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b(no)g(question)g(of)g(a)h(phase)e (tr)-5 b(ansition)7 b Fv(".)227 1295 y(In)28 b(other)g(w)m(ords,)i (since)e(the)h(cost)f(for)g(\015ipping)e(an)m(y)j(giv)m(en)f(spin)g(is) f(in\014nite,)h(the)g(temp)s(erature,)227 1415 y(ev)m(en)36 b(v)m(ery)g(high,)e(can)h(nev)m(er)h(obtain)d(to)h(create)h(lo)s(cal)d (deviations)i(from)f(the)i(ground)f(states,)227 1535 y(as)40 b(the)g(ones)h(describ)s(ed)f(ab)s(o)m(v)m(e)h(in)e(the)h(case) h(of)e(summable)g(p)s(oten)m(tials:)56 b(at)40 b(an)m(y)g(temp)s(era-) 227 1656 y(ture)34 b(the)g(system)h(is)e(\\frozen")h(in)f(one)h(of)f (its)h(ground)f(states,)i(and)f(no)g(critical)d(temp)s(erature)227 1776 y(can)42 b(b)s(e)g(de\014ned.)73 b(Therefore)43 b(one)f(can)g(infer)f(that)h(in\014nitely)e(large)h(lattice)f(systems)k (with)227 1897 y(non-summable)27 b(p)s(oten)m(tials)h(ha)m(v)m(e,)k (indep)s(enden)m(tly)e(of)e(the)i(metho)s(d)e(used)j(to)e(describ)s(e)h (them,)227 2017 y(trivial)i(thermo)s(dynamic)h(b)s(eha)m(viour.)50 b(Nev)m(ertheless,)37 b(v)-5 b(arious)34 b(w)m(a)m(ys)j(of)d(rescaling) f(the)i(ther-)227 2137 y(mo)s(dynamic)d(p)s(oten)m(tials)g(ha)m(v)m(e)j (b)s(een)g(considered)g(in)e(the)h(litterature,)f(in)g(view)h(of)f (obtaining)227 2258 y(non-trivial)k(pseudo-densities)k(in)e(the)h (thermo)s(dynamic)f(limit.)62 b(See)41 b(for)f(example)f([17])h(for)227 2378 y(a)c(di\013eren)m(t)h(scaling)e(of)h(the)h(free)g(energy)h(of)e (gra)m(vitational)e(and)i(electrostatic)g(particle)f(sys-)227 2498 y(tems,)j(or)e([8],)i([25],)f([7],)h(where)g(mean)f(\014eld)f(v)m (ersions)i(of)e(ferromagnetic)f(spin)i(mo)s(dels)f(with)227 2619 y(non-summable)31 b(in)m(teractions)h(ha)m(v)m(e)h(b)s(een)h (studied)f(n)m(umerically)-8 b(.)227 2860 y(In)37 b(the)g(presen)m(t)h (note,)g(w)m(e)f(study)h(non-summable)d(systems)j(b)m(y)f(using)g(a)f (simpler)f(approac)m(h)227 2980 y(whic)m(h)j(is)g(the)h(follo)m(wing.) 57 b(Let)39 b Fs(\026)1495 2995 y Fz(\036;)p Fn(\003)1647 2980 y Fv(denote)g(the)g(Gibbs)e(distribution)f(of)i(the)h(system)g (with)227 3100 y(ferromagnetic)d(pair)h(in)m(teraction)g Fs(\036)g Fv(in)h(\014nite)f(v)m(olume)h(\003.)59 b(Since)38 b(nothing)f(can)h(b)s(e)g(said,)h(in)227 3221 y(the)33 b(sense)h(of)e(w)m(eak)i(con)m(v)m(ergence,)h(ab)s(out)e(the)g (existence)h(of)e(the)h(thermo)s(dynamic)e(limit)1817 3438 y(lim)1860 3500 y Fn(\003)1969 3438 y Fs(\026)2028 3453 y Fz(\036;)p Fn(\003)2159 3438 y Fs(;)227 3684 y Fv(w)m(e)k(\014rst)g Ft(trunc)-5 b(ate)42 b Fv(the)34 b(p)s(oten)m(tial)f Fs(\036)g Fv(b)m(y)i(suppressing)h(in)m(teractions) d(b)s(et)m(w)m(een)k(p)s(oin)m(ts)c(at)h(dis-)227 3805 y(tance)f(larger)e(than)i Fs(N)10 b Fv(:)1334 4007 y Fs(\036)1392 4022 y Fz(N)1459 4007 y Fv(\()p Fs(x)p Fv(\))28 b(:=)1749 3836 y Fm(\()1829 3939 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))130 b(if)32 b Fr(k)p Fs(x)p Fr(k)27 b(\024)i Fs(N)e(;)1829 4083 y Fv(0)270 b(if)32 b Fr(k)p Fs(x)p Fr(k)27 b Fs(>)h(N)f(;)3651 4007 y Fv(\(1\))227 4262 y(and)32 b(then)i(study)f(the)g(double)g(limiting)28 b(pro)s(cedure)1712 4479 y(lim)1677 4541 y Fz(N)7 b Fx(!1)1898 4479 y Fv(lim)1941 4541 y Fn(\003)2050 4479 y Fs(\026)2109 4494 y Fz(\036)2151 4505 y Fl(N)2209 4494 y Fz(;)p Fn(\003)2298 4479 y Fs(:)227 4725 y Fv(Since)31 b Fs(\036)538 4740 y Fz(N)636 4725 y Fv(has)g(\014nite)g(range,)h(the)f(in\014nite-v)m (olume)e(truncated)j(mo)s(del)d(with)i(measure)g Fs(\026)3568 4740 y Fz(\036)3610 4751 y Fl(N)3700 4725 y Fv(=)227 4846 y(lim)362 4861 y Fn(\003)432 4846 y Fs(\026)491 4861 y Fz(\036)533 4872 y Fl(N)590 4861 y Fz(;)p Fn(\003)705 4846 y Fv(is)43 b(w)m(ell)e(de\014ned.)75 b(When)44 b Fs(\036)e Fv(is)g(non-summable,)h(the)g(phenomenon)g(whic)m(h)g(w)m(e) 227 4966 y(observ)m(e)34 b(is)f(the)g(follo)m(wing:)41 b(for)33 b Ft(any)i(temp)-5 b(er)g(atur)g(e,)35 b(the)g(me)-5 b(asur)g(e)35 b Fs(\026)2790 4981 y Fz(\036)2832 4992 y Fl(N)2928 4966 y Ft(b)-5 b(e)g(c)g(omes)34 b(sensitive)g(to)227 5086 y(b)-5 b(oundary)40 b(c)-5 b(onditions)38 b(onc)-5 b(e)40 b Fs(N)50 b Ft(is)40 b(lar)-5 b(ge)39 b(enough)h(\(but)g (\014nite\))p Fv(.)60 b(That)38 b(is,)h(if)e Fs(\036)h Fv(has)h(a)f(\014nite)227 5207 y(set)28 b(of)f(ground)h(state)g (con\014gurations)f(indexed)i(b)m(y)f Fs(s)p Fv(,)h(then)f(the)g (measures)g Fs(\026)3104 5171 y Fz(s)3104 5233 y(\036)3146 5244 y Fl(N)3236 5207 y Fv(=)f(lim)3475 5222 y Fn(\003)3544 5207 y Fs(\026)3603 5171 y Fz(s)3603 5233 y(\036)3645 5244 y Fl(N)3703 5233 y Fz(;)p Fn(\003)227 5327 y Fv(obtained)d(b)m(y)i (taking)f(the)g(thermo)s(dynamic)f(limit)e(along)h(a)i(sequence)j(of)d (b)s(o)m(xes)h(with)f(di\013eren)m(t)227 5448 y(b)s(oundary)38 b(condition)e Fs(s)p Fv(,)k(di\013er)d(once)h Fs(N)49 b Fv(is)37 b(su\016cien)m(tly)i(large:)53 b Fs(\026)2771 5411 y Fz(s)2771 5473 y(\036)2813 5484 y Fl(N)2911 5448 y Fr(6)p Fv(=)37 b Fs(\026)3083 5411 y Fz(s)3116 5388 y Fk(0)3083 5473 y Fz(\036)3125 5484 y Fl(N)3224 5448 y Fv(when)i Fs(s)d Fr(6)p Fv(=)h Fs(s)3725 5411 y Fx(0)3748 5448 y Fv(.)227 5568 y(Moreo)m(v)m(er,)32 b(in)d(the)h(limit)c Fs(N)38 b Fr(!)27 b(1)p Fv(,)j(eac)m(h)h Fs(\026)1862 5532 y Fz(s)1862 5594 y(\036)1904 5605 y Fl(N)1995 5568 y Fv(con)m(v)m(erges)h(w)m(eakly)e(to)f Fs(\016)2907 5583 y Fz(s)2944 5568 y Fv(,)i(the)f(Dirac)e(measure)227 5688 y(concen)m(trated)36 b(on)e(the)i(ground)e(state)h (con\014guration)f Fs(s)p Fv(.)49 b(This)35 b(concen)m(tration)g (phenomenon)227 5809 y(in)43 b(the)i(limit)c Fs(N)58 b Fr(!)48 b(1)43 b Ft(at)j(any)g(\014xe)-5 b(d)45 b(temp)-5 b(er)g(atur)g(e)52 b Fs(T)61 b(>)48 b Fv(0)c(is)g(th)m(us)h(similar)c (to)j(the)h(one)p Black 1977 6069 a(2)p Black eop %%Page: 3 3 3 2 bop Black Black 227 332 a Fv(discussed)36 b(ab)s(o)m(v)m(e)e(for)g (summable)e(in)m(teractions)i(in)f(the)h(limit)d Fs(T)44 b Fr(!)29 b Fv(0,)34 b(and)g(is)g(in)f(agreemen)m(t)227 452 y(with)28 b(Dyson's)i(heuristic)e(description)h(of)f(non-summable)f (in)m(teractions.)42 b(Notice)28 b(that)g(in)g(our)227 572 y(approac)m(h,)j(the)f(in)m(teraction)f(b)s(et)m(w)m(een)j(an)m(y)f (pair)e(of)h(spins)g(is)g(restored)h(in)e(the)i(limit)26 b Fs(N)38 b Fr(!)28 b(1)p Fv(,)227 693 y(and)k(no)h(mean)f(\014eld)g (rescaling)g(is)g(ev)m(er)i(used.)227 933 y(In)e(Section)g(2)g(w)m(e)h (sho)m(w)h(that)e(the)g(scenario)g(presen)m(ted)j(ab)s(o)m(v)m(e)e (indeed)f(o)s(ccurs)h(for)e(the)i(ferro-)227 1054 y(magnetic)26 b(P)m(otts)j(mo)s(del)d(\()p Fs(d)i Fr(\025)g Fv(2\),)g(for)g(v)-5 b(arious)27 b(families)e(of)i(non-summable)g(p)s(oten)m(tials.)40 b(Our)227 1174 y(results)c(include,)g(for)g(instance,)h(all)d(p)s(oten) m(tials)h(with)g(slo)m(w)h(algebraic)f(deca)m(y)i(\(see)g(Theorem)227 1295 y(1\):)1182 1459 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))27 b Fr(\030)1591 1392 y Fv(1)p 1513 1436 205 4 v 1513 1528 a Fr(k)p Fs(x)p Fr(k)1668 1499 y Fz(\013)1858 1459 y Fv(for)32 b(some)130 b(0)27 b Fr(\024)i Fs(\013)f Fr(\024)g Fs(d)17 b(;)830 b Fv(\(2\))227 1677 y(whic)m(h)38 b(are)g(the)h(usual)e (non-summable)g(p)s(oten)m(tials)f(considered)j(in)f(ph)m(ysics.)61 b(Nev)m(ertheless,)227 1797 y(our)38 b(aim)e(is)i(to)g(treat)g(general) g(in)m(teractions,)h(whic)m(h)g(need)h(not)e(b)s(e)g(asymptotically)e (regular)227 1918 y(as)i(in)f(\(2\),)i(but)g(can)f(ha)m(v)m(e)h(an)f (irregular)e(structure.)61 b(W)-8 b(e)39 b(also)e(giv)m(e)h(t)m(w)m(o)g (results)h(for)e Ft(sp)-5 b(arse)227 2038 y Fv(in)m(teractions,)34 b(whic)m(h)g(are)h(of)e(indep)s(enden)m(t)j(in)m(terest.)48 b(In)35 b(Section)f(3)g(w)m(e)h(giv)m(e)f(the)g(inequalit)m(y)227 2159 y(whic)m(h)28 b(allo)m(ws)f(to)h(study)h(this)f(problem)f(via)g (indep)s(enden)m(t)i(long)e(range)h(p)s(ercolation)f(and)h(then)227 2279 y(reform)m(ulate)j(and)i(pro)m(v)m(e)h(all)d(our)h(results)i(in)e (this)g(setting.)44 b(As)33 b(will)e(b)s(e)i(seen,)h(the)f(pro)s(ofs)f (are)227 2399 y(simple)f(geometric)g(argumen)m(ts.)44 b(In)33 b(Section)f(4)g(w)m(e)i(conclude)f(with)f(some)h(general)f (remarks.)227 2640 y(This)23 b(w)m(ork)h(originated)d(with)i(the)g (study)h(of)f(the)g(problem)f(of)h(truncation)f(in)g(indep)s(enden)m(t) j(long)227 2760 y(range)38 b(p)s(ercolation)e([11])i(\(see)h([22)o(],)h ([24)o(],)g([3])e(for)f(other)h(cases)i(treated)e(in)f(the)i (litterature\).)227 2881 y(Therefore,)g(the)e(results)g(presen)m(ted)i (in)d(Section)g(3)g(giv)m(e)h(more)f(particular)f(cases)j(where)g(this) 227 3001 y(problem)31 b(can)i(b)s(e)g(solv)m(ed.)227 3332 y Fu(2)161 b(Long)53 b(Range)g(P)l(otts)g(F)-13 b(erromagnet)227 3551 y Fv(W)-8 b(e)27 b(consider)g(the)h(lattice)d Fw(Z)1286 3514 y Fz(d)1327 3551 y Fv(,)j Fs(d)f Fr(\025)i Fv(2,)e(with)g(the)g(norm)f Fr(k)p Fs(x)p Fr(k)i Fv(=)g(max)2765 3566 y Fz(k)r Fn(=1)p Fz(;:::)n(;d)3049 3551 y Fr(j)p Fs(x)3132 3566 y Fz(k)3175 3551 y Fr(j)p Fv(.)41 b(In)m(teractions)227 3671 y(are)25 b(de\014ned)h(via)e(a)h Fj(ferromagnetic)g(p)s(otential)p Fv(,)i(whic)m(h)e(is)f(an)m(y)i(function)e Fs(\036)j Fv(:)h Fw(Z)3048 3635 y Fz(d)3089 3671 y Fr(nf)p Fv(0)p Fr(g)f(!)g Fv([0)p Fs(;)17 b Fv(+)p Fr(1)p Fv(\))227 3791 y(suc)m(h)34 b(that)e(sup)805 3815 y Fz(x)p Fx(6)p Fn(=0)956 3791 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))27 b Fs(<)h Fv(+)p Fr(1)p Fv(,)k(with)g(the)h(symmetry)1267 3995 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))28 b(=)f Fs(\036)p Fv(\()p Fs(y)t Fv(\))129 b(when)i Fr(k)p Fs(x)p Fr(k)28 b Fv(=)g Fr(k)p Fs(y)t Fr(k)17 b Fs(:)914 b Fv(\(3\))227 4198 y(Let)31 b Fs(N)38 b Fr(2)29 b Fw(N)p Fv(.)42 b(T)-8 b(o)32 b(eac)m(h)g(p)s(oten)m(tial)d Fs(\036)i Fv(can)h(b)s(e)f(asso)s (ciated)h(a)f Fj(truncated)h(p)s(otential)f Fs(\036)3346 4213 y Fz(N)3413 4198 y Fv(,)h(de\014ned)227 4319 y(as)39 b(in)e(\(1\).)61 b(In)39 b(the)g Fs(q)t Fv(-state)g(P)m(otts)g(mo)s (del,)g Fs(q)i Fr(\025)e Fv(2)f(is)g(an)m(y)h(\014xed)h(in)m(teger,)g (and)f(at)f(eac)m(h)i(site)227 4439 y Fs(x)k Fr(2)h Fw(Z)505 4403 y Fz(d)588 4439 y Fv(liv)m(es)d(a)g(spin)g Fs(\033)1177 4454 y Fz(x)1265 4439 y Fr(2)j(f)p Fv(1)p Fs(;)17 b Fv(2)p Fs(;)g(:)g(:)g(:)d(;)j(q)t Fr(g)p Fv(.)72 b(When)43 b Fs(q)48 b Fv(=)c(2)e(it)f(th)m(us)i(reduces)h(to)e(the)h(Ising)227 4559 y(mo)s(del.)64 b(Spin)40 b(con\014gurations)g(are)g(elemen)m(ts)g (of)g(\012)h(=)f Fr(f)p Fv(1)p Fs(;)17 b Fv(2)p Fs(;)g(:)g(:)g(:)e(;)i (q)t Fr(g)2873 4523 y Fi(Z)2923 4500 y Fl(d)2962 4559 y Fv(.)66 b(Consider)41 b(a)f(\014nite)227 4680 y(b)s(o)m(x)29 b(\003)478 4695 y Fz(L)558 4680 y Fv(=)e([)p Fr(\000)p Fs(L;)17 b Fv(+)p Fs(L)p Fv(])1044 4644 y Fz(d)1100 4680 y Fr(\\)e Fw(Z)1249 4644 y Fz(d)1290 4680 y Fv(,)30 b Fs(L)e Fr(\025)g Fv(1.)42 b(F)-8 b(or)28 b Fs(\033)k Fr(2)c Fv(\012)2086 4695 y Fn(\003)2135 4706 y Fl(L)2213 4680 y Fv(=)g Fr(f)p Fv(1)p Fs(;)17 b Fv(2)p Fs(;)g(:)g(:)g(:)e(;)i(q)t Fr(g)2824 4644 y Fn(\003)2873 4655 y Fl(L)2923 4680 y Fv(,)29 b(the)g Fj(truncated)h(P)m(otts)227 4800 y(Hamiltonian)i(with)h (b)s(ounda)m(ry)f(condition)h Fs(\021)e Fr(2)d Fv(\012)33 b(is)f(giv)m(en)h(b)m(y)455 5015 y Fs(H)544 4968 y Fz(\021)536 5042 y(N)s(;)p Fn(\003)664 5053 y Fl(L)714 5015 y Fv(\()p Fs(\033)t Fv(\))27 b(=)h Fr(\000)1161 4921 y Fm(X)1074 5137 y Fx(f)p Fz(x;y)r Fx(g\032)p Fn(\003)1345 5148 y Fl(L)1167 5208 y Fz(x)p Fx(6)p Fn(=)p Fz(y)1408 5015 y Fs(\036)1466 5030 y Fz(N)1533 5015 y Fv(\()p Fs(x)22 b Fr(\000)h Fs(y)t Fv(\))p Fs(\016)t Fv(\()p Fs(\033)1978 5030 y Fz(x)2021 5015 y Fs(;)17 b(\033)2120 5030 y Fz(y)2162 5015 y Fv(\))22 b Fr(\000)2440 4921 y Fm(X)2321 5132 y Fz(x)p Fx(2)p Fn(\003)2457 5143 y Fl(L)2503 5132 y Fz(;y)r Fx(2)p Fn(\003)2656 5109 y Fl(c)2656 5155 y(L)2719 5015 y Fs(\036)2777 5030 y Fz(N)2844 5015 y Fv(\()p Fs(x)h Fr(\000)f Fs(y)t Fv(\))p Fs(\016)t Fv(\()p Fs(\033)3289 5030 y Fz(x)3333 5015 y Fs(;)17 b(\021)3425 5030 y Fz(y)3466 5015 y Fv(\))g Fs(;)227 5397 y Fv(where)35 b Fs(\016)t Fv(\()p Fs(a;)17 b(b)p Fv(\))29 b(=)h(1)j(if)g Fs(a)c Fv(=)h Fs(b)p Fv(,)k(0)f(otherwise.)48 b(W)-8 b(e)34 b(will)d(mainly)h(b)s(e)h(in)m(terested)i(in)e(considering)227 5518 y(the)j Fj(pure)h Fs(s)f Fj(b)s(ounda)m(ry)f(condition)p Fv(,)j(in)d(whic)m(h)i Fs(\021)1987 5533 y Fz(j)2057 5518 y Fv(=)c Fs(s)j Fv(for)g(all)e Fs(j)40 b Fr(2)34 b Fw(Z)2788 5482 y Fz(d)2829 5518 y Fv(.)54 b(W)-8 b(e)36 b(ha)m(v)m(e,)j(with)c(some)227 5638 y(abuse)e(of)f(notation,)1402 5758 y Fs(H)1491 5717 y Fz(s)1483 5783 y(N)s(;)p Fn(\003)1611 5794 y Fl(L)1661 5758 y Fv(\()p Fs(s)p Fv(\))27 b(=)63 b(min)1914 5820 y Fz(\033)r Fx(2)p Fn(\012)2054 5831 y Fh(\003)2095 5849 y Fl(L)2163 5758 y Fs(H)2252 5717 y Fz(s)2244 5783 y(N)s(;)p Fn(\003)2372 5794 y Fl(L)2422 5758 y Fv(\()p Fs(\033)t Fv(\))17 b Fs(:)p Black 1977 6069 a Fv(3)p Black eop %%Page: 4 4 4 3 bop Black Black 227 332 a Fv(Therefore,)48 b(w)m(e)d(also)e(call)f (the)j(pure)g(con\014gurations)e Fs(s)h Fj(ground)g(state)h (con\014gurations)p Fv(.)78 b(On)227 452 y(\012)297 467 y Fn(\003)346 478 y Fl(L)397 452 y Fv(,)34 b(the)g Fj(truncated)h (Gibbs)g(measure)g(at)e(inverse)i(temp)s(erature)g Fs(\014)h(>)30 b Fv(0)j Fj(with)i(pure)f Fs(s)g Fj(b)s(ounda)m(ry)227 572 y(condition)e Fv(is)h(de\014ned)h(b)m(y:)1116 831 y Fs(\026)1175 784 y Fz(\014)s(;s)1175 859 y(\036)1217 870 y Fl(N)1275 859 y Fz(;)p Fn(\003)1344 870 y Fl(L)1394 831 y Fv(\()p Fs(\033)t Fv(\))27 b(:=)1815 764 y(1)p 1697 808 286 4 v 1697 914 a Fs(Z)1771 866 y Fz(\014)s(;s)1764 941 y(\036)1806 952 y Fl(N)1863 941 y Fz(;)p Fn(\003)1932 952 y Fl(L)2009 831 y Fv(exp)2175 751 y Fm(\000)2242 831 y Fr(\000)c Fs(\014)6 b(H)2492 790 y Fz(s)2484 856 y(N)s(;)p Fn(\003)2612 867 y Fl(L)2662 831 y Fv(\()p Fs(\033)t Fv(\))2797 751 y Fm(\001)2859 831 y Fs(;)227 1156 y Fv(where)41 b Fs(Z)590 1108 y Fz(\014)s(;s)583 1183 y(\036)625 1194 y Fl(N)682 1183 y Fz(;)p Fn(\003)751 1194 y Fl(L)841 1156 y Fv(is)e(a)g(normalizing)d(factor.)65 b(Let)39 b Fr(F)49 b Fv(b)s(e)40 b(the)g Fs(\033)t Fv(-algebra)e(on)i (\012)g(generated)g(b)m(y)227 1298 y(cylinder)j(ev)m(en)m(ts.)78 b(W)-8 b(e)44 b(consider)g(the)g(in\014nite-v)m(olume)d(Gibbs)i (measures)i Fs(\026)3184 1251 y Fz(\014)s(;s)3184 1326 y(\036)3226 1337 y Fl(N)3330 1298 y Fv(on)f(\(\012)p Fs(;)17 b Fr(F)10 b Fv(\),)227 1418 y(obtained)29 b(b)m(y)h(taking)f (limits)d(along)i(an)i(increasing)e(sequence)k(of)d(b)s(o)m(xes)2940 1382 y Fn(1)3009 1418 y Fv(\(this)g(limit)d(is)j(to)g(b)s(e)227 1539 y(understo)s(o)s(d)k(in)f(the)h(sense)h(of)e(subsequences\):)1205 1759 y Fs(\026)1264 1711 y Fz(\014)s(;s)1264 1786 y(\036)1306 1797 y Fl(N)1368 1759 y Fv(\()p Fs(A)p Fv(\))c(:=)54 b(lim)1675 1821 y Fz(L)p Fx(!1)1881 1759 y Fs(\026)1940 1711 y Fz(\014)s(;s)1940 1786 y(\036)1982 1797 y Fl(N)2039 1786 y Fz(;)p Fn(\003)2108 1797 y Fl(L)2158 1759 y Fv(\()p Fs(A)p Fv(\))114 b Fr(8)p Fs(A)29 b Fr(2)f(F)e Fs(:)227 2037 y Fv(A)32 b Fj(phase)h(transition)g(o)s(ccurs)h Fv(in)e(the)h(truncated)g(mo)s(del)e(if)g Fs(\026)2463 1990 y Fz(\014)s(;s)2463 2065 y(\036)2505 2076 y Fl(N)2594 2037 y Fr(6)p Fv(=)d Fs(\026)2757 1990 y Fz(\014)s(;s)2853 1966 y Fk(0)2757 2065 y Fz(\036)2799 2076 y Fl(N)2911 2037 y Fv(for)k Fs(s)3106 2001 y Fx(0)3157 2037 y Fr(6)p Fv(=)27 b Fs(s)p Fv(.)227 2278 y(Let)33 b Fs(B)476 2293 y Fz(N)571 2278 y Fv(:=)27 b(\003)769 2293 y Fz(N)836 2278 y Fr(nf)p Fv(0)p Fr(g)p Fv(.)43 b(When)33 b Fs(\036)g Fv(is)f Fj(summable)p Fv(,)h(i.e.)43 b(when)1427 2420 y Fm(X)1434 2633 y Fz(x)p Fx(6)p Fn(=0)1587 2515 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))28 b(:=)62 b(lim)1935 2577 y Fz(N)7 b Fx(!1)2182 2420 y Fm(X)2155 2632 y Fz(x)p Fx(2)p Fz(B)2295 2643 y Fl(N)2370 2515 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))1092 b(\(4\))227 2857 y(exists,)29 b(the)f(un)m(truncated)i(Gibbs)d (measures)h(\(with)g Fs(N)38 b Fv(=)27 b Fr(1)p Fv(\))h Fs(\026)2619 2810 y Fz(\014)s(;s)2619 2884 y(\036)2745 2857 y Fv(are)g(w)m(ell)f(de\014ned,)j(and)e(the)227 3004 y(problem)h(of)g(kno)m(wing)h(if)f Fs(\026)1240 2957 y Fz(\014)s(;s)1240 3032 y(\036)1367 3004 y Fr(6)p Fv(=)f Fs(\026)1530 2957 y Fz(\014)s(;s)1626 2933 y Fk(0)1530 3032 y Fz(\036)1681 3004 y Fv(for)h(some)h Fs(s)2115 2968 y Fx(0)2166 3004 y Fr(6)p Fv(=)e Fs(s)i Fv(dep)s(ends)h(strongly)f (on)g(the)g(temp)s(era-)227 3146 y(ture.)42 b(When)30 b Fs(\036)e Fv(is)h(not)f(summable)g(these)i(measures)f(are)g(not)g (de\014ned,)i(and)e(w)m(e)g(study)h Fs(\026)3556 3099 y Fz(\014)s(;s)3556 3174 y(\036)3598 3185 y Fl(N)3689 3146 y Fv(at)227 3267 y(large)f Fs(N)10 b Fv(.)43 b(W)-8 b(e)31 b(remind)e(that)h(for)g(\014xed)h Fs(N)10 b Fv(,)31 b(in)f(the)g(limit)d(of)j(v)m(ery)i(lo)m(w)d(temp)s(erature,)i Fs(\014)i Fr(!)28 b(1)p Fv(,)227 3387 y(the)41 b(t)m(ypical)f (con\014gurations)h(of)g Fs(\026)1548 3340 y Fz(\014)s(;s)1548 3415 y(\036)1590 3426 y Fl(N)1692 3387 y Fv(concen)m(trate)h(on)f(the)h (ground)f(state)g(con\014guration)f Fs(s)227 3508 y Fv(\([23)o(]\).)48 b(When)35 b(the)f(temp)s(erature)g(is)f(\014xed)i(and)f Fs(N)45 b Fv(b)s(ecomes)34 b(large,)f(w)m(e)i(observ)m(e)h(essen)m (tially)227 3628 y(the)31 b(same)f(phenomenon.)43 b(In)31 b(view)g(of)f(the)h(argumen)m(t)f(of)g(Dyson)h(cited)g(in)f(the)h(In)m (tro)s(duction,)227 3748 y(it)d(is)h(reasonable)g(to)g(b)s(eliev)m(e)g (that)g(at)g(an)m(y)h(\014xed)h Fs(\014)i(>)27 b Fv(0,)j(eac)m(h)g(of)f (the)h(measures)g Fs(\026)3323 3701 y Fz(\014)s(;s)3323 3776 y(\036)3365 3787 y Fl(N)3456 3748 y Fv(concen-)227 3869 y(trates,)g(when)h Fs(N)38 b Fr(!)28 b(1)p Fv(,)h(on)h(a)f(single) g(con\014guration,)g(whic)m(h)h(is)f(the)h(ground)g(state)g Fs(s)p Fv(.)42 b(This)30 b(is)227 3989 y(the)36 b(statemen)m(t)g(of)g (the)g(follo)m(wing)d(conjecture.)55 b(Let)36 b Fs(\016)2293 4004 y Fz(s)2366 3989 y Fv(denote)g(the)h(Dirac)d(mass)i(on)g(\(\012)p Fs(;)17 b Fr(F)10 b Fv(\))227 4109 y(concen)m(trated)42 b(on)e(the)h(ground)g(state)g(con\014guration)e Fs(s)p Fv(,)k(and)d(write)g Fs(\026)2944 4062 y Fz(\014)s(;s)2944 4137 y(\036)2986 4148 y Fl(N)3089 4109 y Fr(\))h Fs(\016)3273 4124 y Fz(s)3350 4109 y Fv(when)h Fs(\026)3672 4062 y Fz(\014)s(;s)3672 4137 y(\036)3714 4148 y Fl(N)227 4230 y Fv(con)m(v)m(erges)35 b(w)m(eakly)e(to)f Fs(\016)1148 4245 y Fz(s)1218 4230 y Fv(in)g(the)h(limit)c Fs(N)38 b Fr(!)27 b(1)p Fv(.)p Black 227 4433 a Fw(Conjecture)37 b(1.)p Black 42 w Fv(\()p Fs(d)27 b Fr(\025)h Fv(2\))35 b Ft(If)f Fs(\036)28 b Fr(\025)g Fv(0)35 b Ft(satis\014es)42 b Fv(\(3\))34 b Ft(and)h(is)f(non-summable,)f(i.e.)1651 4576 y Fm(X)1658 4788 y Fz(x)p Fx(6)p Fn(=0)1812 4671 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))27 b(=)h(+)p Fr(1)17 b Fs(;)1299 b Fv(\(5\))227 5012 y Ft(then)34 b Fs(\026)502 4965 y Fz(\014)s(;s)502 5040 y(\036)544 5051 y Fl(N)634 5012 y Fr(\))27 b Fs(\016)804 5027 y Fz(s)876 5012 y Ft(for)34 b(al)5 b(l)35 b Fs(\014)e(>)28 b Fv(0)34 b Ft(and)h(for)f(al)5 b(l)35 b Fs(s)27 b Fr(2)i(f)p Fv(1)p Fs(;)17 b Fv(2)p Fs(;)g(:)g(:)g(:)d(;)j(q)t Fr(g)p Ft(.)p Black 227 5121 1420 4 v 339 5193 a Fg(1)376 5223 y Fp(Here)27 b(w)n(e)h(extend)g Ff(\026)1015 5183 y Fe(\014)s(;s)1015 5248 y(\036)1055 5256 y Fd(N)1108 5248 y Fe(;)p Fg(\003)1173 5256 y Fd(L)1249 5223 y Fp(to)f(a)g(measure)g(on)g(\(\012)p Ff(;)14 b FD(F)8 b Fp(\))28 b(in)g(the)g(standard)f(w)n(a)n(y:)1087 5438 y Ff(\026)1137 5398 y Fe(\014)s(;s)1137 5463 y(\036)1177 5471 y Fd(N)1230 5463 y Fe(;)p Fg(\003)1295 5471 y Fd(L)1344 5438 y Fp(\()p Ff(A)p Fp(\))c(:=)1653 5359 y Fc(X)1604 5537 y Fe(\033)r Fq(2)p Fg(\012)1736 5545 y Fb(\003)1774 5559 y Fd(L)1836 5438 y Ff(\026)1886 5398 y Fe(\014)s(;s)1886 5463 y(\036)1926 5471 y Fd(N)1980 5463 y Fe(;)p Fg(\003)2045 5471 y Fd(L)2093 5438 y Fp(\()p Ff(\033)s Fp(\)1)2249 5450 y Fe(A)2303 5438 y Fp(\()p Ff(\033)g FD(\001)18 b Ff(s)p Fp(\))83 b FD(8)p Ff(A)23 b FD(2)g(F)f Ff(;)227 5717 y Fp(where)27 b(the)h(con\014guration)e Ff(\033)c FD(\001)c Ff(s)23 b FD(2)h Fp(\012)j(coincides)g(with)h Ff(\033)j Fp(on)d(\003)2243 5729 y Fe(L)2320 5717 y Fp(and)f(with)h Ff(s)g Fp(on)f(\003)2910 5686 y Fe(c)2910 5739 y(L)2960 5717 y Fp(.)p Black Black 1977 6069 a Fv(4)p Black eop %%Page: 5 5 5 4 bop Black Black 227 332 a Fv(Observ)m(e)36 b(that)e Fs(\026)875 284 y Fz(\014)s(;s)875 359 y(\036)917 370 y Fl(N)1010 332 y Fr(\))c Fs(\016)1183 347 y Fz(s)1254 332 y Fv(implies)i Fs(\026)1646 284 y Fz(\014)s(;s)1646 359 y(\036)1688 370 y Fl(N)1750 332 y Fv(\()p Fs(\033)1843 347 y Fn(0)1913 332 y Fv(=)f Fs(s)p Fv(\))g Fr(!)f Fv(1)k(in)g(the)h (limit)c Fs(N)41 b Fr(!)30 b(1)p Fv(,)35 b(i.e.)48 b(a)34 b(phase)227 452 y(transition)j(o)s(ccurs)i(in)f(the)h(truncated)h(mo)s (del)d(for)h(large)g(enough)h Fs(N)10 b Fv(.)62 b(Since)39 b(the)g(system)h(is)227 572 y(ferromagnetic,)26 b(the)h(sequence)1424 492 y Fm(\000)1469 572 y Fs(\026)1528 525 y Fz(\014)s(;s)1528 600 y(\036)1570 611 y Fl(N)1632 572 y Fv(\()p Fs(\033)1725 587 y Fn(0)1792 572 y Fv(=)h Fs(s)p Fv(\))1980 492 y Fm(\001)2025 611 y Fz(N)7 b Fx(\025)p Fn(1)2209 572 y Fv(is)26 b(non-decreasing,)i(but)e(the)h(fact)f(that)g(it)227 706 y(con)m(v)m(erges)h(to)d(1)g(is)h(not)f(trivial.)38 b(Another)26 b(w)m(a)m(y)f(to)g(form)m(ulate)e(the)i(conjecture)h(is:) 39 b Fs(\014)3336 721 y Fz(c)3371 706 y Fv(\()p Fs(\036)3467 721 y Fz(N)3534 706 y Fv(\))27 b Fr(!)h Fv(0)227 826 y(when)37 b Fs(N)45 b Fr(!)33 b(1)p Fv(,)k(where)h Fs(\014)1246 841 y Fz(c)1280 826 y Fv(\()p Fs(\036)1376 841 y Fz(N)1443 826 y Fv(\))e(is)g(the)h(critical)d(in)m(v)m(erse)j(temp)s(erature)g (of)e(the)i(mo)s(del)e(with)227 946 y(p)s(oten)m(tial)c Fs(\036)697 961 y Fz(N)764 946 y Fv(,)h(i.e.)1017 1166 y Fs(\014)1072 1181 y Fz(c)1107 1166 y Fv(\()p Fs(\036)1203 1181 y Fz(N)1269 1166 y Fv(\))c(:=)g(inf)1600 1086 y Fm(\010)1658 1166 y Fs(\014)34 b(>)27 b Fv(0)h(:)f Fs(\026)2040 1119 y Fz(\014)s(;s)2040 1194 y(\036)2082 1205 y Fl(N)2172 1166 y Fr(6)p Fv(=)g Fs(\026)2334 1119 y Fz(\014)s(;s)2430 1096 y Fk(0)2334 1194 y Fz(\036)2376 1205 y Fl(N)2488 1166 y Fv(for)32 b Fs(s)c Fr(6)p Fv(=)f Fs(s)2860 1125 y Fx(0)2884 1086 y Fm(\011)2958 1166 y Fs(:)227 1386 y Fv(The)42 b(conjecture)g(is)f(di\016cult)f(to)h(pro)m(v)m(e)h(in)e (suc)m(h)j(generalit)m(y)-8 b(,)42 b(since)g(w)m(e)g(don't)f(assume)h (an)m(y)227 1507 y(kind)32 b(of)g(regularit)m(y)g(on)g Fs(\036)p Fv(.)43 b(F)-8 b(or)32 b(example,)g(the)h(p)s(oten)m(tial) 1026 1808 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))28 b(=)1346 1638 y Fm(\()1427 1740 y Fs(\017)g(>)f Fv(0)130 b(if)31 b Fr(k)p Fs(x)p Fr(k)d Fv(=)g Fs(k)s Fv(!)k(for)g(some)h Fs(k)e Fr(2)d Fw(N)17 b Fs(;)1427 1885 y Fv(0)300 b(otherwise)17 b Fs(;)227 2110 y Fv(whic)m(h)34 b(will)d(en)m(ter)k(in)d(the)i(family) d(of)i(in)m(teractions)g(considered)i(in)d(Theorem)i(2,)g(satis\014es)g (the)227 2230 y(h)m(yp)s(othesis)h(of)f(the)h(conjecture.)51 b(Usual)34 b(p)s(erturbation)f(tec)m(hniques,)k(suc)m(h)f(as)f(Pirogo)m (v-Sinai)227 2351 y(Theory)26 b([23],)h(are)e(of)g(no)g(use)h(for)f (studying)g(the)h(truncated)g(v)m(ersion)f(of)g(this)g(kind)g(of)g(p)s (oten)m(tial,)227 2471 y(since)35 b(the)h(domain)d(of)i(v)-5 b(alidit)m(y)33 b(for)h(the)i(temp)s(erature)f(shrinks)h(to)f(zero)g (when)h(the)g(range)f(of)227 2592 y(in)m(teraction,)c(here)j Fs(N)10 b Fv(,)33 b(gro)m(ws.)p Black 227 2795 a Fw(Remark)j(1.)p Black 41 w Fv(Observ)m(e)d(that)f(since)g(the)g(Gibbs)f(state)h(of)f (an)m(y)i(one-dimensional)c(mo)s(del)h(with)227 2915 y(\014nite)k(range)h(in)m(teractions)f(is)g(alw)m(a)m(ys)i(unique)f (\(see)h(for)e(example)g(Theorem)i(\(8.39\))e(in)g([14)o(]\),)227 3036 y(Conjecture)e(1)f(is)g Ft(false)37 b Fv(in)31 b(dimension)e(1.)43 b(This)31 b(sho)m(ws)i(that)e(a)g(symmetry)g(of)g(the)g(kind)g(\(3\))g (is)227 3156 y(necessary)-8 b(.)227 3359 y(F)g(or)29 b Fs(d)f Fr(\025)g Fv(2,)i(our)g(\014rst)h(result)f(sho)m(ws)i(that)e (the)g(conjecture)i(is)e(v)-5 b(alid)28 b(under)j(some)f(assumption)227 3480 y(on)i(the)h(sp)s(eed)h(of)e(div)m(ergence)i(of)e(the)h(series)g (\(5\).)p Black 227 3683 a Fw(Theorem)k(1.)p Black 42 w Fv(\()p Fs(d)28 b Fr(\025)h Fv(2\))35 b Ft(If)f Fs(\036)28 b Fr(\025)h Fv(0)35 b Ft(satis\014es)43 b Fv(\(3\))34 b Ft(and)45 b Fv(\(5\))35 b Ft(diver)-5 b(ges)34 b(faster)h(than)g(lo) -5 b(garithmi-)227 3804 y(c)g(al)5 b(ly,)34 b(i.e.)1279 3963 y Fv(lim)17 b(sup)1326 4044 y Fz(N)7 b Fx(!1)1749 3896 y Fv(1)p 1604 3940 340 4 v 1604 4031 a(log)17 b Fr(j)p Fs(B)1849 4046 y Fz(N)1916 4031 y Fr(j)1997 3868 y Fm(X)1970 4080 y Fz(x)p Fx(2)p Fz(B)2110 4091 y Fl(N)2184 3963 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))28 b(=)f(+)p Fr(1)17 b Fs(;)927 b Fv(\(6\))227 4256 y Ft(then)34 b 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Fr(\024)g Fs(\013)g Fr(\024)g Fs(d)p Fv(,)33 b(using)f(the)h(m)m (ultiscale)e(analysis)g(of)i([3)o(].)p Black 1977 6069 a(5)p Black eop %%Page: 6 6 6 5 bop Black Black 227 332 a Fw(Sparse)49 b(In)m(teractions.)97 b Fv(W)-8 b(e)42 b(also)f(giv)m(e)h(t)m(w)m(o)h(results)f(for)g(p)s (oten)m(tials)e(whic)m(h)j(don't)f(ha)m(v)m(e)227 452 y(the)c(symmetry)i(\(3\))o(.)60 b(Namely)-8 b(,)38 b(w)m(e)h(consider)f (in)m(teractions)g(only)f(along)g(directions)g(parallel)227 572 y(to)d(the)i(co)s(ordinate)e(axis)h Fs(e)1247 587 y Fz(i)1275 572 y Fv(,)h Fs(i)c Fv(=)g(1)p Fs(;)17 b(:)g(:)g(:)e(;)i(d) p Fv(,)35 b(where)i Fs(e)2221 587 y Fn(1)2292 572 y Fv(=)32 b(\(1)p Fs(;)17 b Fv(0)p Fs(;)g(:)g(:)g(:)e(;)i Fv(0\),)35 b Fs(e)2992 587 y Fn(2)3064 572 y Fv(=)c(\(0)p Fs(;)17 b Fv(1)p Fs(;)g Fv(0)p Fs(;)g(:)g(:)g(:)e(;)i Fv(0\),)227 693 y(...,)33 b Fs(e)413 708 y Fz(d)481 693 y Fv(=)28 b(\(0)p Fs(;)17 b Fv(0)p Fs(;)g(:)g(:)g(:)e(;)i Fv(1\).)42 b(That)33 b(is,)f(w)m(e)i(are)f(giv)m(en)f(a)h(sequence)i(\()p Fs(\036)2651 708 y Fz(n)2698 693 y Fv(\))2736 708 y Fz(n)p Fx(\025)p Fn(1)2873 693 y Fv(,)d Fs(\036)2990 708 y Fz(n)3065 693 y Fr(\025)c Fv(0,)k(and)3468 657 y Fn(2)767 1001 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))c(=)1088 831 y Fm(\()1168 933 y Fs(\036)1226 948 y Fx(k)p Fz(x)p Fx(k)1470 933 y Fv(if)j Fs(x)i Fv(is)f(parallel)e(to)j(some)49 b Fs(e)2518 948 y Fz(i)2563 933 y Fs(;)17 b(i)27 b Fv(=)h(1)p Fs(;)17 b Fv(2)p Fs(;)g(:)g(:)g(:)e(;)i(d)g(;)1168 1077 y Fv(0)253 b(otherwise)q Fs(:)3651 1001 y Fv(\(7\))227 1303 y(In)23 b Fs(d)28 b Fv(=)f(2,)e(for)e(example,)i(there)f(are)f(only)g(v)m (ertical)f(and)i(horizon)m(tal)e(couplings.)39 b(F)-8 b(or)23 b(p)s(oten)m(tials)227 1423 y(of)32 b(the)h(form)e(\(7\),)h (assumption)g(\(5\))h(of)f(Conjecture)i(1)e(b)s(ecomes:)1693 1566 y Fm(X)1699 1776 y Fz(n)p Fx(\025)p Fn(1)1854 1661 y Fs(\036)1912 1676 y Fz(n)1986 1661 y Fv(=)c(+)p Fr(1)17 b Fs(:)1341 b Fv(\(8\))227 1980 y(The)33 b(\014rst)g(result)g(is)f(for) g(sequences)k(\()p Fs(\036)1686 1995 y Fz(n)1732 1980 y Fv(\))1770 1995 y Fz(n)p Fx(\025)p Fn(1)1940 1980 y Fv(whic)m(h)d(don't)g(con)m(v)m(erge)h(to)e(zero:)p Black 227 2184 a Fw(Theorem)37 b(2.)p Black 42 w Fv(\()p Fs(d)27 b Fr(\025)h Fv(2\))p Ft(.)44 b(If)1679 2304 y Fv(lim)17 b(sup)1737 2383 y Fz(n)p Fx(!1)1995 2304 y Fs(\036)2053 2319 y Fz(n)2127 2304 y Fs(>)28 b Fv(0)17 b Fs(;)1327 b Fv(\(9\))227 2549 y Ft(then)34 b Fs(\026)502 2501 y Fz(\014)s(;s)502 2576 y(\036)544 2587 y Fl(N)634 2549 y Fr(\))27 b Fs(\016)804 2564 y Fz(s)876 2549 y Ft(for)34 b(al)5 b(l)35 b Fs(\014)e(>)28 b Fv(0)34 b Ft(and)h(for)f(al)5 b(l)35 b Fs(s)27 b Fr(2)i(f)p Fv(1)p Fs(;)17 b Fv(2)p Fs(;)g(:)g(:)g(:)d(;)j(q)t Fr(g)p Ft(.)227 2752 y Fv(Notice)26 b(that)h(\(9\))g(implies)d(\(8\),)k(but)g(with)e(no)h(information)d(on) j(the)h(sp)s(eed)g(of)e(div)m(ergence.)43 b(Our)227 2872 y(second)30 b(result)f(is)f(where)i(w)m(e)g(pro)m(v)m(e)g(our)e (conjecture)i(under)g(the)f(general)f(condition)g(\(8\))o(,)i(with)227 2993 y(no)i(assumption)g(on)h(the)g(sp)s(eed)g(of)g(div)m(ergence,)h (but)e(only)g(in)g(dimensions)g(three)h(or)f(more:)p Black 227 3196 a Fw(Theorem)37 b(3.)p Black 42 w Fv(\()p Fs(d)27 b Fr(\025)h Fv(3\))p Ft(.)44 b(If)1693 3239 y Fm(X)1699 3450 y Fz(n)p Fx(\025)p Fn(1)1854 3334 y Fs(\036)1912 3349 y Fz(n)1986 3334 y Fv(=)28 b(+)p Fr(1)17 b Fs(;)1292 b Fv(\(10\))227 3627 y Ft(then)34 b Fs(\026)502 3579 y Fz(\014)s(;s)502 3654 y(\036)544 3665 y Fl(N)634 3627 y Fr(\))27 b Fs(\016)804 3642 y Fz(s)876 3627 y Ft(for)34 b(al)5 b(l)35 b Fs(\014)e(>)28 b Fv(0)34 b Ft(and)h(for)f(al)5 b(l)35 b Fs(s)27 b Fr(2)i(f)p Fv(1)p Fs(;)17 b Fv(2)p Fs(;)g(:)g(:)g(:)d(;)j(q)t Fr(g)p Ft(.)227 3959 y Fu(3)161 b(Indep)t(enden)l(t)50 b(Long)j(Range)h(P)l(ercolation)227 4179 y Fv(T)-8 b(o)38 b(sho)m(w)i(that)e(the)h(truncated)h Fs(q)t Fv(-states)f(P)m(otts)g(mo)s(del)e(exhibits)h(a)g(phase)i (transition,)e(it)g(is)227 4299 y(su\016cien)m(t)33 b(to)g(sho)m(w)g (that)1624 4439 y Fs(\026)1683 4392 y Fz(\014)s(;s)1683 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339 5521 a Fg(2)376 5551 y Fp(The)28 b(reader)f(should)h(pa)n(y)f(atten)n(tion)h(to)g(the)g(follo)n(wing:)37 b(w)n(e)28 b(use)g Ff(n)g Fp(to)f(index)i(elemen)n(ts)f(of)g(the)g (sequence)227 5651 y(\()p Ff(\036)308 5663 y Fe(n)354 5651 y Fp(\))386 5663 y Fe(n)p Fq(\025)p Fg(1)516 5651 y Fp(,)g(whereas)e Ff(N)37 b Fp(is)27 b(used)h(as)f(the)h(parameter)e (of)h(truncation)g(for)h Ff(\036)2577 5663 y Fe(N)2640 5651 y Fp(.)p Black Black 1977 6069 a Fv(6)p Black eop %%Page: 7 7 7 6 bop Black Black 227 332 a Fv(Consider)46 b(the)g(graph)f(\()p Fw(Z)1223 295 y Fz(d)1264 332 y Fs(;)17 b Fr(E)1370 295 y Fz(d)1410 332 y Fv(\),)49 b Fs(d)g Fr(\025)i Fv(1,)d(where)f Fr(E)2233 295 y Fz(d)2319 332 y Fv(is)e(the)h(set)g(of)g(all)d(unorien) m(ted)j(edges)227 452 y Fs(e)30 b Fv(=)g Fr(f)p Fs(x;)17 b(y)t Fr(g)30 b(\032)g Fw(Z)864 416 y Fz(d)929 452 y Fr(\002)23 b Fw(Z)1097 416 y Fz(d)1138 452 y Fv(,)35 b Fs(x)30 b Fr(6)p Fv(=)h Fs(y)t Fv(.)47 b(Edge)35 b(con\014gurations)e (are)h(elemen)m(ts)h Fs(!)e 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4620 y Fn(1)2645 4656 y Fv(\(0)32 b Fr($)g Fs(K)7 b Fv(\))36 b(for)e(a)h(w)m(ell)g(c)m(hosen)227 4776 y Fs(K)7 b Fv(.)43 b(The)34 b(rest)f(of)f(the)h(pro)s(of)f(can)h(b)s(e)g(adapted)f(in)g(a) h(straigh)m(tforw)m(ard)f(w)m(a)m(y)-8 b(.)p 3705 4776 4 66 v 3709 4714 59 4 v 3709 4776 V 3767 4776 4 66 v 227 4971 a(Observ)m(e)44 b(that)e(the)h(only)f(place)g(where)i(w)m(e)f (used)g(the)g(div)m(ergence)h(of)e(the)h(series)g(\(29\))e(w)m(as)227 5092 y(to)36 b(obtain)f Fs(P)734 5056 y Fn(1)773 5092 y Fv(\(0)f Fr($)g(1)p Fv(\))g(=)g(1.)55 b(A)37 b(v)-5 b(arian)m(t)35 b(of)h(Theorem)h(6)f(can)h(therefore)g(b)s(e)f(reform)m (ulated)227 5212 y(under)31 b(a)f(more)f(abstract)i(condition)e(on)h (the)h(sequence)h(\()p Fs(p)2412 5227 y Fz(n)2459 5212 y Fv(\))2497 5227 y Fz(n)p Fx(\025)p Fn(1)2634 5212 y Fv(,)f(whic)m(h)g(can)f(hold)g(also)f(when)227 5333 y(the)k(series)656 5258 y Fm(P)761 5362 y Fz(n)825 5333 y Fs(p)874 5348 y Fz(n)953 5333 y Fv(con)m(v)m(erges:)p Black 227 5536 a Fw(Theorem)41 b(7.)p Black 43 w Fv(\()p Fs(d)33 b Fr(\025)h Fv(3\))p Ft(.)53 b(Assume)38 b(the)g(se)-5 b(quenc)g(e)37 b Fv(\()p Fs(p)2257 5551 y Fz(n)2304 5536 y Fv(\))2342 5551 y Fz(n)p Fx(\025)p Fn(1)2517 5536 y Ft(is)g(such)h(that)g Fs(\022)f Fv(:=)c Fs(P)3346 5500 y Fn(1)3385 5536 y Fv(\(0)g Fr($)g(1)p Fv(\))227 5656 y Ft(satis\014es)39 b Fs(\022)646 5620 y Fn(2)722 5656 y Fs(>)e(p)884 5671 y Fz(c)919 5656 y Fv(\()p Fr(H)q Fv(\))i Ft(and)h(that)g(the)g(one-dimensional)d (in\014nite)i(cluster)h(is)g(unique.)59 b(Then)227 5777 y Fs(P)290 5792 y Fz(N)357 5777 y Fv(\(0)27 b Fr($)h(1)p Fv(\))f Fs(>)g Fv(0)35 b Ft(when)f Fs(N)45 b Ft(is)35 b(lar)-5 b(ge)34 b(enough.)p Black 1952 6069 a Fv(13)p Black eop %%Page: 14 14 14 13 bop Black Black 227 332 a Fu(4)161 b(Final)56 b(Remarks)227 551 y Fv(W)-8 b(e)36 b(ha)m(v)m(e)i(considered)f(the)f(problem)f(of)g (truncation)h(in)f(the)i(long)d(range)j(P)m(otts)f(mo)s(del)f(with)227 671 y(non-summable)42 b(ferromagnetic)g(in)m(teractions,)47 b(via)c(simple)f(p)s(ercolation)g(tec)m(hniques.)80 b(W)-8 b(e)227 791 y(ha)m(v)m(e)37 b(sho)m(wn)g(that)e(for)g(v)-5 b(arious)35 b(families)e(of)i(p)s(oten)m(tials,)h(a)f(phase)i (transition)d(o)s(ccurs)i(in)f(the)227 912 y(truncated)27 b(mo)s(del)e(as)i(so)s(on)f(as)h(the)g(parameter)f(of)g(truncation)g Fs(N)37 b Fv(is)26 b(tak)m(en)h(su\016cien)m(tly)g(large.)227 1032 y(Notice)h(that)g(b)m(y)h(Prop)s(osition)e(1)h(all)e(the)j (existing)f(results)h(on)f(truncation)g(in)f(long)g(range)i(p)s(er-)227 1152 y(colation)e(\([22)o(,)i(24,)g(3)o(]\))g(ha)m(v)m(e)i(their)d (coun)m(terpart)i(in)e(the)h(long)f(range)h(P)m(otts)g(mo)s(del.)41 b(W)-8 b(e)29 b(hop)s(e)227 1273 y(that)g Ft(emb)-5 b(e)g(ddings)p Fv(,)28 b(as)h(those)h(w)m(e)g(used)g(in)e(the)i(pro)s(ofs)e(of)h (Theorems)h(5)f(and)g(6,)g(migh)m(t)f(b)s(e)h(used)227 1393 y(for)k(p)s(ossible)f(generalisations)g(since)i(they)g(don't)g (require)g(an)m(y)g(particular)e(regularit)m(y)g(of)h(the)227 1514 y(p)s(oten)m(tial/edge)27 b(probabilit)m(y)g(at)h(in\014nit)m(y)-8 b(,)28 b(and)h(giv)m(e)g(some)f(insigh)m(t)f(in)m(to)h(new)i(mec)m (hanisms)e(of)227 1634 y(phase)33 b(transitions)f(in)g(systems)i(with)e (long)f(range)i(in)m(teractions.)227 1875 y(Before)g(ending,)f(w)m(e)i (mak)m(e)e(t)m(w)m(o)i(remarks)e(concerning)h(the)g(problem)e(of)i (truncation.)227 2135 y Fw(A)j(Mean)i(Field)e(Limit.)96 b Fv(As)33 b(our)f(results)g(sho)m(w,)i Ft(in\014nite)39 b Fv(systems)33 b(with)f(non-summable)227 2255 y(in)m(teractions)f(ha)m (v)m(e)j(trivial)c(dep)s(endence)k(on)f(the)f(temp)s(erature.)44 b(In)32 b(the)h(ph)m(ysics)h(litterature,)227 2375 y(some)g(metho)s(ds) g(ha)m(v)m(e)h(b)s(een)g(used)h(in)d(view)h(of)g(understanding)h(the)f (prop)s(erties)g(of)g(large)f(but)227 2496 y Ft(\014nite)44 b Fv(systems)39 b(with)e(non-summable)e(in)m(teractions.)57 b(These)39 b(metho)s(ds)e(rely)h(essen)m(tially)e(on)227 2616 y(the)25 b(study)i(of)e(the)h(mean)e(\014eld)h(v)m(ersion)h(of)f (the)h(original)c(mo)s(del.)39 b(Namely)-8 b(,)26 b(since)g(the)f (energy)i(of)227 2736 y(the)h(\014nite)g(system)g(gro)m(ws)h(faster)f (than)g(its)f(size)h(in)f(the)h(limit)c(of)k(large)f(v)m(olumes)g (\003,)i(the)f(mo)s(del)227 2857 y(is)34 b(mo)s(di\014ed)f(\([7,)i(8)o (,)g(25]\))f(b)m(y)i(dividing)c(the)j(total)e(hamiltonian)e(b)m(y)36 b(a)e(w)m(ell)g(c)m(hosen)i(p)s(o)m(w)m(er)g(of)227 2977 y(the)f(v)m(olume)f Fr(j)p Fv(\003)p Fr(j)p Fv(.)50 b(In)35 b(the)g(case)h(of)e(Ising)h(spins)g Fs(\033)2093 2992 y Fz(x)2169 2977 y Fv(=)c Fr(\006)p Fv(1)k(with)g(ferromagnetic)e(in)m (teractions)227 3098 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))27 b(=)h Fr(k)p Fs(x)p Fr(k)702 3061 y Fx(\000)p Fz(\013)806 3098 y Fv(,)33 b(0)28 b Fr(\024)g Fs(\013)g Fr(\024)g Fs(d)p Fv(,)k(this)h(means)f(considering)g(the)h(follo)m(wing)d(formal) g(iden)m(tit)m(y:)1027 3373 y Fs(\014)1105 3278 y Fm(X)1111 3490 y Fz(x)p Fx(6)p Fn(=)p Fz(y)1265 3373 y Fs(\036)p Fv(\()p Fs(x)22 b Fr(\000)h Fs(y)t Fv(\))p Fs(\033)1683 3388 y Fz(x)1726 3373 y Fs(\033)1781 3388 y Fz(y)1851 3373 y Fv(=)k Fs(\014)6 b Fr(j)p Fv(\003)p Fr(j)2139 3331 y Fz(\016)2193 3278 y Fm(X)2199 3490 y Fz(x)p Fx(6)p Fn(=)p Fz(y)2363 3305 y Fs(\036)p Fv(\()p Fs(x)22 b Fr(\000)h Fs(y)t Fv(\))p 2363 3350 362 4 v 2464 3441 a Fr(j)p Fv(\003)p Fr(j)2588 3412 y Fz(\016)2735 3373 y Fs(\033)2790 3388 y Fz(x)2834 3373 y Fs(\033)2889 3388 y Fz(y)2948 3373 y Fs(:)627 b Fv(\(33\))227 3690 y(The)37 b(scaling)e(parameter)g Fs(\016)40 b Fv(m)m(ust)c(b)s(e)g(c)m(hosen)i(in)d(function)h(of)g Fs(\013)g Fv(in)f(order)i(to)e(obtain)g(a)h(w)m(ell)227 3810 y(de\014ned)47 b(thermo)s(dynamic)d(limit)e(for)j(the)h(p)s(oten)m (tial)d Fr(j)p Fv(\003)p Fr(j)2425 3774 y Fx(\000)p Fz(\016)2517 3810 y Fs(\036)p Fv(\()p Fs(x)31 b Fr(\000)h Fs(y)t Fv(\),)47 b(leading)d(to)i(a)f(mean)227 3931 y(\014eld)d(in)m(v)m(erse)h (critical)d(temp)s(erature)h Fs(\014)1740 3895 y Fx(\003)1734 3955 y Fz(c)1780 3931 y Fv(\()p Fs(\013)q Fv(\).)71 b(Then,)45 b(the)e(\\critical)c(in)m(v)m(erse)44 b(temp)s(erature")227 4051 y Fs(\014)282 4066 y Fz(c)317 4051 y Fv(\()p Fs(\013)q(;)17 b Fv(\003\))43 b(of)i(the)g(real)f(system)h(in)f(a)h(\014nite)f(v)m (olume)h(\003)f(can)h(b)s(e)g(infered)g(to)f(go)h(to)f(zero)h(as)227 4172 y Fs(\014)282 4187 y Fz(c)317 4172 y Fv(\()p Fs(\013)q(;)17 b Fv(\003\))36 b Fr(\030)h Fs(\014)779 4135 y Fx(\003)773 4196 y Fz(c)818 4172 y Fv(\()p Fs(\013)q Fv(\))p Fr(j)p Fv(\003)p Fr(j)1081 4135 y Fx(\000)p Fz(\016)1173 4172 y Fv(.)60 b(It)38 b(w)m(as)h(n)m(umerically)d(observ)m(ed)k(\([8],)f ([25]\))f(that)g Fs(\014)3215 4135 y Fx(\003)3209 4196 y Fz(c)3254 4172 y Fv(\()p Fs(\013)q Fv(\))g(dep)s(ends)227 4292 y(w)m(eakly)43 b(on)f Fs(\013)q Fv(,)i(whic)m(h)e(has)g(lead)g (the)g(authors)g(to)g(conjecture)h(that)f(all)e(the)i(systems)i(with) 227 4412 y(0)27 b Fr(\024)h Fs(\013)h Fr(\024)f Fs(d)i Fv(ha)m(v)m(e)h(the)f(same)g(thermo)s(dynamic)f(b)s(eha)m(viour,)h (i.e.)43 b(iden)m(tical)28 b(to)i(the)g(pure)h(mean)227 4533 y(\014eld)25 b(case)h Fs(\013)j Fv(=)e(0.)41 b(This)26 b(\\univ)m(ersal")f(mean)g(\014eld)g(b)s(eha)m(viour)g(w)m(as)i(then)f (giv)m(en)f(an)h(in)m(telligible)227 4653 y(explanation)31 b(b)m(y)j(V)-8 b(ollma)m(yr-Lee)30 b(and)j(Luijten)f([26)o(].)227 4774 y(W)-8 b(e)43 b(no)m(w)g(wish)f(to)g(presen)m(t)i(an)f(argumen)m (t)f(in)g(fa)m(v)m(or)g(of)g(our)h(conjecture,)j(similar)39 b(in)i(some)227 4894 y(w)m(a)m(ys)f(to)f(the)g(strategy)g(of)g([26)o (].)63 b(Apart)38 b(from)g(helping)f(to)i(understand)h(our)f (conjecture,)i(it)227 5014 y(also)29 b(sheds)i(some)e(ligh)m(t)g(on)g (the)h(w)m(eak)h(dep)s(endence)h(in)d Fs(\013)i Fv(observ)m(ed)g(n)m (umerically)d(in)h([8],)i([25)o(].)227 5135 y(F)-8 b(or)41 b(simplicit)m(y)d(w)m(e)43 b(consider)f(the)g(case)g Fs(q)47 b Fv(=)c(2,)h(i.e.)69 b(the)42 b(Ising)g(mo)s(del.)68 b(Remem)m(b)s(er)42 b(that)227 5255 y(Conjecture)d(1)f(claims)e(that)i (when)i Fs(\036)e Fv(is)f(non-summable,)h(then)h Fs(\014)2764 5270 y Fz(c)2799 5255 y Fv(\()p Fs(\036)2895 5270 y Fz(N)2962 5255 y Fv(\))e Fr(!)g Fv(0)h(as)g Fs(N)48 b Fr(!)37 b(1)p Fv(,)227 5375 y(where)c Fs(\014)563 5390 y Fz(c)598 5375 y Fv(\()p Fs(\036)694 5390 y Fz(N)761 5375 y Fv(\))f(is)g(the)g (critical)e(in)m(v)m(erse)k(temp)s(erature)e(of)f(the)i(truncated)g(mo) s(del)e Fs(\036)3369 5390 y Fz(N)3435 5375 y Fv(.)44 b(De\014ne)1368 5613 y Fs(e)1413 5628 y Fz(N)1508 5613 y Fv(:=)1639 5518 y Fm(X)1646 5730 y Fz(x)p Fx(6)p Fn(=0)1799 5613 y Fs(\036)1857 5628 y Fz(N)1924 5613 y Fv(\()p Fs(x)p Fv(\))28 b(=)2214 5518 y Fm(X)2187 5730 y Fz(x)p Fx(2)p Fz(B)2327 5741 y Fl(N)2401 5613 y Fs(\036)p Fv(\()p Fs(x)p Fv(\))17 b Fs(;)p Black 1952 6069 a Fv(14)p Black eop %%Page: 15 15 15 14 bop Black Black 227 332 a Fv(and)32 b(consider)h(the)g(formal)e (iden)m(tit)m(y:)1006 546 y Fs(\014)1083 452 y Fm(X)1089 664 y Fz(x)p Fx(6)p Fn(=)p Fz(y)1244 546 y Fs(\036)1302 561 y Fz(N)1369 546 y Fv(\()p Fs(x)22 b Fr(\000)h Fs(y)t Fv(\))p Fs(\033)1729 561 y Fz(x)1772 546 y Fs(\033)1827 561 y Fz(y)1897 546 y Fv(=)28 b Fs(\014)6 b(e)2107 561 y Fz(N)2190 452 y 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