to)h(a)f(later)g(construction)g(of)h(a) f(con)m(tour)h(functional)e Ft(F)13 b FB(\()p FD(\000)2742 4632 y Fo(\014)2803 4618 y FB(\),)28 b(related)e(to)h(the)f(energy)50 4727 y Ft(E)5 b FB(\()p FD(\000)220 4741 y Fo(\014)282 4727 y FB(\))26 b(as)g(follo)m(ws.)39 b(See)26 b(Theorem)g(1)g(and)g(b) s(elo)m(w)f(it,)i(for)f(a)g(detailed)f(discussion)f(on)i(wh)m(y)f(w)m (e)i(in)m(tro)s(duce)50 4836 y(a)i(new,)f(seemingly)f(arbitrary)h(term) g Ft(e)1392 4803 y Fr(\012)1477 4836 y FB(here.)40 b(One)28 b(should)e(k)m(eep)k(in)d(mind)f(that)j(the)g(constan)m(ts)g Ft(!)i FB(and)50 4946 y Ft(\017)f FB(are)h(small,)e(so)i(w)m(e)f(will)e (still)g(ha)m(v)m(e)k(roughly)d Ft(E)5 b FB(\()p FD(\000)1837 4960 y Fo(\014)1899 4946 y FB(\))31 b(in)e(the)h(exp)s(onen)m(tial)g (on)g(the)g(r.h.s.)1180 5167 y Ft(e)1222 5129 y Fo(\000)p Fq(F)10 b Fr(\()p Fn(\000)1412 5138 y Fp(\014)1467 5129 y Fr(\))1555 5167 y Fs(\024)55 b Ft(e)1723 5129 y Fr(\012)1841 5167 y Ft(e)1883 5125 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)2073 5134 y Fp(\014)2128 5125 y Fr(\)+\()p Fq(!)r Fr(+)p Fq(\017)p Fr(\))p Fo(j)p Fn(\000)2473 5134 y Fp(\014)p 2424 5154 104 4 v 2528 5125 a Fo(j)2554 5167 y Ft(:)898 b FB(\(1)p Ft(:)p FB(30\))50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(17)p eop %%Page: 18 18 18 17 bop 804 340 a FC(2.)51 b(The)33 b(contour)f(functional)g(and)g (its)i(pr)n(oper)-6 b(ties)50 649 y FD(De\014nition)35 b(of)g(the)g(con)m(tour)g(functional)h Ft(F)13 b FB(\()p FD(\000)1880 663 y Fo(\014)1941 649 y FB(\))p FD(.)159 891 y FB(The)39 b(quan)m(tit)m(y)h(pla)m(ying)e(a)i(fundamen)m(tal)e (role)h(in)g(dev)m(eloping)f(of)i(Pirogo)m(v)g(Sinai)d(theory)j(is)f (their)50 1000 y(\\con)m(tour)g(functional")d Ft(F)13 b FB(\()p FD(\000)1082 1014 y Fo(\014)1144 1000 y FB(\))38 b({)g(here)g(considered)f(as)h(functional)e(of)i(con)m(tour)g(systems)g FD(\000)3368 1014 y Fo(\014)3467 1000 y FB(rather)50 1109 y(than)30 b(single)f(con)m(tours)i FD(\000)p FB(.)41 b(This)28 b(is)i(really)f(a)i(k)m(ey)g(concept.)42 b(It)31 b(app)s(eared)e(\014rst)h(in)f(the)i(pap)s(er)e([PS].)i(W)-8 b(e)50 1219 y(will)31 b(use)i(a)h(v)m(arian)m(t)h(\(of)f([Z])g(and)f (subsequen)m(t)g(pap)s(ers\))f(of)i(this)f(quan)m(tit)m(y)-8 b(,)35 b(tec)m(hnically)e(more)h(suitable)50 1328 y(for)d(us.)45 b(It)32 b(could)e(b)s(e)h(also)h(called)f(the)h(\\free)h(energy")f(of)g (the)g(con)m(tour)g(system)g FD(\000)2936 1342 y Fo(\014)2998 1328 y FB(,)g(or)g(\\w)m(ork)g(needed)50 1437 y(to)f(install)d FD(\000)497 1451 y Fo(\014)558 1437 y FB(".)42 b(Hence)31 b(also)f(the)h(notation,)f Ft(F)13 b FB(\()p FD(\000)1836 1451 y Fo(\014)1898 1437 y FB(\),)31 b(in)e(con)m(trast)j(to)f(the)g Fu(ener)-5 b(gy)38 b Ft(E)5 b FB(\()p FD(\000)3177 1451 y Fo(\014)3239 1437 y FB(\))31 b(of)f FD(\000)3471 1451 y Fo(\014)3533 1437 y FB(.)159 1619 y(So,)24 b(w)m(e)g(will)c(de\014ne) i(suc)m(h)g(a)i(quan)m(tit)m(y)f(not)g(only)f(for)g(single)g(con)m 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b(a)h(simple)e(P)m(eierls)h(argumen)m(t)i(can)f (b)s(e)f(used)g(to)i(estimate)g(the)f(probabilit)m(y)d(of)j(the)h(ev)m (en)m(t)g(that)g FD(\000)3649 4345 y Fo(\014)50 4440 y FB(app)s(ears.)55 b(If)35 b(a)h(P)m(eierls)f(condition)f(holds)g(w)m (e)i(can)g(also)f(easily)g(\014nd)f(cluster)h(expansion)f(form)m(ulas)h (for)50 4549 y(these)e(probabilities.)46 b(It)34 b(is)e(a)h(c)m (haracteristic)h(feature)g(of)f(P)m(.)h(S.)f(theory)g(that)h(it)f(dev)m (elops)g(a)h(univ)m(ersal)50 4658 y(mac)m(hinery)29 b(applicable)g(to)i (the)f(cases)i(when)d(suc)m(h)h(a)h(cancellation)f(do)s(es)g Fu(not)40 b FB(tak)m(e)32 b(place.)159 4840 y(It)e(is)g(w)m(orth)m (while)f(to)i(giv)m(e)g(another)f(expression)f(of)i(\(2.1\),)h (clarifying)d(b)s(etter)h(its)g(meaning:)50 5082 y FD(\\Metastable)k (free)h(energies"of)g(the)f(mo)s(del)h(\(1.23\).)46 b(Another)35 b(expression)h(of)f Ft(F)13 b FB(\()p FD(\000)3440 5096 y Fo(\014)3502 5082 y FB(\))p FD(.)p 50 5202 546 4 v 159 5263 a Fl(6)197 5289 y Fz(If)25 b Fh(q)j Fz(denotes)e(the)g (external)h(color)f(of)f Ff(\000)1378 5299 y Fd(\014)1461 5289 y Fz(then)h(w)n(e)g(use)g(the)g(sym)n(b)r(ol)32 b(~)-45 b Fh(q)28 b Fz(to)d(denote)h(all)h(the)f(v)n(ariables)g(from)f Fh(Q)p Fz(.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)k(18:34)1080 b Fz(18)p eop %%Page: 19 19 19 18 bop 159 340 a FB(Denote)33 b(b)m(y)e Ft(f)644 354 y Fq(q)715 340 y FB(the)g(\\metastable)i(free)e(energy")h(of)g(the)f (mo)s(del)g(\(1.23\))i(\(de\014ned)e(if)f(w)m(e)i(\\remo)m(v)m(e)h(all) 50 449 y(con)m(tours)e FD(\000)480 416 y Fq(q)550 449 y FB(from)f(it"\),)h(more)f(precisely)f(the)i(quan)m(tit)m(y)g(\(indep) s(enden)m(t)d(on)i(a)h(c)m(hoice)g(of)g Ft(i)25 b Fs(2)g FB(\003\))1235 618 y Ft(f)1280 632 y Fq(q)1345 618 y FB(=)g Ft(e)1483 632 y Fq(q)1543 618 y Fs(\000)1658 532 y Fm(X)1634 727 y Fq(T)1677 736 y Fv(q)1726 727 y Fo(3)12 b Fq(i)1873 551 y Ft(w)1938 565 y Fq(T)1981 574 y Fv(q)p 1869 597 157 4 v 1869 680 a Fs(j)p Ft(T)p 1894 695 66 4 v 1960 703 a Fq(q)2000 680 y Fs(j)2092 618 y FB(=)55 b Ft(e)2260 632 y Fq(q)2321 618 y Fs(\000)50 b Ft(s)2485 632 y Fq(q)3523 618 y FB(\(2)p Ft(:)p FB(3\))50 857 y(where)30 b Ft(e)355 871 y Fq(q)425 857 y FB(is)g(the)g(mean)h(energy)g(of)f(the) h(constan)m(t)g(con\014guration)f Ft(x)2436 871 y Fr(\003)2490 824 y Fq(q)2556 857 y Fs(\021)25 b Ft(q)33 b FB(and)1552 1029 y Ft(s)1595 1043 y Fq(q)1660 1029 y FB(=)1792 943 y Fm(X)1756 1138 y Fq(T)1799 1147 y Fv(q)1860 1138 y Fo(3)24 b Fq(i)1989 962 y Ft(w)2054 976 y Fq(T)2097 985 y Fv(q)p 1985 1008 157 4 v 1985 1091 a Fs(j)p Ft(T)p 2010 1106 66 4 v 2076 1114 a Fq(q)2116 1091 y Fs(j)2183 1029 y Ft(:)1315 b FB(\(2)p Ft(:)p FB(4\))50 1268 y(The)30 b(quan)m(tit)m(y)g Ft(s)643 1282 y Fq(q)713 1268 y FB(can)h(b)s(e)f (often)h(in)m(terpreted)e(as)i(the)g(free)f(energy)h(of)f(a)h (restricted)f(ensem)m(ble)g(in)f(some)50 1377 y(original)f(mo)s(del)i (\(1.1\),)i(whic)m(h)d(w)m(as)i(\014rst)e(partially)g(expanded)g(to)j (the)e(form)g(\(1.11\).)50 1596 y FD(Theorem)44 b(0.)53 b Fu(The)41 b(contour)h(functional)g(\(2.1\))g(can)f(be)g(also)h(expr) -5 b(esse)g(d)43 b(as)e(fol)5 b(lows,)45 b(by)c(using)f(the)50 1705 y(quantities)33 b Ft(f)511 1719 y Fq(q)550 1705 y Fu(:)42 b(F)-7 b(or)34 b(any)f(extr)-5 b(emal)34 b(system)g FD(\000)1691 1719 y Fo(\014)1777 1705 y FB(=)25 b FD(\000)1936 1661 y Fq(q)1936 1728 y Fo(\014)2030 1705 y Fu(with)33 b(an)g(external)h(color)g Ft(q)h Fu(we)e(have)888 1871 y Ft(F)13 b FB(\()p FD(\000)1057 1885 y Fo(\014)1119 1871 y FB(\))25 b(=)g Ft(E)5 b FB(\()p FD(\000)1445 1885 y Fo(\014)1507 1871 y FB(\))21 b(+)1654 1784 y Fm(X)1661 1980 y Fr(~)-41 b Fq(q)r Fo(6)p Fr(=)p Fq(q)1785 1871 y FB(\()p Ft(f)1871 1885 y Fr(~)f Fq(q)1925 1871 y Fs(\000)20 b Ft(f)2061 1885 y Fq(q)2101 1871 y FB(\))p Fs(j)15 b FB(in)m(t)2291 1885 y Fr(~)-42 b Fq(q)2340 1871 y FD(\000)2403 1885 y Fo(\014)2465 1871 y Fs(j)20 b FB(+)g(\001\()p FD(\000)2775 1885 y Fo(\014)2836 1871 y FB(\))652 b(\(2)p Ft(:)p FB(5\))50 2110 y Fu(wher)-5 b(e)34 b(the)f(additional)i 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b(Put)1593 4469 y Ft(f)35 b FB(=)25 b(min)1920 4483 y Fq(q)r Fo(2)p Fq(Q)2082 4469 y Ft(f)2127 4483 y Fq(q)3523 4469 y FB(\(2)p Ft(:)p FB(8\))50 4609 y(and)30 b(de\014ne)f(useful)g(auxiliary)f(quan)m (tities,)i(for)g(an)m(y)h Ft(q)d Fs(2)d Ft(Q)1223 4749 y(u)1275 4763 y Fq(q)1341 4749 y FB(=)g Ft(f)1482 4763 y Fq(q)1541 4749 y Fs(\000)20 b Ft(f)101 b FB(;)106 b(min)2060 4763 y Fq(q)r Fo(2)p Fq(Q)2252 4749 y Ft(u)2304 4763 y Fq(q)2370 4749 y FB(=)25 b(0)p Ft(:)987 b FB(\(2)p Ft(:)p FB(9\))50 4962 y Fu(Note.)46 b FB(A)26 b(more)g(traditional)f (notation)h(of)g(these)g(quan)m(tities)g(\(\\Pirogo)m(v)h(Sinai)d (parameters"\))j(w)m(ould)d(b)s(e)50 5071 y Ft(a)98 5085 y Fq(q)161 5071 y FB(but)f(w)m(e)h(already)f(used,)h(in)e(accordance)j (with)d([BZE])h(and)g(the)g(references)h(quoted)f(there,)j(the)d (letter)h Ft(a)50 5180 y FB(for)f(a)h(di\013eren)m(t)f(purp)s(ose)e(in) h(\(1.27\).)41 b(A)24 b(crude)e(appro)m(ximation)h(to)h Ft(u)2420 5194 y Fq(q)2483 5180 y FB(is)f(the)g(quan)m(tit)m(y)h 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Fr(\()p Fq(a)p Fr(+)p Fq(\017)p Fr(+)p Fq(!)r Fr(\))p Fo(j)p Fn(\000)p 2103 1713 V 2 w Fo(j)2235 1738 y Fs(\024)56 b Ft(e)2404 1701 y Fo(\000)p Fr(\012)2578 1738 y Ft(b)860 b FB(\(2)p Ft(:)p FB(10\))50 1999 y(where)325 1976 y(~)314 1999 y Ft(L)426 1947 y(:)403 1999 y FB(=)27 b(1)32 b(is)e(another)i(auxiliary)d (constan)m(t)k(\(whic)m(h)e(will)e(b)s(e)h(\014xed)h(b)s(elo)m(w,)h(in) e(\(3.3\),)k(imagine)c(that)61 2092 y(~)50 2115 y Ft(L)25 b FB(=)g(1\))k(and)f(where)h(w)m(e)g(tak)m(e)h(\012)e(and)g Ft(b)h FB(suc)m(h)f(that)h(\(2.10\))i(w)m(ould)d(b)s(e)g(just)g(the)h (relation)f(\(1.27\))j(but)c(not)50 2231 y(with)i Ft(b)h(e)368 2198 y Fo(\000)p Fr(\012)507 2231 y FB(=)614 2208 y(~)603 2231 y Ft(L)g(!)j FB(but)d(with)f(a)i(sligh)m(tly)e(bigger)g(constan)m (t)j Ft(b)e FB(suc)m(h)h(that)1502 2376 y Ft(b)g(e)1614 2338 y Fo(\000)p Fr(\012)1752 2376 y FB(=)1859 2353 y(~)1848 2376 y Ft(L!)s FB(\(1)21 b(+)27 b(~)-52 b Ft(!)s FB(\))50 2518 y(where)31 b(the)h(v)m(alue)f(of)39 b(~)-52 b Ft(!)30 b Fs(\034)e FB(1)k(will)d(b)s(e)i(in)m(tro)s(duced)f(later,)i(in)e (\(2.22\))k(and)d(the)h(term)g(1)21 b(+)28 b(~)-52 b Ft(!)34 b FB(will)29 b(app)s(ear)50 2627 y(in)g(\(2.28\).)43 b(Recall)30 b(also)g(the)h(b)s(ound)d(\(1.28\),)33 b(whic)m(h)c(will)f (b)s(e)h(written)h(no)m(w)g(analogously)g(as)h(\(2.10\))556 2730 y Fm(X)380 2926 y Fn(\000)p 380 2938 V Fo(3)p Fq(i)p Fr(:diam)11 b Fn(\000)p Fr(=)p Fq(N)909 2817 y Ft(e)951 2779 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))1205 2817 y Ft(e)1247 2779 y Fr(\()p Fq(a)p Fr(+)p Fq(\017)p Fr(+)p Fq(!)r Fr(\))p Fo(j)p Fn(\000)p 1554 2791 V 2 w Fo(j)1717 2817 y Fs(\024)25 b Ft(e)1855 2779 y Fo(\000)p Fr(\012)2010 2755 y Ft(b)2049 2769 y Fq(N)p 2010 2796 108 4 v 2043 2868 a FB(~)2033 2891 y Ft(L)2214 2817 y Fs(\024)56 b Ft(K)37 b(\024)2507 2779 y Fq(N)2606 2817 y FB(;)2707 2730 y Fm(X)2741 2925 y Fq(N)2854 2817 y Ft(b)2893 2831 y Fq(N)2987 2817 y FB(=)25 b Ft(b:)330 b FB(\(2)p Ft(:)p FB(11\))50 3056 y(It)30 b(will)e(b)s(e)i(useful)e(to) 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FB({)d(small)f(\\pac)m(king)i(cub)s(e")f Fj(\003)g FB(of)h FD(\000)1314 4648 y Fo(\014)1405 4634 y FB(suc)m(h)f(that)h FD(\000)1870 4648 y Fo(\014)p 1807 4670 125 4 v 1957 4634 a Fs(\032)25 b Fj(\003)30 b FB(.)159 4818 y(This)h(notion)h(suggests)i(the)f(w)m(a)m(y)h(ho)m(w)f(w)m(e)h (will)c(organize)j(our)g(expansions.)47 b(Namely)-8 b(,)34 b(b)m(y)f(c)m(ho)s(osing)50 4927 y Fu(smal)5 b(l)k FB(,)35 b(extremal)e(con)m(tours)g(resp.)47 b(con)m(tour)34 b(systems,)f(as)g (ob)5 b(jects)34 b(to)f(b)s(e)f(expanded.)47 b(Recall)33 b(that)g(for)50 5036 y(extremal)c(small)f FD(\000)725 5050 y Fo(\014)787 5036 y FB(,)h(no)h(small)e(con)m(tours)i(\\smaller)e (than)i FD(\000)2203 5050 y Fo(\014)2264 5036 y FB(")g(exist)f(\(they)h (w)m(ere)g(expanded)f(b)s(efore\).)p 50 5111 546 4 v 159 5172 a Fl(7)197 5198 y Fz(An)n(y)j(large)i(enough)e(v)n(alue)h(of)g (a)g(\\testing)g(constan)n(t")g(\012)h Fx(\024)g Fh(\034)8 b Fx(j)p Ff(\000)p 2120 5211 54 4 v 1 w Fx(j)34 b Fz(w)n(ould)f(do)f(a) h(similar)h(job.)57 b(The)33 b(test)h(could)f(b)r(e)50 5289 y(applied)26 b(directly)i(to)d(con)n(tours)h(instead)g(of)f(cub)r (es,)j(with)e(v)n(alues)g Fh(C)5 b Fx(j)14 b Fz(diam)e Ff(\000)p Fx(j)26 b Fz(or)g Fh(C)5 b Fx(j)p Ff(\000)p 2556 5302 V 1 w Fx(j)26 b Fz(lik)n(e)h(in)f([ZR])h(or)e([Z].)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)k(18:34)1080 b Fz(20)p eop %%Page: 21 21 21 20 bop 50 340 a FD(The)34 b(sk)m(eleton)i(of)f(a)f(system)h(\000) 1284 354 y Fo(\014)1345 340 y FD(.)46 b(P)m(eierls)36 b(condition)g(for)f(a)g(small,)f(extremal)f(\000)3269 354 y Fo(\014)3330 340 y FD(.)159 571 y FB(The)i(forthcoming)g(lo)m(w)m (er)i(b)s(ound)c(for)j Ft(F)13 b FB(\()p FD(\000)1707 585 y Fo(\014)1769 571 y FB(\))36 b(will)d(b)s(e)i(com)m(bined)g(with)f (some)j(en)m(trop)m(y)f(estimates)50 680 y(dev)m(elop)s(ed)31 b(later.)45 b(Recall)31 b(that)h(ev)m(en)h(a)f(small,)f(extremal)h(con) m(tour)g FD(\000)g FB(ma)m(y)g(con)m(tain)g(a)g(lot)g(of)g(in)m (terior,)50 789 y(nonsmall)i(con)m(tours)i(forming,)h(together)g(with)e (the)h(external)g(con)m(tour)h FD(\000)p FB(,)g(an)f(extremal)g(system) g FD(\000)3623 803 y Fo(\014)3685 789 y FB(.)50 898 y(The)30 b(follo)m(wing)f(construction)g(will)f(help)h(to)i(treat)h(their)d(en)m (trop)m(y)i(in)e(a)i(tec)m(hnically)f(con)m(v)m(enien)m(t)h(w)m(a)m(y) -8 b(.)159 1080 y(F)g(or)31 b(an)m(y)37 b(~)-52 b Ft(q)28 b Fs(2)d Ft(Q)p FB(,)30 b(\014x)f(a)i(decomp)s(osition)d Fs(K)1652 1094 y Fr(~)-42 b Fq(q)1716 1080 y FB(of)31 b(the)f(lattice)g(\003)g(in)m(to)g(m)m(utually)e(disjoin)m(t)h(non)36 b(~)-52 b Ft(q)s FB({small)50 1189 y(cub)s(es)32 b(of)i(a)g Fu(smal)5 b(lest)36 b(possible)42 b FB(diameter.)49 b(The)33 b(union,)g(for)g(all)f(p)s(ossible)f(colors)40 b(~)-52 b Ft(q)34 b Fs(2)29 b Ft(Q)p FB(,)35 b(of)e(all)g(cub)s(es)50 1298 y Fj(\003)p FB(,)d(from)f(a)i(corresp)s(onding)d Fs(K)1125 1312 y Fr(~)-42 b Fq(q)1159 1298 y FB(,)30 b(whic)m(h)f(are)h Fu(subsets)38 b FB(of)30 b(in)m(t)2154 1312 y Fr(~)-42 b Fq(q)2204 1298 y FD(\000)2267 1312 y Fo(\014)2328 1265 y Fr(8)2399 1298 y FB(will)27 b(b)s(e)i(called)h (the)g Fu(skeleton)37 b FB(of)30 b FD(\000)3623 1312 y Fo(\014)3685 1298 y FB(.)50 1407 y(The)g(cardinalit)m(y)e(of)j(this)e (system)i(of)f(cub)s(es)g(\(tak)m(en)i(o)m(v)m(er)f(all)f Ft(q)s FB(\))g(will)e(b)s(e)i(denoted)g(b)m(y)g(conn\()p FD(\000)3391 1421 y Fo(\014)3453 1407 y FB(\).)50 1638 y Fu(Notes.)46 b FB(1\))c(Here,)j(the)c(fact)h(that)g(w)m(e)f(are)h(on) f(a)h(\014nite)e(torus)g(and)h(not)g(on)g Fi(Z)2867 1605 y Fq(\027)2949 1638 y FB(causes)h(some)f(minor)50 1747 y(tec)m(hnical)36 b(incon)m(v)m(eniences,)j(whic)m(h)c(are)i(ho)m(w)m (ev)m(er)h(solv)m(able)e(in)f(an)i(ob)m(vious)f(w)m(a)m(y)-8 b(.)61 b(F)-8 b(or)37 b(example)g(it)f(is)50 1856 y(tec)m(hnically)22 b(con)m(v)m(enien)m(t)j(to)f(assume)f(that)h(the)g(diameter)f(of)h(the) f(torus)h(\003)f(is)g(divisible)c(b)m(y)24 b(the)f(diameters)50 1965 y(of)i(all)f(the)h(considered)f(\\smallest")h(cub)s(es.)38 b(W)-8 b(e)26 b(omit)f(a)g(discussion)e(on)i(ho)m(w)g(to)g(p)s(olish)e (the)i(form)m(ulation)50 2074 y(of)30 b(the)h(forthcoming)f (constructions)g(in)f(the)h(situations)f(when)h(suc)m(h)g(an)g (assumption)f(is)g(not)i(v)m(alid.)159 2256 y(2\))g(Of)f(course,)i (complicated)e(con)m(tour)h(systems)g(with)e Fu(nonzer)-5 b(o)39 b FB(conn)o(\()p FD(\000)2714 2270 y Fo(\014)2776 2256 y FB(\))31 b(usually)d(will)g(not)j(\014gure)50 2365 y(at)38 b(the)h(crudest)e(appro)m(ximations)g(of)h(the)g(mo)s (del.)62 b(Nev)m(ertheless)39 b(they)f(pla)m(y)g(a)g(decisiv)m(e)f (role)h(in)f(the)50 2474 y(more)j(delicate)f(tec)m(hnical)h(asp)s(ects) g(of)g(the)g(theory)-8 b(,)43 b(and)c(they)h(should)d(b)s(e)i(treated)i (carefully!)67 b(Here)50 2583 y(w)m(e)32 b(presen)m(t)g(a)g(treatmen)m (t)h(whic)m(h)e(someho)m(w)h(deviates)g(from)f(the)h(treatmen)m(t)h(of) f([ZR].)g(It)g(brings)d(some)50 2692 y(simpli\014cations,)20 b(related)i(to)g(more)g(systematic)g(usage)h(of)e(the)h(concept)h(of)f 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3738 y Ft(F)13 b FB(\()p FD(\000)891 3752 y Fo(\014)953 3738 y FB(\))26 b Fs(\025)e Ft(E)5 b FB(\()p FD(\000)1279 3752 y Fo(\014)1341 3738 y FB(\))21 b Fs(\000)f FB(\()p Ft(!)j FB(+)d Ft(")p FB(\))p Fs(j)p FD(\000)1859 3752 y Fo(\014)p 1796 3773 V 1921 3738 a Fs(j)h FB(+)f(\012)15 b(conn)o(\()p FD(\000)2423 3752 y Fo(\014)2485 3738 y FB(\))21 b Fs(\000)f Ft(u)2684 3752 y Fq(q)2724 3738 y Fs(j)15 b FB(in)m(t)g FD(\000)2951 3752 y Fo(\014)3012 3738 y Fs(j)440 b FB(\(2)p Ft(:)p FB(16\))50 3937 y(and)30 b(this)f(will)f(yield)g(\(2.15\))33 b(for)d Fu(smal)5 b(l)41 b FB(systems)30 b FD(\000)1846 3951 y Fo(\014)1907 3937 y FB(,)h(b)s(ecause)f(smallness)f(of)h FD(\000)2866 3951 y Fo(\014)2958 3937 y FB(implies)1404 4130 y Ft(u)1456 4144 y Fq(q)1496 4130 y Fs(j)15 b FB(in)m(t)g FD(\000)1723 4144 y Fo(\014)1784 4130 y Fs(j)26 b(\024)f Ft(u)1983 4144 y Fq(q)2023 4130 y Fs(j)p Fj(\003)p Fs(j)g(\024)g FB(\012)p Ft(:)1121 b FB(\(2)p Ft(:)p FB(17\))159 4395 y(Recall)30 b(the)g(expression)g(\(2.5\))i(of)e Ft(F)13 b FB(\()p FD(\000)1517 4409 y Fo(\014)1579 4395 y FB(\))31 b(for)f(a)h(system)f FD(\000)2224 4409 y Fo(\014)2316 4395 y FB(with)f(external)h(color)g Ft(q)k FB(namely)880 4605 y Ft(F)13 b FB(\()p FD(\000)1049 4619 y Fo(\014)1111 4605 y FB(\))26 b(=)f Ft(E)5 b FB(\()p FD(\000)1438 4619 y Fo(\014)1500 4605 y FB(\))20 b(+)1646 4519 y Fm(X)1654 4714 y Fr(~)-42 b Fq(q)s Fo(6)p Fr(=)p Fq(q)1777 4605 y FB(\()p Ft(u)1870 4619 y Fr(~)g Fq(q)1925 4605 y Fs(\000)20 b Ft(u)2068 4619 y Fq(q)2108 4605 y FB(\))p Fs(j)15 b FB(in)m(t)2298 4619 y Fr(~)-41 b Fq(q)2348 4605 y FD(\000)2411 4619 y Fo(\014)2472 4605 y Fs(j)21 b FB(+)e(\001\()p FD(\000)2782 4619 y Fo(\014)2844 4605 y FB(\))598 b(\(2)p Ft(:)p FB(18\))50 4896 y(\(notice)31 b(that)g Ft(f)595 4910 y Fq(q)655 4896 y Fs(\000)20 b Ft(f)797 4910 y Fr(~)-42 b Fq(q)855 4896 y FB(=)25 b Ft(u)1003 4910 y Fq(q)1064 4896 y Fs(\000)20 b Ft(u)1213 4910 y Fr(~)-42 b Fq(q)1247 4896 y FB(\))31 b(where)e(the)i(\\error)f(term")h(\001\()p FD(\000)2437 4910 y Fo(\014)2499 4896 y FB(\))f(satis\014es)1399 5088 y Fs(j)p FB(\001\()p FD(\000)1598 5102 y Fo(\014)1659 5088 y FB(\))p Fs(j)c(\024)f FB(\()p Ft(!)f FB(+)c Ft(")p FB(\))p Fs(j)p FD(\000)2213 5102 y Fo(\014)p 2150 5124 V 2275 5088 a Fs(j)p FB(\))p Ft(:)p 50 5202 546 4 v 159 5263 a Fl(8)197 5289 y Fz(F)-6 b(or)25 b(a)h(small)g(system)g Ff(\000)911 5253 y Fg(q)911 5312 y Fd(\014)969 5289 y Fz(,)g(the)g(sets)h(in)n(t)1398 5301 y Fl(~)-39 b Fg(q)1443 5289 y Ff(\000)1496 5253 y Fg(q)1496 5312 y Fd(\014)1579 5289 y Fz(ma)n(y)25 b(con)n(tain)g(suc)n(h)h(squares)h(only)f(for)k(~) -44 b Fh(q)23 b Fx(6)p Fz(=)f Fh(q)s Fz(.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(21)p eop %%Page: 22 22 22 21 bop 159 340 a FB(So)38 b(it)f(really)g(su\016ces)g(to)i(estimate) f(from)g(b)s(elo)m(w)f(the)h(middle)d(term)j(in)e(\(2.18\).)66 b(Ignore)38 b(the)g(term)50 449 y Ft(u)102 463 y Fq(q)157 381 y Fm(P)259 476 y Fr(~)-42 b Fq(q)s Fo(2)p Fq(Q)415 449 y Fs(j)15 b FB(in)m(t)569 463 y Fr(~)-41 b Fq(q)619 449 y FD(\000)682 463 y Fo(\014)743 449 y Fs(j)25 b FB(whic)m(h)f(w)m (as)h(just)g(estimated)g(in)f(\(2.17\))j(and)e(notice)g(that)g(the)h (\(colored,)g(b)m(y)f(v)m(arious)57 566 y(~)-52 b Ft(q)38 b Fs(2)c Ft(Q)p FB(\))j(cub)s(es)e(of)i(the)f(sk)m(eleton)h(mak)m(e)g (a)g(disjoin)m(t)e(decomp)s(osition)f(of)j(a)g(suitable)d(\\coarse)k (grained")50 675 y Fu(subset)44 b FB(of)36 b(the)f(corresp)s(onding)f (in)m(t)1305 689 y Fr(~)-42 b Fq(q)1355 675 y FD(\000)1418 689 y Fo(\014)1479 675 y FB(.)56 b(Imagine)35 b(that)h(the)g(\\v)m (olume)g(energy)f(excess)i Ft(u)3269 689 y Fr(~)-42 b Fq(q)3303 675 y Fs(j)p Fj(\003)p Fs(j)36 b FB(of)f(an)m(y)50 784 y(suc)m(h)30 b(cub)s(e)g(is)f(con)m(v)m(erted)j(in)m(to)e(the)h(v)m (alue)f(\012")g(.)41 b(Then)30 b(w)m(e)g(ha)m(v)m(e)i(the)f(ob)m(vious) e(b)s(ound)1279 875 y Fm(X)1279 1070 y Fr(~)-41 b Fq(q)r Fo(2)p Fq(Q)1431 961 y Ft(u)1489 975 y Fr(~)f Fq(q)1523 961 y Fs(j)15 b FB(in)m(t)1678 975 y Fr(~)-42 b Fq(q)1727 961 y FD(\000)1790 975 y Fo(\014)1851 961 y Fs(j)26 b Ft(>)f FB(\012)15 b(conn\()p FD(\000)2364 975 y Fo(\014)2425 961 y FB(\))p Ft(:)992 b FB(\(2)p Ft(:)p FB(19\))50 1350 y Fu(Note.)46 b FB(The)24 b(estimate)h(ab)s(o)m(v)m(e,)h(namely)e(the)g (\\con)m(v)m(ersion)h(of)f(the)g(v)m(olume)g(energy)h(excess)f(of)h (the)f(in)m(terior)50 1459 y(of)h FD(\000)211 1473 y Fo(\014)298 1459 y FB(to)h(a)f(sk)m(eleton)h(energy)g(capable)f(to)h(b) s(eat)g(its)e(en)m(trop)m(y")j(con)m(v)m(enien)m(tly)e(replaces)g(and)g (strengthens)50 1568 y(an)30 b(older)g(estimate,\\Main)h(Lemma")g(of)f ([Z84].)42 b(It)31 b(already)f(app)s(eared)f(in)g([HZ])i(and)f([ZR].)50 1799 y FD(Enric)m(hed)35 b(con)m(tour)h(system)e(\000)1285 1766 y Fk(\003)1285 1821 y Fo(\014)1346 1799 y FD(,)h(in)m(terpreted)g (as)g(a)f(tree.)159 2026 y FB(Belo)m(w,)29 b(w)m(e)f(will)e(expand)h (the)h(small,)f(extremal)h(con)m(tour)h(systems)e FD(\000)2580 2040 y Fo(\014)2642 2026 y FB(.)39 b(Recall)28 b(that)g(these)h (systems)50 2135 y(ha)m(v)m(e)h(alw)m(a)m(ys)f(one)g(external)g(con)m (tour,)h(denoted)f(b)s(elo)m(w)f(alw)m(a)m(ys)h(b)m(y)g(sym)m(b)s(ol)f FD(\000)p FB(,)h(but)f(they)h(ma)m(y)g(con)m(tain)50 2244 y(\(though)h(not)h(t)m(ypically\))e(man)m(y)i(in)m(terior)e Fu(nonsmal)5 b(l)42 b FB(con)m(tours.)159 2426 y(In)31 b(principle,)e(w)m(e)k(could)e(ha)m(v)m(e)i(t)m(w)m(o)g(alternativ)m (es)f(no)m(w,)h(either)e(to)i(expand)e(sim)m(ultaneously)f(all)h(the)50 2535 y(extremal)25 b(systems)f FD(\000)815 2549 y Fo(\014)901 2535 y FB(with)g(the)h(same,)h(\\small")e(external)h(con)m(tour)g FD(\000)p FB(,)h(or)f(to)g(expand)f(all)g(the)h(systems)50 2644 y FD(\000)113 2658 y Fo(\014)210 2644 y FB(together,)40 b(with)35 b(a)i(\014xed)f(external)g(con)m(tour)h FD(\000)p FB(.)59 b(The)36 b(latter)h(approac)m(h)f(w)m(ould)g(require)f(a)h(use) h(of)50 2753 y(a)f(factorized,)j(\\external")e(con)m(tour)g(functional) d(\(2.7\))k Ft(F)2079 2767 y Fq(f)2127 2753 y FB(\()p FD(\000)p FB(\).)58 b(While)35 b(the)i(latter,)h(more)e(traditional)50 2862 y(approac)m(h)29 b(is)e(simpler)f(conceptionally)i(the)h(sim)m (ultaneous)e(expansion)g(of)i(the)f(systems)h FD(\000)3215 2876 y Fo(\014)3305 2862 y FB(is)e(actually)50 2971 y(more)j(natural)g (tec)m(hnically)-8 b(,)30 b(and)g(it)g(giv)m(es)g(sharp)s(er)f(con)m (trol)i(of)g(the)f(expansions.)159 3153 y(W)-8 b(e)28 b(will)c(rely)j(here)g(on)f(the)i(notion)e(of)h(a)h(small)d(con)m(tour) j(system,)g(and)f(on)g(the)g(expression)f(\(2.5\))i(and)50 3262 y(the)g(b)s(ound)e(\(2.15\).)42 b(They)27 b(will)e(b)s(e)i(com)m (bined)g(with)g(the)h(observ)m(ation,)h(elab)s(orated)e(no)m(w,)i (namely)e(that)50 3371 y(\\adding)i(of)i(the)f(cub)s(es)g(of)g(the)h (sk)m(eleton)g(mak)m(es)g(an)m(y)g(extremal)f(system)h FD(\000)2699 3385 y Fo(\014)2790 3371 y FB(almost)g(connected".)159 3555 y(Denote)j(b)m(y)e FD(\000)664 3522 y Fk(\003)664 3577 y Fo(\014)758 3555 y FB(the)h(con)m(tour)g(system)f FD(\000)1616 3569 y Fo(\014)1710 3555 y FB(enric)m(hed)f(b)m(y)i(the)f (collection)h(of)f(all)g(the)g(cub)s(es)g(\(ha)m(ving)50 3674 y(appropriate)d(colors\))h(of)h(its)e(sk)m(eleton.)41 b(It)30 b(will)e(b)s(e)h(useful)g(to)h(construct)h(a)f(structure)g(of)g (a)h Fu(tr)-5 b(e)g(e)37 b FB(on)30 b FD(\000)3623 3641 y Fk(\003)3623 3697 y Fo(\014)3685 3674 y FB(,)50 3783 y(with)37 b(a)h(ro)s(ot)g FD(\000)p FB(,)i(for)e(an)m(y)g(simple)e (extremal)j(con)m(tour)f(system)g FD(\000)2390 3797 y Fo(\014)2490 3783 y FB(ha)m(ving)f(an)h(external)g(con)m(tour)h FD(\000)p FB(.)50 3892 y(Suc)m(h)c(a)i(tree)f(will)e(b)s(e)h(de\014ned) g(as)h(a)g(suitable)f(extension)g(of)i(a)f(forest)g Fs(T)2617 3906 y Fn(\000)2666 3915 y Fp(\014)2760 3892 y FB(constructed)h(no)m(w) f(on)g(the)50 4001 y(v)m(ertex)31 b(family)e(of)i(all)e(con)m(tours)i FD(\000)f FB(resp.)40 b FD(\000)1584 4015 y Fq(i)1644 4001 y FB(of)31 b FD(\000)1811 4015 y Fo(\014)1898 4001 y FB(=)24 b(\()p FD(\000)p Ft(;)15 b Fs(f)p FD(\000)2239 4015 y Fq(i)2271 4001 y Fs(g)p FB(\),)31 b(and)f(all)f(cub)s(es)h Fj(\003)g FB(of)g(its)g(sk)m(eleton.)159 4183 y(Consider)37 b(\014rst)i(an)g(auxiliary)e(tree)j Fs(T)1508 4197 y Fq(aux)1675 4183 y FB(whose)f(v)m(ertices)h(are)f(all)f(con)m(tours)i (of)g FD(\000)3137 4197 y Fo(\014)3237 4183 y FB(and)f(also)g(all)50 4292 y Fu(components)k Ft(C)c FB(of)34 b(in)m(t)14 b FD(\000)955 4306 y Fo(\014)1017 4292 y FB(,)34 b(and)f(whose)g(\(pro)m (visionally\))e(ordered)i(b)s(onds)f(are)h(c)m(haracterized)i(just)d (as)50 4401 y(\\go)c(inside".)37 b(Then)26 b(\\forget")i(the)f(v)m (ertices)g(of)f(the)h(t)m(yp)s(e)g(\\comp)s(onen)m(t")g(and)f(from)g (the)h(ordered)e(b)s(onds)50 4511 y(\()p FD(\000)148 4478 y Fo(0)174 4511 y Ft(;)15 b FD(\000)277 4478 y Fo(00)323 4511 y FB(\))21 b(th)m(us)f(arising)f(from)i Fs(T)1109 4525 y Fq(aux)1257 4511 y FB(k)m(eep)g(only)f(those)h(b)s(onds)e (satisfying)g(a)i(relation)f(of)h(commensurabilit)m(y)1388 4688 y(dist)o(\()p FD(\000)p 1570 4703 63 4 v -40 x Fo(0)1658 4688 y Ft(;)15 b FD(\000)p 1698 4703 V 1762 4648 a Fo(00)1808 4688 y FB(\))26 b Fs(\024)f FB(2)15 b(diam)f FD(\000)p 2237 4703 V -40 x Fo(00)2347 4688 y Ft(:)1105 b FB(\(2)p Ft(:)p FB(20\))50 4865 y(Th)m(us)32 b(w)m(e)i(get)h(some)f(forest)g Fs(T)1110 4879 y Fn(\000)1159 4888 y Fp(\014)1251 4865 y FB(of)g(trees)g(on)f(the)h(collection)f(of)h(con)m(tours)g(of)g FD(\000)2922 4879 y Fo(\014)2983 4865 y FB(,)h(not)f(necessarily)e(a)50 4974 y(connected)f(one)g(if)e(the)i(in)m(terior)e(con)m(tours)i(of)f FD(\000)1735 4988 y Fo(\014)1827 4974 y FB(are)h(v)m(ery)f(\\sparse".) 159 5157 y(T)-8 b(o)36 b(mak)m(e)h(a)f(tree)g(from)g(suc)m(h)f Fs(T)1290 5171 y Fn(\000)1339 5180 y Fp(\014)1397 5157 y FB(,)i(denoted)f(b)s(elo)m(w)f(b)m(y)h Fs(T)2275 5124 y Fk(\003)2253 5183 y Fn(\000)2302 5192 y Fp(\014)2396 5157 y FB(and)f(ha)m(ving)g(as)h(set)g(of)g(v)m(ertices)h(the)50 5289 y(family)30 b FD(\000)392 5256 y Fk(\003)392 5312 y Fo(\014)453 5289 y FB(,)j(w)m(e)f(ha)m(v)m(e)h(to)g(add)e(some)h (other)h(\(y)m(et)g(unordered\))e(b)s(onds)f(\(\\forget")k(no)m(w)e (the)g(ordering)f(of)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)e(18:34)1080 b Fz(22)p eop %%Page: 23 23 23 22 bop 50 340 a FB(previously)30 b(constructed)i(b)s(onds,)f(just)h (k)m(eep)h(in)e(mind)f(that)i FD(\000)g FB(will)e(b)s(e)h(the)i(ro)s (ot\))g(going)f(also)g Fu(thr)-5 b(ough)50 449 y(the)32 b(cubes)k FB(of)29 b(the)g(sk)m(eleton.)41 b(These)29 b(new)g(b)s(onds)e(of)i(the)h(t)m(yp)s(e)f(con)m(tour)h Fs(j)f FB(cub)s(e)g(resp.)39 b(cub)s(e)29 b Fs(j)g FB(cub)s(e)f(will)50 558 y(b)s(e)34 b(unordered)g(and)g(de\014ned)g(as)h(follo)m(ws.)54 b(Among)35 b(the)h(ob)5 b(jects,)37 b(either)d(con)m(tours)i FD(\000)3097 525 y Fo(0)3157 558 y FB(or)f(cub)s(es)f Fj(\003)h FB(in)50 667 y(the)30 b(same)h(comp)s(onen)m(t)g Ft(C)37 b FB(of)30 b(in)m(t)15 b FD(\000)1291 681 y Fo(\014)1382 667 y FB(w)m(e)31 b(consider)f(a)g(relation)g(of)g(commensurabilit)m(y) 902 879 y Ft(AcB)91 b Fs(\()-15 b(\))86 b FB(dist)o(\()p Ft(A;)15 b(B)5 b FB(\))26 b Fs(\024)f FB(2)15 b(min)o(\(diam)f Ft(A;)h FB(diam)g Ft(B)5 b FB(\))p Ft(:)620 b FB(\(2)p Ft(:)p FB(21\))50 1090 y(Clearly)-8 b(,)26 b(when)f(adding)f(all)h (these)h(b)s(onds)e(to)j(the)f(ab)s(o)m(v)m(e)h(forest)f Fs(T)2325 1104 y Fn(\000)2374 1113 y Fp(\014)2458 1090 y FB(w)m(e)g(get)h(a)f Fu(conne)-5 b(cte)g(d)30 b(gr)-5 b(aph)7 b FB(.)41 b(Then)50 1200 y(w)m(e)c(tak)m(e)h(a)f(subtree)f(of)h (this)e(graph,)j(b)m(y)e(p)s(ossibly)e(erasing)i(some)h(of)g(the)g (newly)e(added)h(b)s(onds.)57 b(The)50 1309 y(construction)32 b(of)h(suc)m(h)g(a)g(tree)g(is)f(of)h(course)g(non)m(unique,)f(its)g (only)g(aim)g(is)g(to)h(formalize)f(the)h(prop)s(ert)m(y)50 1418 y(of)d(\\almost)h(connectedness")h(of)e(the)h(enric)m(hed)e (system)i FD(\000)2111 1385 y Fk(\003)2111 1440 y Fo(\014)2172 1418 y FB(.)50 1716 y FD(Bounds)36 b(on)f(the)f(deca)m(y)i(of)f Ft(e)1205 1684 y Fo(\000)p Fq(F)10 b Fr(\()p Fn(\000)1395 1693 y Fp(\014)1449 1684 y Fr(\))1517 1716 y FD(for)35 b(extremal)f(con)m(tour)i(systems)e(\000)2951 1730 y Fo(\014)3012 1716 y FD(.)159 1951 y FB(No)m(w)d(w)m(e)g(will)d (transcript)h(\(2.10\))j(in)m(to)f(an)f(analogous)h(inequalit)m(y)d (for)j(the)f(sum)g(o)m(v)m(er)h Fu(systems)39 b FD(\000)3592 1965 y Fo(\014)1526 2095 y Fm(X)1502 2290 y Fn(\000)1551 2299 y Fp(\014)1617 2290 y Fq(\023)12 b(i)1697 2181 y Ft(e)1739 2139 y Fo(\000)p Fq(F)e Fr(\()p Fn(\000)1929 2148 y Fp(\014)1984 2139 y Fr(\)+)p Fq(a)p Fo(j)p Fn(\000)2178 2148 y Fp(\014)p 2129 2168 104 4 v 2232 2139 a Fo(j)50 2496 y FB(where)25 b(the)h(sym)m(b)s(ol)f FD(\000)830 2510 y Fo(\014)906 2496 y Ft(\023)15 b(i)27 b FB(\(\\)p Ft(i)g FB(is)e(incompatible)f(with)g FD(\000)2037 2510 y Fo(\014)2099 2496 y FB("\))i(means)g(that)h Ft(i)f FB(o)m(v)m(erlaps)g(the)g(supp)s(ort)e FD(\000)3623 2510 y Fo(\014)p 3560 2532 125 4 v 3685 2496 a FB(.)159 2685 y(Actually)-8 b(,)28 b(w)m(e)g(will)d(still)h(ha)m(v)m(e)j(to)f(lo)m(w) m(er)g(the)g(constan)m(t)h Ft(a)e FB(\(whic)m(h)g(tests)h(the)g (\\correlation)g(length")g(of)50 2798 y(the)i(gas)h(with)f(fugacities)g Ft(e)1015 2765 y Fo(\000)p Fq(F)10 b Fr(\()p Fn(\000)1205 2774 y Fp(\014)1259 2765 y Fr(\))1292 2798 y FB(\))31 b(sligh)m(tly)-8 b(,)29 b(see)i(b)s(elo)m(w)f(for)g(details.)39 b(T)-8 b(ak)m(e)531 3041 y Ft(a)579 3003 y Fo(0)629 3041 y FB(=)25 b Ft(a)c Fs(\000)27 b FB(~)-52 b Ft(!)154 b FB(where)k(~)-52 b Ft(!)28 b Fs(\031)d Ft(!)e FB(+)d Ft(\030)34 b(e)1948 3003 y Fo(\000)p Fr(\012)2213 3041 y FB(and)182 b Ft(\030)29 b FB(=)c(max)2876 3055 y Fq(q)2946 2967 y Fm(\000)2999 2979 y Ft(u)3051 2993 y Fq(q)p 2999 3020 93 4 v 3012 3103 a FB(\012)3102 2967 y Fm(\001)3156 2961 y Fw(1)p 3155 2971 34 4 v 3155 3008 a Fv(d)3204 3041 y FB(;)248 b(\(2)p Ft(:)p FB(22\))50 3278 y(the)31 b(constan)m(t)i Ft(\030)i FB(b)s(eing)29 b(prop)s(ortional)g(to)j(the)g (maximal)d(in)m(v)m(erse)j(v)m(alue)f(of)g(a)g(diameter)g(of)h(a)f(cub) s(e)g(in)e(a)50 3387 y(sk)m(eleton,)i(see)g(more)g(precisely)e(b)s(elo) m(w.)50 3622 y FD(Theorem)34 b(2.)52 b Fu(F)-7 b(or)34 b(any)f Ft(i)26 b Fs(2)f FB(\003)32 b Fu(we)h(have)g(the)g(bound)g (\(compar)-5 b(e)35 b(\(2.10\))f(and)g(\(2.11\)\))1312 3794 y Fm(X)1080 3990 y Fn(\000)1129 3999 y Fp(\014)1196 3990 y Fq(\023)12 b(i)p Fr(:diam)f Fn(\000)1500 3999 y Fp(\014)1554 3990 y Fr(=)p Fq(N)1722 3881 y Ft(e)1764 3843 y Fo(\000)p Fq(F)f Fr(\()p Fn(\000)1954 3852 y Fp(\014)2008 3843 y Fr(\))2073 3881 y Ft(e)2115 3839 y Fq(a)2154 3811 y Fp(0)2180 3839 y Fo(j)p Fn(\000)2251 3848 y Fp(\014)p 2202 3868 104 4 v 2304 3839 a Fo(j)2388 3881 y Fs(\024)2560 3819 y Ft(b)2599 3833 y Fq(N)p 2560 3860 108 4 v 2594 3932 a FB(~)2583 3955 y Ft(L)3477 3881 y FB(\(2)p Ft(:)p FB(23\))50 4204 y Fu(wher)-5 b(e)34 b Ft(b)346 4218 y Fq(N)447 4204 y Fu(de)-5 b(cay)34 b(ge)-5 b(ometrical)5 b(ly)34 b(like)64 b Ft(b)1488 4218 y Fq(N)1583 4204 y Fs(\024)57 b Ft(K)39 b(\024)1879 4171 y Fq(N)2014 4204 y Fu(and)1190 4342 y Fm(X)1223 4537 y Fq(N)1336 4429 y Ft(b)1375 4443 y Fq(N)1469 4429 y FB(=)25 b Ft(b)191 b 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b(b)s(onds)f(b)s(et)m(w)m (een)j(these)f(ob)5 b(jects)42 b(are)f(as)50 1717 y(describ)s(ed)20 b(ab)s(o)m(v)m(e,)26 b(and)j(~)-52 b Ft(!)25 b FB(will)20 b(estimate)j(the)g(sum)f(of)h(w)m(eigh)m(ts)g(of)f(these)h(b)s(onds.)37 b(Let)23 b(us)f(write)f FD(\000)3408 1684 y Fo(0)3459 1717 y Fs(!)k FD(\000)3638 1684 y Fo(00)3685 1717 y FB(,)50 1826 y FD(\000)113 1793 y Fo(0)163 1826 y Fs(!)h Fj(\003)p FB(,)k Fj(\003)25 b Fs(!)g Fj(\003)689 1793 y Fo(0)744 1826 y FB(etc.)42 b(if)29 b(there)h(could)f(b)s(e)g(a)i(b)s(ond)d(b)s (et)m(w)m(een)j(these)f(t)m(w)m(o)i(ob)5 b(jects)31 b(\(con)m(tours)g (or)f(cub)s(es)50 1950 y(of)g(the)h(sk)m(eleton\))g(in)e(some)i(tree)g Fs(T)1283 1902 y Fn(\000)1332 1911 y Fp(\014)1260 1981 y Fk(\003)1390 1950 y FB(,)g(in)e(the)i(direction)e(\\from)h(the)g(ro)s (ot".)159 2154 y(The)c(sums)g(o)m(v)m(er)i(the)g(trees)f Fs(T)1199 2106 y Fn(\000)1248 2115 y Fp(\014)1176 2185 y Fk(\003)1333 2154 y FB(will)d(b)s(e)j(no)m(w)g(estimated)g(b)m(y)g 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4890 y Fs(\024)55 b Ft(K)2213 4852 y Fo(00)2259 4890 y Fs(j)p Fj(\003)p Fs(j)31 b Ft(\024)2463 4852 y Fr(diam)11 b Fk(\003)2714 4890 y FB(+)20 b(2)p Ft(de)2939 4852 y Fo(\000)p Fr(\012+)6 b(~)-42 b Fq(!)3157 4890 y FB(+)25 b Fs(\024)32 b FB(~)-52 b Ft(!)71 b FB(\(2)p Ft(:)p FB(27\))50 5180 y(where)32 b(again)i(the)f(\014rst)f(part)h(of)h(the)f(estimate)h(b)s (ounds)d(the)i(w)m(eigh)m(ts)h(of)f(con)m(tours)h FD(\000)3087 5147 y Fo(0)o(0)3166 5180 y FB(stic)m(k)m(ed)g(to)g(the)50 5289 y(cub)s(e)f Fj(\003)h FB(and)g(it)g(uses)g(the)g(imp)s(ortan)m(t)g (fact)h(that)g(the)g(con)m(tours)f(are)h(nonsmall)d(i.e.)53 b(ha)m(v)m(e)35 b(a)g(diameter)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(24)p eop %%Page: 25 25 25 24 bop 50 340 a FB(at)33 b(least)f(diam)14 b Fj(\003)p FB(.)46 b(Then)31 b(w)m(e)h(ha)m(v)m(e)i(to)f(c)m(ho)s(ose)g(suitable) 38 b(~)-53 b Ft(!)35 b FB(suc)m(h)d(that)h(the)f(\014nal)f (inequalities)e(in)i(these)50 449 y(last)26 b(t)m(w)m(o)j(b)s(ounds)24 b(are)j(really)f(v)m(alid.)39 b(Let)27 b(us)f(emphasize)g(once)i(again) f(that)g(in)f(b)s(oth)g(inequalities)e(it)i(w)m(as)50 558 y(rather)k(imp)s(ortan)m(t)g(that)h(the)f(size)g(of)h(p)s(ossible)d FD(\000)1787 525 y Fo(00)1863 558 y FB(is)i Fu(big)8 b FB(,)30 b(at)h(least)g Ft(M)40 b FB(resp.)g(1)p Ft(=)p FB(2)15 b(diam)h FD(\000)3237 525 y Fo(0)3262 558 y FB(.)159 740 y(No)m(w)39 b(w)m(e)h(com)m(bine)e(these)h(estimates)h(and)e(our)g (general)h(tree)g(estimate)h(\(1.8\).)68 b(In)38 b(fact,)k(w)m(e)d (need)50 849 y(a)44 b(generalization)f(of)g(\(1.8\))j(\(already)d(men)m (tioned)g(there\))h(for)f Fu(site)i(dependent)53 b FB(quan)m(tities)43 b Ft(\020)3441 863 y Fq(i)3471 849 y FB(,)k(here)50 958 y(denoted)27 b(as)h Ft(\020)539 972 y Fn(\000)588 954 y Fp(0)644 958 y FB(and)f Ft(\020)858 974 y Fk(\003)919 958 y FB(.)39 b(The)27 b(\014nal)f(estimate)i(pro)s(duced)e(from)g(suc) m(h)h(a)h(generalized)f(estimate)h(\(1.8\))h(is)50 1067 y(the)24 b(follo)m(wing)f(one,)j(with)c(the)j(sum)e(b)s(eing)g(o)m(v)m (er)i(all)e(extremal)h(systems)g FD(\000)2613 1081 y Fo(\014)2699 1067 y FB(ha)m(ving)f FD(\000)h FB(as)h(their)e(external) 50 1176 y(con)m(tour)31 b(\(notice)g(that)g(2)21 b Fs(\000)e Ft(e)1079 1143 y Fo(\000)6 b Fr(~)-42 b Fq(!)1214 1176 y Fs(\031)25 b FB(1,)31 b(w)m(e)g(include)d(here)j(also)f(the)h(case)g (of)f(an)h(empt)m(y)g(tree\))297 1309 y Fm(X)161 1504 y Fn(\000)210 1513 y Fp(\014)277 1504 y Fq(\023)12 b(i)p Fr(:)p Fn(\000)412 1513 y Fp(\014)465 1504 y Fo(3)p Fn(\000)633 1309 y Fm(Y)579 1504 y Fn(\000)628 1486 y Fp(0)652 1504 y Fo(2)p Fn(\000)750 1513 y Fp(\014)820 1395 y Ft(e)862 1358 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)1052 1330 y Fp(0)1078 1358 y Fr(\))1141 1395 y Ft(e)1183 1358 y Fr(\()p Fq(a)p Fr(+)p Fq(!)r Fr(+)p Fq(\017)p Fr(\))p Fo(j)p Fn(\000)1539 1330 y Fp(0)p 1490 1370 74 4 v 1565 1358 a Fo(j)1652 1395 y Ft(e)1694 1358 y Fo(\000)6 b Fr(~)-42 b Fq(!)r Fo(j)p Fn(\000)1870 1330 y Fp(0)p 1821 1370 V 1894 1358 a Fo(j)2012 1309 y Fm(Y)1966 1520 y Fk(\003)p Fo(2)p Fn(\000)2121 1498 y Fc(\003)2121 1540 y Fp(\014)2190 1395 y Ft(e)2232 1358 y Fo(\000)p Fr(\012)2401 1395 y Fs(\024)25 b FB(\(2)c Fs(\000)f Ft(e)2731 1358 y Fo(\000)6 b Fr(~)-42 b Fq(!)2841 1395 y FB(\))30 b Ft(e)2948 1358 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))3173 1395 y Ft(e)3215 1358 y Fr(\()p Fq(a)p Fr(+)p Fq(!)r Fr(+)p Fq(\017)p Fr(\))p Fo(j)p Fn(\000)p 3522 1370 50 4 v 1 w Fo(j)50 1726 y FB(and)g(from)f(this)h(w)m(e)h(\014nally)d(get)j(the)g (relation,)f(ob)m(viously)f(pro)m(ving)g(\(2.23\))k(b)m(y)d(\(2.10\)) 482 1859 y Fm(X)251 2054 y Fn(\000)300 2063 y Fp(\014)366 2054 y Fq(\023)13 b(i)p Fr(:diam)d Fn(\000)670 2063 y Fp(\014)724 2054 y Fr(=)p Fq(N)890 1945 y Ft(e)932 1907 y Fo(\000)p Fq(F)g Fr(\()p Fn(\000)1122 1916 y Fp(\014)1176 1907 y Fr(\))1239 1945 y Ft(e)1281 1903 y Fq(a)1320 1876 y Fp(0)1346 1903 y Fo(j)p Fn(\000)1417 1912 y Fp(\014)p 1368 1932 104 4 v 1470 1903 a Fo(j)1552 1945 y Fs(\024)25 b FB(\(1)c(+)27 b(~)-52 b Ft(!)s FB(\))p Ft(e)1977 1907 y Fr(\012)2226 1859 y Fm(X)2049 2054 y Fn(\000)13 b Fq(\023)f(i)p Fr(:diam)e Fn(\000)p Fr(=)p Fq(N)2550 1945 y Ft(e)2592 1907 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\)+\()p Fq(a)p Fr(+)p Fq(!)r Fr(+)p Fq(\017)p Fr(\))p Fo(j)p Fn(\000)p 3173 1919 50 4 v 3 w Fo(j)3251 1945 y Ft(:)201 b FB(\(2)p Ft(:)p FB(28\))166 2468 y FC(3.)51 b(Exp)-6 b(ansion)32 b(of)h(small)f(extremal)g(subsystems)e FD(\000)2292 2482 y Fo(\014)2387 2468 y FC(in)k(an)f(abstra)n(ct)e(P)m(.S.)j(model) 50 2705 y FB(This)d(is)i(the)g(main)g(step)g(of)g(Pirogo)m(v)h(Sinai)d (theory)-8 b(,)35 b(if)e(w)m(e)h(understand)d(the)j(latter)f(just)g(as) h(a)f(metho)s(d)50 2814 y(of)i(\(lo)m(w)g(temp)s(erature\))g(cluster)f (expansions.)52 b(Before)36 b(going)f(to)g(our)g(actual)g(situation,)g (namely)f(to)h(a)50 2923 y(mo)s(del)22 b(with)g(\\matc)m(hing)i (compatible")g(con)m(tour)g(systems)f(and)g(with)f(external)i(cluster)e (\014eld)h(w)m(e)g(explain)50 3032 y(brie\015y)30 b(the)i(\\mono)s(c)m (hromatic")i(v)m(ersion)d(of)h(the)h(problem,)e(arising)f(for)i(the)g (case)i Fs(j)p Ft(Q)p Fs(j)28 b FB(=)g(1.)46 b(This)30 b(is)i(of)50 3141 y(course)c(unin)m(teresting)f(from)g(the)i(p)s(oin)m (t)e(of)h(view)g(of)g(phase)g(transitions,)f(but)h(it)f(is)h(still)e (imp)s(ortan)m(t)h(as)h(a)50 3250 y(source)i(of)h(some)g(standard)e (cluster)h(expansion)f(tec)m(hnology:)50 3483 y FD(A)35 b(tec)m(hnical)g(preparation:)47 b(On)34 b(partially)h(expanded)g(p)s (olymer)f(mo)s(dels.)159 3715 y FB(W)-8 b(e)22 b(will)c(no)m(w)i (rephrase)g(the)h(second)g(section)f(of)h([BZE].)g(The)g(notations)f (will)e(b)s(e)i(tak)m(en)i(in)d(accordance)50 3824 y(with)25 b(that)h(pap)s(er)f(\(for)h(an)g(easy)h(comparison\))f(e.g.)40 b(clusters)25 b(\(here)h(also)g(called)g(c)m(hains\))f(denoted)i(b)m(y) e(us)50 3933 y(later)k(as)h Ft(T)42 b FB(resp.)e Fi(T)31 b FB(are)f(no)m(w)f(pro)m(visionally)e(denoted)j(b)m(y)f Ft(G)g FB(resp.)40 b Ft(I)c FB(and)29 b(con)m(tours)h(\(here)g(called)e (just)50 4042 y(p)s(olymers\))k FD(\000)i FB(b)m(y)g Ft(P)13 b FB(.)51 b(Later)35 b(w)m(e)f(will)d(return)i(to)i(our)e (usual)g(notations.)51 b(W)-8 b(e)35 b(presen)m(t)f(here)g(a)g(sligh)m (tly)50 4151 y(less)28 b(abstract)h(\(in)e(the)i(requiremen)m(ts)f(on)g (the)h(constan)m(ts)g Ft(a)g FB(and)f Ft(b)g FB(resp.)39 b Ft(c)29 b FB(;)g(these)g(are)g Fu(not)38 b FB(considered)50 4260 y(to)33 b(b)s(e)f(general)h(functions)f(of)h(p)s(olymers)e(resp.) 47 b(clusters)32 b(here\))h(but)f(also)h(a)g(sligh)m(tly)e(more)i (general)g(\(in)50 4370 y(the)g(requiremen)m(ts)f(on)g(cluster)g(w)m (eigh)m(ts)h(for)g(clusters)f(touc)m(hing)h(the)f(p)s(olymers\))g(v)m (arian)m(t)h(of)g(Theorem)50 4479 y(2)e(of)f(that)h(pap)s(er.)40 b(The)30 b(pro)s(of)f(is)h(completely)g(analogous)g(to)i(that)e(giv)m (en)h(there,)g(so)f(w)m(e)h(omit)f(it.)159 4660 y(Consider)h(a)i (\\mono)s(c)m(hromatic")h(p)s(olymer)d(mo)s(del)h(with)f(external)i (cluster)f(\014eld)f(i.e.)48 b(the)33 b(mo)s(del)f(of)50 4770 y(the)f(t)m(yp)s(e)f(\(1.11\))j(ho)m(w)m(ev)m(er)f(with)e Fs(j)p Ft(Q)p Fs(j)c FB(=)f(1)31 b(and)f Ft(e)1770 4784 y Fq(q)1836 4770 y Fs(\021)c FB(0.)42 b(Giv)m(en)30 b Fu(complex)43 b FB(fugacities)30 b Fs(f)p Ft(w)3178 4784 y Fq(P)3240 4770 y Fs(g)h FB(and)f Fs(f)p Ft(w)3603 4784 y Fq(G)3664 4770 y Fs(g)50 4879 y FB(de\014ne)f(the)i(partition)e (function)g(in)g(a)i(v)m(olume)f Ft(M)35 b Fs(\032)25 b FB(\003)766 5183 y Ft(Z)7 b FB(\()p Ft(M)j FB(\))26 b(=)1361 5097 y Fm(X)1125 5296 y Fo(f)p Fq(P)1208 5305 y Fw(1)1244 5296 y Fq(;:::)10 b(;P)1410 5305 y Fv(n)1455 5296 y Fo(g)1492 5305 y Fv(c)1526 5296 y Fr(:)p Fq(P)1593 5305 y Fv(i)1622 5296 y Fo(2P)1855 5070 y Fq(n)1819 5097 y Fm(Y)1819 5291 y Fq(i)p Fr(=1)1952 5183 y Ft(w)2017 5197 y Fq(P)2063 5206 y Fv(i)2142 5183 y FB(exp)2296 5001 y Fm(0)2296 5164 y(@)2493 5097 y(X)2375 5296 y Fq(G)p Fk(@)p Fr(\()p Fo([)2566 5305 y Fv(i)2595 5296 y Fq(P)p 2595 5308 57 4 v 2651 5316 a Fv(i)2680 5296 y Fr(\))2708 5278 y Fv(c)2788 5183 y Ft(w)2853 5197 y Fq(G)2914 5001 y Fm(1)2914 5164 y(A)3523 5183 y FB(\(3)p Ft(:)p FB(1\))50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(25)p eop %%Page: 26 26 26 25 bop 50 340 a FB(where)33 b(the)h(\014rst)f(sum)f(is)h(o)m(v)m(er) i(all)d(families)g Fs(f)p Ft(P)1722 354 y Fr(1)1764 340 y Ft(;)15 b(:)g(:)g(:)31 b(;)15 b(P)2038 354 y Fq(n)2089 340 y Fs(g)2134 354 y Fq(c)2205 340 y FB(of)33 b(pairwise)f(compatible) h(p)s(olymers)f(in)g Fs(P)50 449 y FB(and)k(the)h(second)f(sum)894 381 y Fm(P)990 476 y Fq(G)1066 449 y Ft(w)1131 463 y Fq(G)1229 449 y FB(is)g(o)m(v)m(er)i(all)e(c)m(hains)g Ft(G)g FB(touc)m(hing)g(the)h(complemen)m(t)g(of)g(the)g(union)e(of)50 558 y(supp)s(orts)j(of)i(all)f(p)s(olymer)g Ft(P)1103 572 y Fq(i)1133 558 y FB(,)k(the)d(latter)g(prop)s(ert)m(y)g(b)s(eing)e (denoted)i(as)h Ft(G)g Fj(@)g FB(\()p Fs([)3056 572 y Fq(i)3086 558 y Ft(P)p 3086 573 71 4 v 3157 581 a Fq(i)3187 558 y FB(\))3222 525 y Fq(c)3260 558 y FB(.)69 b(In)39 b([BZE])50 667 y(and)d(other)g(references,)j(only)d(a)h(sum)e(o)m(v)m (er)j(those)f Ft(G)f FB(whic)m(h)f(are)i(compatible)f(with)f(all)g Ft(P)3264 681 y Fq(i)3331 667 y FB(is)h(usually)50 776 y(considered.)i(The)26 b(quan)m(tities)f Ft(w)1192 790 y Fq(G)1279 776 y FB(are)i(often)f(assumed)g(to)h(b)s(e)e(translation)g (in)m(v)m(arian)m(t.)39 b(It)27 b(is)e(reasonable)50 885 y(to)31 b(assume)f(the)h(latter)f(prop)s(ert)m(y)g(only)g(for)g (those)h Ft(G)e FB(in)g(\(3.1\))k(whic)m(h)c(are)h(compatible)g(with)f (all)g Ft(P)3520 899 y Fq(i)3551 885 y FB(.)159 1067 y(In)m(v)m(estigate)j(the)f(T)-8 b(a)m(ylor)30 b(series)g(of)g(the)h (logarithm)e(of)i(partition)e(function)g(\(3.1\))550 1298 y(log)16 b Ft(Z)7 b FB(\()p Ft(M)j FB(\))26 b(=)1146 1212 y Fm(X)1041 1411 y Fq(I)5 b Fo(2I)t Fr(\()p Fo(P)g([G)t Fr(\))1397 1298 y Ft(w)1462 1312 y Fq(I)1777 1298 y FB(with)271 b Ft(w)2291 1312 y Fq(I)2358 1298 y FB(=)25 b Ft(n)2509 1312 y Fq(I)2579 1212 y Fm(Y)2566 1407 y Fq(P)10 b Fo(2)p Fq(I)p 2671 1419 38 4 v 2723 1298 a Ft(w)2790 1257 y Fq(I)2822 1266 y Fv(P)2788 1325 y Fq(P)2908 1212 y Fm(Y)2895 1407 y Fq(G)p Fo(2)p Fq(I)p 3001 1419 V 3053 1298 a Ft(w)3120 1257 y Fq(I)3152 1266 y Fv(G)3118 1325 y Fq(G)3523 1298 y FB(\(3)p Ft(:)p FB(2\))50 1616 y(where)41 b(the)g(sum)g(is)g(o)m(v)m (er)h(the)g(collection)f Fs(I)50 b FB(=)44 b Fs(I)7 b FB(\()p Fs(P)35 b([)27 b(G)5 b FB(\))42 b(of)g(all)e(m)m(ulti-indices)e Ft(I)49 b FB(\(in)m(teger)42 b(v)m(alued)50 1726 y(functions)21 b(on)h Fs(P)11 b([)t(G)5 b FB(\))23 b(and)e Ft(n)1031 1740 y Fq(I)1095 1726 y FB(are)h(corresp)s(onding)e(partial)h(deriv)m (ativ)m(es)h(of)h(log)16 b Ft(Z)29 b FB(at)23 b Fs(f)p Ft(w)3078 1740 y Fq(P)3164 1726 y FB(=)i(0)p Ft(;)15 b(w)3410 1740 y Fq(G)3498 1726 y FB(=)25 b(0)p Fs(g)p FB(.)159 1907 y(The)i Fu(support)37 b FB(of)28 b(a)f(m)m(ulti-index)e Ft(I)35 b FB(is)26 b(de\014ned)g(as)h Ft(I)p 1908 1922 48 4 v 33 w FB(=)e Fs(f)p Ft(P)38 b FB(:)26 b Ft(I)2309 1921 y Fq(P)2395 1907 y Ft(>)f FB(0)p Fs(g)14 b([)g(f)p Ft(G)25 b FB(:)h Ft(I)2903 1921 y Fq(G)2989 1907 y Ft(>)f FB(0)p Fs(g)p FB(.)41 b(A)27 b(collection)50 2016 y Fs(P)h([)20 b(G)35 b FB(is)30 b(called)f(a)i Fu(cluster)41 b FB(if)29 b(the)h(graph)g(with)f(b)s(onds)g Ft(P)43 b(\023)31 b(P)2243 1983 y Fo(0)2299 2016 y FB(and)e Ft(P)44 b(\023)30 b(G)g FB(is)f Fu(conne)-5 b(cte)g(d)9 b FB(.)159 2198 y(It)37 b(can)g(b)s(e)f(easily)f(sho)m(wn)h(that)i(nonzero)e Ft(n)1708 2212 y Fq(I)1786 2198 y FB(app)s(ear)g(only)g(for)g(clusters) g(i.e.)60 b(for)36 b(connected,)j(\\in-)50 2307 y(decomp)s(osable")d(m) m(ultiindices)d Ft(I)7 b FB(.)59 b(The)36 b Fu(de)-5 b(composability)48 b FB(of)36 b(a)h(m)m(ultiindex)d(with)h(a)i(supp)s (ort)e Fs(P)c([)24 b(G)50 2416 y FB(=)i(\()p Fs(P)245 2430 y Fr(1)307 2416 y Fs([)21 b(G)443 2430 y Fr(1)484 2416 y FB(\))52 b Fs([)f FB(\()p Fs(P)781 2430 y Fr(2)843 2416 y Fs([)21 b(G)979 2430 y Fr(2)1020 2416 y FB(\))31 b(means)g(that)h(an)m(y)g(p)s(olymer)d(resp.)43 b(c)m(hain)31 b(from)f(the)i(\014rst)e(system)i(is)e(com-)50 2526 y(patible)i(with)g (an)m(y)i(one)f(from)g(the)h(second)f(one.)51 b(Since)32 b(all)g(the)i(c)m(hains)f Ft(G)g FB(are)g(m)m(utually)f(compatible,)50 2635 y(incompatibilit)m(y)23 b(can)k(only)f(o)s(ccur)h(b)s(et)m(w)m (een)g(t)m(w)m(o)h(p)s(olymers)d(or)i(b)s(et)m(w)m(een)g(a)g(p)s (olymer)e(and)h(a)h(c)m(hain,)g(and)50 2744 y(it)j(is)f(usually)f (de\014ned)h(b)m(y)i(the)f(requiremen)m(t)g(that)h(the)f(supp)s(orts)f (of)h(these)h(ob)5 b(jects)31 b(\\o)m(v)m(erlap".)159 2928 y(No)m(w)g(de\014ne)f(auxiliary)e(constan)m(ts)j Ft(L)1520 2876 y(:)1497 2928 y FB(=)25 b(1)31 b(and)1857 2905 y(~)1846 2928 y Ft(L)1956 2876 y(:)1933 2928 y FB(=)25 b(1)31 b(as)f(follo)m(ws,)g(with)f(a)i(\(small\))e Ft(\016)35 b FB(giv)m(en)30 b(b)s(elo)m(w)990 3202 y Ft(L)25 b FB(=)1184 3140 y Fs(\000)15 b FB(log)q(\(1)21 b Fs(\000)f Ft(\016)s FB(\))p 1184 3181 475 4 v 1399 3264 a Ft(\016)1669 3202 y(;)76 b(E)31 b FB(=)1975 3140 y Ft(e)2017 3107 y Fq(\016)2077 3140 y Fs(\000)20 b FB(1)p 1975 3181 239 4 v 2072 3264 a Ft(\016)2225 3202 y(;)2397 3179 y FB(~)2386 3202 y Ft(L)25 b FB(=)g Ft(L)2631 3164 y Fr(2)2672 3202 y Ft(E)5 b(:)754 b FB(\(3)p Ft(:)p FB(3\))50 3445 y(In)29 b(the)i(pro)s(of)f(of) g(the)h(statemen)m(ts)h(\(3.9\))g(and)e(\(3.10\))i(b)s(elo)m(w)e(it)g (will)e(b)s(e)h(imp)s(ortan)m(t)h(that)933 3653 y(~)922 3676 y Ft(L)c(>)e(L)152 b FB(and)f(1)21 b(+)e Ft(LE)5 b FB(\()p Ft(e)1995 3611 y Fv(Lx)p 1996 3621 80 4 v 2022 3655 a Fw(~)2014 3666 y Fv(L)2111 3676 y Fs(\000)20 b FB(1\))26 b Fs(\024)f Ft(e)2446 3638 y Fq(x)2523 3676 y Ft(;)45 b(x)26 b Fs(\025)f FB(0)p Ft(:)686 b FB(\(3)p Ft(:)p FB(4\))50 3961 y FD(Theorem)34 b(3.)47 b(On)34 b(deca)m(y)h(of)g(the)g(terms)f(in)h(\(3.2\).)52 b Fu(L)-5 b(et)33 b Ft(\016)j Fu(be)c(chosen)i(such)f(that)g(the)g(bounds)661 4181 y Fs(j)p Ft(w)751 4195 y Fq(P)812 4181 y Fs(j)p Ft(e)879 4144 y Fq(a)p Fo(j)p Fq(P)p 940 4156 57 4 v 10 w Fo(j)1048 4181 y Fs(\024)25 b Ft(\016)n(;)64 b Fs(j)p Ft(w)1361 4195 y Fq(G)1422 4181 y Fs(j)26 b(\024)f Ft(\016)161 b Fu(and)e FB(\()p Ft(e)2149 4144 y Fo(j)p Fq(w)2224 4153 y Fv(G)2277 4144 y Fo(j)2324 4181 y Fs(\000)19 b FB(1\))34 b(\()p Ft(e)2605 4144 y Fq(b)p Fo(j)p Fq(G)p 2659 4156 V Fo(j)2762 4181 y Fs(\000)20 b FB(1\))26 b Fs(\024)f Ft(\016)428 b FB(\(3)p Ft(:)p FB(5\))50 4394 y Fu(hold)34 b(for)f(any)g Ft(P)39 b Fs(2)25 b(P)40 b Fu(and)33 b Ft(G)25 b Fs(2)g(G)5 b Fu(.)42 b(Assume)32 b(mor)-5 b(e)g(over)35 b(that)f(ther)-5 b(e)33 b(ar)-5 b(e)34 b(some)f Ft(a)26 b(>)f(b)g(>)g FB(0)33 b Fu(such)f(that)29 4576 y(1\))47 b(Polymer)33 b(fugacities)g(satisfy)g(the)g(condition:)43 b(for)33 b(any)h Ft(i)25 b Fs(2)g Ft(M)43 b Fu(we)32 b(have)2709 4543 y Fr(9)1494 4805 y Fm(X)1403 5000 y Fq(P)10 b Fo(2P)5 b Fr(:)p Fq(P)p 1586 5012 V 10 w Fo(3)p Fq(i)1732 4891 y Fs(j)p Ft(w)1822 4905 y Fq(P)1883 4891 y Fs(j)p Ft(e)1950 4854 y Fq(a)p Fo(j)p Fq(P)p 2011 4866 V 10 w Fo(j)2119 4891 y Fs(\024)2270 4830 y Ft(b)p 2258 4870 62 4 v 2269 4943 a FB(~)2258 4966 y Ft(L)2331 4891 y(:)1167 b FB(\(3)p Ft(:)p FB(6\))p 50 5107 546 4 v 159 5172 a Fl(9)197 5198 y Fz(One)29 b(w)n(ould)f(lik)n(e)h(to)g(ha)n(v)n (e)1023 5180 y(~)1013 5198 y Fh(L)d Fz(=)g Fh(L)g Fz(=)g(1)i(here)i (and)e(ev)n(erywhere.)44 b(Ho)n(w)n(ev)n(er,)29 b(w)n(e)f(do)g(not)h (kno)n(w)e(whether)i(our)f(result)50 5289 y(can)d(b)r(e)i(really)f (sharp)r(ened)h(in)f(suc)n(h)g(a)g(w)n(a)n(y)-6 b(,)24 b(lik)n(e)j(\(see)g([KP]\))f(in)g(the)h(case)f(of)f(ordinary)g(p)r (olymer)h(mo)r(dels)g(with)g Fh(w)3463 5301 y Fg(G)3542 5289 y Fx(\021)21 b Fz(0.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(26)p eop %%Page: 27 27 27 26 bop 29 340 a Fu(2\))47 b(Chain)33 b(fugacities)g Ft(w)898 354 y Fq(G)991 340 y Fu(satisfy)h(the)f(condition:)43 b(for)33 b(any)g Ft(i)26 b Fs(2)f Ft(M)1511 516 y Fm(X)1424 711 y Fq(G)p Fo(2G)t Fr(:)p Fq(G)p 1599 723 57 4 v -1 w Fo(3)p Fq(i)1744 602 y Fs(j)p Ft(w)1834 616 y Fq(G)1896 602 y Fs(j)p Ft(e)1963 565 y Fq(b)p Fo(j)p Fq(G)p 2017 577 V Fo(j)2125 602 y Fs(\024)g Ft(a)20 b Fs(\000)g Ft(b:)1079 b 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(y)f(include)e(all)h(the)50 4455 y(new,)g(non)m(trivial)f(clusters)h Ft(I)j Fs(6)p FB(=)25 b Ft(G)k FB(in)f(the)i(\014nal)f(estimate)h (\(3.11\).)43 b(This)27 b(will)g(b)s(e)i(our)h(main)e(philosoph)m(y)50 4564 y(also)i(in)f(all)g(the)i(forthcoming)f(estimates,)h(for)f(sums)g (of)g(more)h(complicated)f(clusters.)159 4818 y(It)k(will)e(b)s(e)i (useful)f(to)i(form)m(ulate)f(also)g(the)h(follo)m(wing)e(ob)m(vious)h (consequence)h(of)f(\(3.11\).)56 b(Consider)50 4930 y(new)26 b(auxiliary)e(w)m(eigh)m(ts)46 b(~)-64 b Ft(w)993 4944 y Fq(P)1079 4930 y FB(=)25 b Ft(w)1240 4944 y Fq(P)1301 4930 y Ft(e)1343 4897 y Fq(\013)p Fo(j)p Fq(P)p 1412 4909 V 10 w Fo(j)1521 4930 y FB(and)44 b(~)-63 b Ft(w)1759 4944 y Fq(G)1845 4930 y FB(=)25 b Ft(w)2006 4944 y Fq(G)2067 4930 y Ft(e)2109 4897 y Fq(\013)p Fo(j)p Fq(G)p 2178 4909 V Fo(j)2288 4930 y FB(and)h(rewrite)g(\(3.11\))i(using)e(the)g(ob) m(vious)50 5039 y(inequalit)m(y)j Fs(j)p Ft(I)p 497 5054 48 4 v 7 w Fs(j)c(\024)g(j)p Ft(I)7 b Fs(j)31 b FB(where)f Fs(j)p Ft(I)7 b Fs(j)26 b FB(=)1300 4971 y Fm(P)1396 5067 y Fq(P)10 b Fo(2)p Fq(I)p 1501 5079 38 4 v 1558 5039 a Ft(I)1598 5053 y Fq(P)1658 5039 y Fs(j)p Ft(P)p 1683 5054 71 4 v 14 w Fs(j)20 b FB(+)1891 4971 y Fm(P)1987 5067 y Fq(G)p Fo(2)p Fq(I)p 2093 5079 38 4 v 2149 5039 a Ft(I)2189 5053 y Fq(G)2251 5039 y Fs(j)p Ft(G)p 2276 5054 72 4 v -1 w Fs(j)31 b FB(and)f Fs(j)p Ft(I)p 2605 5054 48 4 v 7 w Fs(j)25 b FB(=)2798 4971 y Fm(P)2894 5067 y Fq(P)10 b Fo(2)p Fq(I)p 2999 5079 38 4 v 3056 5039 a Fs(j)p Ft(P)p 3081 5054 71 4 v 14 w Fs(j)20 b FB(+)3289 4971 y Fm(P)3385 5067 y Fq(G)p Fo(2)p Fq(I)p 3491 5079 38 4 v 3547 5039 a Fs(j)p Ft(G)p 3572 5054 72 4 v Fs(j)p FB(:)p 50 5202 546 4 v 159 5263 a Fl(10)230 5289 y Fz(\\Old")26 b(clusters)i(are)d(just)h(c)n(hains)h Fh(G)f Fz(con)n(taining)f Fh(i)p Fz(,)i(\\new")e(clusters)j(con)n(tain) e(a)f(p)r(olymer)g(con)n(taining)h Fh(i)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)j(18:34)1080 b Fz(27)p eop %%Page: 28 28 28 27 bop 50 340 a FD(Corollary)35 b(1.)52 b Fu(L)-5 b(et)33 b(the)g(weights)h Ft(w)1340 354 y Fq(P)1433 340 y Fu(and)f Ft(w)1674 354 y Fq(G)1768 340 y Fu(satisfy)h(the)f (assumptions)612 543 y Fm(X)521 738 y Fq(P)10 b Fo(2P)5 b Fr(:)p Fq(P)p 704 750 57 4 v 10 w Fo(3)p Fq(i)850 629 y Fs(j)p Ft(w)940 643 y Fq(P)1001 629 y Fs(j)p Ft(e)1068 592 y Fr(\()p Fq(a)p Fr(+)p Fq(\013)p Fr(\))p Fo(j)p Fq(P)p 1288 604 V 11 w Fo(j)1396 629 y Fs(\024)1547 568 y Ft(b)p 1536 608 62 4 v 1547 681 a FB(~)1536 704 y Ft(L)1767 629 y Fu(and)2171 543 y Fm(X)2084 738 y Fq(G)p Fo(2G)t Fr(:)p Fq(G)p 2259 750 57 4 v -2 w Fo(3)p Fq(i)2404 629 y Fs(j)p Ft(w)2494 643 y Fq(G)2555 629 y Fs(j)p Ft(e)2622 592 y Fr(\()p Fq(b)p Fr(+)p Fq(\013)p Fr(\))p Fo(j)p Fq(G)p 2835 604 V 1 w Fo(j)2944 629 y Fs(\024)25 b Ft(a)c Fs(\000)e Ft(b)239 b FB(\(3)p Ft(:)p FB(12\))50 972 y Fu(wher)-5 b(e)34 b Ft(\013)25 b Fs(\025)g FB(0)p Fu(.)42 b(Then)33 b(we)g(have)g(the)g(estimate)900 1140 y Fm(X)704 1340 y Fq(I)5 b Fo(2C)s Fr(\()p Fo(P)g([G)t Fr(\):)41 b Fq(I)18 b Fo(3)p Fq(i)1244 1227 y Fs(j)p Ft(w)1334 1241 y Fq(I)1376 1227 y Fs(j)33 b Ft(e)1476 1189 y Fq(\013)p Fo(j)p Fq(I)p 1545 1201 38 4 v 5 w Fo(j)1731 1227 y Fs(\024)2024 1140 y Fm(X)1827 1340 y Fq(I)5 b Fo(2C)s Fr(\()p Fo(P)g([G)t Fr(\):)42 b Fq(I)17 b Fo(3)p Fq(i)2367 1227 y Fs(j)p Ft(w)2457 1241 y Fq(I)2499 1227 y Fs(j)33 b Ft(e)2599 1189 y Fq(\013)p Fo(j)p Fq(I)5 b Fo(j)2854 1227 y Fs(\024)58 b Ft(a:)421 b FB(\(3)p Ft(:)p FB(13\))159 1647 y(This)29 b(Corollary)g(will)f(b)s(e)j(no)m(w)f(mo)s(di\014ed)f(to)j(b)s(e)e (useful)f(also)i(in)e(the)i(cases,)h(studied)d(b)s(elo)m(w,)i(when)e(a) 50 1756 y Fu(se)-5 b(quential)35 b FB(expansion)24 b(of)i(p)s(olymers)e (from)h Fs(P)33 b FB(is)24 b(p)s(erformed.)38 b(Though)24 b(the)i(idea)e(of)i(an)f(expansion)g(of)g Fs(P)50 1866 y FB(in)k(more)h(than)g(one)h(step)f(lo)s(oks)g(unnecessary)g(and)f (arbitrary)g(here)h(\(in)g(the)g(case)i(of)e(p)s(olymer)f(mo)s(dels\)) 50 1975 y(the)h(same)h(idea)f(will)e(b)s(e)i(indisp)s(ensable)c(later,) k(in)f(the)i(more)f(complex)g(situation)g(of)g(P)m(.S.)h(theory)-8 b(.)50 2230 y FD(Sequen)m(tial)35 b(v)m(ersion)g(of)g(the)g(ab)s(o)m(v) m(e)g(expansion.)159 2486 y FB(Let)j(us)f(decomp)s(ose)h(the)h(family)d Fs(P)45 b FB(in)m(to)38 b(sub)s(classes,)g Fs(P)46 b FB(=)37 b Fs([)2409 2453 y Fo(1)2409 2508 y Fq(n)p Fr(=1)2550 2486 y Fs(P)2613 2500 y Fq(n)2701 2486 y FB(,)j(where)d(t)m(ypically)g Fs(P)3478 2500 y Fq(n)3565 2486 y FB(is)g(a)50 2595 y(collection)g(of)h (all)e(p)s(olymers)g(from)i Fs(P)45 b FB(whic)m(h)36 b(ha)m(v)m(e)j(a)f(\(suitably)e(de\014ned\))h(\\size")h Ft(n)p FB(.)62 b(Then)37 b(w)m(e)h(ma)m(y)50 2704 y(apply)30 b(the)i(ab)s(o)m(v)m(e)h(result)e(inductiv)m(ely;)f(the)i(\014rst)g (step)f(b)s(eing)g(the)h(cluster)f(expansion,)g(in)g(the)h(st)m(yle)g (of)50 2813 y(ab)s(o)m(v)m(e)f(theorem,)f(of)g(the)g(\\mixed")f(mo)s (del)f(emplo)m(ying)h(all)f(the)i(clusters)f Ft(G)g FB(of)h(the)f (original)f(mo)s(del)g(but)50 2922 y Fu(only)39 b(those)44 b FB(p)s(olymers)35 b Ft(P)49 b FB(whic)m(h)35 b(are)i(from)f(the)h (subfamily)d Fs(P)2293 2936 y Fr(1)2334 2922 y FB(.)58 b(The)36 b(second)h(step)f(will)e(then)i(b)s(e)g(to)50 3031 y(allo)m(w,)29 b(and)g(\014x)g(temp)s(orarily)-8 b(,)29 b(a)h(compatible)f(collection)g(of)h(p)s(olymers)e(from)h Fs([)2835 3045 y Fq(k)r Fo(\025)p Fr(3)2972 3031 y Fs(P)3035 3045 y Fq(k)3110 3031 y FB(and)g(to)h(p)s(erform)50 3140 y(the)j(cluster)f(expansion)f(of)i(the)g(\\rest")h(of)f(the)g (con\014gurations)f(\(already)g(partially)f(expanded)h(b)m(y)h(the)50 3249 y(\014rst)f(step,)i(con)m(taining)f(still)e(nonexpanded)h(p)s (olymers)g(from)g Fs(P)2315 3263 y Fr(2)2389 3249 y FB(and)h(also)g (\\\014rst)g(lev)m(el")g(clusters)g Ft(G)p FB(\).)50 3358 y(Then)24 b(another)h(\\mixed)f(mo)s(del")g(of)h(the)g(t)m(yp)s(e) g(ab)s(o)m(v)m(e)h(will)c(b)s(e)i(obtained,)i(with)d(p)s(olymers)g (from)h Fs([)3464 3372 y Fq(k)r Fo(\025)p Fr(3)3602 3358 y Fs(P)3665 3372 y Fq(k)50 3467 y FB(and)29 b(one)g(can)h(expand)f(it)g (again)g(\(\014xing)g(no)m(w)g(a)h(compatible)f(family)f(of)i(p)s (olymers)d(from)i Fs([)3258 3481 y Fq(k)r Fo(\025)p Fr(4)3396 3467 y Fs(P)3459 3481 y Fq(k)3504 3467 y FB(\))g(etc.)159 3649 y(It)f(is)f(imp)s(ortan)m(t)g(to)i(require)e(that)i(the)f(con)m (trol)g(o)m(v)m(er)i(these)e(expansions)f(should)f(not)j(b)s(e)e(lost)h (during)50 3758 y(these)i(expansion)e(pro)s(cesses)i(\(requiring)d(an)j (in\014nite)d(n)m(um)m(b)s(er)h(of)i(steps)g(in)e(the)i(thermo)s (dynamic)e(limit)50 3867 y(\003)42 b Fs(!)f Fi(Z)352 3834 y Fq(\027)393 3867 y FB(\))f(more)h(sp)s(eci\014cally)c(that)k (the)g(b)s(ounds)d(\(3.13\))k(surviv)m(e)d(\(with)g(an)h(appropriate)g (c)m(hoice)h(of)50 3977 y(constan)m(ts\))d(suc)m(h)e(a)h(sequen)m(tial) f(pro)s(cess:)53 b(Denote)38 b(b)m(y)f Fs(G)2088 3991 y Fq()28 b Fs(j)p Ft(w)1002 1980 y Fq(P)1063 1966 y Fs(j)p Ft(e)1130 1933 y Fr(\()p Fq(\013)p Fr(+)p Fq(a)1300 1942 y Fv(n)1346 1933 y Fr(\))p Fo(j)p Fq(P)p 1396 1945 V 10 w Fo(j)1479 1966 y FB(,)33 b(the)g(assumptions)e(of)i(the)g(\014rst)f(part)g(of)h(\(3.15\))i(are)e (actually)50 2075 y(w)m(eak)m(er)f(\(if)e(summed)g(together)i(for)f (all)f Ft(n)p FB(\))g(than)h(that)h(of)f(\(3.6\).)44 b(So)30 b(w)m(e)i(come)g(to)f(a)g(conclusion)f(that)h(a)50 2184 y(sequen)m(tial)c(construction)g(of)h(the)g(cluster)e(expansion)h (giv)m(es)h(ev)m(en)g(a)g(sligh)m(tly)e(more)i(detailed)f(con)m(trol)h (of)50 2293 y(its)i(con)m(v)m(ergence)i(than)e(the)h(usual)e(expansion) g(\\at)j(once".)50 2534 y FD(T)-9 b(o)m(w)m(ards)35 b(a)g(cluster)g (expansion)g(of)g(\\p)s(olyc)m(hromatic",)g(abstract)g(P)m(.S.)h(mo)s (dels.)159 2774 y FB(W)-8 b(e)32 b(return)e(bac)m(k)i(no)m(w)f(to)h (our)e(previous)g(situation)g(of)h(an)g(abstract)h(P)m(.S.)g(mo)s(del,) e(and)g(to)i(previous)50 2883 y(notations.)57 b(T)-8 b(ak)m(e)37 b(again)e(a)i(mo)s(del)d(of)i(the)g(t)m(yp)s(e)g(\(1.23\).) 59 b(Assume)35 b(that)h(for)g(eac)m(h)g Ft(q)j FB(some)d(collection)50 2992 y Fs(H)128 2959 y Fq(q)193 2992 y Fs(\032)25 b(A)362 2959 y Fq(q)432 2992 y FB(of)30 b Fu(smal)5 b(l)34 b(contours)39 b FD(\000)25 b FB(=)g FD(\000)1402 2959 y Fq(q)1472 2992 y FB(is)30 b(selected,)h(with)e(the)h(prop)s(erties)f(describ)s(ed)f (ab)s(o)m(v)m(e)k(i.e.)40 b(suc)m(h)50 3101 y(that)30 b(an)m(y)g(extremal)g(con)m(tour)g(system)g FD(\000)1487 3115 y Fo(\014)1578 3101 y FB(with)e(the)i(external)g(con)m(tour)g FD(\000)f FB(con)m(tains)h(no)g(con)m(tour)g(from)50 3217 y Fs([)111 3231 y Fq(q)146 3213 y Fp(0)171 3231 y Fo(2)p Fq(Q)281 3217 y Fs(H)359 3184 y Fq(q)394 3157 y Fp(0)453 3217 y FB(other)g(than)f FD(\000)p FB(.)40 b(Assume)29 b(that)h(the)g(c)m(hoice)g(of)f Fs(fH)2211 3184 y Fq(q)2252 3217 y Fs(g)h FB(is)e(non)m(trivial)f(i.e.)41 b Fs([)3048 3231 y Fq(q)r Fo(2)p Fq(Q)3194 3217 y Fs(H)3272 3184 y Fq(q)3337 3217 y Fs(6)p FB(=)25 b Fs(;)p FB(;)30 b(then)50 3326 y(w)m(e)j(will)e(sa)m(y)i(that)h(our)e(mo)s(del)g(is)g (not)h(y)m(et)i(fully)30 b(expanded.)48 b(T)m(ypically)31 b(w)m(e)j(tak)m(e)g(for)f Fs(H)3189 3293 y Fq(q)3262 3326 y FB(the)g(class)g(of)50 3436 y(all)c(smallest)h(existing)f(small) g(con)m(tours)i FD(\000)1530 3403 y Fq(q)1600 3436 y FB(of)g(the)f(mo)s(del.)159 3617 y(If)h Fs([)312 3631 y Fq(q)r Fo(2)p Fq(Q)458 3617 y Fs(H)536 3584 y Fq(q)603 3617 y FB(=)c Fs(;)32 b FB(holds)e(\(for)h(a)h(giv)m(en)g(\014nite)e (torus)i(\003\))g(and)f(moreo)m(v)m(er)i(if)d Fs(R)2838 3584 y Fq(q)2910 3617 y FB(is)g(already)h Fu(empty)41 b FB(for)50 3726 y(all)32 b Ft(q)i Fs(2)d Ft(Q)i FB(minimizing)d(the)k (v)m(alue)g Ft(f)1370 3740 y Fq(q)1443 3726 y FB(w)m(e)g(call)f(our)h (mo)s(del)e Fu(ful)5 b(ly)36 b(expande)-5 b(d)37 b(in)k FB(\003.)51 b(This)32 b(is)h(of)h(course)50 3836 y(our)h(goal,)k(and)c (b)s(elo)m(w)g(w)m(e)i(explain)d(ho)m(w)i(to)h(reac)m(h)g(it.)57 b(First)35 b(w)m(e)i(form)m(ulate)f(and)f(analyze)i(our)e(basic)50 3945 y(expansion)f(step)h({)g(Theorem)f(4.)55 b(A)35 b(more)g(detailed)f(discussion)f(\(and)h(mo)s(di\014cation\))g(of)h (the)g(sc)m(heme)50 4054 y(\(1.22\),)j(and)d(ho)m(w)f(it)h(c)m(hanges)h (during)c(the)j(application)e(of)i(expansion)f(steps,)i(will)c(b)s(e)j (then)f(giv)m(en)h(in)50 4163 y(the)30 b(forthcoming)g(c)m(hapter.)159 4345 y(The)k(follo)m(wing)g(result)g(will)f(b)s(e)h(a)h(crucial)f(one.) 56 b(W)-8 b(e)36 b(already)f(form)m(ulated)f(ab)s(o)m(v)m(e)i(its)f (preparatory)50 4454 y(v)m(arian)m(t)22 b(\(Corollary)f(2\))i(for)e (the)h(\\mono)s(c)m(hromatic")i(case)f(of)f(p)s(olymer)e(mo)s(dels.)37 b(The)21 b(idea)g(of)h(a)h(partially)50 4563 y(expanded)k Fu(polymer)40 b FB(mo)s(del)28 b(w)m(as)h(already)f(used)f(there)i(and) f(no)m(w)g(w)m(e)h(ha)m(v)m(e)h(to)f(adapt)g(it)f(to)h(the)g(sp)s (eci\014c)50 4672 y(situation)42 b(of)h(P)m(.S.)g(theory)-8 b(.)80 b(Ho)m(w)m(ev)m(er,)48 b(the)c(expansion)d(of)j(con)m(tour)f (systems)g(ha)m(ving)g(the)g(activit)m(y)50 4781 y Ft(e)92 4748 y Fo(\000)p Fq(F)10 b Fr(\()p Fn(\000)282 4757 y Fp(\014)336 4748 y Fr(\))410 4781 y FB(will)39 b(require)h 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30 30 29 bop 159 340 a FB(It)45 b(is)f(v)m(ery)h(imp)s(ortan)m(t)f(to)h (realize)g(the)g(idea)f(that)i(b)s(efore)e(in)m(v)m(estigating)h(the)g (expandabilit)m(y)e(of)50 449 y(some)d FD(\000)350 463 y Fo(\014)453 449 y FB(=)h(\()p FD(\000)p Ft(;)15 b Fs(f)p FD(\000)811 463 y Fq(i)843 449 y Fs(g)40 b FB(w)m(e)h(need)f(to)h (expand)e(carefully)g(the)h(mo)s(del)f(\\with)h(con)m(tours)g(smaller)f (than)50 558 y FD(\000)p FB(".)50 b(Actually)-8 b(,)34 b(one)f(cannot)h(ev)m(en)g(compute)g(exactly)g Ft(F)13 b FB(\()p FD(\000)2160 572 y Fo(\014)2221 558 y FB(\))34 b(b)s(efore)f(doing)f(it!)49 b(Another)34 b(imp)s(ortan)m(t)50 667 y(observ)m(ation)g(is)e(that)i(nonexpandable)e(con)m(tours)i FD(\000)1893 634 y Fq(q)1963 667 y Fs(2)c(R)2131 634 y Fq(q)2205 667 y FB(ma)m(y)k(surviv)m(e)e(in)h(the)g(mo)s(del)g(forev) m(er)h(\(for)50 785 y(unstable)25 b Ft(q)s FB(\))i(but)f Fu(only)36 b FB(if)26 b(they)h(are)g(not)g(\\pac)m(k)m(ed")i(b)m(y)e(a) g(bigger)g(con)m(tour)g FD(\000)2709 752 y Fq(q)2744 724 y Fp(0)2800 785 y FB(ha)m(ving)g(a)g(\\more)h(stable")50 894 y(external)f(color)h Ft(q)662 861 y Fo(0)714 894 y FB(than)f Ft(q)s FB(.)40 b(Namely)-8 b(,)28 b(suc)m(h)g(a)f (\\conglomerate")j(could)c(b)s(e)h(p)s(ossibly)e(expanded)h(in)g(some) 50 1003 y(later)i(step.)40 b(It)28 b(is)f(imp)s(ortan)m(t)g(to)i (realize)f(then)g(\(Theorem)g(2\))h(that)f(the)h(en)m(trop)m(y)g(of)f (the)g(large)g(con)m(tours)50 1120 y FD(\000)113 1087 y Fq(q)183 1120 y FB(inside)g(suc)m(h)i(a)h FD(\000)785 1087 y Fq(q)820 1060 y Fp(0)880 1120 y FB(cannot)g(b)s(eat)f(the)h (energy)g(densit)m(y)e(excess)j(of)e(the)h(unstable)e(color)h Ft(q)j FB(inside.)50 1349 y FD(Notations.)52 b FB(In)32 b(the)h(form)m(ulation)e(of)i(the)f(forthcoming)g(result)g(w)m(e)h(use) f(the)h(concept)g(of)g(a)g(\()p Fs(H)3461 1305 y Fq(q)3460 1372 y Fo(\014)3521 1349 y Ft(;)15 b Fs(T)3634 1316 y Fq(q)3674 1349 y FB(\))50 1459 y(cluster)32 b(in)m(tro)s(duced)f(ab)s (o)m(v)m(e.)49 b(The)32 b(\\n)m(uclei")h(of)g(a)g(cluster)f(are)h (extremal)g(con)m(tour)g(systems)g FD(\000)3395 1473 y Fo(\014)3486 1459 y FB(=)c FD(\000)3649 1416 y Fq(q)3649 1482 y Fo(\014)50 1569 y FB(suc)m(h)f(that)i(the)f(external)g(con)m (tour)h FD(\000)1347 1536 y Fq(q)1415 1569 y FB(of)g(the)f(system)g(is) f(from)g Fs(H)2354 1536 y Fq(q)2394 1569 y FB(.)40 b(Recall)29 b(that)g FD(\000)2990 1583 y Fo(\014)3080 1569 y FB(are)h(formed)e(suc) m(h)50 1678 y(that)j(the)f(external)h(con)m(tour)g FD(\000)1149 1645 y Fq(q)1219 1678 y FB(is)e(p)s(ossibly)f(\\\014lled")h(b)m(y)h(a)h (system)g(of)f(con)m(tours)h(from)f Fs([)3229 1692 y Fr(~)-42 b Fq(q)r Fo(2)p Fq(Q)3369 1678 y Fs(R)3452 1645 y Fr(~)g Fq(q)3486 1678 y FB(.)159 1859 y(When)34 b(summing)e(also)i(o) m(v)m(er)h(all)e(suc)m(h)h(new)f(clusters)h(with)e(n)m(uclei)h(w)m(e)i (will)c(con)m(v)m(ert)36 b(the)e(estimates)50 1969 y(\(1.25\),)h (\(2.10\))f(w)m(e)f(already)e(ha)m(v)m(e)j(for)d(the)i(\\old")f (cluster)f(w)m(eigh)m(ts)i Ft(w)2503 1983 y Fq(T)2592 1969 y 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Fs(j)p Ft(e)923 5165 y Fq(\013)p Fr(+)p Fq(d)1064 5174 y Fv(N)5 b Fw(+1)1200 5165 y Fo(j)p Fq(T)p 1222 5177 V 10 w Fo(j)1321 5203 y FB(+)1412 5116 y Fm(X)1419 5312 y Fe(T)l Fo(3)p Fq(i)1558 5203 y Fs(j)p Ft(w)1648 5217 y Fe(T)1696 5203 y Fs(j)32 b Ft(e)1795 5165 y Fq(\013)p Fr(+)p Fq(d)1936 5174 y Fv(N)5 b Fw(+1)2073 5165 y Fo(j)p Fe(T)p 2095 5177 43 4 v -10 w Fo(j)2221 5203 y Fs(\024)57 b Ft(a)2397 5217 y Fq(N)2524 5203 y FB(=)g Ft(a)2700 5217 y Fq(N)7 b Fr(+1)2882 5203 y Fs(\000)20 b Ft(b)3012 5217 y Fq(N)7 b Fr(+1)3173 5203 y Ft(:)279 b FB(\(3)p Ft(:)p FB(25\))50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(31)p eop %%Page: 32 32 32 31 bop 50 340 a Fu(Pr)-5 b(oof)36 b(of)f(The)-5 b(or)g(em)36 b(4.)47 b FB(It)33 b(is)f(clear)g(that)i(the)e(b)s(ounds)f (\(3.23\){\(3.25\))38 b(are)33 b(lik)m(e)f(in)f(Corollary)g(2)i(ab)s(o) m(v)m(e.)50 449 y(So)i(it)h(su\016ces)f(to)i(clarify)-8 b(,)36 b(and)f(this)g(will)e(b)s(e)i(an)h(imp)s(ortan)m(t)f(step,)i 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3106 1046 a Fv(q)3142 1026 y Fk(@)p Fr(\003)3249 996 y Fp(\014)3249 1036 y Fv(q)3317 901 y Ft(w)3382 915 y Fq(T)3425 924 y Fv(q)3466 901 y FB(\))283 1202 y(=)495 1116 y Fm(X)379 1315 y Fo(f)p Fn(\000)465 1324 y Fv(i)494 1315 y Fo(g)p Fr(:)p Fn(\000)601 1324 y Fv(i)630 1315 y Fo(2R)787 1116 y Fm(Y)832 1310 y Fq(i)949 1202 y Ft(w)1014 1216 y Fn(\000)1063 1225 y Fv(i)1157 1202 y Ft(Z)1219 1219 y Fo(f)p Fn(\000)1305 1228 y Fv(i)1334 1219 y Fo(g)1376 1202 y FB(\(\003\))334 b(where)151 b Ft(w)2292 1216 y Fn(\000)2371 1202 y FB(=)25 b Ft(e)2509 1165 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))2976 1202 y FB(and)354 b(\(3)p Ft(:)p FB(26\))110 1475 y Ft(Z)172 1491 y Fo(f)p Fn(\000)258 1500 y Fv(i)288 1491 y Fo(g)330 1475 y FB(\(\003\))56 b(=)645 1389 y Fm(Y)685 1580 y Fq(q)777 1475 y FB(exp)o(\()p Fs(\000)p Ft(e)1063 1489 y Fq(q)1104 1475 y Fs(j)p FB(\003)1192 1437 y Fo(\014)1192 1497 y Fq(q)1254 1475 y Fs(j)p FB(\))1383 1389 y Fm(X)1360 1601 y Fo(f)p Fn(\000)1446 1570 y Fv(j)1446 1622 y 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768 y Ft(F)13 b FB(\()p FD(\000)754 782 y Fo(\014)816 768 y FB(\))55 b(=)25 b(maxs)q(\()p Ft(F)13 b FB(\()p FD(\000)1412 782 y Fo(\014)1474 768 y FB(\))30 b Ft(;)46 b(E)5 b FB(\()p FD(\000)1780 782 y Fo(\014)1842 768 y FB(\))20 b Fs(\000)g FB(\()p Ft(!)k FB(+)c Ft(\017)p FB(\))p Fs(j)p FD(\000)p 2292 783 63 4 v Fs(j)g FB(+)g Ft(u)2549 782 y Fr(~)-42 b Fq(q)2584 768 y Fs(j)15 b FB(in)m(t)2738 782 y Fr(~)-41 b Fq(q)2788 768 y FD(\000)2851 782 y Fo(\014)2912 768 y Fs(j)21 b(\000)e FB(\012\))p Ft(:)349 b FB(\(4)p Ft(:)p FB(3\))50 978 y(It)29 b(is)e(clear)i(that)g (if)e Ft(F)13 b FB(\()p FD(\000)p FB(\))30 b(\(and)e(other)h(terms)f (in)g(\(4.3\)\))i(dep)s(end)d(on)i(some)g(parameter)g Ft(\025)f FB(then)g(w)m(e)i(ha)m(v)m(e)1385 1247 y Fs(j)1421 1185 y Ft(@)1494 1162 y FB(\026)1474 1185 y Ft(F)13 b FB(\()p FD(\000)1643 1199 y Fo(\014)1705 1185 y FB(\))p 1421 1226 320 4 v 1527 1309 a Ft(@)5 b(\025)1751 1247 y Fs(j)56 b(\024)f(j)1994 1185 y Ft(@)5 b(F)13 b FB(\()p FD(\000)2216 1199 y 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y(and)f(the)h(expansion)f(w)m(e)h(obtain)g(in)e(suc)m(h)i(a)g(w)m (a)m(y)h(will)d(in)g(principle)f(a\013ect,)33 b(in)c(a)j(complicated)e (w)m(a)m(y)-8 b(,)33 b(all)50 2226 y(future)c(expansions)g(of)i(ob)5 b(jects)31 b(\\con)m(taining")g(suc)m(h)f(a)h(partially)d(expanded)i FD(\000)2843 2240 y Fo(\014)2904 2226 y FB(.)159 2408 y(W)-8 b(e)22 b(in)m(tro)s(duce)e(here)h(the)g(follo)m(wing)f(mo)s (di\014cation)g(of)h(Theorem)g(4.)38 b(It)21 b(k)m(eeps)g(the)h(notion) e(of)i(smallness)50 2517 y(with)k(the)i(roughly)f(the)h(same)g(meaning) f(\(with)g(sligh)m(tly)f(di\013eren)m(t)h(quan)m(tities)g Ft(f)2861 2484 y Fq(q)2928 2517 y FB(and)g Ft(u)3154 2531 y Fq(q)3222 2517 y FB(than)h(b)s(efore\))50 2626 y(but)i(it)g(\\smo)s(othens")g(in)g(a)g(con)m(v)m(enien)m(t)i(w)m(a)m (y)f(the)g(expansion)e(algorithm)h(of)g(Theorem)g(4,)i(at)f(the)f(same) 50 2735 y(time)g(rede\014ning)e(the)j(concept)g(of)g(the)f (\\remainder")g Fs(R)p FB(,)h(of)f(con)m(tours)h(whic)m(h)e(cannot)i(b) s(e)f(expanded.)159 2917 y(The)j(systems)h FD(\000)753 2931 y Fo(\014)849 2917 y FB(whic)m(h)e(cannot)j(b)s(e)e(expanded)g (are)i(not)f(treated,)i(as)e(b)s(efore,)h(as)f(conglomerates)50 3026 y(of)28 b(con)m(tours)g(from)g Fs(R)g FB(but)f(as)h Fu(new)i(elementary)i(obje)-5 b(cts)7 b FB(.)41 b(W)-8 b(e)29 b(denote)f(b)m(y)g Fs(P)36 b FB(the)28 b(collection)f(of)h(all)f (suc)m(h)50 3135 y(\\partially)36 b(expanded")h FD(\000)991 3149 y Fo(\014)1052 3135 y FB(.)61 b(T)-8 b(o)38 b(an)m(y)g(suc)m(h)f FD(\000)1733 3149 y Fo(\014)1831 3135 y FB(w)m(e)h(assign)e(a)i(new) 2488 3102 y Fr(13)2602 3135 y FB(energy)g Ft(E)2967 3149 y Fq(r)r(est)3103 3135 y FB(\()p FD(\000)3201 3149 y Fo(\014)3263 3135 y FB(\))f Ft(>)g(E)5 b FB(\()p FD(\000)3613 3149 y Fo(\014)3674 3135 y FB(\))50 3244 y(whic)m(h)31 b(is)g(computed,)h(as)h(a)f(new)g Ft(E)5 b FB(\()p FD(\000)p FB(\),)34 b(from)d(the)i(r.h.s.)45 b(of)32 b(\(2.1\))i(if)d(w)m(e)i (replace)f Ft(F)13 b FB(\()p FD(\000)3137 3258 y Fo(\014)3199 3244 y FB(\))32 b(on)g(its)f(l.h.s.)50 3353 y(b)m(y)f(the)h(new)f(quan) m(tit)m(y)g Ft(F)941 3367 y Fq(r)r(est)1078 3353 y FB(\()p FD(\000)1176 3367 y Fo(\014)1238 3353 y FB(\))g(whose)h(exp)s(onen)m (tial)e(is)g(giv)m(en)i(b)m(y)f(the)h(relation)1215 3576 y Ft(e)1257 3538 y Fo(\000)p Fq(F)10 b Fr(\()p Fn(\000)1447 3547 y Fp(\014)1502 3538 y Fr(\))1560 3576 y FB(=)25 b Ft(e)1698 3538 y Fo(\000)1771 3522 y Fr(\026)1755 3538 y Fq(F)10 b Fr(\()p Fn(\000)1888 3547 y Fp(\014)1942 3538 y Fr(\))1995 3576 y FB(+)20 b Ft(e)2128 3538 y Fo(\000)p Fq(F)2231 3547 y Fv(r)r(est)2355 3538 y Fr(\()p Fn(\000)2432 3547 y Fp(\014)2486 3538 y Fr(\))2519 3576 y Ft(:)979 b FB(\(4)p Ft(:)p FB(5\))50 3785 y(Then)30 b(w)m(e)i(ha)m(v)m(e)h(the)f (follo)m(wing)e(result,)g(whic)m(h)h(will)d(con)m(v)m(enien)m(tly)k (replace)g(Theorem)f(4)h(in)e(ev)m(erything)50 3894 y(what)i(follo)m (ws)f(and)g(concerns)h(the)g Fu(phase)k(diagr)-5 b(am)41 b FB(of)32 b(the)g(mo)s(del.)44 b(\(Only)31 b(for)h(a)g(detailed)f (analysis)g(of)50 4004 y(Gibbs)24 b(states)k(of)e(a)h(\014xed)e (Hamiltonian,)h(it)f(is)g(still)f(recommended)i(to)h(sta)m(y)g(in)e (the)h(setting)h(of)f(Theorem)50 4113 y(4.\))45 b(Extremalit)m(y)31 b(of)h FD(\000)873 4127 y Fo(\014)966 4113 y FB(will)d(b)s(e)i(mean)m (t,)i(from)e(no)m(w)g(on,)h(in)f(the)g(sense)h(that)g(no)g(con)m(tour)g (system)g FD(\000)3649 4080 y Fo(0)3649 4135 y(\014)50 4222 y FB(with)d(a)i Fu(\014nite)37 b FB(energy)31 b Ft(E)930 4236 y Fq(r)r(est)1066 4222 y FB(\()p FD(\000)1164 4189 y Fo(0)1164 4244 y(\014)1226 4222 y FB(\))f(can)h(b)s(e)f (compatibly)f(placed)h(inside)e FD(\000)2658 4236 y Fo(\014)2719 4222 y FB(.)50 4456 y FD(Theorem)40 b(5)h(\(Smo)s(othening)g(of)g (Theorem)f(4\).)53 b Fu(Assume)37 b(a)h(situation)h(like)e(in)g(The)-5 b(or)g(em)40 b(4)e(but)50 4565 y(r)-5 b(elaxing)32 b(the)f(condition)h (that)g(contours)h(fr)-5 b(om)32 b Fs(H)26 b(\032)f(A)30 b Fu(ar)-5 b(e)32 b(smal)5 b(l.)42 b(\(Just)31 b(suppose)h(that)g Fs(H)g Fu(is)f(a)g(subset)50 4674 y(of)k Fs(A)p 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b(fr)-5 b(om)38 b(\(3.18\))f(such)g(that)g Fs(f)p FD(\000)3277 5020 y Fo(\014)3339 4967 y Fq(i)3369 5006 y Fs(g)g Fu(can)f(be)p 50 5111 546 4 v 159 5172 a Fl(13)230 5198 y Fz(As)29 b(the)h(system)f Ff(\000)796 5208 y Fd(\014)883 5198 y Fz(will)h(b)r(e)g(\\partially)f(expanded")f(no)n(w,)i(the)f(energy)g (of)g(the)g(\\remainder")g(will)h(increase.)46 b(All)50 5289 y(this)26 b(will)h(b)r(e)g(done)e(in)h(suc)n(h)h(a)e(w)n(a)n(y)g (that)g(partition)h(functions)g(of)f(in)n(terest)j(remain)d(the)h (same.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)j(18:34)1080 b Fz(34)p eop %%Page: 35 35 35 34 bop 50 340 a Fu(compatibly)33 b(place)-5 b(d)34 b(into)e(it.)42 b(De\014ne)31 b(the)h(values)g Ft(w)1865 354 y Fe(T)1944 340 y Fu(of)g(new)g(cluster)g(weights)h(analogously)h (as)e(befor)-5 b(e,)50 449 y(with)28 b Ft(F)13 b FB(\()p FD(\000)411 463 y Fo(\014)472 449 y FB(\))27 b Fu(r)-5 b(eplace)g(d)29 b(by)1012 426 y FB(\026)991 449 y Ft(F)13 b FB(\()p FD(\000)1160 463 y Fo(\014)1222 449 y FB(\))27 b Fu(everywher)-5 b(e.)41 b(Then)27 b(we)g(get)g(a)g(model)h Fs(R)p Fu({e)-5 b(quivalent)28 b(to)f(the)h(original)50 558 y(one.)42 b(In)31 b(particular)j(the)e(r)-5 b(elation)33 b(\(we)f(do)h(not)f(intr)-5 b(oduce)33 b(any)f(new)g(notations)i(for)e (quantities)g(like)f Ft(f)3670 525 y Fq(o)3660 581 y(q)50 670 y Fu(and)52 b FB(~)-64 b Ft(w)291 684 y Fq(T)349 670 y Fu(,)32 b(and)i(symbol)f Ft(T)938 684 y Fq(q)1011 670 y Fu(her)-5 b(e)34 b(denotes)f(both)h(the)f(\\old")g(and)g(\\new")g (clusters)g Ft(T)2966 684 y Fq(q)3039 670 y Fu(and)g Fi(T)3274 684 y Fq(q)3348 670 y Fu(\))101 879 y Ft(Z)7 b FB(\(\003\))25 b(=)768 793 y Fm(X)424 1005 y Fo(f)p Fn(\000)510 1014 y Fv(i)540 1005 y Fo(g[f)p Fn(\000)712 975 y Fv(j)712 1026 y Fp(\014)766 1005 y Fo(g)p Fr(:)p Fn(\000)873 1014 y Fv(i)902 1005 y Fo(2R)p Fq(;)p Fn(\000)1084 975 y Fv(j)1084 1026 y Fp(\014)1138 1005 y Fo(2P)1291 793 y Fm(Y)1337 987 y Fq(i)1423 879 y Ft(w)1488 893 y Fn(\000)1537 902 y Fv(i)1585 793 y Fm(Y)1627 987 y Fq(j)1717 879 y Ft(w)1782 909 y Fn(\000)1831 878 y Fv(j)1831 930 y Fp(\014)1905 793 y Fm(Y)1945 984 y Fq(q)2036 879 y FB(exp\()p Fs(\000)p Ft(f)2336 842 y Fo(\016)2326 902 y Fq(q)2377 879 y Fs(j)p FB(\003)2465 893 y Fq(q)2505 879 y Fs(j)p FB(\))49 b(exp\()2965 793 y Fm(X)2788 1005 y Fq(T)2831 1014 y Fv(q)2867 1005 y Fq(\023)p Fo(f)p Fn(\000)2980 1014 y Fv(i)3009 1005 y Fo(g[f)p Fn(\000)3181 975 y Fv(j)3181 1026 y Fp(\014)3236 1005 y Fo(g)3307 879 y FB(~)-64 b Ft(w)3353 893 y Fq(T)3396 902 y Fv(q)3437 879 y FB(\))51 b(\(4)p Ft(:)p FB(6\))50 1211 y Fu(is)32 b(valid,)i(with)f Ft(w)656 1225 y Fn(\000)735 1211 y FB(=)25 b Ft(e)873 1178 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))1130 1211 y Fu(for)34 b(any)f FD(\000)25 b Fs(2)g(R)32 b Fu(and)i(with)f Ft(w)2172 1225 y Fn(\000)2221 1234 y Fp(\014)2305 1211 y FB(=)25 b Ft(e)2443 1178 y Fo(\000)p Fq(E)2553 1187 y Fv(r)r(est)2677 1178 y Fr(\()p Fn(\000)2754 1187 y Fp(\014)2809 1178 y Fr(\))2874 1211 y Fu(for)33 b(any)h FD(\000)3257 1225 y Fo(\014)3343 1211 y Fs(2)25 b(P)7 b Fu(.)50 1441 y(Notes.)46 b FB(1\))32 b(Of)e(course)h(in)f (practise)g(w)m(e)i(will)c(b)s(e)i(in)m(terested)h(in)f(v)m(ery)h(sp)s (ecial)e(c)m(hoices)j(of)f Fs(H)q FB(,)g(see)g(b)s(elo)m(w.)50 1550 y(Ho)m(w)m(ev)m(er,)39 b(it)c(is)f(imp)s(ossible)e(to)k(describ)s (e)e Fs(H)i FB(precisely)e(as)i(family)e(of)h(small)f(con)m(tours)i(if) f(w)m(e)h(w)m(an)m(t)g(to)50 1659 y(apply)e(Theorem)h(5)h(sim)m (ultaneously)d(also)j(to)g(p)s(erturbations)d(of)j(a)g(giv)m(en)f(mo)s (del)f(\(of)i(a)g(t)m(yp)s(e)g(\(1.23\)\).)50 1768 y(Namely)-8 b(,)30 b(small)e(con)m(tours)i(of)f(the)g(original)f(mo)s(del)g(ma)m(y) i(b)s(ecome)f(nonsmall)f(in)g(the)h(p)s(erturb)s(ed)e(mo)s(del.)50 1877 y(2\))40 b(Notice)g(the)f(di\013erence)g(w.r.)67 b(to)40 b(the)f(situation)f(in)g(\(3.17\))k({)e(where)e(energies)h Ft(E)5 b FB(\()p FD(\000)p FB(\)\))41 b(w)m(ere)f(used)50 1986 y(for)32 b(all)f(nonexpanded)g(con)m(tours)i FD(\000)28 b Fs(2)g(R)p FB(.)46 b(No)m(w,)33 b(the)g(systems)f FD(\000)2366 2000 y 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2494 104 4 v 1489 2465 a Fo(j)1549 2507 y FB(are)g(smaller)e(than) h(1)h(for)g(large)f(enough)h Ft(C)40 b FB(and)33 b(deca)m(y)h(v)m(ery) 50 2616 y(quic)m(kly)-8 b(.)39 b(So)27 b(it)f(is)h(no)g(problem,)f(b)m (y)h(p)s(ossibly)e(strengthening)h(our)h(assumptions)e(on)i Ft(c;)15 b(g)s FB(,)30 b(to)d(rewrite)g(all)50 2725 y(the)k(b)s(ounds)d (considered)h(b)s(efore)h(so)h(that)g(they)g(w)m(ould)e(remain)h(v)m (alid)f(also)h(for)h(the)f(deriv)m(ativ)m(es)h(of)g(the)50 2834 y(w)m(eigh)m(ts)g Ft(w)439 2848 y Fn(\000)488 2857 y Fp(\014)576 2834 y FB(and)f Ft(w)818 2848 y Fq(T)876 2834 y FB(.)40 b(So,)31 b(inserting)d(inequalities)g(lik)m(e)1172 3073 y Fs(j)15 b FB(in)m(t)g FD(\000)1399 3087 y Fo(\014)1461 3073 y Fs(j)p Ft(w)1551 3087 y Fn(\000)1600 3096 y Fp(\014)1658 3073 y Ft(e)1700 3031 y Fr(\()p Fq(a)p Fo(\000)p Fq(C)t Fr(\))p Fo(j)p Fn(\000)1979 3040 y Fp(\014)p 1930 3060 V 2035 3031 a Fo(j)2086 3073 y Fs(\024)25 b Ft(w)2247 3087 y Fn(\000)2296 3096 y Fp(\014)2355 3073 y Ft(e)2397 3031 y 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Ft(E)5 b FB(\()p FD(\000)2989 3713 y Fo(\014)3051 3699 y FB(\))p Ft(e)3128 3666 y Fo(\000)p Fq(F)10 b Fr(\()p Fn(\000)3318 3675 y Fp(\014)3373 3666 y Fr(\))3442 3699 y FB(follo)m(ws)50 3820 y(from)30 b(our)g (assumptions)e(on)j(su\016cien)m(t)e(smallness)g(of)1988 3784 y Fq(@)p 1966 3799 V 1966 3851 a(@)t(\025)2063 3820 y Ft(e)2105 3787 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)2295 3796 y Fp(\014)2350 3787 y Fr(\))2383 3820 y FB(.)50 4056 y FD(On)34 b(the)h(c)m(hoice)h(of)f Fs(H)q FD(,)g Fs(R)f FD(and)h Fs(P)43 b FD(in)35 b(the)f(expansion)i(pro)s(cedures)g (of)f(Theorem)f(5.)159 4293 y FB(The)e(partially)f(expanded)h(ob)5 b(jects)34 b FD(\000)1503 4307 y Fo(\014)1597 4293 y FB(do)f(not)g(disapp)s(ear)e(from)h(our)g(mo)s(del)g(no)m(w.)48 b(They)33 b(surviv)m(e)50 4402 y(in)38 b(it)g(but)h(are)g(treated,)k (from)38 b(no)m(w)i(on,)h(as)e(conglomerates)i FD(\000)2318 4416 y Fo(\014)2418 4402 y FB(nev)m(er)f(separated)f(in)m(to)g(single)f (con-)50 4511 y(tours)d(again.)57 b(Of)36 b(course,)h(if)e Ft(E)1183 4525 y Fq(r)r(est)1319 4511 y FB(\()p FD(\000)1417 4525 y Fo(\014)1479 4511 y FB(\))g(=)f Fs(1)h FB(then)h(suc)m(h)f(a)h (con)m(tour)h(system)f FD(\000)2991 4525 y Fo(\014)3088 4511 y FB(still)d(\\e\013ectiv)m(ely)50 4620 y(disapp)s(ears")c(from)g (the)i(expanded)f(mo)s(del.)1588 4587 y Fr(14)159 4802 y FB(W)-8 b(e)26 b(mak)m(e)f(an)f(agreemen)m(t)i(that)f(the)g(class)f Fs(P)32 b FB(of)25 b(all)e(these)i(systems)g FD(\000)2576 4816 y Fo(\014)2661 4802 y FB(will)d(b)s(e)i(added)g(to)h(the)f(family) 50 4911 y Fs(R)30 b FB(\(of)g(con)m(tours)h(not)f(used)g(as)g(external) g(con)m(tours)h(of)f(some)g FD(\000)2253 4925 y Fo(\014)2315 4911 y FB(\))g(in)f(future)g(expansion)g(steps.)40 b(This)29 b(is)p 50 5020 546 4 v 159 5082 a Fl(14)230 5107 y Fz(Nonexpandabilit)n (y)g(of)g Ff(\000)1001 5117 y Fd(\014)1087 5107 y Fz(i.e.)47 b(the)30 b(ev)n(en)n(t)f Fh(E)1632 5116 y Fg(r)r(est)1760 5107 y Fz(\()p Ff(\000)1843 5117 y Fd(\014)1901 5107 y Fz(\))e Fh(<)h Fx(1)h Fz(is)h(rather)g(a)f(delicate)i(thing.)45 b(In)29 b(spite)i(of)e(the)g(fact)50 5198 y(that)h(it)i(is)f(de\014ned) g(in)h(someho)n(w)d(arti\014cial)j(w)n(a)n(y)e(\(arbitrariness)i(in)g (the)f(c)n(hoice)g(of)f(\(4.3\))h(and)f(in)h(the)g(c)n(hoice)h(of)e (\012,)i(for)50 5289 y(example\))26 b(it)g(giv)n(es)g(some)g (information)e(on)i(the)g(tendency)g(to)g(phase)g(transitions)h(in)f(v) n(olumes)g(bigger)f(than)h(in)n(t)13 b Ff(\000)3480 5299 y Fd(\014)3537 5289 y Fz(.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(36)p eop %%Page: 37 37 37 36 bop 50 340 a FB(a)27 b(little)f(bit)g(more)h(abstract)h(approac)m (h)f(than)g(in)f(the)h(iterations)f(of)h(Theorem)g(4,)h(where)f(future) f(families)50 449 y Fs(R)k FB(w)m(ere)h(just)f(subsets)f Fs(R)21 b(n)f(H)31 b FB(of)g(the)f(previous)f(families)g Fs(R)p FB(.)159 631 y(All)35 b(the)i(elemen)m(ts)g FD(\000)918 645 y Fo(\014)1016 631 y FB(of)f(suc)m(h)h(a)g(newly)e(formed)h(family) f Fs(P)44 b FB(can)37 b(b)s(e)f(then)g(included,)g(as)h(in)m(terior)50 753 y(subsystems,)f(in)m(to)g(more)g(complex)g(systems)1701 729 y Fm(f)1684 753 y FD(\000)1747 767 y Fo(\014)1844 753 y FB(created)h(in)e(the)h(next)g(expansion)f(steps;)k(in)c(suc)m(h) g(a)50 862 y(w)m(a)m(y)e(that)f(w)m(e)g(\\wrap")g(the)g(collections)f FD(\000)1551 876 y Fo(\014)1644 862 y FB(b)m(y)h(external)g(con)m (tours)g(tak)m(en)h(from)e(some)i(newly)d(c)m(hosen)50 971 y(family)24 b Fs(H)j FB(of)f(\\small")f(con)m(tours.)40 b(Then,)26 b(in)e(a)i(new)g(application)e(of)i(Theorem)g(5,)h(these)f (bigger)g(systems)66 1066 y Fm(f)50 1089 y FD(\000)113 1103 y Fo(\014)204 1089 y FB(will)i(b)s(e)i(\(partially\))f(expanded,)h (to)s(o.)41 b(And)30 b(so)g(on.)159 1271 y(In)i(suc)m(h)h(a)g(setting,) h(whic)m(h)d(is)h(di\013eren)m(t)g(from)h(those)g(used)f(in)g(Theorem)g (4,)i(it)f(is)f(not)h(necessary)g(to)50 1380 y(care)28 b(whether)f(the)g(elemen)m(ts)h(of)g Fs(H)g FB(are)g(small)d(in)i(the)g (sense)h(of)f(\(2.13\).)42 b(Ho)m(w)m(ev)m(er,)30 b(the)e(latter)g (notion)f(is)50 1489 y(still)h(crucial)h(when)h(w)m(e)g(w)m(an)m(t)i (to)f(in)m(terpret)f(the)g(result)f(of)i(these)g(expansions.)39 b(See)31 b(the)f(next)h(c)m(hapter.)50 1738 y Fu(A)-5 b(dditional)43 b(notes.)k FB(1\))40 b(F)-8 b(or)41 b(v)m(ery)f(big)e (nonsmall)g FD(\000)1900 1752 y Fo(\014)2001 1738 y FB(w)m(e)i(ha)m(v)m (e)h Ft(E)2430 1752 y Fq(r)r(est)2566 1738 y FB(\()p FD(\000)2664 1752 y Fo(\014)2726 1738 y FB(\))g Fs(\031)f Ft(E)5 b FB(\()p FD(\000)3083 1752 y Fo(\014)3145 1738 y FB(\).)69 b(\(Imagine)39 b(a)50 1847 y(con)m(tour)44 b(app)s(earing)d(around)h(a)i(b)s(oundary)d(of)i(a)g(v)m(ery)h(large)f (\014nite)f(v)m(olume)h Ft(M)10 b FB(,)47 b(if)42 b(w)m(e)h(imp)s(ose)f (an)50 1959 y(unstable)h(b)s(oundary)f(color)i Ft(q)i FB(around)d(it.\))82 b(Then)43 b(the)h(expanded)f(part)h Ft(e)2793 1926 y Fo(\000)2866 1910 y Fr(\026)2850 1926 y Fq(F)10 b Fr(\()p Fn(\000)2983 1935 y Fp(\014)3038 1926 y Fr(\))3114 1959 y FB(is)43 b(just)h(a)g(small)50 2068 y(correction)27 b(to)h(the)f(\\basic")g(situation)f(with)g Ft(E)1694 2082 y Fq(r)r(est)1830 2068 y FB(\()p FD(\000)1928 2082 y Fo(\014)1990 2068 y FB(\).)40 b(The)26 b(latter)i(situation)d (cannot)j(b)s(e)e(expanded.)50 2178 y(A)39 b(detailed)f(analysis)f(of)i (what)g(exactly)g(happ)s(ens)e(under)h(unstable)f(b)s(oundary)g (conditions)g(in)h(\014nite)50 2287 y(v)m(olumes)32 b(cannot)g(b)s(e)g (done)g(with)e(the)j(help)d(of)j(expansions)e(only)-8 b(.)45 b(In)31 b(fact)i(w)m(e)g(do)f(not)g(address)g(suc)m(h)f(a)50 2396 y(question)c(here,)i(in)m(v)m(estigation)f(of)h(what)f(happ)s(ens) e(precisely)h(under)g(unstable)g(b)s(oundary)f(condition)h(in)50 2505 y(\014nite)34 b(v)m(olumes)h(is)g(not)g(a)h(goal)g(of)g(this)e (article.)56 b(In)34 b(thermo)s(dynamical)g(limit)f(the)j(situation)e (is)h(more)50 2614 y(trivial)27 b(ev)m(en)j(for)f(unstable)f(b)s (oundary)f(conditions.)39 b(See)29 b([Z])g(where)g(it)g(is)f(pro)m(v)m (en)h(\(for)h(more)f(restricted)50 2723 y(mo)s(dels\))f(that)i(no)f (other)h(translation)e(in)m(v)m(arian)m(t)h(Gibbs)f(states,)j(than)e (those)h(corresp)s(onding)d(to)j(stable)50 2832 y(b)s(oundary)e (conditions,)h(exist)h(in)f(the)i(thermo)s(dynamic)e(limit.)159 3014 y(2\))35 b(In)e(general,)i(there)f(is)f(alw)m(a)m(ys)h(a)g(lot)g (of)g(freedom)g(in)f(the)h(c)m(hoice)g(of)g Fs(fH)2808 2981 y Fq(q)2849 3014 y Fs(g)p FB(,)h(and)e(there)h(could)f(b)s(e)50 3123 y(also)25 b(other)h(v)m(arian)m(ts)g(of)g(the)g(notion)f(of)h (\\smallness")e(of)i(a)g(con)m(tour)g(system.)40 b(See)25 b([ZR],)h(where)f(the)h(\014nal)50 3232 y(pro)s(duct)c(\(the)h(fully)e (expanded)h(mo)s(del\))g(lea)m(v)m(es)j(less)d(con)m(tours)h(left)g (unexpanded)e(than)i(in)f(our)g(approac)m(h)50 3341 y(here)33 b(\(assuming,)h(ho)m(w)m(ev)m(er,)h(sharp)s(er)d(P)m(eierls)h(b)s (ounds)e(in)h(principle\).)47 b(An)m(yw)m(a)m(y)-8 b(,)36 b(the)e(freedom)f(in)g(the)50 3450 y(c)m(hoice)d(of)f(the)h(families)d Fs(H)985 3417 y Fq(q)1054 3450 y FB(do)s(es)h(not)i(a\013ect)h(the)e (ph)m(ysics)f(of)h(the)h(problem)d(i.e.)40 b(the)30 b(determination)e (of)50 3559 y(\014nal,)36 b Fu(stable)42 b FB(v)m(alues)36 b(of)g(colors)f Ft(q)i Fs(2)c Ft(Q)p FB(.)56 b(F)-8 b(or)36 b(the)g(stable)f(v)m(alues)h Ft(q)s FB(,)g(all)f(the)g(con)m(tours)h FD(\000)3232 3526 y Fq(q)3308 3559 y FB(are)g(\014nally)50 3668 y(expanded,)30 b(in)f(an)m(y)h(of)h(the)g(ab)s(o)m(v)m(e)g(men)m (tioned)f(approac)m(hes.)478 3985 y FC(5.)51 b(Itera)-6 b(ted)31 b(exp)-6 b(ansions.)48 b(Constr)n(uction)32 b(of)h(the)g(phase)f(dia)n(gram)159 4294 y FB(First)44 b(w)m(e)h(ha)m(v)m(e)h(to)f(mo)s(dify)e(and)h(mak)m(e)h(more)g(precise) f(the)h(expansion)e(plan)h(\(1.22\),)50 b(originally)50 4403 y(form)m(ulated)41 b(in)g(setting)h(of)g(Theorem)f(4.)76 b(W)-8 b(e)43 b(will)d(no)m(w)h(c)m(ho)s(ose,)46 b(in)41 b(the)h Ft(n)p FB({th)g(step)g(of)g(expansion)50 4512 y(b)m(y)d(Theorem)f(5,)k(the)d(family)e Fs(H)1217 4526 y Fq(n)1305 4512 y FB(as)i(the)h(collection)e(of)h(all)f(con)m(tours)h (from)g(the)g(original)e(family)g Fs(A)50 4621 y FB(\(of)e(all)e(con)m (tours)j(of)e(the)h(starting)g(mo)s(del)e(\(1.23\)\))k(ha)m(ving)d(a)h (diameter)g(equal)f(to)h Ft(n)p FB(.)53 b(Th)m(us)34 b(w)m(e)h(get)g(a)50 4730 y(decomp)s(osition)1652 4858 y Fs(A)25 b FB(=)g Fs([)1907 4872 y Fq(n)1956 4858 y Fs(H)2033 4872 y Fq(n)2082 4858 y Ft(:)1416 b FB(\(5)p Ft(:)p FB(1\))50 5050 y(W)-8 b(e)31 b(put)f Fs(R)450 5064 y Fq(n)525 5050 y Fs(\021)25 b([)682 5017 y Fo(1)682 5072 y Fq(n)758 5050 y Fs(H)835 5064 y Fq(m)935 5050 y FB(and)k(de\014ne)h(recursiv)m(ely)f(the)i(new)e(families)g(of)h(con) m(tour)h(systems)845 5289 y Fs(P)908 5303 y Fq(n)983 5289 y Fs(\021)25 b(f)31 b FD(\000)1218 5303 y Fo(\014)1305 5289 y FB(=)25 b(\()p FD(\000)p Ft(;)15 b Fs(f)p FD(\000)1647 5252 y Fq(i)1647 5312 y Fo(\014)1709 5289 y Fs(g)p FB(\))56 b(:)g FD(\000)25 b Fs(2)g(H)2177 5303 y Fq(n)2226 5289 y Ft(;)46 b FD(\000)2360 5252 y Fq(i)2360 5312 y Fo(\014)2447 5289 y Fs(2)24 b([)2593 5250 y Fq(n)p Fo(\000)p Fr(1)2593 5313 y(1)2736 5289 y Fs(P)2799 5303 y Fq(k)2843 5289 y Fs(g)p Ft(:)610 b FB(\(5)p Ft(:)p FB(2\))50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(37)p eop %%Page: 38 38 38 37 bop 50 340 a FB(Th)m(us,)46 b(in)d(the)h Ft(n)p FB({th)f(step)h(of)g(expansion)f(w)m(e)h(consider)f(a)h(mo)s(del)f (\(1.23\))j(with)c(a)i(a)h(family)d Fs(H)3532 307 y Fo(\014)3531 362 y Fq(n)3637 340 y FB(of)50 449 y(systems)29 b FD(\000)449 463 y Fo(\014)539 449 y FB(formed)g(from)g(an)g(external)g(con)m(tour)g (from)g Fs(H)2153 463 y Fq(n)2202 449 y FB(,)h(and)e(from)h(some)h (other)f(in)m(terior)f(ob)5 b(jects)50 558 y FD(\000)113 525 y Fo(0)113 581 y(\014)206 558 y Fs(2)33 b(P)370 519 y Fo(\014)363 582 y Fq(n)p Fo(\000)p Fr(1)540 558 y FB(whic)m(h)g(w)m (ere)i(already)g Fu(partial)5 b(ly)44 b FB(expanded)34 b(b)s(efore.)54 b(\(Hence)35 b(the)g(notation)g(\\P")h(of)f(the)50 667 y(family)30 b Fs(P)392 681 y Fq(n)p Fo(\000)p Fr(1)534 667 y FB(,)i(whic)m(h)e(w)m(as)i(empt)m(y)g(in)e(Theorem)h(4)g(and)g (whic)m(h)f(w)m(e)i(no)m(w)f(add)g(to)h Fs(R)3006 681 y Fq(n)3056 667 y FB(.\))43 b(Some)32 b(of)f(these)50 776 y(systems)f FD(\000)450 743 y Fo(0)450 799 y(\014)542 776 y FB(w)m(ere)h(actually)f(expanded)f(fully)f(i.e.)41 b(left)30 b(with)f(the)h(energy)h Ft(E)2748 790 y Fq(r)r(est)2885 776 y FB(\()p FD(\000)2983 743 y Fo(0)2983 799 y(\014)3044 776 y FB(\))26 b(=)f Fs(1)p FB(.)50 1013 y Fu(Notes.)46 b FB(1\))c(A)g(somewhat)f(arbitrary)-8 b(,)43 b(but)e(con)m(v)m(enien)m (t)h(fact)g(is)f(that)g(with)f(our)h(c)m(hoice)h(of)f(m)m(utually)50 1122 y(disjoin)m(t)24 b(families)f Fs(H)772 1136 y Fq(n)847 1122 y FB(\(of)j(external)f(con)m(tours)h(of)g(systems)f FD(\000)2183 1136 y Fo(\014)2270 1122 y FB(whic)m(h)f(ha)m(v)m(e)j(to)f (b)s(e)f(expanded)g(in)f(the)i Ft(n)p FB({)50 1231 y(th)i(step\))g(an)g (attempt)h(to)g(expand)e(some)i FD(\000)1570 1245 y Fo(\014)1659 1231 y FB(will)c(b)s(e)i(nev)m(er)i(b)s(e)e(rep)s(eated)h(again,)h(in)e (future)g(expansion)50 1340 y(steps.)49 b(Alw)m(a)m(ys)34 b(new)f(p)s(ortions)f Fs(H)1266 1354 y Fq(n)1345 1340 y FB(=)e Fs([)1507 1354 y Fq(q)1547 1340 y Fs(H)1625 1307 y Fq(q)1624 1362 y(n)1706 1340 y FB(of)k Fs(A)29 b FB(=)h Fs([)2077 1354 y Fq(n)2126 1340 y Fs(H)2203 1354 y Fq(n)2286 1340 y FB(will)h(b)s(e)h(used)h(as)g(external)h(con)m (tours)g(of)50 1449 y(ob)5 b(jects)31 b FD(\000)424 1463 y Fo(\014)485 1449 y FB(,)g(partially)d(expanded)i(in)f(the)h Ft(n)p FB(-)h(th)f(use)g(of)g(Theorem)g(5.)159 1631 y(2\))38 b(The)e(systems)i Fs(R)f FB(resp.)60 b Fs(A)37 b FB(are)g(more)h (complicated)e(no)m(w,)k(than)d(in)e(Theorem)i(4.)62 b(The)36 b(family)50 1740 y Fs(A)123 1754 y Fq(n)211 1740 y FB(of)k(allo)m(w)m(ed)f(con)m(tours)h(in)e(a)i(mo)s(del)e (expanded)h(in)f(the)h Ft(n)p FB({th)h(expansion)e(step)h(\(b)m(y)h (Theorem)f(5\))50 1849 y(lo)s(oks)29 b(lik)m(e)h(follo)m(ws:)40 b Fs(A)856 1863 y Fq(n)930 1849 y FB(=)25 b Fs(A)19 b(n)h FB(\()p Fs([)1279 1816 y Fq(n)1279 1872 y(m)p Fr(=1)1440 1849 y Fs(H)1517 1863 y Fq(m)1586 1849 y FB(\))g Fs([)f(P)1784 1863 y Fq(n)1864 1849 y FB(where)30 b Fs(A)g FB(is)f(the)h(starting)g (family)e(of)j(con)m(tours)f(and)50 1958 y Fs(P)113 1972 y Fq(n)200 1958 y FB(is)36 b(the)i(collection)f(of)h(systems)f FD(\000)1391 1972 y Fo(\014)1490 1958 y FB(created)h(and)f(partially)f (expanded)g(in)g(previous)g(steps.)62 b(The)50 2067 y(\\con)m(tours")32 b FD(\000)571 2081 y Fo(\014)662 2067 y FB(from)e(the)h(family)d Fs(P)1374 2081 y Fq(n)1454 2067 y FB(carry)j(the)f(energy)h Ft(E)2204 2081 y Fq(r)r(est)2341 2067 y FB(\()p FD(\000)2439 2081 y Fo(\014)2500 2067 y FB(\))g(no)m(w,)g(not)f Ft(E)5 b FB(\()p FD(\000)3112 2081 y Fo(\014)3174 2067 y FB(\).)159 2249 y(3\))29 b(One)f(has)g(also)g(to)h(stress)f(that)h(the)f (construction)g(of)g(a)h(sk)m(eleton)g(of)f(an)m(y)g FD(\000)2877 2263 y Fo(\014)2964 2249 y FB(=)d(\()p FD(\000)p Ft(;)15 b Fs(f)p FD(\000)3306 2216 y Fq(i)3306 2272 y Fo(\014)3368 2249 y Fs(g)p FB(\))29 b(where)50 2363 y FD(\000)e Fs(2)g(H)305 2377 y Fq(n)386 2363 y FB(and)k FD(\000)627 2330 y Fq(i)627 2386 y Fo(\014)715 2363 y Fs(2)c(P)866 2377 y Fq(n)948 2363 y FB(go)s(es,)33 b(of)e(course,)i (also)e(inside)f(of)i(the)f(in)m(terior)g(systems)g FD(\000)3014 2330 y Fq(i)3014 2386 y Fo(\014)3076 2363 y FB(.)44 b(The)31 b(argumen)m(t)50 2472 y(remains)d(the)i(same)f(as)h(b)s(efore,)g(the)f (b)s(ound)f(\(2.23\))j(b)s(eing)d(rephrased)g(without)h(an)m(y)h(c)m (hange)g(\(actually)-8 b(,)50 2581 y(with)29 b Ft(E)324 2595 y Fq(r)r(est)461 2581 y FB(\()p FD(\000)559 2595 y Fo(\014)620 2581 y FB(\))d Ft(>)g(E)5 b FB(\()p FD(\000)948 2595 y Fo(\014)1010 2581 y FB(\)\))31 b(also)f(here,)h(when)f(w)m(e)h (emplo)m(y)f(systems)h FD(\000)2602 2548 y Fq(i)2602 2604 y Fo(\014)2694 2581 y FB(partially)d(expanded)i(b)s(efore)50 2709 y(and)g(then)g(left)g(unexpanded)e(with)h(the)i(remaining)d(w)m (eigh)m(t)j Ft(e)2218 2673 y Fo(\000)p Fq(E)2328 2682 y Fv(r)r(est)2452 2673 y Fr(\()p Fn(\000)2529 2646 y Fp(0)2529 2691 y(\014)2584 2673 y Fr(\))2617 2709 y FB(.)159 2891 y(4\))38 b(When)g(w)m(e)g(discussed)d(the)j(energy/en)m(trop)m(y)h (con)m(test)h(inside)35 b FD(\000)2548 2905 y Fo(\014)2647 2891 y FB(in)h(\(2.23\),)42 b(w)m(e)c(had)f(to)h(a)m(v)m(oid)50 3000 y(that)f(small)d(con)m(tours)j(w)m(ould)e(app)s(ear)g(as)h(in)m (terior)f(con)m(tours)i(of)f FD(\000)2435 3014 y Fo(\014)2496 3000 y FB(.)58 b(The)36 b(same)g(care)h(is,)g(of)f(course,)50 3109 y(needed)27 b(here)h(for)f(the)h(mo)s(di\014ed)e(expansion)g(pro)s (cedure.)39 b(Small)26 b(con)m(tours)i(do)g(not)g(disapp)s(ear)e(no)m (w)h(but)50 3218 y(trivialize)h(\(add)i(an)h(in\014nite)d(energy)i (to\))i(an)m(y)f(conglomerate)g FD(\000)2300 3232 y Fo(\014)2392 3218 y FB(con)m(taining)f(them.)159 3400 y(5\))25 b(The)f(family)e Fs(P)779 3414 y Fq(n)853 3400 y FB(gro)m(ws)j(with)e Ft(n)p FB(.)38 b(It)25 b(is)e(our)h(aim)f(to)i(sho)m(w)g(that)f(for)g (some)h(\\stable")g Ft(q)s FB(,)h(an)m(y)e(elemen)m(t)50 3509 y(of)k(this)e(\(complicated!\))40 b(family)26 b(has)i(the)f (energy)h Ft(E)1878 3523 y Fq(r)r(est)2015 3509 y FB(\()p FD(\000)2113 3523 y Fo(\014)2175 3509 y FB(\))d(=)g Fs(1)p FB(,)j(ev)m(en)h(in)d(the)i(limit)d Ft(n)g Fs(!)g(1)p FB(.)40 b(Then)50 3618 y(only)30 b(the)h(con\014guration)f(\\constan)m (t)i(color)f Ft(q)s FB(")g(will)d(surviv)m(e.)41 b(\(Belo)m(w)32 b(w)m(e)f(dev)m(elop)f(corollaries)g(of)h(suc)m(h)50 3727 y(an)g(imp)s(ortan)m(t)g(fact.\))46 b(Recall)31 b(also)g(that)h(up)f(to)h(no)m(w,)g(all)f(expansions)f(w)m(ere)i(done)f (sim)m(ultaneously)f(on)50 3836 y(all)g(tori)i(\003.)45 b(So)31 b(it)g(is)g(necessary)h(to)h(mak)m(e)g(a)f(comparison;)f(for)h (example)f(w)m(e)i(w)m(ould)d(lik)m(e)h(to)i(sho)m(w)e(that)50 3945 y(there)f(is)g(at)h(least)g(one)f Ft(q)e Fs(2)d Ft(Q)30 b FB(for)g(whic)m(h)g(w)m(e)g(\014nally)f(get)i(stabilit)m(y)e (on)h(an)m(y)h Fu(lar)-5 b(ge)34 b(enough)j FB(torus)31 b(\003:)50 4182 y FD(The)j(fully)i(expanded)f(mo)s(del)f(in)h(a)f (\014xed)h(torus)h FB(\003)p FD(.)159 4418 y FB(Denote)26 b(b)m(y)f Ft(f)641 4385 y Fq(n)631 4440 y(q)715 4418 y FB(the)g(quan)m(tit)m(y)h Ft(f)1280 4385 y Fq(o)1270 4440 y(q)1344 4418 y FB(from)f(the)g Ft(n)p FB({th)g(application)e(of)j (Theorem)e(5,)j(as)e(describ)s(ed)e(ab)s(o)m(v)m(e.)50 4530 y(In)m(tro)s(duce)30 b(also)g(the)h(follo)m(wing)d(notations)j (\(for)f(an)m(y)h(\003,)g(w)m(e)g(do)f(not)h(mark)f(the)g(dep)s (endence)g(on)g(it\))619 4748 y Ft(f)674 4711 y Fq(n)748 4748 y FB(=)25 b(min)996 4762 y Fq(q)r Fo(2)p Fq(Q)1157 4748 y Ft(f)1212 4711 y Fq(n)1202 4771 y(q)1261 4748 y Ft(;)106 b(f)1447 4711 y Fo(1)1437 4771 y Fq(q)1549 4748 y FB(=)25 b(lim)1686 4803 y Fq(n)1786 4748 y Ft(f)1841 4711 y Fq(n)1831 4771 y(q)1890 4748 y Ft(;)76 b(f)2046 4711 y Fo(1)2148 4748 y FB(=)25 b(min)2395 4762 y Fq(q)r Fo(2)p Fq(Q)2587 4748 y Ft(f)2642 4711 y Fo(1)2632 4771 y Fq(q)2774 4748 y FB(=)g(lim)2911 4803 y Fq(n)3011 4748 y Ft(f)3066 4711 y Fq(n)3115 4748 y Ft(:)383 b FB(\(5)p Ft(:)p FB(3\))50 4998 y(W)-8 b(e)31 b(will)d(see)j(b)s(elo)m(w)f(that)h (the)f(quan)m(tit)m(y)h Ft(f)1555 4965 y Fo(1)1661 4998 y FB(equals)f(to)h(the)g(free)f(energy)h(of)g(\(1.23\))h(for)e(\003)c Fs(!)f Fi(Z)3445 4965 y Fq(\027)3486 4998 y FB(.)159 5180 y(F)-8 b(or)29 b(a)g(comparison)f(of)h(the)f(results)g(of)g(v)m (arious)g(expansion)g(steps)g(it)g(is)g(crucial)f(to)j(realize)e(that)h (once)50 5289 y(a)g(con)m(tour)h FD(\000)517 5256 y Fq(q)586 5289 y FB(with)e(a)h(diameter)g(diam)14 b FD(\000)1519 5256 y Fq(q)1584 5289 y FB(=)25 b Ft(n)k FB(b)s(ecomes)g(nonsmall)e(in) h(the)h Ft(n)p FB({th)g(expansion)f(step)h(i.e.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)g(18:34)1080 b Fz(38)p eop %%Page: 39 39 39 38 bop 50 340 a FB(once)31 b(w)m(e)g(ha)m(v)m(e)g(the)g(relation,)f (for)g(the)g(smallest)g(\\pac)m(king)h(cub)s(e")f Fj(\003)25 b Fs(\033)g FD(\000)p 2592 355 63 4 v -40 x Fq(q)2725 340 y FB(of)31 b FD(\000)2892 307 y Fq(q)1675 534 y Ft(u)1727 496 y Fq(n)1727 556 y(q)1777 534 y Fs(j)p Fj(\003)p Fs(j)25 b Ft(>)g FB(\012)1438 b(\(5)p Ft(:)p FB(4\))50 730 y(then)24 b(suc)m(h)g(a)h(colour)f Ft(q)j Fu(never)g(attains)h(a)f(minimum)32 b FB(of)25 b(the)f(v)m(alues)h Fs(f)p Ft(f)2492 697 y Fq(m)2482 753 y(q)2560 730 y Ft(;)15 b(q)29 b Fs(2)c Ft(Q)p Fs(g)f FB(later,)i(for)e(an)m(y)h Ft(m)g(>)g(n)p FB(.)50 840 y(Really)-8 b(,)30 b(all)f(the)i(future)e(quan)m(tities)h Ft(f)1377 807 y Fq(m)1367 862 y(q)1476 840 y FB(di\013er)f(from)h Ft(f)1981 807 y Fq(n)1971 862 y(q)2060 840 y FB(v)m(ery)h(little:)50 1070 y FD(Prop)s(osition.)54 b Fu(L)-5 b(et)32 b Ft(f)873 1038 y Fq(n)863 1093 y(q)953 1070 y Fu(be)f(the)i(quantity)f Ft(f)1626 1038 y Fq(o)1616 1093 y(q)1697 1070 y Fu(compute)-5 b(d)34 b(fr)-5 b(om)33 b(The)-5 b(or)g(em)34 b(5)e(at)g(a)g(moment)h (when)g(con-)50 1183 y(tour)h(functionals)g Ft(F)13 b FB(\()p FD(\000)881 1197 y Fo(\014)943 1183 y FB(\))33 b Fu(of)g(extr)-5 b(emal)35 b(systems)f FD(\000)1883 1197 y Fo(\014)1944 1183 y Fu(,)f(of)h(a)f(size)g FB(diam)15 b FD(\000)2645 1197 y Fo(\014)2732 1183 y FB(=)26 b Ft(n)33 b Fu(wer)-5 b(e)33 b(just)g(\(partial)5 b(ly\))50 1292 y(expande)-5 b(d.)43 b(L)-5 b(et)31 b Ft(f)687 1259 y Fq(m)677 1314 y(q)786 1292 y Fu(be)f(an)h(analogous)h(quantity)f (constructe)-5 b(d)32 b(in)f(a)f(later,)i Ft(m)p Fu({th)g(step;)62 b Ft(m)25 b(>)g(n)p Fu(.)40 b(Then)1063 1501 y Fs(j)p Ft(f)1143 1464 y Fq(n)1133 1524 y(q)1212 1501 y Fs(\000)20 b Ft(f)1358 1464 y Fq(m)1348 1524 y(q)1426 1501 y Fs(j)26 b(\024)f Ft(\015)1620 1515 y Fq(n)1892 1501 y Fu(wher)-5 b(e)94 b Ft(\015)2256 1515 y Fq(n)2331 1501 y Fs(\031)25 b Ft(e)2469 1464 y Fo(\000)p Fq(C)t(an)2672 1501 y Ft(:)826 b FB(\(5)p Ft(:)p FB(5\))50 1820 y Fu(Pr)-5 b(oof.)47 b FB(Analyze)f(the)f(terms)g(in)f(\(2.23\))k(whic)m(h)c(mak)m(e)i (principal)c(con)m(tributions)i(to)i(the)g(terms)f Ft(w)3663 1834 y Fe(T)50 1929 y FB(formed)32 b(b)m(y)h(subsequen)m(t)g (applications)e(of)j(Theorem)f(5.)49 b(Notice)34 b(that)g(the)f(cores)h (of)g(the)f(new)g(clusters)50 2038 y(ha)m(v)m(e)e(diameters)e(of)h(the) h(order)e Fs(\025)c FB(diam)14 b FD(\000)p FB(;)30 b(the)g(smaller)f (systems)h(b)s(eing)e(already)i(partially)e(expanded)50 2147 y(b)s(efore.)40 b(It)31 b(is)e(on)m(y)i(here)f(where)g(one)h (needs)f(the)g(b)s(ound)e(\(1.28\).)159 2378 y(Then)44 b(w)m(e)h(easily)e(get)j(the)f(follo)m(wing)e(useful)g(consequence,)49 b(for)44 b(an)m(y)h(expansion)f(pro)s(cess)g(of)h(an)50 2487 y(abstract)31 b(P)m(.S.)g(mo)s(del)e(\(1.23\))k(whic)m(h)c(w)m(as) h(organized)h(according)f(to)h(sc)m(heme)g(desrib)s(ed)d(ab)s(o)m(v)m (e.)50 2718 y FD(Corollary.)52 b Fu(A)26 b(color)h Ft(q)i Fu(for)d(which)h(some)g(contour)g FD(\000)1952 2685 y Fq(q)2018 2718 y Fu(was)g(classi\014e)-5 b(d)27 b(nonsmal)5 b(l)28 b(at)e(le)-5 b(ast)27 b(once)f(during)50 2827 y(the)33 b(expansion)h(pr)-5 b(ocess)34 b(wil)5 b(l)33 b(never)f(be)-5 b(come)33 b(stable)h(later.)50 3058 y(Pr)-5 b(oof.)47 b FB(Denote)34 b(b)m(y)d Fj(\003)h FB(the)g(smallest)f(pac)m (king)g(cub)s(e)h(of)f FD(\000)2111 3025 y Fq(q)2152 3058 y FB(.)44 b(Clearly)-8 b(,)32 b(a)g(di\013erence)f(of)h(t)m(w)m(o) h(quan)m(tities)50 3167 y(where)28 b(the)g(\014rst)g(one)g(\(the)h Ft(n)p FB({th)f(v)m(alue)h Ft(u)1507 3181 y Fq(q)1547 3167 y FB(\))g(is)e Fs(\025)e FB(\012)p Ft(=)p Fs(j)p Fj(\003)p Fs(j)k FB(while)d(the)i(second)h(one)g(\(future)e(correction) i(to)50 3283 y(suc)m(h)h(a)h Ft(u)383 3297 y Fq(q)423 3283 y FB(\))f(is)g(lik)m(e)g Ft(e)789 3250 y Fo(\000)p Fq(C)t(a)13 b Fr(diam)e Fk(\003)1215 3283 y FB(only)29 b(cannot)i(b)s(e)f(equal)g(to)h(0)g(\(for)f(a)h(reasonable)f(c)m(hoice) i(of)e(\012)25 b Fs(\031)g Ft(\034)10 b FB(\).)159 3514 y(Therefore,)25 b(a)g(non)m(trivial)d(result)i(of)g(the)h(decomp)s (osition)e(\(4.5\))j(alw)m(a)m(ys)f(signalizes)d(that)j(the)g(external) 50 3623 y(color)i(of)f FD(\000)431 3637 y Fo(\014)519 3623 y FB(will)e(b)s(ecome)k(\014nally)c Fu(unstable)34 b FB(in)26 b(the)g(thermo)s(dynamical)f(limit.)38 b(Ev)m(en)26 b(if)g(suc)m(h)h(a)g(system)50 3732 y FD(\000)113 3746 y Fo(\014)201 3732 y FB(b)s(ecomes)g(\014nally)d(small)h(then)i(it)f (will)e(not)j(b)s(e)f(necessary)h(to)g(expand)f(it)g(\(fully\),)g (except)i(that)f(it)f(will)50 3855 y(app)s(ear)k(as)g(a)h(subsystem)e (of)i(some)g(bigger)f(system)1899 3832 y Fm(f)1882 3855 y FD(\000)1945 3869 y Fo(\014)2037 3855 y FB(ha)m(ving)g(a)g(stable)h (external)f(color.)159 4037 y(So)42 b(w)m(e)g(see)h(that)g(the)f (colors)g Ft(q)j FB(for)d(whic)m(h)f(some)h(nonsmall)e FD(\000)2455 4004 y Fq(q)2537 4037 y FB(app)s(ear)i(during)e(the)i (expansion)50 4146 y(pro)s(cedures)27 b(are)j(the)f(sure)f (\\candidates)h(for)g(unstabilit)m(y")e(in)h(the)h(limit.)38 b(But)29 b(ho)m(w)g(is)f(it)h(with)e(p)s(ossible)50 4255 y(\\candidates)j(for)g(stabilit)m(y")g(in)f Ft(Q)h FB(?)40 b(It)31 b(is)e(natural)h(to)h(lo)s(ok)f(for)g(them)g(at)h(the)g (collection)827 4435 y Fm(f)812 4458 y Ft(Q)884 4472 y Fq(n)959 4458 y FB(=)25 b Fs(f)p Ft(q)j Fs(2)d Ft(Q)g FB(:)h Ft(f)1458 4420 y Fq(n)1448 4480 y(q)1532 4458 y Fs(\024)f Ft(f)1683 4420 y Fq(n)1752 4458 y FB(+)20 b Ft(\015)1890 4472 y Fq(n)1939 4458 y Fs(g)92 b FB(where)151 b Ft(\015)2507 4472 y Fq(n)2582 4458 y Fs(\031)24 b Ft(e)2719 4420 y Fo(\000)p Fq(C)t(an)2922 4458 y Ft(:)576 b FB(\(5)p Ft(:)p FB(6\))50 4677 y(One)33 b(can)h(actually)f(manage)i(that)1316 4654 y Fm(f)1301 4677 y Ft(Q)1373 4691 y Fq(n)1456 4677 y FB(deca)m(ys)f(monotonically)f(with)f Ft(n)e Fs(!)h(1)p FB(,)j(for)g(a)g(suitable)e(c)m(hoice)50 4786 y(of)e Ft(\015)200 4800 y Fq(n)250 4786 y FB(.)41 b(Moreo)m(v)m(er,)32 b(if)e Ft(f)879 4753 y Fq(n)869 4809 y(q)953 4786 y FB(=)25 b Ft(f)1104 4753 y Fq(n)1183 4786 y FB(is)k(a)i(temp)s(orary)f(minim)m (um)d(then)j Ft(f)2463 4753 y Fo(1)2453 4809 y Fq(q)2570 4786 y FB(satis\014es)g(a)h(relation,)f(b)m(y)g(\(5.5\))1584 4983 y Ft(f)1639 4946 y Fo(1)1629 5006 y Fq(q)1741 4983 y Fs(\024)25 b Ft(f)1892 4946 y Fo(1)1988 4983 y FB(+)e(~)-48 b Ft(\015)2126 4997 y Fq(n)3523 4983 y FB(\(5)p Ft(:)p FB(7\))50 5180 y(with)31 b(another)k(~)-48 b Ft(\015)641 5194 y Fq(n)690 5180 y FB(.)46 b(So,)32 b(w)m(e)h(conclude)e(that)i (the)f(limit)e(of)i(\(5.6\))i(is)d(really)g(the)h(collection)g(of)g (all)f(v)m(alues)50 5289 y Ft(q)d Fs(2)d Ft(Q)p FB(,)30 b(for)g(whic)m(h)f(w)m(e)i(ha)m(v)m(e)h(the)e(limit)f(relation)g Ft(f)1837 5256 y Fo(1)1827 5312 y Fq(q)1939 5289 y FB(=)c Ft(f)2090 5256 y Fo(1)2166 5289 y FB(.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)k(18:34)1080 b Fz(39)p eop %%Page: 40 40 40 39 bop 50 340 a FD(Comparison)34 b(of)h(fully)g(expanded)g(mo)s (dels)g(in)g(di\013eren)m(t)f(tori,)h(and)g(in)g Fi(Z)2969 307 y Fq(\027)3010 340 y FD(.)159 586 y FB(The)h(limit)e(of)i(the)h (expansion)e(pro)s(cedures)g(of)i(Theorem)f(5,)i(discussed)d(ab)s(o)m (v)m(e)j(will)33 b(b)s(e)j(called)g(the)50 695 y Fu(ful)5 b(ly)32 b(expande)-5 b(d)35 b(model)9 b FB(.)42 b(Colors)30 b(realizing)f(the)h(minim)m(um)e(of)i Fs(f)p Ft(f)2366 662 y Fo(1)2356 718 y Fq(q)2443 695 y Fs(g)h FB(will)d(b)s(e)h(called)h (the)h Fu(stable)37 b FB(ones.)159 877 y(Notice)j(that)g(all)e(these)h (constructions)g(w)m(ere)h(still)d(done)i(not)g(on)g Fi(Z)2602 844 y Fq(\027)2682 877 y FB(but)f Fu(simultane)-5 b(ously)43 b(on)e(al)5 b(l)50 986 y(tori)41 b FB(\003.)i(Of)30 b(course,)i(one)f(w)m(ould)f(lik)m(e)g(to)i(compare)f(no)m(w)g(all)f (these)i(quan)m(tities,)f(for)f(di\013eren)m(t)h(tori)f(\003)i(of)50 1095 y(arbitrarily)h(large)j(size.)57 b(An)35 b(easy)i(insp)s(ection)c (of)j(the)g(expansions)f(then)g(sho)m(ws)h(that)g(the)g(quan)m(tities) 50 1207 y Ft(f)105 1174 y Fr(\003)95 1230 y Fq(q)189 1207 y FB(resp.)k Ft(f)472 1174 y Fr(\003)522 1147 y Fp(0)462 1230 y Fq(q)580 1207 y FB(constructed)31 b(in)e(\003)h(resp.) 40 b(in)29 b(\003)1667 1174 y Fo(0)1723 1207 y FB(di\013er)g(as)i(\(a)g (v)m(ery)g(little)e(quan)m(tit)m(y)i(indeed!\))1128 1465 y Fs(j)p Ft(f)1208 1428 y Fr(\003)1198 1488 y Fq(q)1282 1465 y Fs(\000)20 b Ft(f)1428 1428 y Fr(\003)1478 1400 y Fp(0)1418 1488 y Fq(q)1506 1465 y Fs(j)56 b(\024)g Ft(e)1756 1428 y Fo(\000)p Fq(C)t(\034)20 b Fr(min)n Fo(f)p Fr(diam)11 b(\003)p Fq(;)p Fr(diam)g(\003)2540 1400 y Fp(0)2564 1428 y Fo(g)2606 1465 y Ft(:)892 b FB(\(5)p Ft(:)p FB(8\))159 1770 y(Then)24 b(w)m(e)i(can)f(lo)s(ok)g(for)g(a)g(p) s(ossible)e(co)s(existence)j(of)f Fu(sever)-5 b(al)35 b FB(stable)25 b Ft(q)j FB(in)c(a)h(thermo)s(dynamical)f(limit.)50 1880 y(In)36 b(\014nite)g(v)m(olumes,)i(a)f(\\co)s(existence)h(of)f (phases")g(do)s(es)g(not)g(ha)m(v)m(e)h(a)f(sharply)e(de\014ned)g (meaning,)j(but)50 1989 y(still)28 b(the)j(notion)e(of)i(\\almost)g (stabilit)m(y")e(\(5.6\))k(has)d(some)g(practical)g(sense,)h(see)g(b)s (elo)m(w.)159 2170 y(Of)26 b(course,)h(the)g(co)s(existence)h(of)e (phases)g(is)g(in)f(principle)e(a)k Fu(very)i(delicate)34 b FB(phenomenon,)26 b(something)50 2280 y(whic)m(h)e(can)j(nev)m(er)f (b)s(e)f(decided)g(with)g(a)h(full)e(accuracy)j(\(for)f(nonsymmetric)e (mo)s(dels\))h(on)h(\014nite)f(v)m(olumes)50 2389 y(of)35 b(an)m(y)g(size.)54 b(W)-8 b(e)36 b(ma)m(y)f(b)s(e)f(able)h(to)g (expand)f(all)g(the)h(con)m(tour)h(systems)f FD(\000)2710 2345 y Fq(q)2710 2411 y Fo(\014)2806 2389 y FB(in)e(a)i(giv)m(en)g (torus)g(\003)f(but)50 2498 y(still)28 b(a)j(sligh)m(t)e(instabilit)m (y)f(of)j(suc)m(h)f(an)g(\\almost)h(stable")f Ft(q)k FB(can)c(b)s(e)g(found)f(in)g(bigger)h(\\univ)m(erses")g(\003)3603 2465 y Fo(0)3629 2498 y FB(.)159 2680 y(W)-8 b(e)29 b(will)d(see)j (that)f(if)f Ft(q)k FB(is)d(stable)f(in)g(the)i(thermo)s(dynamic)d (limit,)h(then)h(a)g(Gibbs)f(state)i(\\of)g(the)g(t)m(yp)s(e)50 2789 y Ft(q)s FB(")34 b(alw)m(a)m(ys)h(exists)e(for)h(the)g(mo)s(del)f (\(1.23\),)k(on)d(tori)g(of)g Fu(any)43 b FB(size.)51 b(By)35 b(the)f(latter)g(w)m(e)h(mean)f(the)g(\014nite)50 2898 y(v)m(olume)g(Gibbs)f(state)j(conditioned)e(at)h(\\in\014nit)m(y") f(\(see)h(the)g(remark)g(made)f(in)g(the)h(next)g(section\))g(b)m(y)50 3007 y(the)30 b(v)m(alue)g Ft(q)s FB(.)41 b(This)28 b(giv)m(es)i(the)h (usual)d(limit)g Ft(q)s FB({th)i(Gibbs)f(state)i(\(if)e(in)m(terpreted) h(in)f(suc)m(h)g(a)i(w)m(a)m(y)-8 b(,)32 b(on)e(the)50 3116 y(corresp)s(onding)e(spin)h(con\014guration)g(space\))j(in)d(the)h (thermo)s(dynamic)f(limit)f(\003)e Fs(!)f Fi(Z)3057 3083 y Fq(\027)3098 3116 y FB(.)159 3298 y(A)i(conclusion)f(is)g(that)i (there)f(is)g(alw)m(a)m(ys)h(a)f(nonempt)m(y)g(collection)g(of)h Fu(stable)34 b FB(colors)27 b Ft(q)j FB(\(namely)d(those)50 3407 y(minimizing)g(the)j(v)m(alue)h Ft(f)969 3374 y Fo(1)959 3429 y Fq(q)1045 3407 y FB(\))g(for)f(whic)m(h)f(\\ev)m (erything)i(is)e(expanded")h(and)g(on)g(an)m(y)h(torus)f(\003,)h Fu(only)i(one)50 3519 y(con\014gur)-5 b(ation)30 b(of)f(a)g(type)g Ft(q)i Fu(survives)e(al)5 b(l)29 b(the)g(expansions)35 b FB(\(w)m(e)27 b(do)f(not)g(lo)s(ok)f(here)h(at)h(systems)f FD(\000)3446 3533 y Fo(\014)3533 3519 y FB(with)50 3628 y Ft(E)117 3642 y Fq(r)r(est)253 3628 y FB(\()p FD(\000)351 3642 y Fo(\014)413 3628 y FB(\))g(=)f Fs(1)p FB(\);)30 b(namely)g(the)h(constan)m(t)g(con\014guration)f(\\)p Ft(q)k FB(ev)m(erywhere".)159 3810 y(Let)d(us)e(conclude)h(this)f (discussion)f(and)i(state)i(some)e(imp)s(ortan)m(t)g(consequences)h(:) 50 4056 y FD(Extremal)i(Gibbs)i(states)f(for)g(stable)g(colors)i Ft(q)s FD(.)52 b FB(T)-8 b(ak)m(e)32 b(the)e(Hamiltonian)e(\(1.23\),)33 b(select)d(a)h(giv)m(en)50 4165 y(stable)f Ft(q)e Fs(2)d Ft(Q)30 b FB(and)g(de\014ne)f(a)i(new)f(auxiliary)e(v)m(alue)j (\\in\014nitesimally)26 b(bigger)k(than")g Ft(e)3095 4179 y Fq(q)322 4453 y Ft(e)364 4416 y Fo(\003)364 4476 y Fq(q)432 4453 y FB(=)25 b Ft(e)570 4467 y Fq(q)661 4453 y FB(+)793 4392 y Ft(C)858 4406 y Fq(q)898 4392 y FB(\(\003\))p 793 4432 239 4 v 856 4516 a Fs(j)p FB(\003)p Fs(j)1194 4453 y FB(where)145 b(lim)1533 4512 y Fr(\003)p Fo(!)p Fe(Z)1704 4494 y Fv(\027)1794 4392 y Ft(C)1859 4406 y Fq(q)1899 4392 y FB(\(\003\))p 1794 4432 V 1857 4516 a Fs(j)p FB(\003)p Fs(j)2069 4453 y FB(=)25 b(0)92 b(and)144 b(lim)2554 4512 y Fr(\003)p Fo(!)p Fe(Z)2725 4494 y Fv(\027)2774 4453 y Ft(C)2839 4467 y Fq(q)2879 4453 y FB(\(\003\))26 b(=)f Fs(1)p Ft(:)273 b FB(\(5)p Ft(:)p FB(9\))50 4768 y(A)36 b(probabilit)m(y)d(measure)j Ft(P)1055 4719 y Fq(C)1107 4728 y Fv(q)1143 4719 y Fr(\(\003\))1042 4781 y Fo(\003)1254 4768 y FB(,)h(de\014ned)e(b)m(y)g(suc)m(h)h(a)g(c)m (hanged)g(Hamiltonian)e(\(using)h(the)h(v)m(alue)g Ft(e)3668 4735 y Fo(\003)3668 4790 y Fq(q)50 4880 y FB(instead)42 b(of)g Ft(e)533 4894 y Fq(q)616 4880 y FB(for)g(this)g(particular)f (color)h Ft(q)s FB(,)k(ev)m(erything)c(else)g(in)g(\(1.23\))i(left)f (unc)m(hanged\))f(on)h(the)50 4989 y(con\014guration)29 b(space)i(of)f(\(1.23\))i(will)27 b(b)s(e)i(called)g(an)h Fu(extr)-5 b(emal)34 b Ft(q)s Fu({th)f(Gibbs)f(state)37 b FB(of)30 b(the)h(mo)s(del)d(\(1.23\),)50 5098 y(on)33 b(a)g(torus)g(\003.)48 b(Of)33 b(course,)h(there)f(is)f(some)h(am)m (biguit)m(y)g(\(the)g(c)m(hoice)h(of)f Ft(e)2672 5065 y Fo(\003)2672 5121 y Fq(q)2714 5098 y FB(\))g(in)f(the)h(de\014nition) e(of)i(this)50 5207 y(notion)d(but)f(this)h(asm)m(biguit)m(y)f(disapp)s (ears)f(with)i(\003)25 b Fs(!)g Fi(Z)2064 5174 y Fq(\027)2105 5207 y FB(:)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)k(18:34)1080 b Fz(40)p eop %%Page: 41 41 41 40 bop 50 346 a FD(Theorem)46 b(7.)53 b Fu(The)43 b(limit)1028 313 y Fr(15)1105 346 y Fu(,)i(for)e FB(\003)h Fs(!)f Fi(Z)1639 313 y Fq(\027)1680 346 y Fu(,)i(of)d Ft(P)1940 297 y Fq(C)1992 306 y Fv(q)2029 297 y Fr(\(\003\))1927 359 y Fo(\003)2182 346 y Fu(does)i(not)f(depend,)j(under)d(\(5.9\),)j (on)d(the)50 455 y(choice)36 b(of)g Ft(C)507 469 y Fq(q)547 455 y FB(\(\003\))p Fu(.)52 b(T)-7 b(r)i(anslate)g(d)38 b(to)e(a)h(spin)f(language,)h(it)e(is)h(e)-5 b(qual)36 b(to)g(the)g(extr)-5 b(emal)38 b Ft(q)s Fu({th)e(Gibbs)g(state)50 564 y(on)d Fi(Z)245 531 y Fq(\027)286 564 y Fu(.)50 808 y FD(Residual)i(con)m(tours.)159 1052 y FB(F)-8 b(or)21 b(a)g(mo)s(del)e(\(1.23\))j(w)m(e)f(in)m(tro)s(duce)e(another)i (notion,)h(c)m(haracteristic)f(for)f(this)f(approac)m(h)i(to)g(Pirogo)m (v{)50 1161 y(Sinai)28 b(theory)j(and)e(ha)m(ving)h(no)h(analogy)g(in)e (the)h(theory)h(of)f(p)s(olymer)f(mo)s(dels.)159 1343 y(This)22 b(will)g(b)s(e)h(relev)m(an)m(t)i(for)f(a)h(detailed)e(study) h(of)g(structure)g(of)g(Gibbs)f(states)i(on)f(an)m(y)h(torus)f(\003,)h (under)50 1452 y(an)m(y)35 b(constrain)m(t)g Fs(C)5 b FB(.)53 b(W)-8 b(e)36 b(are)f(concerned)g(here)f(with)f(a)i(\014xed)f (Hamiltonian)f(and)h(so)h(it)f(is)g(m)m(uc)m(h)h(more)50 1561 y(con)m(v)m(enien)m(t)d(to)g Fu(r)-5 b(eturn)35 b(to)f(the)f(setting)h(of)g(The)-5 b(or)g(em)35 b(4)44 b FB(and)31 b(to)h(forget)g(an)m(y)g(partial)e(expansions)g(made)50 1670 y(later.)40 b(An)30 b(analogy)h(of)f(suc)m(h)g(a)g(construction)g (in)f(the)h(con)m(text)i(of)f(Theorem)e(5)i(can)f(b)s(e)g(also)g(made,) g(but)50 1779 y(it)g(is)f(a)i(little)e(bit)g(arti\014cial,)h(less)f (useful)g(then)h(the)g(more)h(direct)f(approac)m(h)h(of)f(Theorem)g(4.) 159 1961 y(So,)37 b(w)m(e)f(tak)m(e)h(here)f(again)f(the)h(sc)m(heme)g (\(1.22\))i(and)d(w)m(e)h(use)f(rep)s(eatedly)g(\(in\014nitely)e(man)m (y)j(times\))50 2070 y(Theorem)31 b(4.)44 b(The)31 b(limit)e(will)g(b)s (e)h(again)i(called)e(a)i Fu(ful)5 b(ly)33 b(expande)-5 b(d)36 b(model)9 b FB(.)45 b(It)31 b(is)g(imp)s(ortan)m(t)f(to)i (notice)50 2179 y(that)i(the)g(quan)m(tities)f Ft(f)888 2146 y Fo(1)878 2201 y Fq(q)999 2179 y FB(are)h(the)g(same)g(as)g(b)s (efore)g(for)f(all)g(stable)h Ft(q)s FB(,)g(and)g(stabilit)m(y)e(of)i Ft(q)j FB(means)d(the)50 2288 y(same)c Fu(in)h(both)i(appr)-5 b(oaches)7 b FB(.)44 b(This)27 b(follo)m(ws)i(from)g(Theorems)g(5)g (and)g(4;)h(namely)f(b)s(oth)g(results)f(pro)s(duce)50 2397 y(a)j(mo)s(del)e Fs(R)p FB({equiv)m(alen)m(t)i(to)g(the)f (original)f(one.)159 2579 y(F)-8 b(or)36 b(nonstable)f Ft(q)s FB(,)i(the)f(quan)m(tities)f Ft(f)1490 2546 y Fo(1)1480 2601 y Fq(q)1601 2579 y FB(are)h(not)g(the)g(same)g(as)g(b)s (efore.)56 b(Ho)m(w)m(ev)m(er,)39 b(the)c(quan)m(tities)50 2691 y Ft(u)102 2658 y Fo(1)102 2714 y Fq(q)206 2691 y FB(=)27 b Ft(f)359 2658 y Fo(1)349 2714 y Fq(q)457 2691 y Fs(\000)20 b Ft(f)603 2658 y Fo(1)711 2691 y FB(are)32 b(of)g(the)g(same)g(order)f(in)f(b)s(oth)h(approac)m(hes.)45 b(\(W)-8 b(e)34 b(omit)d(a)h(detailed)f(discussion.\))50 2800 y(F)-8 b(or)33 b(these)g(v)m(alues)f(of)h Ft(q)s FB(,)g(there)g(are)f(non)m(trivial)f(families)f Fs(R)2148 2767 y Fo(1)2148 2823 y Fq(q)2254 2800 y Fs(\032)f(A)2427 2767 y Fq(q)2499 2800 y FB(of)j(con)m(tours)h FD(\000)c FB(=)g FD(\000)3228 2767 y Fq(q)3300 2800 y FB(\\surviving)50 2912 y(all)h(the)h(expansions".)42 b(Let)32 b(us)e(call)g(the)i(elemen) m(ts)f(of)g Fs(R)2034 2879 y Fo(1)2138 2912 y FB(=)26 b Fs([)2296 2926 y Fq(q)2336 2912 y Fs(R)2413 2879 y Fo(1)2413 2935 y Fq(q)2521 2912 y FB(\(the)32 b(union)d(is)h(o)m(v)m (er)i Ft(q)e FB(:)c Ft(u)3428 2879 y Fo(1)3428 2935 y Fq(q)3532 2912 y Ft(>)g FB(0\))50 3024 y(the)k Fu(r)-5 b(esidual)34 b(contours)39 b FB(of)31 b(the)f(mo)s(del)f(\(1.23\).)43 b(Of)30 b(course)g(there)g(is)g(again)g(some)h(arbitrariness)d(in)h (the)50 3133 y(de\014nition)f(of)i(all)g(these)g(notions;)g(they)h(dep) s(end)e(on)h(the)g(concrete)i(c)m(hoice)f(of)g(\012)f(and)g(other)g (things.)159 3315 y(Residual)39 b(con)m(tours)i(are)g(imp)s(ortan)m(t)f (when)g(in)m(v)m(estigating)h(unstable)e(b)s(oundary)g(conditions)g(of) i(a)50 3424 y(t)m(yp)s(e)33 b(\\unstable)g(color)h Ft(q)i FB(outside)d(a)g(giv)m(en)h(v)m(olume)f Ft(M)41 b Fs(\032)30 b FB(\003".)50 b(More)34 b(general)g(b)s(oundary)d(conditions)50 3533 y(in)e(spin)g(mo)s(dels)g(cannot)i(b)s(e)f(directly)f(addressed)h (in)f(full)f(generalit)m(y)j(in)e(the)i(language)g(of)g(\(1.23\).)43 b(The)50 3643 y(setting)e(\(1.23\))h(allo)m(ws)e(only)g(a)h(form)m (ulation)f(of)g(someho)m(w)i(more)e(restricted)h(questions)e(lik)m(e)h (ab)s(o)m(v)m(e;)50 3752 y(most)i(imp)s(ortan)m(t)f(case)i(b)s(eing)e (that)h(of)g(an)g(ensem)m(ble)g(where)f(a)h(giv)m(en)g(\\distan)m(t")h (region)f(\003)3413 3766 y Fo(1)3532 3752 y 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y(to)k(lo)s(ok)f(at)i(suc)m(h)e(situations.)56 b(If)35 b(\003)24 b Fs(n)g FB(\003)1477 4493 y Fo(1)1589 4479 y FB(is)35 b(relativ)m(ely)g(small)f(sub)m(v)m(olume)h(of)h(\003)g (then)f(astronomical)50 4588 y(con)m(tours)c(are)g(excluded)e(b)m(y)i (de\014nition,)d(and)i(problem)f(has)i(a)g(direct)f(description)e(in)i (the)h(language)g(of)50 4697 y(\(1.23\).)43 b(This)28 b(is)i(a)h(most)f(natural)g(situation,)f(lik)m(e)h(if)f(\003)21 b Fs(n)f FB(\003)2130 4711 y Fo(1)2238 4697 y FB(is)29 b(a)i(\014nite)e(sub)m(v)m(olume)h(of)g(\003)c Fs(\021)f Fi(Z)3438 4664 y Fq(\027)3478 4697 y FB(.)159 4879 y(If)39 b(w)m(e)h(fully)d(expand)i(the)g(latter)h(situation)f(then)g(residual)e (con)m(tours)j(separating)f(the)h(region)f(\\of)50 4988 y(t)m(yp)s(e)32 b Ft(q)s FB(")g(around)f(\003)750 5002 y Fo(1)859 4988 y FB(from)g(the)h(big)f(\\bubbles")f(\(of)i(\\phases")h (ha)m(ving)e(another,)i(more)f(stable)f(color\))p 50 5111 546 4 v 159 5172 a Fl(15)230 5198 y Fz(W)-6 b(e)38 b(omit)f(a)g(more)g(detailed)i(discussion)g(of)e(these)i(notions)e (here.)71 b(Ho)n(w)n(ev)n(er,)41 b(an)c(appropriately)g(general)h(and) 50 5289 y(exhaustiv)n(e)26 b(reference)h(on)f(these)h(general)f (questions)h(is)f(still)i(missing,)f(in)f(the)g(con)n(text)g(of)f (abstract)h(P)n(.S.)g(mo)r(dels.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)j(18:34)1080 b Fz(41)p eop %%Page: 42 42 42 41 bop 50 340 a FB(do)26 b(app)s(ear)g(for)g(large)g(enough)g(\003) 12 b Fs(n)g FB(\003)1326 354 y Fo(1)1403 340 y FB(,)28 b(and)d(their)g(statistics)i(has)f(to)g(b)s(e)g(analyzed.)39 b(This)25 b(problem)f(w)m(as)50 449 y(addressed)g(in)f([Z])i(but)g(its) f(fuller)f(treatmen)m(t)j(is)e(also)h(missing.)37 b(Ho)m(w)m(ev)m(er,) 28 b(this)c(is)g(a)h(problem)f(essen)m(tially)50 558 y(complemen)m(tary)33 b(to)g(what)g(w)m(e)g(call)f(the)h(P)m(.S.)g(tec) m(hnology)-8 b(.)49 b(Detailed)33 b(statistics)g(of)g(residual)d(con)m (tours)50 667 y(\(ha)m(ving)c(expanded)f(the)h(rest)g(of)g(the)g(mo)s 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FC(6.)51 b(Cluster)32 b(exp)-6 b(ansion)32 b(of)h(conditioned)f (models.)49 b(Str)n(ucture)31 b(of)i(Gibbs)g(st)-6 b(a)g(tes)50 1871 y FB(Here)40 b(again,)i(it)d(is)g(more)h(con)m(v)m(enien)m(t)g(to) h(sta)m(y)f(in)e(the)i(setting)g(of)g(Theorem)f(4.)69 b(W)-8 b(e)40 b(start)h(with)d(an)50 1980 y(imp)s(ortan)m(t)27 b(remark)g(that)h(an)g(analogous)g(construction)f(\(to)i(that)f(of)g (the)g(full)e(expansion)g(of)i Ft(Z)7 b FB(\(\003\)\))28 b(can)50 2089 y(b)s(e)i(done)i Fu(also)i(in)g(the)f(conditione)-5 b(d)36 b(ensemble)d(\(1.24\))s FB(,)h(with)c(a)i(\014xed)e(limitation)f Fs(C)37 b FB(i.e.)43 b(at)32 b(a)g(presence)50 2198 y(of)e(some)h (\014xed)f(conditioning)e(in)h(the)i(ensem)m(ble)f(\(1.23\).)159 2380 y(T)-8 b(o)26 b(understand)e(this,)i(analyze)g(in)f(what)g(sense)h (w)m(e)g(actually)g(needed)f(the)h(translation)f(in)m(v)m(ariance)h(of) 50 2489 y(the)c(mo)s(del)e(\(1.23\))k(resp.)38 b(more)21 b(generally)g(ho)m(w)h(m)m(uc)m(h)g(\\lo)s(cal")g(w)m(as)g(the)g 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b(not)h(e\013ect)h(construction)e(of)h(clusters)f(outside)g(it.)50 b(More)34 b(precisely)-8 b(,)34 b(if)50 3325 y(suc)m(h)j(a)i (limitation)c(is)i(expressed)h(in)e(terms)i(of)g(con)m(tours)h(from)e (some)h(subsystem)f Fs(S)45 b(\032)38 b(A)f FB(then)h(the)50 3435 y(v)m(alues)25 b(of)h Ft(w)482 3449 y Fe(T)555 3435 y FB(for)g(clusters)f(from)g(ob)5 b(jects)27 b Fu(compatible)34 b FB(with)24 b(an)m(y)i(con)m(tour)h(from)e Fs(S)33 b FB(are)26 b(indep)s(enden)m(t)e(of)50 3544 y Fs(S)7 b FB(.)45 b(So,)32 b(while)e(the)i(estimates)h(of)f(the)g(con)m(v)m (ergence)j(of)d(clusters)f(remain)g(essen)m(tially)f(unc)m(hanged)i(ev) m(en)50 3653 y(at)f(the)g(presence)g(of)f Fs(C)5 b FB(,)31 b(the)g(translation)f(in)m(v)m(ariancy)g(of)h(the)f(newly)g(formed)g Ft(w)2845 3667 y Fq(T)2933 3653 y FB(will)e(b)s(e)i(lost)g(only)g(for) 50 3762 y(clusters)37 b(whic)m(h)g(touc)m(h)h(the)h(system)f Fs(S)7 b FB(.)63 b(In)38 b(practise,)i Fs(C)j FB(will)35 b(b)s(e)i(often)i(form)m(ulated)e(as)i(a)f(condition)50 3871 y(on)32 b(\\)p Ft(q)s FB({v)m(alued)g(spins)f(at)i(in\014nit)m (y".)44 b(Then)31 b(the)i(iterativ)m(e)g(application)d(of)j(Theorem)f (4)g(giv)m(es)h(a)f(cluster)50 3980 y(expansion,)d(for)h Ft(q)k FB(stable,)c(of)g(an)h(extremal)f(Gibbs)f(state)j(\\of)f(a)g(t)m (yp)s(e)f Ft(q)s FB(")h(in)e(the)h(limit)e(\003)e Fs(!)f Fi(Z)3390 3947 y Fq(\027)3431 3980 y FB(.)159 4162 y(So)32 b(it)g(has)g(a)h(prefect)g(sense)f(to)i(apply)-8 b(,)32 b(iterativ)m(ely)-8 b(,)33 b(the)g(expansions)e(of)i(Theorem)f(4)h(at)g (a)g(presence)50 4271 y(of)38 b(some)h(\014xed)e(limitation)f Fs(C)5 b FB(,)41 b(expressed)c(in)g(terms)h(of)g(con)m(tours)h(from)f (some)g Fs(S)7 b FB(.)64 b(W)-8 b(e)40 b(ma)m(y)e(expand)50 4380 y(iterativ)m(ely)23 b(all)g(the)h(small)e(extremal)i(systems)f FD(\000)1739 4394 y Fo(\014)1824 4380 y FB(\(arising)f(during)g(the)i (iterativ)m(e)g(pro)s(cess\))f(compatible)50 4489 y(with)29 b Fs(C)5 b FB(,)31 b(but)e Fu(only)34 b(those)f(systems)7 b FB(.)43 b(Of)29 b(course,)i(also)f(systems)h FD(\000)2355 4503 y Fo(\014)2446 4489 y FB(bigger)f(than)g(con)m(tours)h(from)f Fs(S)7 b FB(.)159 4671 y(This)38 b(clearly)i(requires)f(some)h (adaptation)h(of)f(the)h(sc)m(heme)g(\(1.22\),)k(b)m(y)40 b(reducing)f(it)h(only)g(to)h(the)50 4780 y(ob)5 b(jects)41 b Fu(untouche)-5 b(d)51 b FB(b)m(y)40 b Fs(C)5 b FB(.)70 b(Con)m(tour)40 b(systems)h(incompatible)d(with)h(some)h(con)m(tour)h (of)g(a)f(family)f Fs(S)50 4889 y FB(\(in)28 b(whic)m(h)g(the)i (limitation)d Fs(C)35 b FB(is)28 b(form)m(ulated\))i(are)f(left)h (unexpanded,)e(of)h(course.)41 b(Ho)m(w)m(ev)m(er,)32 b(for)d(stable)50 4998 y(colors)21 b Ft(q)j FB(suc)m(h)d(a)h (restriction)e(creates)j(only)d(v)m(ery)i(minor,)g(easily)e(con)m (trollable)h(c)m(hanges)h(\(when)f(compared)50 5107 y(to)31 b(the)f(full)f(expansion)g(in)g(the)h(unconditioned)e(ensem)m(ble)i (\(1.23\)\).)159 5289 y(It)i(is)g(imp)s(ortan)m(t)f(to)i(realize)g (that)f(suc)m(h)h(expansions)e(actually)h(will)d(giv)m(e)k(a)g(v)m(ery) g(compact)g Fu(formula)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)c(18:34)1080 b Fz(42)p eop %%Page: 43 43 43 42 bop 50 340 a Fu(for)35 b(Gibbs)g(pr)-5 b(obability)43 b FB(of)33 b(an)g(ev)m(en)m(t)h(\\an)f(island)e Fs(D)k FB(with)d(external)g(color)h Ft(q)j FB(app)s(ears)c(in)f(the)i Ft(q)s FB({)g(sea",)50 449 y(b)s(oth)f(in)f(the)h(full)f(ensem)m(ble)h (\(1.23\))i(resp.)47 b(in)31 b(the)i(conditioned)e(ensem)m(ble)h (\(1.24\).)49 b(Let)33 b(us)f(denote)h(b)m(y)50 558 y Ft(P)13 b FB(\()p Fs(D)s FB(\))36 b(the)g(probabilit)m(y)e(of)i(suc)m (h)f(an)h(ev)m(en)m(t,)j(namely)c(that)i(there)f(is)f(no)h(con)m(tour)g (encircling)e Fs(D)s FB(.)57 b(The)50 667 y(kno)m(wledge)30 b(of)h(this)e(quan)m(tit)m(y)i(is)e(esp)s(ecially)g(useful)f(in)h(the)i (follo)m(wing)e(t)m(w)m(o)i(cases:)53 849 y(i\))45 b(if)29 b(the)h(external)g(color)g Ft(q)j FB(of)d(the)h(con)m(tour)f(system)h Fs(D)h FB(is)d Fu(stable)38 b FB(and)29 b(all)g(the)h(in)m(terior)f (con)m(tours)i(of)f Fs(D)159 958 y FB(are)h(residual)d(in)h(the)h(mo)s (del)g(\(1.23\))28 1140 y(ii\))44 b(if)25 b Fs(D)j FB(=)d FD(\000)g FB(is)g(a)h(single)f(residual)e(con)m(tour)k({or)f(a)g (collection)f(of)h(residual)e(con)m(tours{)j(with)d(an)i(unstable)159 1249 y(external)k(color)f Ft(q)s FB(,)h(in)f(a)h(v)m(olume)f(\003)h (whic)m(h)f(is)f(not)i(\\to)s(o)h(big")f(\(resp.)40 b(under)28 b(additional)g(constrain)m(ts)159 1358 y(outside)36 b Fs(D)k FB(whic)m(h)c(mak)m(e)i(the)f(probabilit)m(y)d(of)k(suc)m(h)e (ev)m(en)m(t)j(an)e(in)m(teresting)f(quan)m(tit)m(y)-8 b(,)40 b(ev)m(en)e(if)e(the)159 1467 y(instabilit)m(y)28 b(of)i(the)h(external)f(color)h Ft(q)i FB(w)m(ould)c(b)s(e)h(otherwise) f(strongly)h(felt)g(in)f(the)i(exterior)f(of)h Fs(D)s FB(\).)159 1649 y(W)-8 b(e)31 b(prefer)f(to)h(write)f Fs(D)j FB(instead)c(of)i FD(\000)1503 1663 y Fo(\014)1594 1649 y FB(b)s(ecause)g(w)m(e)g(do)f(not)h(assume)f(its)f(\\extremalit)m (y")j(here.)159 1831 y(T)-8 b(ak)m(e)33 b(a)e(limitation)e Fs(C)37 b FB(\\ha)m(ving)31 b(the)h(v)m(alue)f Ft(q)j FB(at)e(in\014nit)m(y")e(in)g(the)h(ensem)m(ble)g(\(1.24\).)46 b(In)m(tro)s(duce,)31 b(in)50 1940 y(addition)20 b(to)i(the)g (probabilit)m(y)d(\(1.17\),)26 b(here)21 b(denoted)h(as)f Ft(P)2083 1907 y Fq(q)2124 1940 y FB(\()p Fs(D)s FB(\))g(also)h(the)g (quan)m(tit)m(y)f Ft(P)3035 1896 y Fq(q)3022 1967 y(r)r(ef)3140 1940 y FB(\()p Fs(D)s FB(\))h(measuring)50 2057 y(the)29 b(probabilit)m(y)-8 b(,)27 b(in)h(the)h(ensem)m(ble)g(\(1.24\),)i(of)f (the)f(ev)m(en)m(t)h(that)g(\\there)f(is)f(a)i(free)f(place)g(around)f Fs(D)p 3462 2072 73 4 v 31 w FB(and)50 2166 y(moreo)m(v)m(er)36 b Fs(D)p 446 2181 V 37 w FB(has)e(color)g Ft(q)s FB(".)53 b(More)35 b(precisely)-8 b(,)35 b(in)e(the)h(latter)h(case)h(w)m(e)e (in)m(v)m(estigate)i(the)f(ev)m(en)m(t)g(that)g(a)50 2275 y(con\014guration)29 b(\()p Fs(\021)h FB(compatible)f(con)m(tour)h (system\))g Fs(f)p FD(\000)1974 2289 y Fq(i)2005 2275 y Fs(g)g FB(with)e(a)i(prescrib)s(ed)d(external)j(color)g Ft(q)i FB(is)d(suc)m(h)50 2384 y(that)34 b(dist)n(\()p FD(\000)494 2398 y Fq(i)525 2384 y Ft(;)15 b Fs(D)p 565 2399 V 3 w FB(\))30 b Fs(\025)g FB(2)j(for)g(eac)m(h)h FD(\000)1294 2398 y Fq(i)1324 2384 y FB(.)49 b(W)-8 b(e)35 b(will)30 b(see)k(that)f(the)h(quan)m(tit)m(y)f Ft(F)13 b FB(\()p Fs(D)s FB(\))34 b(is)e(really)g(\\tailored")h(to)50 2493 y(answ)m(ering)c(the)i(ab)s(o)m(v)m(e)g(probabilistic)c (questions.)40 b(Namely)-8 b(,)31 b(w)m(e)g(will)d(ha)m(v)m(e)j(the)g (relation)1280 2705 y Ft(P)1351 2668 y Fq(q)1391 2705 y Fs(D)s FB(\))1547 2654 y Ft(:)1524 2705 y FB(=)25 b Ft(P)1691 2662 y Fq(q)1678 2733 y(r)r(ef)1796 2705 y FB(\()p Fs(D)s FB(\))46 b(exp\()p Fs(\000)p Ft(F)13 b FB(\()p Fs(D)s FB(\)\))50 2925 y(more)29 b(precisely)f(in)f(place)i(of) g Ft(F)13 b FB(\()p Fs(D)s FB(\))30 b(w)m(e)f(ha)m(v)m(e)h(to)g(tak)m (e)h(quan)m(tit)m(y)e Ft(F)2394 2939 y Fo(1)2472 2925 y FB(\()p Fs(D)s FB(\))g(describ)s(ed)e(b)s(elo)m(w.)39 b(The)29 b(term)50 3034 y Ft(P)121 3001 y Fo(\016)108 3059 y Fr(\003)163 3034 y FB(\()p Fs(D)s FB(\))d(will)d(b)s(e)i(appro)m (ximately)g(equal)h(to)g(1)h(in)d(man)m(y)i(cases)h(of)f(in)m(terest.) 39 b(Then)25 b(w)m(e)i(really)d(get)j(a)g(useful)50 3143 y(form)m(ula)i({)i(assuming)e(that)i(w)m(e)g(establish)e(the)h(P)m (eierls)g(condition)f(for)h Ft(F)13 b FB(\()p Fs(D)s FB(\):)50 3378 y FD(Con)m(tour)35 b(functional)g Ft(F)1022 3345 y Fo(1)1099 3378 y FB(\()p Fs(D)s FB(\))g FD(and)g(the)g (probabilit)m(y)g Ft(P)2267 3345 y Fq(q)2307 3378 y FB(\()p Fs(D)s FB(\))p FD(.)47 b(The)35 b(P)m(eierls)g(argumen)m(t.)159 3613 y FB(It)g(remains)g(to)h(giv)m(e)g(a)g(precise)f(de\014nition)e (of)i(the)h(\\limit")e(con)m(tour)j(functional)c Ft(F)3118 3627 y Fo(1)3196 3613 y FB(\()p Fs(D)s FB(\),)k(as)f(com-)50 3722 y(pared)e(to)i(\\temp)s(orary")f(\(inductiv)m(ely)f(de\014ned\))g (con)m(tour)h(functionals)f Ft(F)13 b FB(\()p FD(\000)2796 3736 y Fo(\014)2857 3722 y FB(\))36 b(used)e(in)f(Theorem)i(4.)50 3831 y(While)f(the)i(latter)g(quan)m(tities)g(w)m(ere)g(constructed)g 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b(These)50 4486 y(quan)m(tities)30 b(are)g(translation)g (in)m(v)m(arian)m(t)g(except)h(of)g(the)f(situation)f(when)h Ft(T)43 b(\023)30 b Fs(D)s FB(.)50 4721 y FD(The)k(limit)g(con)m(tour)i (functional)f Ft(F)1456 4688 y Fo(1)1533 4721 y FB(\()p Fs(D)s FB(\))h FD(of)f(a)f(con)m(tour)i(system)e Fs(D)s FD(.)159 4956 y FB(Let)d Fs(D)i FB(b)s(e)c(a)i(con)m(tour)g(system)g (with)e(an)h(external)g(color)h Ft(q)s FB(.)40 b(Put)433 5190 y Ft(F)504 5152 y Fo(1)581 5190 y FB(\()p Fs(D)s FB(\))25 b(=)g Ft(E)5 b FB(\()p Fs(D)s FB(\))21 b(+)1172 5103 y Fm(X)1180 5299 y Fr(~)-42 b Fq(q)s Fo(6)p Fr(=)p Fq(q)1304 5089 y Fm(\020)1358 5190 y FB(\()p Ft(e)1441 5204 y Fr(~)g Fq(q)1496 5190 y Fs(\000)20 b Ft(e)1629 5204 y Fq(q)1669 5190 y FB(\))p Fs(j)15 b FB(in)m(t)1859 5204 y Fr(~)-41 b Fq(q)1909 5190 y Fs(D)s(j)20 b(\000)2217 5103 y Fm(X)2118 5298 y Fq(T)2166 5307 y Fw(~)-36 b Fv(q)2197 5298 y Fo(\032)p Fr(in)n(t)2346 5307 y Fw(~)g Fv(q)2390 5298 y Fo(D)2463 5190 y Ft(w)2530 5152 y Fo(1)2528 5212 y Fq(T)2576 5221 y Fw(~)g Fv(q)2662 5190 y FB(+)2853 5103 y Fm(X)2753 5298 y Fq(T)2796 5307 y Fv(q)2832 5298 y Fo(\032)p Fr(in)n(t)2981 5307 y Fw(~)g Fv(q)3025 5298 y Fo(D)3099 5190 y Ft(w)3166 5152 y Fo(1)3164 5212 y Fq(T)3207 5221 y Fv(q)3247 5089 y Fm(\021)3302 5190 y Ft(:)196 b FB(\(6)p Ft(:)p FB(1\))50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(43)p eop %%Page: 44 44 44 43 bop 50 340 a FB(Clearly)-8 b(,)25 b Ft(F)450 307 y Fo(1)527 340 y FB(\()p Fs(D)s FB(\))h(is)f(almost)g(equal)g(to)h(the) f(\\temp)s(orary")h Ft(F)13 b FB(\()p Fs(D)s FB(\))26 b(used)f(in)f(Theorem)h(4.)39 b(The)25 b(di\013erence,)50 449 y(caused)30 b(b)m(y)h(\\future")f(corrections)g(to)i Ft(F)13 b FB(\()p Fs(D)s FB(\),)31 b(is)e(v)m(ery)i(small,)e(of)h(the)h (order)f(\(compare)h(\(2.6\)\))1204 664 y Fs(j)p Ft(F)1300 627 y Fo(1)1378 664 y FB(\()p Fs(D)s FB(\))20 b Fs(\000)g Ft(F)13 b FB(\()p Fs(D)s FB(\))p Fs(j)26 b(\024)f Ft(C)7 b(e)2107 627 y Fo(\000)p Fq(C)2220 600 y Fp(0)2245 627 y Fq(\034)19 b Fr(diam)11 b Fo(D)2530 664 y Ft(:)968 b FB(\(6)p Ft(:)p FB(2\))50 940 y FD(Theorem)40 b(8.)52 b Fu(L)-5 b(et)38 b Fs(E)8 b FB(\()p Fs(D)s FB(\))38 b Fu(be)f(an)h(event,)g(in)f(an)h(ensemble)f(given)g(by)h(the)f (partition)j(function)d(\(1.24\))50 1049 y(\(conditione)-5 b(d)34 b(by)f(\\color)g Ft(q)j Fu(at)d(in\014nity"\))g(that)g(a)g (given)f(contour)i(system)f Fs(D)i Fu(appe)-5 b(ars)35 b(as)e(a)g(subsystem)50 1158 y(of)g(the)g(contour)g(system)h Ft(x)993 1172 y Fr(\003)1047 1158 y Fu(.)42 b(Then)33 b(the)g(pr)-5 b(obability)34 b(of)f(this)g(event)g(is)f(e)-5 b(qual)33 b(to)1028 1373 y Ft(P)1099 1330 y Fq(q)1086 1400 y Fr(\003)1140 1373 y FB(\()p Fs(E)8 b FB(\()p Fs(D)t FB(\)\))26 b(=)f(exp\()p Fs(\000)p Ft(F)1848 1336 y Fo(1)1925 1373 y FB(\()p Fs(D)s FB(\)\))15 b(exp)q(\()2333 1287 y Fm(X)2293 1482 y Fq(T)10 b Fr(:)p Fq(T)g(\023)p Fo(D)2571 1373 y FB(\026)-64 b Ft(w)2619 1336 y Fo(1)2617 1396 y Fq(T)2697 1373 y FB(\))791 b(\(6)p Ft(:)p FB(3\))50 1656 y Fu(wher)-5 b(e)52 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y(hold)h(for)h(more)h(complicated)f(constrain)m(ts)g Fs(C)35 b FB(\(in\015uencing)28 b(then)i(the)h(v)m(alues)f(of)h Ft(w)2971 2339 y Fq(T)3058 2325 y FB(for)g Ft(T)13 b(\023)p Fs(C)5 b FB(\).)50 2558 y FD(Conclusion.)159 2791 y FB(Let)42 b(us)f(shortly)f(summarize)h(the)g(additional)f(information)f(w)m(e)j (deriv)m(ed)f(in)f(these)i(last)f(sections.)50 2900 y(Denote)49 b(b)m(y)e Ft(Z)591 2867 y Fq(r)631 2900 y FB(\(\003\))h(the)f (partition)f(function)g(o)m(v)m(er)j(all)d(con\014gurations)g(whic)m(h) g(ha)m(v)m(e)j(at)f(least)f(one)50 3009 y(\\astronomically)42 b(large")g(con)m(tour)i FD(\000)p FB(.)76 b(\(Ob)m(viously)-8 b(,)44 b(these)f(con\014gurations)f(are)g(negligible)e(in)h(v)m(ery)50 3118 y(large)33 b(v)m(olumes)f(\003.\))48 b(Consider)30 b(\014nally)h(the)i Fu(ful)5 b(l)34 b(ensemble)40 b FB(with)31 b(the)i(partition)e(function)g Ft(Z)3339 3132 y Fq(f)7 b(ull)3476 3118 y FB(\(\003\))30 b(=)50 3228 y Ft(Z)7 b FB(\(\003\))20 b(+)g Ft(Z)432 3195 y Fq(r)472 3228 y FB(\(\003\).)42 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b(se)-5 b(e)33 b(also)g(\(3.24\),)h(with)g Ft(\013)26 b Fs(\031)f Ft(\034)1501 4813 y Fm(X)1466 5008 y Fq(T)10 b Fr(:)p Fq(T)p 1540 5020 53 4 v 9 w Fo(3)p Fq(i)1682 4899 y Ft(w)1749 4862 y Fo(1)1747 4922 y Fq(T)1859 4899 y Ft(e)1901 4862 y Fq(\013)p Fo(j)p Fq(T)p 1970 4874 V 10 w Fo(j)2107 4899 y Fs(\034)25 b FB(1)p Ft(:)1230 b FB(\(6)p Ft(:)p FB(5\))p 50 5111 546 4 v 159 5172 a Fl(16)230 5198 y Fz(In)27 b(particular,)i(the)e(result)i(is)f(v)n(alid)f(for)f(an)n(y)h (mo)r(del)g(of)f(a)h(t)n(yp)r(e)g(\(1.1\))g(satisfying)h(\(1.5\).)39 b(It)27 b(is)h(not)e(the)i(aim)f(of)f(this)50 5289 y(article)h(to)e(in) n(v)n(estigate)i(in)f(detail)h(the)f(relation)g(of)f(these)i(abstract)f (P)n(.S.)g(mo)r(dels)g(to)g(concrete)h(spin)f(mo)r(dels.)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)j(18:34)1080 b Fz(44)p eop %%Page: 45 45 45 44 bop 159 340 a Fu(2\))40 b(F)-7 b(or)41 b(\\stable")f(values)g(of) g Ft(q)s Fu(,)i(i.e.)63 b(such)40 b(that)h Ft(f)1971 307 y Fo(1)1961 362 y Fq(q)2086 340 y FB(=)d Ft(f)2250 307 y Fo(1)2326 340 y Fu(,)j(a)g(tr)-5 b(anslation)42 b(invariant,)h(extr)-5 b(emal)50 449 y Ft(q)s Fu({th)37 b(Gibbs)g(state)g(on)g Fi(Z)930 416 y Fq(\027)1007 449 y Fu(exists.)54 b(A)n(lmost)37 b(any)g(its)g(con\014gur)-5 b(ation)38 b(belongs)f(to)g(the)g(class)g(of)g(\\)p Ft(q)s Fu({)f(like)50 558 y(con\014gur)-5 b(ations")34 b Ft(X)765 525 y Fq(q)758 581 y(p)806 558 y FB(\(\003\))26 b(=)f Fs(f)p Ft(x)1158 572 y Fr(\003)1238 558 y FB(:)g Fs(j)p FB(\003)1376 572 y Fq(q)1417 558 y Fs(j)g Ft(>)g(p)p Fs(j)p FB(\003)p Fs(jg)34 b Fu(wher)-5 b(e)33 b Ft(p)g Fu(is)f(r)-5 b(ather)35 b(close)e(to)g FB(1)p Fu(,)g Ft(p)25 b Fs(!)g FB(1)33 b Fu(as)g Ft(\034)i Fs(!)26 b(1)p Fu(.)159 743 y(3\))39 b(On)g(any)g(\014nite)g(torus)g FB(\003)p Fu(,)i(the)e(support)h(of)f(the)g(Gibbs)g(state)h(is)e (containe)-5 b(d,)42 b(with)e(an)f(accur)-5 b(acy)50 856 y Fs(\031)25 b Ft(e)188 823 y Fo(\000)p Fq(C)t(\034)20 b Fr(diam)11 b(\003)595 856 y FB(+)17 b Ft(e)725 823 y Fo(\000)p Fq(a)p Fo(j)p Fr(\003)p Fo(j)919 856 y Fu(,)31 b(in)g(the)h(union)f(of)g(those)i Ft(X)1915 823 y Fq(q)1908 878 y(p)1955 856 y FB(\(\003\))f Fu(for)g(which)g Ft(q)i Fu(is)d(\\almost")h(stable,)g(such)f(that)50 979 y Ft(f)105 946 y Fo(1)95 1001 y Fq(q)206 979 y Fs(\024)25 b Ft(f)357 946 y Fo(1)446 979 y FB(+)13 b Ft(a)p Fu(.)41 b(In)29 b(any)h(subvolume)f(of)h(a)f(size)g Fs(\024)1825 943 y Fq(C)t(\034)p 1825 958 98 4 v 1854 1010 a(a)1963 979 y Fu(\(wher)-5 b(e)30 b Ft(C)36 b Fu(is)29 b(a)h(universal)f (constant\))i(and)f(under)50 1091 y(such)g(a)h(boundary)h(color)g Ft(q)s Fu(,)e(the)h(conditione)-5 b(d)32 b(Gibbs)e(pr)-5 b(obability)33 b(has)e(analogous)h(pr)-5 b(operties)33 b(as)e(in)f(2\).)159 1273 y(4\))j(The)g(pr)-5 b(obability,)34 b(in)f(the)g(ensemble)g(\(1.23\),)h(of)e(the)h(event)g(\\system)g Fs(D)i Fu(appe)-5 b(ars")35 b(is)d(given)g(by)245 1465 y Ft(P)316 1422 y Fq(q)303 1492 y Fr(\003)358 1465 y FB(\()p Fs(D)s FB(\))25 b(=)g(exp\()p Fs(\000)p Ft(E)5 b FB(\()p Fs(D)s FB(\)\))49 b(exp\()33 b Ft(\030)1413 1479 y Fr(\003)1467 1465 y FB(\()p Fs(D)s FB(\))20 b(+)g Ft(\020)1761 1479 y Fr(\003)1815 1465 y FB(\()p Fs(D)s FB(\))33 b(\))224 b Fu(se)-5 b(e)32 b(mor)-5 b(e)34 b(pr)-5 b(e)g(cisely)34 b(\(6.3\))336 b FB(\(6)p Ft(:)p FB(6\))143 1666 y Fu(wher)-5 b(e)159 b Fs(j)p Ft(\030)590 1680 y Fr(\003)644 1666 y FB(\()p Fs(D)s FB(\))p Fs(j)26 b(\024)f Ft(\017)32 b Fs(jD)p 1028 1681 73 4 v 3 w(j)126 b Fu(and)h Ft(\020)1562 1680 y Fr(\003)1616 1666 y FB(\()p Fs(D)s FB(\))58 b Fs(\024)f FB(\()p Ft(f)2035 1628 y Fo(1)2025 1688 y Fq(q)2132 1666 y Fs(\000)20 b Ft(f)2278 1628 y Fo(1)2354 1666 y FB(\))33 b Fs(j)15 b FB(in)m(t)g Fs(D)s(j)20 b FB(+)53 b Ft(e)2870 1628 y Fo(\000)p Fq(C)t(\034)20 b Fr(diam)11 b(\003)3523 1666 y FB(\(6)p Ft(:)p FB(7\))50 1830 y Fu(wher)-5 b(e)37 b Ft(q)i Fu(denotes)d(the)h(external)g(color)g (of)f Fs(D)s Fu(.)52 b(While)36 b(the)g(term)g Ft(\030)2401 1844 y Fr(\003)2456 1830 y FB(\()p Fs(D)s FB(\))g Fu(can)g(be)g (interpr)-5 b(ete)g(d)38 b(as)e(some)50 1939 y(smal)5 b(l)34 b(\\surface)g(tension)-9 b(")34 b(of)f Fs(D)j Fu(the)e(term)h Ft(\020)1636 1953 y Fr(\003)1723 1939 y Fu(in)f(\(6.7\))g(accounts)g(also)h(the)f(missing)g(\\astr)-5 b(onomical")50 2049 y(contours)29 b(in)e(\(6.3\))i(and,)g(much)f(mor)-5 b(e)29 b(importantly,)j(the)c(possible)g(instability)h(of)f Ft(q)s Fu(:)39 b(Then)28 b Ft(f)3317 2016 y Fo(1)3307 2071 y Fq(q)3402 2049 y Fs(\000)9 b Ft(f)3537 2016 y Fo(1)3639 2049 y Ft(>)50 2161 y FB(0)p Fu(,)37 b(and)f(for)h(very)f (lar)-5 b(ge)37 b(volumes)f FB(\003)g Fu(the)g(term)g Ft(\020)1775 2175 y Fr(\003)1830 2161 y FB(\()p Fs(D)s FB(\))g Fu(incr)-5 b(e)g(ases)37 b(line)-5 b(arly)37 b(with)g Fs(j)15 b FB(in)m(t)g Fs(D)s(j)36 b Fu(and)g(formula)50 2270 y(\(6.6\))26 b(loses)h(any)f(useful)f(information.)42 b(Big)25 b(\\bubbles")f(of)i(stable)g(phases)h(may)g(then)f(appe)-5 b(ar)28 b(in)d(volumes)50 2379 y Ft(M)35 b Fs(\032)25 b FB(\003)p Fu(,)33 b(under)g(a)g(\014xe)-5 b(d)33 b(external)h(color)g Ft(q)1558 2395 y Fr(\003)p Fo(n)p Fq(M)1725 2379 y Fu(.)42 b(We)32 b(do)h(not)h(study)f(their)g(statistics)h(her)-5 b(e.)159 2561 y(5\))31 b(The)g(events)g(consider)-5 b(e)g(d)33 b(in)d(\(6.6\))i(ar)-5 b(e)32 b(r)-5 b(ar)g(e)32 b(for)g(stable)f Ft(q)s Fu(,)g(and)h(also)g(for)f(instable)h Ft(q)h Fu(in)e(tori)h(such) 50 2674 y(that)i FB(\()p Ft(f)326 2641 y Fo(1)316 2696 y Fq(q)423 2674 y Fs(\000)20 b Ft(f)569 2641 y Fo(1)645 2674 y FB(\))p Fs(j)p FB(\003)p Fs(j)27 b(\034)f Ft(\034)f FB(diam)14 b(\003)p Fu(.)42 b(Then)34 b(they)f(have)g(a)h(pr)-5 b(obability)34 b(de)-5 b(caying)34 b(like)e Ft(e)3090 2641 y Fo(\000)p Fq(C)t(\034)7 b Fo(jD)p 3266 2653 59 4 v 2 w(j)3351 2674 y Fu(.)43 b(T)-7 b(ypical)50 2786 y(con\014gur)i(ation)37 b(in)f(the)g(Gibbs)f(state)i(corr)-5 b(esponding)38 b(to)e(\(1.23\))h(on)g FB(\003)e Fu(then)h(looks,)i(for) e(some)g(of)g(these)50 2895 y Ft(q)s Fu(,)30 b(like)g(a)h(\\se)-5 b(a)31 b(of)f(color)i Ft(q)h Fu(with)e(r)-5 b(ar)g(e)32 b(and)f(smal)5 b(l)31 b(islands)h Fs(D)s Fu(")e(\(bor)-5 b(der)g(e)g(d)32 b(by)f(external)g(contours)h(of)e Fs(D)s Fu(\).)159 3077 y(6\))j(The)g(pr)-5 b(operties)35 b(of)e(the)g FD(phase)i(diagram)c Fu(of)i(the)g(model)1317 3269 y Fs(f)g Ft(H)f Fs(!)26 b(f)p Ft(q)i Fs(2)d Ft(Q)g FB(:)h Ft(f)2023 3232 y Fo(1)2013 3292 y Fq(q)2124 3269 y FB(=)f Ft(f)2275 3232 y Fo(1)2351 3269 y Fs(gg)1082 b FB(\(6)p Ft(:)p FB(8\))50 3462 y Fu(can)33 b(be)f(derive)-5 b(d)33 b(fr)-5 b(om)34 b(the)f(pr)-5 b(operties)35 b(of)e(the)g(mapping,)h (constructe)-5 b(d)34 b(with)f(the)g(help)h(of)f(The)-5 b(or)g(em)34 b(5)1400 3655 y Fs(f)f Ft(H)97 b Fs(!)58 b(f)p Ft(f)1900 3617 y Fo(1)1890 3677 y Fq(q)1976 3655 y Ft(;)15 b(q)29 b Fs(2)c Ft(Q)p Fs(gg)p Ft(:)1164 b FB(\(6)p Ft(:)p FB(9\))50 3850 y Fu(T)-7 b(o)30 b(get)g(a)g(useful)f (conse)-5 b(quence)30 b(fr)-5 b(om)31 b(it,)f(we)g(str)-5 b(engthen)31 b(the)f(assumptions)i(\(1.13\),)g(\(1.14\))f(as)f(fol)5 b(lows.)50 3959 y(L)-5 b(et)31 b(al)5 b(l)32 b(the)g(weights)g(of)g (\(1.23\))h(depend)f(on)g(another)h(\(ve)-5 b(ctor\))33 b(par)-5 b(ameter)33 b Ft(\025)f Fu(such)f(that)i(the)f(estimates)1224 4113 y Fm(X)1204 4308 y Fq(T)10 b Fo(3)p Fq(x)1347 4317 y Fv(i)1391 4199 y Fs(j)1454 4138 y Ft(@)p 1427 4178 107 4 v 1427 4262 a(@)5 b(\025)1544 4199 y(w)1609 4213 y Fq(T)1667 4199 y Fs(j)32 b Ft(e)1766 4162 y Fq(c)p Fo(j)p Fq(T)p 1821 4174 53 4 v 10 w Fo(j)1990 4199 y Fs(\024)58 b Ft(\017)25 b Fs(\034)g FB(1)h Fs(\034)57 b Ft(c)922 b FB(\(6)p Ft(:)p FB(10\))50 4499 y Fu(and)33 b(\(r)-5 b(e)g(cal)5 b(l)34 b(that)g Ft(w)757 4513 y Fn(\000)836 4499 y FB(=)25 b Ft(e)974 4466 y Fo(\000)p Fq(E)s Fr(\()p Fn(\000)p Fr(\))1199 4499 y Fu(\))1068 4653 y Fm(X)1036 4848 y Fn(\000)p Fr(:)p Fn(\000)p 1106 4860 50 4 v Fo(3)p Fq(i)1278 4739 y Fs(j)1340 4678 y Ft(@)p 1314 4718 107 4 v 1314 4802 a(@)5 b(\025)1431 4739 y(w)1496 4753 y Fn(\000)1550 4739 y Fs(j)33 b(j)15 b FB(in)m(t)g FD(\000)p Fs(j)33 b Ft(e)1935 4702 y Fq(g)r Fo(j)p Fn(\000)p 1994 4714 50 4 v 1 w Fo(j)2127 4739 y Fs(\024)58 b Ft(!)j Fs(\034)25 b FB(1)g Fs(\034)h Ft(g)757 b FB(\(6)p Ft(:)p FB(11\))50 5028 y Fu(ar)-5 b(e)33 b(valid,)h(with)f (appr)-5 b(opriately)37 b(str)-5 b(onger)34 b(r)-5 b(e)g(quir)g(ements) 34 b(on)f Ft(";)15 b(!)36 b Fu(than)e(befor)-5 b(e.)42 b(Then)1462 5263 y Fs(j)1525 5202 y Ft(@)p 1498 5242 107 4 v 1498 5325 a(@)5 b(\025)1648 5263 y FB(\()p Ft(f)1738 5226 y Fo(1)1728 5286 y Fq(q)1835 5263 y Fs(\000)20 b Ft(e)1968 5277 y Fq(q)2008 5263 y FB(\))p Fs(j)59 b(\024)e Ft(")50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(45)p eop %%Page: 46 46 46 45 bop 50 340 a Fu(and)40 b(this)g(condition)g(implies,)h(by)e (implicit)h(function)f(the)-5 b(or)g(em,)43 b(the)d(existence)f(and)h (di\013er)-5 b(entiability)50 451 y(of)40 b(the)h(manifolds)i(of)d (coexistence)h Ft(M)1448 463 y Fr(~)1432 479 y Fq(Q)1533 451 y FB(=)e Fs(f)p Ft(q)k Fs(2)1892 428 y FB(~)1871 451 y Ft(Q)d Fs(\()-15 b(\))39 b Ft(f)2244 418 y Fo(1)2234 474 y Fq(q)2359 451 y FB(=)g Ft(f)2524 418 y Fo(1)2601 451 y Fs(g)p Fu(,)j(if)e(a)h(suitable)g(\\de)-5 b(gener)g(acy)50 569 y(r)g(emoving)27 b(condition)-9 b(")27 b(on)g Ft(e)1034 583 y Fq(q)1074 569 y FB(\()p Ft(\025)p FB(\))g Fu(is)g(impose)-5 b(d.)41 b(Se)-5 b(e)26 b([PS],)g([Z].)g(In)g(case)h(of)f(\(almost\))j (line)-5 b(ar)27 b(dependence)50 678 y(of)k(the)f(quantities)h Ft(e)759 692 y Fq(q)830 678 y Fu(and)g Ft(E)5 b FB(\()p FD(\000)p FB(\))32 b Fu(on)e Ft(\025)c FB(=)f(\()p Ft(\025)1631 692 y Fr(1)1672 678 y Ft(;)15 b(:)g(:)g(:)i(;)e(\025)1927 692 y Fq(n)1976 678 y FB(\))26 b Fs(2)f Fi(R)2183 645 y Fq(n)2268 678 y Fu(wher)-5 b(e)32 b Fs(j)p Ft(Q)p Fs(j)26 b FB(=)f Ft(n)15 b FB(+)g(1)31 b Fu(this)g(condition)g(can)50 787 y(be)39 b(formulate)-5 b(d)42 b(in)e(terms)h(of)f(the)g (invertibility)g(of)g(the)g(matrix)h(of)f(partial)i(derivatives)e Fs(f)p Ft(@)5 b(f)3377 754 y Fo(1)3367 809 y Fq(q)3454 787 y Ft(=@)g(\025)3605 801 y Fq(i)3636 787 y Fs(g)p Fu(,)50 896 y(complemente)-5 b(d)44 b(by)e(a)g(column)h(of)f(values)g FB(1)p Fu(.)70 b(If)42 b(the)g(matrix)i(norm)f(of)f(such)g(an)h (inverse)e(matrix)j(is)50 1005 y(smal)5 b(ler)45 b(enough)f(than)i Ft(\034)10 b Fu(,)46 b(then)f(we)f(can)g(conclude)h(that)g(the)f(phase) i(diagr)-5 b(am)46 b(\(6.8\))f(of)f(the)g(model)50 1114 y(\(1.23\))34 b(is)f(a)g(smal)5 b(l)33 b(perturbation)i(of)d(the)h (\\zer)-5 b(o)34 b(temper)-5 b(atur)g(e")34 b(diagr)-5 b(am)791 1336 y Fs(f)33 b Ft(H)g Fs(!)25 b(f)p Ft(q)j Fs(2)d Ft(Q)g FB(:)h Ft(e)1484 1350 y Fq(q)1550 1336 y FB(=)e Ft(e)p Fs(gg)192 b Fu(wher)-5 b(e)159 b Ft(e)26 b FB(=)f(min)2672 1350 y Fr(~)-42 b Fq(q)s Fo(2)p Fq(Q)2860 1336 y Ft(e)2908 1350 y Fr(~)g Fq(q)2943 1336 y Ft(:)509 b FB(\(6)p Ft(:)p FB(12\))1615 1744 y FC(References)-128 1980 y FB([BZK])46 b(A.)34 b(Bo)m(vier,)i(M.)e(Zahradn)-10 b(\023)-35 b(\020k,)33 b(Cluster)f(Expansions)g(and)i(Pirogo)m(v)g (Sinai)e(Theory)h(for)h(Long)g(Range)159 2089 y(Spin)28 b(Systems,)i(to)h(b)s(e)f(submitted)f(to)i(MPRF)-139 2271 y([LMP])46 b(J.L.)39 b(Leb)s(o)m(witz,)i(A.)e(Mazel,)j(E.)d (Presutti,)h(\\Liquid-v)m(ap)s(or)d(phase)h(transitions)g(for)g (systems)h(with)159 2380 y(\014nite-range)30 b(in)m(teractions",J.)h (Statist.)41 b(Ph)m(ys.)f FD(94\(5/6\))p FB(,)31 b(955{1025)j(\(1999\)) -80 2562 y([PS])76 b(S.A.)38 b(Pirogo)m(v,)i(Y)-8 b(a.G.)39 b(Sinai,)f(\\Phase)g(diagrams)f(of)g(classical)g(lattice)h(systems",)i (Theor.)62 b(Math.)159 2671 y(Ph)m(ys.)40 b FD(25,)c(26)31 b FB(\()f(1975,)j(1976\))f(,)f(1185{1192,)j(39{49)-18 2853 y([S])76 b(Y)-8 b(a.G.)32 b(Sinai)c(\\Theory)j(of)f(Phase)h(T)-8 b(ransitions.)39 b(Rigorous)29 b(Results",)h(\()h(1982\),)i(P)m (ergamon)e(Press)-23 3035 y([Z])76 b(M.)33 b(Zahradn)-10 b(\023)-35 b(\020k)30 b(\\An)j(alternativ)m(e)f(v)m(ersion)g(of)h (Pirogo)m(v{Sinai)e(theory",)j(Comm.)e(Math.)48 b(Ph)m(ys.)32 b FD(93)159 3144 y FB(\()f(1984)h(\))e(,)h(559{581)-90 3325 y([ZR])76 b(M.)24 b(Zahradn)-10 b(\023)-35 b(\020k,)23 b(\\A)i(short)e(course)h(in)e(the)i(Pirogo)m(v{Sinai)f(theory",)j(1997) g(,)f(Rendicon)m(ti)e(Mat.)39 b(Appl.)159 3435 y(\(7\))31 b FD(18)g FB(\(1998\),)i(411-486)-65 3616 y([BI])77 b(Ch.)63 b(Borgs,)41 b(J.)e(Z.)f(Im)m(brie)f(\\A)h(Uni\014ed)f(Approac)m(h)h(to) h(Phase)f(Diagrams)h(in)e(Field)g(Theory)g(and)159 3725 y(Statistical)30 b(Mec)m(hanics",)h(Comm.)41 b(Math.)g(Ph)m(ys.)f FD(123)31 b FB(,)g(305{324)i(\(1989\).)-89 3907 y([VM])47 b(V)-5 b(.A.)24 b(Malyshev,)h(Cluster)d(expansions)h(in)f(lattice)i(mo) s(dels)e(of)i(statistical)f(ph)m(ysics)g(and)g(the)g(quan)m(tum)159 4016 y(theory)31 b(of)f(\014elds,)f(Russ.)40 b(Math.)h(Surv)m(eys)30 b FD(35)h FB(,)f(1{62)i(\(1980\))-104 4198 y([MM])47 b(V)-5 b(.A.)34 b(Malyshev,)h(R.A.)f(Minlos,)f(\\Gibbs)g(Random)g (Fields,)g(Cluster)f(Expansions",)i(Klu)m(w)m(er)e(Aca-)159 4307 y(demic)e(Publishers,)d(Dordrec)m(h)m(t,)k(Boston,)h(London)e(\()g (1991\))-70 4489 y([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 4598 y(Ph)m(ys.)c FD(103)p FB(,)32 b(491)f({)g(498)h(\(1986\))-49 4780 y([ES])45 b(E.)23 b(Seiler,)g(\\Gauge)i(theories)e(as)h(a)f(problem)f (of)i(constructiv)m(e)f(quan)m(tum)g(\014eld)f(theory)h(and)g (statistical)159 4889 y(mec)m(hanics",)31 b(Lecture)g(Notes)g(in)e(Ph)m (ysics)h FD(159)p FB(,)h(Springer)d(V)-8 b(erlag)31 b(\(1982\))-72 5071 y([D)m(V])47 b(R.)26 b(L.)g(Dobrushin,)e(\\Estimates)j(of)f (Semiin)m(v)m(arian)m(ts)e(for)i(the)g(Ising)f(Mo)s(del)g(at)h(Lo)m(w)h (T)-8 b(emp)s(eratures",)159 5180 y(59)32 b({)f(81,)h(T)-8 b(opics)30 b(in)g(Statistical)g(Ph)m(ysics,)h(AMS)f(T)-8 b(ranslation)30 b(Series)g(2,)h FD(177)p FB(,)h(AMS,)f(Adv)m(ances)h (in)159 5289 y(the)e(Mathematical)i(Sciences)e({)h(32,)g(\(1995\))50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)e(18:34)1080 b Fz(46)p eop %%Page: 47 47 47 46 bop -119 340 a FB([BZE])46 b(A.)31 b(Bo)m(vier,)g(M.)g(Zahradn) -10 b(\023)-35 b(\020k,)28 b(\\A)j(simple)d(inductiv)m(e)h(approac)m(h) i(to)g(the)g(problem)d(of)j(con)m(v)m(ergence)i(of)159 449 y(cluster)d(expansions)f(of)h(p)s(olymer)f(mo)s(dels",)h(J.)g (Stat.)41 b(Ph)m(ys.)p FD(100)p FB(,)32 b(765-778)h(\(2000\).)-62 631 y([ZD])46 b(M.)23 b(Zahradn)-10 b(\023)-35 b(\020k,)22 b(\\)h(Con)m(tour)g(Metho)s(ds)f(and)g(Pirogo)m(v)h(Sinai)e(Theory)h (for)g(Con)m(tin)m(uous)f(Spin)g(Mo)s(dels",)159 740 y(Amer.Math.)41 b(So)s(c.)g(T)-8 b(ransl.)39 b(\(2\))32 b FD(198)p FB(,)f(\(2000\))-62 922 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 1031 y(sion)33 b(of)i(the)f(Pirogo)m(v-Sinai)f (theory",)k(in)c(\\Mathematical)i(Problems)e(of)i(Statistical)f(Mec)m (hanics",)159 1140 y(R.L.)c(Dobrushin,)f(Ed.)40 b(\(D.)31 b(Reidel)e(Publishing)d(Compan)m(y)-8 b(,)31 b(Dordrec)m(h)m(t,)h (1986\).)-57 1322 y([DS])46 b(E.)32 b(I.)h(Dinaburg,)f(Y)-8 b(a.)48 b(G.)33 b(Sinai,)e(\\An)h(analysis)f(of)i(ANNI)g(mo)s(del)e(b)m (y)i(P)m(eierls)e(con)m(tour)i(metho)s(d)f(",)159 1431 y(Comm.)40 b(Math.)h(Ph)m(ys.)g FD(98)p FB(,119)32 b({144)g(\(1985\)) -57 1613 y([DS])46 b(E.)32 b(I.)g(Dinaburg,)f(Y)-8 b(a.)46 b(G.)32 b(Sinai,)f(\\Con)m(tour)h(mo)s(dels)f(with)f(in)m(teraction)i (and)f(their)g(applications)f(",)159 1722 y(Selecta)h(Math.)41 b(So)m(v.)g FD(7)p FB(,291)33 b({)d(315,)i(\(1988\))-17 1904 y([JI])46 b(J.)36 b(Z.)g(Im)m(brie,\\Phase)f(diagrams)h(and)f (cluster)g(expansions)g(for)h Ft(P)13 b FB(\()p Ft(\036)p FB(\))2663 1918 y Fr(2)2741 1904 y FB(mo)s(dels",)37 b(Comm.)57 b(Math.)159 2013 y(Ph)m(ys.)40 b FD(82)p FB(,)31 b(261)h({304)g(,)f(305)g({344)h(\(1981\))-82 2195 y([MP])46 b(Y)m(.)28 b(M.)h(P)m(ark,\\Extension)f(of)f(Pirogo)m(v-Sinai)g(theory) h(of)g(phase)f(transitions)f(to)i(in\014nite)e(range)i(in)m(ter-)159 2304 y(actions)i(I.)h(Cluster)e(expansion)g(",)i(Comm)m(un.)40 b(Math.)h(Ph)m(ys.)f FD(114)p FB(,)32 b(187{218,)i(\(1988\))-82 2485 y([MP])46 b(Y)m(.)g(M.)g(P)m(ark,)j(\\Extension)c(of)h(Pirogo)m (v-Sinai)e(theory)h(of)h(phase)f(transitions)e(to)j(in\014nite)d(range) 159 2595 y(in)m(teractions.)d(I)s(I.)30 b(Phase)h(diagram)e(",)i(Comm)m (un.)40 b(Math.)h(Ph)m(ys.)g FD(114)p FB(,)31 b(219{241)j(\(1988\))-101 2776 y([MS])76 b(R.A.)30 b(Minlos,)f(Y)-8 b(a.G.)31 b(Sinai)c(\\Phase)j (separation)g(phenomenon)e(at)j(lo)m(w)e(temp)s(eratures",)h(Mat.)42 b(Sb.)159 2885 y FD(73)35 b FB(,)c(\()g(1967)h(\))e(,)h(375{448)-101 3067 y([MS])76 b(R.A.)25 b(Minlos,)g(Y)-8 b(a.)40 b(G.)26 b(Sinai)d(\\The)i(phase)f(separation)h(phenomenon)f(at)i(lo)m(w)f(temp) s(erature)g(in)e(some)159 3176 y(lattice)31 b(mo)s(dels)e(of)h(a)h (gas",)h(T)-8 b(rudy)29 b(Mosk.)41 b(Math.)g(Obsc)m(h.)f FD(19)66 b FB(\()31 b(1968\))h(,)f(113{178)-132 3358 y([NOZ])46 b(F)-5 b(.)25 b(Nardi,)g(E.)f(Olivieri,)f(M.)i(Zahradn)-10 b(\023)-35 b(\020k,\\On)22 b(the)j(Ising)e(mo)s(del)g(with)g(strongly)h (anisotropic)g(external)159 3467 y(\014eld",)30 b(Jour.)40 b(Stat.)h(Ph)m(ys.)f FD(97)31 b FB(,)g(\(1999\),)i(87-144)-61 3649 y([ZA])46 b(M.)25 b(Zahradn)-10 b(\023)-35 b(\020k,)24 b(\\Analyticit)m(y)g(of)h(Lo)m(w-T)-8 b(emp)s(erature)24 b(Phase)h(Diagrams)f(of)h(Lattice)h(Spin)c(Mo)s(dels",)159 3758 y(Journ.)29 b(Stat.)i(Ph)m(ys.)g FD(47)p FB(,)g(725{755)i (\(1988\))-61 3940 y([HZ])46 b(P)m(.Holic)m(k)q(\023)-47 b(y)33 b(,)f(M.)g(Zahradn)-10 b(\023)-35 b(\020k,\\Strati\014ed)29 b(Lo)m(w)j(T)-8 b(emp)s(erature)31 b(Phases)g(of)h(Strati\014ed)e(Spin) f(Mo)s(dels.)159 4049 y(A)h(general)h(P)m(.S.)f(Approac)m(h)h(",)g (submitted)e(to)i(JSP)-82 4231 y([BS])76 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 4340 y(ground)29 b(states",)k(Journ.)c(Stat.)i(Ph)m(ys.)f FD(54)p FB(,)h(203)h({)f(232)h(\(1989\))-150 4522 y([GKK])46 b(K.)34 b(Ga)m(w)m(edzki,)i(R.)e(Kotec)m(k)q(\023)-47 b(y,)38 b(A.)c(Kupiainen,)e(\\Coarse)j(graining)d(approac)m(h)j(to)f (\014rst)g(order)f(phase)159 4631 y(transitions",)c(J.Stat.Ph)m(ys.)42 b FD(47)p FB(,701)32 b({724)g(\(1987\))-51 4813 y([IS])76 b(S.)25 b(N.)h(Isak)m(o)m(v,)j(\\Nonanalytic)c(features)h(of)g(the)g (\014rst)f(order)g(phase)g(transitions)g(in)f(the)i(Ising)e(mo)s(del",) 159 4922 y(Comm.)40 b(Math.)h(Ph)m(ys.)g FD(95)p FB(,)31 b(427)h({433)g(\(1984\))-75 5104 y([SL])76 b(J.)28 b(Sla)m(wn)m(y)-8 b(,)28 b(\\Lo)m(w-temp)s(erature)i(prop)s(erties)c(of)j(classical)e (lattice)h(systems:)40 b(phase)28 b(transitions)e(and)159 5213 y(phase)32 b(diagrams",)h(Phase)f(T)-8 b(ransitions)30 b(and)i(Critical)e(Phenomena,)j(V)-8 b(ol.)47 b(11)33 b(\(ed.)46 b(C.)33 b(Dom)m(b)f(and)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)d(18:34)1080 b Fz(47)p eop %%Page: 48 48 48 47 bop 159 340 a FB(J.)30 b(L.)h(Leb)s(o)m(witz\),)f(Academic)h (Press,)f(London{New)g(Y)-8 b(ork)31 b(\(1985\))-159 522 y([BKL])76 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)159 631 y(con)m(tin)m(uous)26 b(systems:)39 b(Extension)25 b(of)i(P)m(.S.)f(theory",)j(Comm.)39 b(Math.)h(Ph)m(ys.) f FD(101)p FB(,)28 b(501{538)h(\(1985\))-129 813 y([BKL])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-)159 922 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 1031 y(V)g(erw.)41 b(Geb.)g FD(67)p FB(,)31 b(121{138)i(\(1984\)) -56 1213 y([HS])46 b(W)-5 b(.)44 b(Holszt)m(ynski)f(and)f(J.)h(Sla)m (wn)m(y)-8 b(,)47 b(\\P)m(eierls)c(condition)f(and)g(the)i(n)m(um)m(b)s (er)e(of)h(ground)g(states)h(",)159 1322 y(Comm)m(un.)c(Math.)h(Ph)m (ys)30 b FD(61)p FB(,)h(177{190,)j(\(1978\))-72 1504 y([BK])46 b(C.)26 b(Borgs,)i(R.)f(Kotec)m(k)q(\023)-47 b(y,)30 b(\\Surface)c(induced)f(\014nite)g(size)h(e\013ects)i(for)e (\014rst)g(order)g(phase)g(transitions",)159 1613 y(Journ.)j(Stat.)i (Ph)m(ys.)g FD(79)p FB(,)g(43{115,\(1995\))-72 1795 y([BK])46 b(C.)37 b(Borgs,)j(R.)e(Kotec)m(k)q(\023)-47 b(y,)42 b(\\A)c(Rigorous)f(Theory)g(of)h(Finite)e(Size)h(Scaling)f(at)j(First)d (Order)h(Phase)159 1904 y(T)-8 b(ransitions",)29 b(Journ.)g(Stat.)j(Ph) m(ys.)e FD(61)p FB(,79{119,)k(\(1990\))-34 2085 y([BI])46 b(Ch.)83 b(Borgs,)49 b(J.)c(Z.)f(Im)m(brie,)k(\\A)d(uni\014ed)d (approac)m(h)k(to)f(phase)f(diagrams)h(in)e(\014eld)g(theory)i(and)159 2195 y(statistical)30 b(mec)m(hanics",)h(Comm.)40 b(Math.)i(Ph)m(ys.)e FD(123)p FB(,)31 b(305)h({)f(328,)h(\(1989\))-90 2376 y([DM])47 b(D.)39 b(G.)h(Martirosian,)g(\\P)m(erio)s(dic)e(Gibbs)f (states)j(for)f(classical)f(lattice)h(mo)s(dels",)i(So)m(v.J.Con)m (temp.)159 2485 y(Math.)g(Anal.Comm.)f(Math.)h(Ph)m(ys.)g FD(14)p FB(,)31 b(19)g({)g(38,)g(\(1979\))-34 2667 y([R])76 b(D.)30 b(Ruelle,)e(\\Thermo)s(dynamic)g(formalism",)g(Addison)g(W)-8 b(esley)g(,)31 b(Reading,)e(Massac)m(h)m(usetts)i(\(1978\))-34 2849 y([R])76 b(D.)31 b(Ruelle,)e(\\Statistical)h(Mec)m(hanics",)i (Benjamin,)e(New)g(Y)-8 b(ork)31 b(\(1969\))1 3031 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 FD(32)p FB(,)159 3140 y(477-481)33 b(\(1936\).)-57 3322 y([DS])46 b(R.)41 b(L.)f(Dobrushin)f(and)h(S.)g(B.)h(Shlosman,)h (\\Completely)e(analytic)h(in)m(teractions.)71 b(Constructiv)m(e)159 3431 y(description",)29 b(J.)h(Stat.)42 b(Ph)m(ys.)e FD(46)p FB(,)31 b(983)h({)e(1014)j(\(1987\))-109 3613 y([EFS])46 b(A.)22 b(v)m(an)f(En)m(ter,)j(R.)d(F)-8 b(ernandez,)24 b(A.)e(D.)g(Sok)m(al,)h(\\Regularit)m(y)f(prop)s(erties)d(and)i (pathologies)g(of)h(p)s(osition)159 3722 y(space)31 b(renormalization)e (group)h(transformations:)40 b(Scop)s(e)30 b(and)g(limitations)e(of)j (Gibbsian)d(theory",)159 3831 y(J.)i(Stat.)42 b(Ph)m(ys.)e FD(72)p FB(,)31 b(879)h({)e(1167)i(\(1993\))-94 4013 y([FR])77 b(J.)38 b(F)-8 b(r\177)-45 b(ohlic)m(h,)40 b(\\Mathematical)g(asp)s(ects)f(of)f(the)h(ph)m(ysics)e(of)i (disordered)d(systems",)41 b(Critical)c(Phe-)159 4122 y(nomena,)e(Random)f(Systems,)g(Gauge)h(theories)f(\(Les)g(Houc)m(hes)h (84;)i(ed.)51 b(b)m(y)34 b(K.)g(Osterw)m(alder)f(and)159 4231 y(R.)d(Stora\),)i(North)e(Holland)f(I)s(I,)h(725)h({)g(893)h (\(1986\))-70 4413 y([GE])46 b(H.)38 b(O.)g(Georgii,)i(\\Gibbs)c (measures)i(and)f(Phase)h(T)-8 b(ransitions",)38 b(de)g(Gruyter)g (Studies)e(in)g(Mathe-)159 4522 y(matics,)31 b FD(9)f FB(\(1988\))705 4740 y(MFF)i({)e(UK,)h(Sok)m(olo)m(vsk\023)-45 b(a)31 b(83,)h(186)f(00)h(Prague,)f(Czec)m(h)g(Republic)1123 4876 y(email)60 b Fa(mzahrad@karlin.mff.cuni)o(.cz)50 5507 y Fw(5)p Fv(=september)r(=)p Fw(2001;)29 b(18:34)1080 b Fz(48)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0109051146907--