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4742 y Fx(l)2721 4727 y Fo(\033)2776 4742 y Fx(m)2870 4727 y Fo(>)2963 4613 y Fn(\014)2963 4672 y(\014)2963 4732 y(\014)3051 4727 y Fr(=)56 b Fo(O)s Fr(\()3369 4660 y(1)p 3309 4704 172 4 v 3309 4724 a Fp(p)p 3391 4724 89 4 v 3391 4809 a Fo(N)3489 4727 y Fr(\))3437 4903 y(\(1.3\))-24 5052 y Fq(wher)-5 b(e)35 b(the)f(sums)h(run)g(over)g (non-c)-5 b(oincident)33 b(indic)-5 b(es)33 b(within)i(e)-5 b(ach)34 b(exp)-5 b(e)g(ctation.)-24 5433 y Fm(Remarks)p Fr(:)1783 5682 y(2)p eop %%Page: 3 3 3 2 bop 96 171 a Fr(1.)48 b(A)30 b(little)d(algebra)h(that)h(uses)i (translation)c(in)m(v)-5 b(ariance)29 b(sho)m(ws)i(that)e(the)h (previous)g(form)m(ula,)220 320 y(when)h(the)f(in)m(teraction)f(co)s (e\016cien)m(ts)i(are)f(set)g(to)g(b)s(e)g(equal)f(to)2567 281 y Fw(1)p 2553 297 64 4 v 2553 354 a Fx(N)2655 320 y Fr(\(Curie-W)-8 b(eiss\))30 b(giv)m(es)g(the)220 469 y(w)m(ell)i(kno)m(wn)i(Wic)m(k)f(rule)f Fo(<)c(\033)1333 484 y Fw(1)1372 469 y Fo(\033)1427 484 y Fw(2)1467 469 y Fo(\033)1522 484 y Fw(3)1562 469 y Fo(\033)1617 484 y Fw(4)1685 469 y Fo(>)17 b Fr(=)43 b(3)28 b Fo(<)f(\033)2132 484 y Fw(1)2172 469 y Fo(\033)2227 484 y Fw(2)2295 469 y Fo(>)2371 433 y Fw(2)2410 469 y Fr(.)96 851 y(2.)48 b(Higher)41 b(order)g(relations)f(in)m(v)m(olving)g(2)p Fo(n)p Fr(-p)s(oin)m(ts)h(connected)i(correlations,)f(with)f Fo(n)i Fp(\025)g Fr(1,)220 1001 y(can)31 b(b)s(e)f(found,)h(together)g (with)f(a)f(more)h(precise)h(statemen)m(t)g(of)f(the)g(results)h(in)e (Sections)i(2)220 1150 y(and)i(3.)96 1532 y(3.)48 b(It)33 b(is)f(p)s(erhaps)i(w)m(orth)f(to)f(men)m(tion)g(that)g(a)h(Wic)m(k-t)m (yp)s(e)g(factorization)e(prop)s(ert)m(y)i(is)f(struc-)220 1682 y(turally)j(di\013eren)m(t)h(from)f(the)i(factorization)d(prop)s (ert)m(y)j(of)e(pure)i(states)g(\(see)g(e.g.)55 b([MPV)q(],)220 1831 y(I)s(I)s(I.1\):)50 b(the)36 b(\014rst)g(describ)s(es)h(the)f (reduction)f(of)g(the)i(Gibbs)e(state)h(to)f(the)h(t)m(w)m(o)g(p)s(oin) m(t)f(cor-)220 1980 y(relation)c(function)i(lik)m(e)g(in)f(the)h (Gaussian)g(case,)h(the)g(latter)e(do)s(esn't)i(hold)e(for)h(the)g (Gibbs)220 2130 y(state)28 b(but)g(only)f(for)f(the)i(extremal)f (states)h(in)f(whic)m(h)h(the)f(former)g(can)g(b)s(e)h(decomp)s(osed.) 42 b(In)220 2279 y(eac)m(h)36 b(of)d(them)h(there)h(is)f(a)g(complete)g (factorization)e(of)i(the)h(correlation)d(functions)j(when)220 2429 y(the)e(thermo)s(dynamic)e(limit)e(is)j(p)s(erformed.)-24 2661 y(Our)e(strategy)h(is)e(the)i(follo)m(wing:)39 b(from)29 b(the)i(study)g(of)e(the)i(\015uctuations)f(of)f(the)i(in)m(tensiv)m(e) g(quan-)-24 2811 y(tities)46 b(\(basically)g(the)i(energy)g(p)s(er)f (particle\))f(w)m(e)j(deduce)f(factorization)d(prop)s(erties)j(for)e (the)-24 2960 y(correlation)30 b(function)h(that)g(pro)m(vide)h(a)f (Wic)m(k-lik)m(e)g(rule)g(for)g(our)g(non)h(translationally-in)m(v)-5 b(arian)m(t)-24 3110 y(in)m(teractions.)49 b(Similar)31 b(results)j(w)m(ere)i(obtained)e(in)g([G)o(])h(and)f([A)m(C)q(])h(for)f (SK.)g(W)-8 b(e)35 b(w)m(an)m(t)g(to)f(stress)-24 3259 y(the)47 b(fact)g(that)f(our)h(approac)m(h)g(do)s(esn't)g(rely)g(on)f (the)i(computation)d(of)h(the)h(solution)e(of)h(the)-24 3408 y(mo)s(del)32 b(\(still)e(not)j(a)m(v)-5 b(ailable)30 b(at)j(least)g(on)g(rigorous)e(grounds\))j(but)f(only)f(on)h(those)h(b) s(ounds)g(o)m(v)m(er)-24 3558 y(the)42 b(\015uctuation)f(of)g(the)h (energy)h(coming)d(from)g(equiv)-5 b(alence)42 b(of)f(ensem)m(bles)h (\(micro)s(canonical)-24 3707 y(and)34 b(canonical\))e(ideas.)47 b(The)34 b(main)e(tec)m(hnical)h(to)s(ol)f(w)m(e)j(use)f(is)f(the)h (prop)s(ert)m(y)h(of)e(orthogonalit)m(y)-24 3857 y(of)j(the)g(in)m (teraction)f(matrix:)48 b(it)35 b(allo)m(ws)g(us)h(to)g(sho)m(w)h (\014rst)f(the)g(extensivit)m(y)h(of)f(the)g(energy)h(and)-24 4006 y(second)d(to)e(pro)s(duce)i(the)f(exp)s(ected)h(1/)1454 3922 y Fp(p)p 1537 3922 89 4 v 84 x Fo(N)43 b Fr(b)s(ound)33 b(on)f(the)h(\015uctuations.)123 4156 y(The)47 b(pap)s(er)f(is)f (organized)h(as)g(follo)m(ws:)69 b(in)46 b(the)g(coming)f(Section)g(2)h (w)m(e)h(review)g(the)g(w)m(ell)-24 4305 y(kno)m(wn)36 b(Wic)m(k)f(factorization)d(prop)s(erties)i(of)g(the)h(Curie-W)-8 b(eiss)34 b(mo)s(del)f(through)h(an)h(analysis)e(of)-24 4455 y(the)45 b(energy)g(\015uctuations)e(and)h(of)g(the)g(high)f(temp) s(erature)h(expansion.)78 b(W)-8 b(e)44 b(w)m(an)m(t)h(to)e(stress)-24 4604 y(the)h(fact)f(that)g(all)e(the)i(results)h(w)m(e)g(presen)m(t)h (are)e(obtained,)i(including)c(the)j(existence)h(of)d(the)-24 4753 y(thermo)s(dynamical)f(limit,)h(without)g(making)g(use)i(of)f(the) g(exact)h(solution)d(of)i(the)g(mo)s(del.)73 b(In)-24 4903 y(Section)36 b(3)f(w)m(e)h(apply)f(the)h(same)f(metho)s(ds)g(to)g (the)h(generalizations)e(to)h(the)h(MPR)f(mo)s(dels)g(and)-24 5052 y(w)m(e)f(obtain)d(a)i(prop)s(erly)f(w)m(eigh)m(ted)h(Wic)m(k)g (form)m(ula.)1783 5682 y(3)p eop %%Page: 4 4 4 3 bop -24 171 a Fs(2)161 b(Remarks)54 b(on)f(the)g(Curie-W)-13 b(eiss)52 b(mo)t(del)-24 419 y Fr(T)-8 b(o)26 b(w)m(arm)f(up)g(w)m(e)h (b)s(egin)f(this)g(pap)s(er)g(with)g(a)g(full)e(description)i(of)g(the) g(high-temp)s(erature)f(\()p Fo(\014)34 b(<)27 b Fr(1\))-24 568 y(regime)42 b(of)g(the)h(Curie-W)-8 b(eiss)43 b(mo)s(del)e(of)h (statistical)e(mec)m(hanics)j(along)f(with)g(a)g(discussion)h(of)-24 718 y(the)35 b(factorization)d(prop)s(erties)i(of)f(the)i(correlation)d (functions)i(whic)m(h)h(can)f(b)s(e)g(obtained)g(in)f(this)-24 867 y(regime.)123 1016 y(The)e(basic)f(setup)h(is)f(a)g(probabilit)m(y) e(space)k(\(\006)1859 1031 y Fx(N)1926 1016 y Fo(;)17 b Fp(F)2042 1031 y Fx(N)2109 1016 y Fo(;)g Fr(P)2236 1031 y Fx(N)2303 1016 y Fr(\))30 b(de\014ned)h(as)g(follo)m(ws:)41 b(the)31 b(sample)-24 1166 y(space)g(\006)304 1181 y Fx(N)402 1166 y Fr(is)e(the)h(con\014guration)f(space,)i(i.e.)43 b(\006)1773 1181 y Fx(N)1868 1166 y Fr(=)27 b Fp(f\000)p Fr(1)p Fo(;)17 b Fr(1)p Fp(g)2290 1130 y Fx(N)2387 1166 y Fr(whose)31 b(elemen)m(ts)f(are)g(sequences)-24 1315 y Fo(\033)39 b Fr(=)33 b Fo(\033)234 1330 y Fw(1)291 1315 y 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Fo(<)g(h)1815 4295 y Fx(N)1910 4280 y Fo(>)1986 4239 y Fw(2)2053 4280 y Fp(j)f Fr(=)g Fp(O)s Fr(\()2361 4212 y(1)p 2341 4257 89 4 v 2341 4348 a Fo(N)2440 4280 y Fr(\))911 b(\(2.13\))-24 4515 y(An)29 b(easy)g(computation)d(using)i(the)h(explicit)d(form)h(of) h(the)g(Hamiltonian)d(sho)m(ws)k(that)f(the)h(former)-24 4664 y(relation)j(b)s(ecomes,)h(once)h(the)f(limit)d(is)j(considered)h (and)f(the)g(translation)e(in)m(v)-5 b(ariance)33 b(has)g(b)s(een)-24 4814 y(tak)m(en)h(in)m(to)e(accoun)m(t)h(\(see)h(also)d(b)s(elo)m(w\):) 1175 5049 y Fo(<)d(\033)1334 5064 y Fw(1)1374 5049 y Fo(\033)1429 5064 y Fw(2)1469 5049 y Fo(\033)1524 5064 y Fw(3)1563 5049 y Fo(\033)1618 5064 y Fw(4)1686 5049 y Fo(>)17 b Fr(=)44 b(3)27 b Fo(<)h(\033)2134 5064 y Fw(1)2173 5049 y Fo(\033)2228 5064 y Fw(2)2296 5049 y Fo(>)2372 5008 y Fw(2)3389 5049 y Fr(\(2.14\))-24 5284 y(whic)m(h)h(expresses)h(the)e(v)-5 b(anishing)27 b(of)g(the)h(4-p)s (oin)m(ts)f(truncated)h(correlations.)41 b(This)28 b(result)g(comes)-24 5433 y(from)38 b(general)g(facts)g(ab)s(out)g(the)h(Gibbs)f(state)h (and)g(holds)f(for)g(almost)f(all)f(temp)s(eratures.)62 b(On)1783 5682 y(5)p eop %%Page: 6 6 6 5 bop -24 171 a Fr(the)40 b(other)f(hand)g(it)e(can)i(b)s(e)g(impro)m (v)m(ed)g(to)g(hold)f(for)g(all)f(high)h(temp)s(eratures,)i(where)g (one)f(can)-24 320 y(also)34 b(get)g(similar)d(results)k(for)e(more)h (general)f(2)p Fo(n)p Fr(-p)s(oin)m(ts)h(truncated)h(correlations,)e (with)h Fo(n)d Fp(\025)g Fr(2.)-24 469 y(T)-8 b(o)38 b(this)f(purp)s(ose)i(w)m(e)g(can)f(compute)f(the)i(partition)c (function)i(on)h(that)f(regime)g(\(see)i(e.g.[Th)q(]\))-24 619 y(b)m(y)34 b(\014rst)f(decoupling)f(the)h(spins)g(in)e(the)i (Hamiltonian)c(through)k(the)g(elemen)m(tary)f(iden)m(tit)m(y:)890 880 y Fo(e)959 839 y Fx(a)996 815 y Fi(2)1042 839 y Fx(b)1104 880 y Fr(=)1289 813 y(1)p 1218 857 191 4 v 1218 877 a Fp(p)p 1301 877 108 4 v 82 x Fr(2)p Fo(\031)1435 744 y Fn(Z)1535 771 y Fj(1)1490 970 y(\0001)1637 880 y Fr(exp)1802 739 y Fn(\022)1876 880 y 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Fx(N)572 2051 y Fr(\()p Fo(\014)6 b Fr(\))82 b(=)h(2)999 2010 y Fx(N)1107 1984 y Fo(e)1152 1947 y Fj(\000)1217 1917 y Fl(\014)p 1217 1932 38 3 v 1220 1973 a Fi(2)p 1093 2029 191 4 v 1093 2048 a Fp(p)p 1176 2048 108 4 v 83 x Fr(2)p Fo(\031)1310 1916 y Fn(Z)1410 1942 y Fj(1)1365 2141 y(\0001)1512 2051 y Fo(e)1557 2010 y Fj(\000)p Fx(x)1652 1987 y Fi(2)1686 2010 y Fx(=)p Fw(2)1777 1881 y Fn(")1835 2051 y Fr(cosh)2037 1881 y Fn( )2116 1892 y(r)p 2215 1892 109 4 v 2239 1984 a Fo(\014)p 2225 2029 89 4 v 2225 2120 a(N)2324 2051 y(x)2379 1881 y Fn(!#)2516 1903 y Fx(N)2617 2051 y Fo(dx)791 2396 y Fr(=)g(2)999 2355 y Fx(N)1083 2396 y Fo(e)1128 2355 y Fj(\000)1193 2324 y Fl(\014)p 1193 2340 38 3 v 1196 2381 a Fi(2)1261 2217 y Fn(s)p 1360 2217 189 4 v 1410 2328 a Fo(N)p 1370 2373 169 4 v 1370 2464 a Fr(2)p Fo(\031)t(\014)1565 2260 y Fn(Z)1665 2287 y Fj(1)1621 2486 y(\0001)1767 2396 y Fr(exp)1932 2255 y Fn(\032)2007 2396 y Fo(N)2112 2255 y Fn(\024)2165 2396 y Fp(\000)2262 2328 y Fo(y)2314 2292 y 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5323 y Fr(1)p 1473 5367 142 4 v 1473 5387 a Fp(p)p 1556 5387 59 4 v 72 x Fo(\031)1658 5255 y Fn(Z)1758 5281 y Fw(+)p Fj(1)1713 5480 y Fw(0)1904 5390 y Fo(e)1949 5349 y Fj(\000)p Fx(y)2045 5390 y Fo(y)2097 5349 y Fj(\000)2162 5322 y Fi(1)p 2161 5334 31 3 v 2161 5375 a(2)2206 5390 y Fo(dy)42 b Fr(+)c Fo(O)s Fr(\()2606 5323 y(1)p 2587 5367 89 4 v 2587 5459 a Fo(N)2685 5390 y Fr(\))666 b(\(2.21\))1783 5682 y(6)p eop %%Page: 7 7 7 6 bop -24 171 a Fr(whic)m(h)33 b(giv)m(es)g(the)g(theorem)g(with)f (the)h(observ)-5 b(ation)32 b(\000\(0\))27 b(=)2267 99 y Fp(p)p 2350 99 59 4 v 72 x Fo(\031)t Fr(.)43 b Fp(})123 320 y Fr(W)-8 b(e)33 b(will)e(no)m(w)j(deal)f(with)f(the)i(limiting)29 b(b)s(eha)m(viour)k(of)g(the)g(energy)i(densit)m(y)f(and)f(connected) -24 469 y(correlations)e(at)h(high)g(temp)s(erature)g(deriving)g(some)g (easy)h(consequences)j(of)c(the)h(ab)s(o)m(v)m(e)g(result.)-24 619 y(F)-8 b(or)34 b(notational)d(simplicit)m(y')g(sak)m(e)36 b(let)d Fo(<)d Fp(\001)g Fo(>)k Fr(denote)g(the)h(thermal)d(a)m(v)m (erage)j(corresp)s(onding)f(to)-24 768 y(\014xed)g Fo(\014)k Fr(and)33 b Fo(N)583 732 y Fw(1)623 768 y Fr(,)f(i.e.)43 b(giv)m(en)33 b Fo(A)28 b Fr(:)g(\006)1331 783 y Fx(N)1426 768 y Fp(!)f Fo(I)-25 b(R)q Fr(,)948 1024 y Fo(<)28 b(A)f(>)h Fr(=)1397 870 y Fn(P)1502 974 y Fx(\033)r Fj(2)p Fw(\006)1642 985 y Fl(N)1721 945 y Fo(A)p Fr(\()p Fo(\033)t Fr(\))34 b(exp)17 b(\()p Fp(\000)p Fo(\014)6 b Fp(H)2388 960 y Fx(N)2456 945 y Fr(\()p Fo(\033)t Fr(\)\))p 1397 1001 1232 4 v 1878 1092 a Fo(Z)1945 1107 y Fx(N)2012 1092 y Fr(\()p Fo(\014)g Fr(\))2638 1024 y Fp(\001)723 b Fr(\(2.22\))-24 1269 y(De\014ne)33 b(moreo)m(v)m(er)h(the)f(energy)g(densit)m(y)1446 1512 y Fo(h)1502 1527 y Fx(N)1569 1512 y Fr(\()p Fo(\033)t Fr(\))28 b(=)1845 1445 y Fp(H)1929 1460 y Fx(N)1997 1445 y Fr(\()p Fo(\033)t Fr(\))p 1845 1489 287 4 v 1944 1581 a Fo(N)2141 1512 y(;)1221 b Fr(\(2.23\))-24 1745 y(whic)m(h,)50 b(b)m(y)d(\(2.9\),)i(tak)m(es)f(v)-5 b(alues)46 b(in)f(the)h(in)m(terv) -5 b(al)45 b([)p Fp(\000)2081 1706 y Fw(1)p 2081 1722 36 4 v 2081 1779 a(2)2127 1745 y Fo(;)2212 1706 y Fw(1)p 2181 1722 99 4 v 2181 1779 a(2)p Fx(N)2289 1745 y Fr(],)50 b(and)c(consider)g(its)g(distribution)-24 1894 y(function)32 b(at)h(\(in)m(v)m(erse\))h(temp)s(erature)e Fo(\014)6 b Fr(,)1323 2118 y Fo(F)1386 2133 y Fx(N)1453 2118 y Fr(\()p Fo(t)p Fr(\))28 b(=)p Fo(<)f(\037)1844 2133 y Fj(f)p Fx(h)1920 2144 y Fl(N)1990 2133 y Fj(\024)11 b Fx(t)p Fj(g)2161 2118 y Fo(>)27 b(:)1098 b Fr(\(2.24\))-24 2341 y(The)31 b(follo)m(wing)26 b(expression)31 b(for)d(the)i(Laplace)f (transform,)g(or)f(c)m(haracteristic)h(function,)h(of)f Fo(F)3476 2356 y Fx(N)3572 2341 y Fr(is)-24 2491 y(easily)j(obtained)g (from)g(\(2.6\))g(and)g(\(2.22\):)698 2757 y Fo(')762 2772 y Fx(N)829 2757 y Fr(\()p Fo(\025)p Fr(\))27 b(=)1093 2621 y Fn(Z)1209 2757 y Fo(e)1254 2716 y Fj(\000)p Fx(\025t)1397 2757 y Fo(dF)1511 2772 y Fx(N)1578 2757 y Fr(\()p Fo(t)p Fr(\))g(=)p Fo(<)h(e)1941 2716 y Fj(\000)p Fx(\025h)2078 2727 y Fl(N)2167 2757 y Fo(>)p Fr(=)2357 2683 y Fo(Z)2424 2698 y Fx(N)2507 2602 y Fn(\000)2553 2683 y Fo(\014)g Fr(+)2755 2644 y Fx(\025)p 2744 2660 64 4 v 2744 2718 a(N)2817 2602 y Fn(\001)p 2357 2734 506 4 v 2474 2825 a Fo(Z)2541 2840 y Fx(N)2608 2825 y Fr(\()p Fo(\014)6 b Fr(\))2889 2757 y Fp(\001)472 b Fr(\(2.25\))-24 3002 y(On)34 b(the)g(other)g(hand,)g(b)m(y)g(Prop)s(osition)e(2.3)h(and)h (the)g(mean)f(v)-5 b(alue)33 b(theorem)g(w)m(e)i(ha)m(v)m(e)g(that,)e (for)-24 3151 y(0)28 b Fp(\024)g Fo(\014)33 b(<)28 b Fr(1)k(and)h Fo(N)43 b Fr(large)31 b(enough)761 3338 y Fo(Z)828 3353 y Fx(N)912 3257 y Fn(\000)957 3338 y Fo(\014)d Fr(+)1159 3299 y Fx(\025)p 1148 3315 64 4 v 1148 3372 a(N)1221 3257 y Fn(\001)p 761 3389 506 4 v 879 3480 a Fo(Z)946 3495 y Fx(N)1013 3480 y Fr(\()p Fo(\014)6 b Fr(\))1304 3412 y(=)28 b(exp)1573 3271 y Fn(\022)1673 3344 y Fo(\025)p 1657 3389 89 4 v 1657 3480 a(N)1755 3412 y(G)1832 3371 y Fj(0)1832 3436 y Fx(N)1899 3412 y Fr(\()p Fo(\014)1998 3371 y Fj(\003)2037 3412 y Fr(\))2075 3271 y Fn(\023)2176 3412 y Fr(=)g(1)22 b(+)g Fp(O)s Fr(\()p Fo(\025)17 b(N)2731 3371 y Fj(\000)p Fw(1)2825 3412 y Fr(\))526 b(\(2.26\))-24 3663 y(for)35 b(some)f Fo(\014)435 3627 y Fj(\003)509 3663 y Fr(suc)m(h)i(that)e(0)d Fp(\024)h Fo(\014)1194 3627 y Fj(\003)1257 3663 y Fp(\000)24 b Fo(\014)37 b Fp(\024)31 b Fo(\025=)-5 b(N)10 b Fr(.)50 b(W)-8 b(e)35 b(ma)m(y)f(no)m(w)i(use)f(a)g(w)m(ell)f(kno)m(wn)i (theorem)e(of)-24 3813 y(probabilit)m(y)24 b(theory)i(\(see)h(e.g.)42 b([S]\))25 b(whic)m(h)i(sa)m(ys)g(that)e Fo(F)2055 3828 y Fx(N)2148 3813 y Fr(con)m(v)m(erges)j(w)m(eakly)f(to)e Fo(F)39 b Fr(if)25 b(and)h(only)f(if)-24 3962 y Fo(')40 3977 y Fx(N)108 3962 y Fr(\()p Fo(\025)p Fr(\))i Fp(!)g Fo(')p Fr(\()p Fo(\025)p Fr(\))32 b(for)g(an)m(y)h Fo(\025)f Fr(\(where)i Fo(')p Fr(\()p Fo(\025)p Fr(\))e(is)f(the)i(c)m (haracteristic)f(function)g(of)g Fo(F)14 b Fr(\).)43 b(Noting)31 b(that)-24 4112 y Fo(')p Fr(\()p Fo(\025)p Fr(\))d(=)f(1)33 b(is)f(the)h(c)m(haracteristic)f(function)g(of)g(the)h (distribution)e(function)h Fo(G)p Fr(\()p Fo(t)p Fr(\))c(=)f Fo(\037)3206 4127 y Fw([0)p Fx(;)p Fj(1)p Fw(\))3383 4112 y Fr(\()p Fo(t)p Fr(\))33 b(w)m(e)-24 4261 y(ha)m(v)m(e)h (obtained)e(the)h(follo)m(wing)d(result)-24 4489 y Fm(Prop)s(osition)36 b(2.4)49 b Fq(F)-7 b(or)41 b Fr(0)g Fp(\024)h Fo(\014)47 b(<)42 b Fr(1)g Fq(and)f Fo(N)52 b Fp(!)42 b(1)f Fq(the)i(ener)-5 b(gy)42 b(densities)f Fo(h)3041 4504 y Fx(N)3151 4489 y Fq(c)-5 b(onver)g(ge)41 b(in)-24 4639 y(distribution)35 b(to)g(a)g(r)-5 b(andom)34 b(variable)g Fo(h)h Fq(which)f(is)g Fo(\016)t Fq(-distribute)-5 b(d)35 b(at)g Fo(x)28 b Fr(=)g(0)p Fq(.)-24 4867 y Fr(Moreo)m(v)m(er,)39 b(since)d(the)h(range)f(of)f Fo(h)h Fr(is)g(the)g(in)m(terv)-5 b(al)35 b([)p Fp(\000)p Fr(1)p Fo(=)p Fr(2)p Fo(;)17 b Fr(1)p Fo(=)p Fr(2)p Fo(N)10 b Fr(],)36 b(the)h(random)e(v)-5 b(ariables)34 b Fo(h)3570 4831 y Fx(n)3570 4892 y(N)-24 5016 y Fr(are)f(uniformly)d(in)m (tegrable)i(for)g(eac)m(h)h Fo(n)28 b Fp(2)g Fo(I)-25 b(N)10 b Fr(,)33 b(i.e.)43 b(for)32 b(some)h Fo(\017)28 b(>)f Fr(0)1305 5240 y(sup)1347 5321 y Fx(N)1480 5240 y Fo(<)g Fp(j)p Fo(h)1667 5255 y Fx(N)1734 5240 y Fp(j)1762 5198 y Fx(n)p Fw(+)p Fx(\017)1920 5240 y Fo(>)55 b(<)g Fp(1)p Fo(:)1080 b Fr(\(2.27\))p -24 5342 1465 4 v 89 5403 a Fh(1)126 5433 y Fz(A)28 b(more)f(consisten)n(t)g(notation)g(w)n (ould)g(b)r(e)h(E)1568 5445 y Fg(N)s(;\014)1687 5433 y Fz(\()p Ff(\001)p Fz(\),)g(so)f(that)h(E)2178 5445 y Fg(N)2240 5433 y Fz(\()p Ff(\001)p Fz(\))c Ff(\021)f Fz(E)2510 5445 y Fg(N)s(;)p Fh(0)2621 5433 y Fz(\()p Ff(\001)p Fz(\).)1783 5682 y Fr(7)p eop %%Page: 8 8 8 7 bop -24 171 a Fr(By)33 b(another)g(w)m(ell)e(kno)m(wn)j(theorem)e (of)f(probabilit)m(y)g(theory)i(\(see)g([S]\))g(the)f(b)s(ound)h (\(2.27\))e(along)-24 320 y(with)i(Prop)s(osition)d(2.4)j(imply)d(that) 894 542 y Fo(<)d(h)1053 500 y Fx(n)1053 566 y(N)1148 542 y Fo(>)h Fp(!)f Fo(<)h(h)1539 500 y Fx(n)1614 542 y Fo(>;)211 b(N)38 b Fp(!)28 b(1)p Fo(;)114 b(n)27 b Fp(2)i Fo(I)-25 b(N)5 b(:)668 b Fr(\(2.28\))-24 763 y(But)33 b(w)m(e)g(can)f(sa)m(y)h(more.)42 b(Indeed,)34 b(b)m(y)f(virtue)f(of)f (\(2.27\))h(the)g(expansion)g(of)g(the)g(function)g Fo(')3438 778 y Fx(N)3505 763 y Fr(\()p Fo(\025)p Fr(\))-24 913 y(in)g(p)s(o)m(w)m(ers)i(of)e Fo(\025)p Fr(,)h(i.e.)1158 1103 y Fo(')1222 1118 y Fx(N)1290 1103 y Fr(\()p Fo(\025)p Fr(\))27 b(=)1590 979 y Fj(1)1554 1009 y Fn(X)1559 1218 y Fx(n)p Fw(=0)1698 1103 y Fr(\()p Fp(\000)p Fr(1\))1900 1062 y Fx(n)1957 1036 y Fo(\025)2014 1000 y Fx(n)p 1957 1080 104 4 v 1966 1172 a Fo(n)p Fr(!)2098 1103 y Fo(<)h(h)2258 1062 y Fx(n)2258 1128 y(N)2353 1103 y Fo(>;)933 b Fr(\(2.29\))-24 1348 y(con)m(v)m(erges)26 b(absolutely)e(in)f(a)g(domain)f(of)i(the)g (complex)f Fo(\025)p Fr(-plane)g(whic)m(h)i(con)m(tains)f(the)g(p)s (oin)m(t)f Fo(\025)k Fr(=)h(0)-24 1497 y(and)f(can)f(b)s(e)g(tak)m(en)h (indep)s(enden)m(t)g(of)f Fo(N)10 b Fr(.)42 b(Therefore,)28 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Fr(Similarly)-8 b(,)695 391 y(V)g(ar)16 b Fo(h)919 406 y Fx(N)1070 391 y Fr(=)99 b Fo(<)28 b(h)1405 350 y Fw(2)1405 416 y Fx(N)1500 391 y Fo(>)f Fp(\000)h Fo(<)g(h)1868 406 y Fx(N)1963 391 y Fo(>)2039 350 y Fw(2)2078 391 y Fr(=)2232 324 y(1)p 2192 368 128 4 v 2192 460 a Fo(N)2280 431 y Fw(2)2346 251 y Fn(\022)2430 324 y Fo(@)2486 288 y Fw(2)2543 324 y Fr(log)16 b Fo(Z)2752 339 y Fx(N)p 2430 368 390 4 v 2546 460 a Fo(@)5 b(\014)2663 431 y Fw(2)2829 251 y Fn(\023)1070 651 y Fr(=)1278 583 y(1)p 1239 628 128 4 v 1239 719 a Fo(N)1327 690 y Fw(4)1452 556 y Fn(X)1393 768 y Fx(i)2245 666 y Fx(c)2280 651 y Fo(;)-24 915 y Fr(where)-24 1111 y Fo(<)h(\033)135 1126 y Fx(i)164 1111 y Fo(\033)219 1126 y Fx(j)256 1111 y Fo(\033)311 1126 y Fx(l)337 1111 y Fo(\033)392 1126 y Fx(k)463 1111 y Fo(>)539 1126 y Fx(c)601 1111 y Fr(=)g Fo(<)f(\033)863 1126 y Fx(i)892 1111 y Fo(\033)947 1126 y Fx(j)984 1111 y Fo(\033)1039 1126 y Fx(l)1065 1111 y Fo(\033)1120 1126 y Fx(k)1191 1111 y Fo(>)83 b Fp(\000)g Fo(<)28 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y(alw)m(a)m(ys)f(true)f(that)g Fo(<)d(\033)876 1646 y Fx(i)936 1631 y Fo(>)p Fr(=)g(0)j(\(otherwise)g(w)m(e)h(w)m(ould)f (ha)m(v)m(e)h(had)g(to)e(write)h(15)g(terms!\).)50 b(More)-24 1780 y(generally)-8 b(,)32 b(for)g Fo(n)c(<)f(N)r(=)p Fr(2)33 b(w)m(e)g(can)g(write)g(the)g Fo(n)p Fr(-th)f Fk(momen)m(t)g Fr(of)g Fo(h)2458 1795 y Fx(N)2558 1780 y Fr(as)601 2005 y Fo(<)c(h)761 1963 y Fx(n)761 2029 y(N)856 2005 y Fo(>)83 b Fr(=)1202 1937 y(\()p Fp(\000)p Fr(1\))1404 1901 y Fx(n)p 1184 1982 287 4 v 1184 2073 a Fo(Z)1251 2088 y Fx(N)1334 2073 y Fo(N)1422 2044 y Fx(n)1497 1864 y Fn(\022)1580 1937 y Fo(@)1636 1901 y Fx(n)1684 1937 y Fo(Z)1751 1952 y Fx(N)p 1580 1982 238 4 v 1617 2073 a Fo(@)5 b(\014)1734 2044 y Fx(n)1828 1864 y Fn(\023)3389 2005 y Fr(\(2.33\))1015 2274 y(=)1184 2207 y(\()p Fp(\000)p Fr(1\))1386 2171 y Fx(n)p 1184 2252 250 4 v 1223 2343 a Fo(N)1311 2314 y Fw(2)p Fx(n)1621 2180 y Fn(X)1460 2390 y Fx(i)1484 2399 y Fi(1)1518 2390 y Fx()28 b(:)-24 2537 y Fr(The)34 b Fo(n)p 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Fx(c)1328 3270 y Fr(is)31 b(called)g(the)i(2)p Fo(n)p Fr(-p)s(oin)m(ts)e Fk(connected)j Fr(\(or)d Fk(truncated)p Fr(\))i(corre-)-24 3420 y(lation,)e(and)i(is)f(de\014ned)i(recursiv)m (ely)f(b)m(y)-24 3698 y Fo(<)28 b(\033)135 3713 y Fx(i)159 3722 y Fi(1)215 3698 y Fp(\001)17 b(\001)g(\001)e Fo(\033)403 3713 y Fx(i)427 3721 y Fl(n)474 3698 y Fo(\033)529 3713 y Fx(j)558 3722 y Fi(1)613 3698 y Fp(\001)i(\001)g(\001)e Fo(\033)801 3713 y Fx(j)830 3721 y Fl(n)904 3698 y Fo(>)980 3713 y Fx(c)1015 3698 y Fr(=)p Fo(<)27 b(\033)1249 3713 y Fx(i)1273 3722 y Fi(1)1329 3698 y Fp(\001)17 b(\001)g(\001)e Fo(\033)1517 3713 y Fx(i)1541 3721 y Fl(n)1588 3698 y Fo(\033)1643 3713 y Fx(j)1672 3722 y Fi(1)1727 3698 y Fp(\001)i(\001)g(\001)e Fo(\033)1915 3713 y Fx(j)1944 3721 y Fl(n)2018 3698 y Fo(>)28 b Fp(\000)2345 3604 y Fn(X)2252 3804 y Fi(partitions)17 b(of)2226 3861 y Fl(i)2248 3876 y Fi(1)2283 3861 y Fl(;j)2327 3876 y Fi(1)2362 3861 y Fl(;:::)o(;i)2478 3869 y(n)2521 3861 y(;j)2565 3869 y(n)2634 3498 y 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Fl(n)1947 5253 y Fo(>)2023 5268 y Fx(c)2057 5253 y Fr(=)28 b Fp(O)s Fr(\()p Fo(C)23 b(N)2462 5211 y Fj(\000)p Fw(1)2557 5253 y Fr(\))794 b(\(2.35\))-24 5433 y Fq(as)35 b Fo(N)j Fp(!)27 b(1)p Fq(.)1783 5682 y Fr(9)p eop %%Page: 10 10 10 9 bop -24 171 a Fs(3)161 b(The)54 b(orthogonal)f(mo)t(del)-24 419 y Fr(W)-8 b(e)31 b(shall)d(no)m(w)j(extend)g(the)f(results)h (obtained)e(in)g(the)i(previous)f(Section)g(to)f(a)h(more)f(in)m (teresting)-24 568 y(class)k(of)f(\(non-translationally)d(in)m(v)-5 b(arian)m(t\))31 b(in)m(teractions)1226 833 y Fp(H)1310 848 y Fx(N)1377 833 y Fr(\()p Fo(\033)t Fr(\))d(=)f Fp(\000)1730 765 y Fr(1)p 1730 810 49 4 v 1730 901 a(2)1855 738 y Fn(X)1806 959 y Fx(i;)12 b(j)t Fj(2)p Fe(Z)1992 936 y Fi(2)1992 982 y Fl(N)2065 833 y Fo(J)2119 848 y Fx(ij)2196 833 y Fo(\033)2251 848 y Fx(i)2296 833 y Fo(\033)2351 848 y Fx(j)3389 833 y Fr(\(3.36\))-24 1175 y(where)46 b Fm(Z)338 1190 y Fx(N)451 1175 y Fr(is)e(the)h(in)m(teger)g(lattice)e Fm(Z)1454 1190 y Fx(N)1571 1175 y Fr(=)48 b Fm(Z)17 b Fr(mo)s(d)f Fo(N)55 b Fr(and)45 b Fo(J)57 b Fr(=)49 b(\()p Fo(J)2647 1190 y Fx(ij)2707 1175 y Fr(\))c(is)f(a)h(symmetric)f(real) -24 1324 y(orthogonal)28 b Fo(N)f Fp(\002)16 b Fo(N)41 b Fr(matrix,)29 b(i.e.)42 b(of)29 b(the)h(form)e Fo(J)37 b Fr(=)28 b Fo(O)s(LO)2204 1288 y Fx(T)2287 1324 y Fr(with)i Fo(L)g Fr(a)f(diagonal)e(matrix)i(whose)-24 1474 y(elemen)m(ts)j(are)g Fp(\006)p Fr(1)g(and)g Fo(O)i Fr(a)d(generic)h(orthogonal)e(matrix)g(c) m(hosen)j(at)f(random)e(w.r.t.)44 b(the)32 b(Haar)-24 1623 y(measure)46 b(on)f(the)h(orthogonal)e(group.)81 b(The)47 b(kno)m(wledge)f(of)f(the)h(eigen)m(v)-5 b(alues)45 b(of)g Fo(J)54 b Fr(imp)s(oses)-24 1772 y(simple)28 b(b)s(ounds)h(on)g (the)g(energy)h(of)f(an)m(y)g(spin)g(con\014guration.)41 b(Here,)31 b(due)e(to)g(orthogonalit)m(y)-8 b(,)27 b(the)-24 1922 y(p)s(ossible)32 b(eigen)m(v)-5 b(alues)33 b(are)f(+1)p Fo(;)17 b Fp(\000)p Fr(1)33 b(so)f(that)1370 2185 y Fp(\000)1457 2117 y Fo(N)p 1457 2162 89 4 v 1477 2253 a Fr(2)1584 2185 y Fp(\024)c(H)1773 2200 y Fx(N)1840 2185 y Fr(\()p Fo(\033)t Fr(\))g Fp(\024)2118 2117 y Fo(N)p 2118 2162 V 2138 2253 a Fr(2)2216 2185 y Fp(\001)1145 b Fr(\(3.37\))-24 2434 y(An)33 b(imp)s(ortan)m(t)e(example)h(for)g(our)g(purp)s(oses)i (is)e(giv)m(en)h(b)m(y)h(the)f Fq(sine)h(mo)-5 b(del)31 b Fr(where)1157 2705 y Fo(J)1211 2720 y Fx(i;j)1319 2705 y Fr(=)1602 2638 y(2)p 1432 2683 390 4 v 1432 2702 a Fp(p)p 1515 2702 307 4 v 81 x Fr(2)p Fo(N)i Fr(+)22 b(1)1848 2705 y(sin)1984 2565 y Fn(\022)2127 2638 y Fr(2)p Fo(\031)t(ij)p 2068 2683 V 2068 2774 a Fr(2)p Fo(N)32 b Fr(+)22 b(1)2384 2565 y Fn(\023)3389 2705 y Fr(\(3.38\))-24 2977 y(whic)m(h)33 b(satis\014es)978 3165 y Fo(J)25 b(J)1120 3124 y Fx(T)1203 3165 y Fr(=)j(Id)98 b(and)1805 3041 y Fx(N)1765 3071 y Fn(X)1780 3281 y Fx(i)p Fw(=1)1926 3165 y Fo(J)1980 3180 y Fx(ii)2059 3165 y Fr(=)2203 3041 y Fx(N)2163 3071 y Fn(X)2178 3281 y Fx(i)p Fw(=1)2323 3165 y Fo(J)2386 3124 y Fw(2)2377 3190 y Fx(ii)2457 3165 y Fr(=)28 b(1)p Fo(:)752 b Fr(\(3.39\))-24 3451 y Fm(Remark)37 b(2)49 b Fq(One)d(might)h(also)f(c)-5 b(onsider)45 b(inter)-5 b(action)47 b(matric)-5 b(es)46 b(with)g(zer)-5 b(o)47 b(diagonal)e(terms,)-24 3601 y(r)-5 b(e)g(c)g(overing)33 b(ortho)-5 b(gonality)33 b(in)g(lar)-5 b(ge)33 b Fo(N)44 b Fq(limit.)g(This)33 b(amounts)g(to)h(c)-5 b(onsider)33 b(the)g(shifte)-5 b(d)33 b(Hamil-)-24 3750 y(tonian)1376 3874 y Fr(~)1347 3900 y Fp(H)1431 3915 y Fx(N)1499 3900 y Fr(\()p Fo(\033)t Fr(\))27 b(=)h Fp(H)1849 3915 y Fx(N)1916 3900 y Fr(\()p Fo(\033)t Fr(\))22 b(+)2181 3832 y(1)p 2181 3877 49 4 v 2181 3968 a(2)2240 3900 y Fo(;)1122 b Fr(\(3.40\))-24 4103 y Fq(so)39 b(that)g(the)g(aver)-5 b(age)37 b(ener)-5 b(gy)39 b(is)f(e)-5 b(qual)39 b(to)g(zer)-5 b(o)38 b(\(inste)-5 b(ad)38 b(of)h Fp(\000)p Fr(1)p Fo(=)p Fr(2)p Fq(\),)g(and)f(may)h(b)-5 b(e)38 b(c)-5 b(onvenient)-24 4252 y(for)35 b(p)-5 b(articular)35 b(purp)-5 b(oses)34 b(\(se)-5 b(e)34 b(also)h(R)-5 b(emark)34 b(1\).)-24 4541 y Fd(3.1)136 b(Mean)41 b(\014eld)g(prop)t(erties)g(at)h(an)l(y)f (temp)t(erature:)60 b(the)41 b(second)g(or-)283 4691 y(der)-24 4904 y Fr(The)30 b(results)f(of)g(this)f(section)h(are)g(for) f(a)h(c)m(hoice)g(of)f(the)h(matrix)f Fo(J)38 b Fr(kno)m(wn)30 b(as)f Fq(sine)h(mo)-5 b(del)28 b Fr(de\014ned)-24 5054 y(b)m(y)1130 5226 y Fo(J)1184 5241 y Fx(i;j)1291 5226 y Fr(=)1575 5159 y(2)p 1405 5203 390 4 v 1405 5223 a Fp(p)p 1488 5223 307 4 v 81 x Fr(2)p Fo(N)k Fr(+)22 b(1)1821 5226 y(sin)1957 5086 y Fn(\022)2100 5159 y Fr(2)p Fo(\031)t(ij)p 2040 5203 V 2040 5294 a Fr(2)p Fo(N)33 b Fr(+)22 b(1)2356 5086 y Fn(\023)2458 5226 y Fo(:)904 b Fr(\(3.41\))1758 5682 y(10)p eop %%Page: 11 11 11 10 bop -24 171 a Fm(Prop)s(osition)36 b(3.6)49 b Fq(L)-5 b(et)33 b(indic)-5 b(ate)32 b(with)g Fo(D)1594 186 y Fx(r)1632 171 y Fr(\()p Fo(N)10 b Fr(\))33 b Fq(the)f Fr(fat)e(diagonal)g Fq(of)i(dimension)f Fo(r)36 b Fq(i.e.)43 b(the)33 b(set)-24 320 y(of)38 b(p)-5 b(oints)37 b(of)g Fm(Z)568 335 y Fx(N)674 320 y Fq(in)g(which)g(at)h(le)-5 b(ast)37 b(two)h(of)f(the)h(indic)-5 b(es)37 b(c)-5 b(oincide.)52 b(L)-5 b(et)38 b(the)f(b)-5 b(ar)38 b(indic)-5 b(ate)37 b(the)-24 469 y(c)-5 b(omplementary)30 b(set)p 794 389 149 4 v 31 w Fo(D)878 435 y Fx(r)875 496 y(N)942 469 y Fq(.)44 b(F)-7 b(or)29 b(every)i(p)-5 b(ositive)30 b(inverse)g(temp)-5 b(er)g(atur)g(e)31 b Fo(\014)37 b Fq(exc)-5 b(ept,)31 b(at)g(most,)g(a)g(set)-24 619 y(of)e(zer)-5 b(o)28 b(L)-5 b(eb)g(esgue)29 b(me)-5 b(asur)g(e,)29 b(the)g(Gibbs)g(state)g Fo(<)f(>)g Fq(exp)-5 b(e)g(ctations)28 b(ful\014l)5 b(l)29 b(the)g(fol)5 b(lowing)27 b(r)-5 b(elation:)-4 763 y Fn(\014)-4 823 y(\014)-4 883 y(\014)95 810 y Fr(1)p 56 855 128 4 v 56 946 a Fo(N)144 917 y Fw(2)388 783 y Fn(X)210 1014 y Fx(i;)12 b(j;)e(l)q(;)i(m)g Fj(2)p 500 955 211 3 v Fx(D)558 1023 y Fi(4)592 1014 y Fw(\()p Fx(N)7 b Fw(\))876 878 y Fo(J)930 893 y Fx(i;j)1010 878 y Fo(J)1064 893 y Fx(l)q(;m)1200 878 y Fo(<)28 b(\033)1359 893 y Fx(i)1387 878 y Fo(\033)1442 893 y Fx(j)1479 878 y Fo(\033)1534 893 y Fx(l)1561 878 y Fo(\033)1616 893 y Fx(m)1710 878 y Fo(>)g Fp(\000)876 1205 y(\000)1031 1138 y Fr(1)p 992 1182 128 4 v 992 1274 a Fo(N)1080 1245 y Fw(2)1453 1111 y Fn(X)1146 1341 y Fx(i;)11 b(j)t Fj(2)p 1281 1283 211 3 v Fx(D)1339 1350 y Fi(2)1373 1341 y Fw(\()p Fx(N)c Fw(\))q(;)20 b Fx(l)q(;)11 b(m)p Fj(2)p 1693 1283 V Fx(D)1751 1350 y Fi(2)1786 1341 y Fw(\()p Fx(N)c Fw(\))1920 1205 y Fo(J)1974 1220 y Fx(i;j)2054 1205 y Fo(J)2108 1220 y Fx(l)q(;m)2244 1205 y Fo(<)28 b(\033)2403 1220 y Fx(i)2431 1205 y Fo(\033)2486 1220 y Fx(j)2551 1205 y Fo(><)f(\033)2785 1220 y Fx(l)2812 1205 y Fo(\033)2867 1220 y Fx(m)2961 1205 y Fo(>)3054 1091 y Fn(\014)3054 1151 y(\014)3054 1210 y(\014)3142 1205 y Fr(=)56 b Fo(O)s Fr(\()3460 1138 y(1)p 3400 1182 172 4 v 3400 1202 a Fp(p)p 3482 1202 89 4 v 3482 1287 a Fo(N)3580 1205 y Fr(\))-24 1553 y(Conceptually)37 b(the)g(pro)s(of)e(relies)h(on)h(the)g(equiv)-5 b(alence)36 b(of)g(the)h(micro)s(canonical)d(and)i(canonical)-24 1703 y(ensem)m(ble)i(whic)m(h)f(in)f(one)h(of)g(its)f(form)m(ulations)f (sa)m(ys)j(that)f(the)g(energy)h(densit)m(y)g(has)f(v)-5 b(anishing)-24 1852 y(\015uctuations)43 b(with)f(resp)s(ect)i(to)e(the) g(Gibbs)g(measure)h(in)f(the)h(thermo)s(dynamical)c(limit.)70 b(The)-24 2002 y(theorem)33 b(is)f(structured)i(in)e(sev)m(eral)h (lemmata:)-24 2259 y Fm(Lemma)k(3.2)49 b Fq(F)-7 b(or)32 b(every)h(p)-5 b(ositive)32 b(temp)-5 b(er)g(atur)g(e)33 b Fo(\014)38 b Fq(but)33 b(at)g(most)f(a)h(set)g(of)f(zer)-5 b(o)32 b(L)-5 b(eb)g(esgue)33 b(me)-5 b(a-)-24 2409 y(sur)g(e,)35 b(the)g(internal)g(ener)-5 b(gy)34 b(density)h(has)f(zer)-5 b(o)35 b(quadr)-5 b(atic)34 b(\015uctuations,)i(i.e.)1136 2668 y Fp(j)28 b Fo(<)f(h)1351 2626 y Fw(2)1351 2692 y Fx(N)1446 2668 y Fo(>)h Fp(\000)g Fo(<)g(h)1815 2683 y Fx(N)1910 2668 y Fo(>)1986 2626 y Fw(2)2053 2668 y Fp(j)f Fr(=)g Fp(O)s Fr(\()2361 2600 y(1)p 2341 2645 V 2341 2736 a Fo(N)2440 2668 y Fr(\))911 b(\(3.42\))-24 3174 y Fk(Pro)s(of)32 b Fr(By)h(the)g(de\014nition)f(of)g(free)h (energy)h(densit)m(y)1072 3433 y Fp(\000)p Fo(\014)6 b(f)1258 3448 y Fx(N)1325 3433 y Fr(\()p Fo(\014)g Fr(\))55 b(=)1678 3365 y(1)p 1658 3410 V 1658 3501 a Fo(N)1773 3433 y Fr(log)1916 3338 y Fn(X)1966 3547 y Fx(\033)2076 3433 y Fr(exp)2225 3392 y Fj(\000)p Fx(\014)s(H)5 b Fw(\()p Fx(\033)r Fw(\))2516 3433 y Fo(;)846 b Fr(\(3.43\))1316 3734 y Fo(d)p 1285 3779 112 4 v 1285 3870 a(d\014)1406 3802 y Fr(\()p Fo(\014)6 b(f)1553 3817 y Fx(N)1620 3802 y Fr(\()p Fo(\014)g Fr(\)\))55 b(=)28 b Fo(<)f(h)2113 3817 y Fx(N)2208 3802 y Fo(>)h Fr(;)1050 b(\(3.44\))910 4031 y Fo(d)961 3995 y Fw(2)p 879 4075 151 4 v 879 4167 a Fo(d\014)991 4138 y Fw(2)1040 4098 y Fr(\()p Fo(\014)6 b(f)1187 4113 y Fx(N)1254 4098 y Fr(\()p Fo(\014)g Fr(\)\))55 b(=)g Fp(\000)p Fo(N)10 b Fr(\()p Fo(<)28 b(h)1978 4057 y Fw(2)1978 4123 y Fx(N)2073 4098 y Fo(>)g Fp(\000)g Fo(<)g(h)2442 4113 y Fx(N)2537 4098 y Fo(>)2613 4057 y Fw(2)2652 4098 y Fr(\))g(;)644 b(\(3.45\))-24 4323 y(The)39 b(function)e Fo(\014)6 b(f)678 4338 y Fx(N)745 4323 y Fr(\()p Fo(\014)g Fr(\))37 b(is)g(b)s(ounded)i(and)f(con)m(v)m (ex)h(with)f(b)s(ounded)g(deriv)-5 b(ativ)m(e)37 b(\(the)h(b)s(ounded-) -24 4472 y(ness)k(comes)e(from)e(the)i(b)s(ounds)h(on)f(the)g (Hamiltonian)c([DEGGI)o(])k(due)g(to)g(orthogonalit)m(y)e(and)-24 4622 y(con)m(v)m(exit)m(y)-8 b(,)31 b(and)e(is)e(pro)m(v)m(ed)j(on)e(v) m(ery)h(general)f(grounds)h(in)e([R]\).)42 b(The)29 b Fo(N)38 b Fp(!)27 b(1)h Fr(limit)d(of)i Fo(\014)6 b(f)3407 4637 y Fx(N)3474 4622 y Fr(\()p Fo(\014)g Fr(\),)-24 4771 y(whic)m(h)29 b(again)d(b)m(y)j(con)m(v)m(exit)m(y)h(alw)m(a)m(ys) e(exists,)i(at)d(least)h(along)e(subsequences)32 b([R],)d(is)e(itself)g (con)m(v)m(ex)-24 4921 y(and)j(has)g(alw)m(a)m(ys)h(righ)m(t)e(and)g (left)g(deriv)-5 b(ativ)m(es)30 b(whic)m(h)g(coincide)g(except)h(at)e (most)g(on)h(a)g(coun)m(table)-24 5070 y(set)40 b(of)e(p)s(oin)m(ts.)62 b(In)m(tegrating)38 b(the)i(\(3.45\))e(in)g(an)m(y)h Fo(\014)45 b Fr(in)m(terv)-5 b(al)37 b(the)i(p)s(ositivit)m(y)f(of)g (the)h(left)f(hand)-24 5220 y(side)33 b(and)g(the)g(fundamen)m(tal)e (theorem)i(of)f(calculus)g(yield)g(the)h(lemma.)1758 5682 y(11)p eop %%Page: 12 12 12 11 bop -24 171 a Fm(Lemma)37 b(3.3)49 b Fq(Case)35 b Fo(i)28 b Fr(=)f Fo(j)6 b Fq(;)35 b(for)f(the)h(sine)g(inter)-5 b(action)34 b Fo(J)44 b Fq(the)35 b(fol)5 b(lowing)33 b(r)-5 b(esult)36 b(holds:)1049 429 y Fp(j)1126 362 y Fr(1)p 1087 407 128 4 v 1087 498 a Fo(N)1175 469 y Fw(2)1242 335 y Fn(X)1241 547 y Fx(i;l)q(;m)1405 429 y Fo(J)1459 444 y Fx(i;i)1530 429 y Fo(J)1584 444 y Fx(l)q(;m)1720 429 y Fo(<)28 b(\033)1879 444 y Fx(l)1905 429 y Fo(\033)1960 444 y Fx(m)2055 429 y Fo(>)f Fp(j)55 b(\024)2446 362 y Fr(1)p 2384 407 172 4 v 2384 426 a Fp(p)p 2467 426 89 4 v 85 x Fo(N)3389 429 y Fr(\(3.46\))-24 936 y Fk(Pro)s(of)491 1085 y Fp(j)568 1018 y Fr(1)p 529 1062 128 4 v 529 1154 a Fo(N)617 1125 y Fw(2)685 991 y Fn(X)683 1203 y Fx(i;l)q(;m)847 1085 y Fo(J)901 1100 y Fx(i;i)973 1085 y Fo(J)1027 1100 y Fx(l)q(;m)1163 1085 y Fo(<)27 b(\033)1321 1100 y Fx(l)1348 1085 y Fo(\033)1403 1100 y Fx(m)1497 1085 y Fo(>)h Fp(j)f Fr(=)h Fp(j)1837 1018 y Fr(1)p 1798 1062 V 1798 1154 a Fo(N)1886 1125 y Fw(2)1952 991 y Fn(X)1984 1203 y Fx(k)r(;l)2112 1085 y Fo(J)2166 1100 y Fx(l)q(;k)2278 1085 y Fo(<)g(\033)2437 1100 y Fx(l)2463 1085 y Fo(\033)2518 1100 y Fx(m)2613 1085 y Fo(>)f Fp(j)h(\024)2948 1018 y Fr(1)p 2887 1062 172 4 v 2887 1082 a Fp(p)p 2970 1082 89 4 v 85 x Fo(N)3096 1085 y Fr(;)266 b(\(3.47\))-24 1363 y(where)41 b(the)e(\014rst)g (equalit)m(y)g(comes)g(from)f(orthogonalit)m(y)f(whic)m(h)i(en)m(tails) 2761 1288 y Fn(P)2866 1392 y Fx(i)2911 1363 y Fo(J)2965 1378 y Fx(i;i)3076 1363 y Fr(=)f(1,)i(and)f(the)-24 1512 y(second)34 b(the)f(inequalit)m(y)f(from)f(the)i(sup)g(b)s(ounds)h Fo(J)1883 1527 y Fx(i;j)1990 1512 y Fp(\024)2149 1473 y Fw(1)p 2106 1489 122 4 v 2106 1499 a Fj(p)p 2164 1499 64 3 v 2164 1556 a Fx(N)2237 1512 y Fr(,)f Fo(<)28 b(\033)2456 1527 y Fx(l)2482 1512 y Fo(\033)2537 1527 y Fx(m)2632 1512 y Fo(>)p Fp(\024)g Fr(1)-24 1770 y Fm(Lemma)37 b(3.4)49 b Fq(Case)35 b Fo(j)e Fr(=)28 b Fo(l)r Fq(;)35 b(for)f(the)h(sine)g (inter)-5 b(action)34 b Fo(J)44 b Fq(the)35 b(fol)5 b(lowing)33 b(r)-5 b(esult)36 b(holds:)1184 1961 y Fr(1)p 1144 2006 128 4 v 1144 2097 a Fo(N)1232 2068 y Fw(2)1304 1934 y Fn(X)1299 2144 y Fx(i;j;m)1470 2028 y Fo(J)1524 2043 y Fx(i;j)1604 2028 y Fo(J)1658 2043 y Fx(j;m)1800 2028 y Fo(<)27 b(\033)1958 2043 y Fx(i)1987 2028 y Fo(\033)2042 2043 y Fx(m)2137 2028 y Fo(>)g Fr(=)2401 1961 y(1)p 2381 2006 89 4 v 2381 2097 a Fo(N)3389 2028 y Fr(\(3.48\))-24 2535 y Fk(Pro)s(of)174 2717 y Fr(1)p 134 2761 128 4 v 134 2852 a Fo(N)222 2824 y Fw(2)294 2689 y Fn(X)289 2899 y Fx(i;j;m)460 2784 y Fo(J)514 2799 y Fx(i;j)594 2784 y Fo(J)648 2799 y Fx(j;m)790 2784 y Fo(<)h(\033)949 2799 y Fx(i)977 2784 y Fo(\033)1032 2799 y Fx(m)1127 2784 y Fo(>)f Fr(=)1411 2717 y(1)p 1372 2761 V 1372 2852 a Fo(N)1460 2824 y Fw(2)1526 2689 y Fn(X)1545 2899 y Fx(i;m)1687 2784 y Fo(\016)1730 2799 y Fx(i;m)1868 2784 y Fo(<)g(\033)2026 2799 y Fx(i)2055 2784 y Fo(\033)2110 2799 y Fx(m)2205 2784 y Fo(>)g Fr(=)2489 2717 y(1)p 2449 2761 V 2449 2852 a Fo(N)2537 2824 y Fw(2)2604 2689 y Fn(X)2664 2899 y Fx(i)2764 2784 y Fr(1)55 b(=)3029 2717 y(1)p 3010 2761 89 4 v 3010 2852 a Fo(N)3136 2784 y(:)p Fp(})149 b Fr(\(3.49\))-24 3099 y Fk(Pro)s(of)32 b(of)g(Prop)s(osition)f(2.1)-24 3249 y Fr(De\014ning)1081 3451 y Fo(A)56 b Fr(=)1341 3276 y Fn(\014)1341 3336 y(\014)1341 3396 y(\014)1341 3456 y(\014)1341 3515 y(\014)1424 3383 y Fr(1)p 1384 3428 128 4 v 1384 3519 a Fo(N)1472 3490 y Fw(2)1540 3356 y Fn(X)1538 3568 y Fx(i;l)q(;m)1703 3451 y Fo(J)1757 3466 y Fx(i;i)1828 3451 y Fo(J)1882 3466 y Fx(l)q(;m)2018 3451 y Fo(<)27 b(\033)2176 3466 y Fx(l)2203 3451 y Fo(\033)2258 3466 y Fx(m)2353 3451 y Fo(>)2429 3276 y Fn(\014)2429 3336 y(\014)2429 3396 y(\014)2429 3456 y(\014)2429 3515 y(\014)2506 3451 y Fo(;)856 b Fr(\(3.50\))-24 3723 y(and)1066 4055 y Fo(B)61 b Fr(=)1332 3880 y 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4573 y Fo(>)h Fp(\000)1217 4833 y Fr(1)p 1177 4878 128 4 v 1177 4969 a Fo(N)1265 4940 y Fw(2)1639 4806 y Fn(X)1332 5037 y Fx(i;)11 b(j)t Fj(2)p 1467 4978 211 3 v Fx(D)1525 5046 y Fi(2)1559 5037 y Fw(\()p Fx(N)c Fw(\))q(;)19 b Fx(l)q(;)12 b(m)p Fj(2)p 1879 4978 V Fx(D)1937 5046 y Fi(2)1972 5037 y Fw(\()p Fx(N)7 b Fw(\))2106 4901 y Fo(J)2160 4916 y Fx(i;j)2240 4901 y Fo(J)2294 4916 y Fx(l)q(;m)2430 4901 y Fo(<)27 b(\033)2588 4916 y Fx(i)2617 4901 y Fo(\033)2672 4916 y Fx(j)2737 4901 y Fo(><)g(\033)2971 4916 y Fx(l)2997 4901 y Fo(\033)3052 4916 y Fx(m)3147 4901 y Fo(>)3223 4786 y Fn(\014)3223 4846 y(\014)3223 4906 y(\014)1662 5181 y Fp(\024)h(j)g Fo(<)f(h)1982 5140 y Fw(2)1982 5205 y Fx(N)2077 5181 y Fo(>)h Fp(\000)g Fo(<)g(h)2446 5196 y Fx(N)2541 5181 y Fo(>)2617 5140 y Fw(2)2684 5181 y Fp(j)21 b Fr(+)h(6)p Fo(A)h Fr(+)f(4)p Fo(B)32 b(:)133 b Fr(\(3.52\))-24 5430 y(The)34 b(previous)f(lemmata)d (pro)m(vide)j(the)g(claim.)1758 5682 y(12)p eop %%Page: 13 13 13 12 bop -24 171 a Fd(3.2)136 b(High)63 b(temp)t(erature)h(expansion)g (of)f(the)h(free)g(energy)f(for)g(the)283 320 y(orthogonal)46 b(mo)t(del)-24 534 y Fr(W)-8 b(e)30 b(can)f(no)m(w)h(try)f(to)f(mimic)e (the)k(pro)s(cedure)g(used)g(in)e(the)h(previous)h(to)e(decouple)i(the) f(spins.)43 b(T)-8 b(o)-24 683 y(this)22 b(end,)i(let)e Fo(B)k Fr(b)s(e)c(an)g(orthogonal)e(matrix)g(suc)m(h)j(that)f Fo(B)2108 647 y Fx(T)2163 683 y Fo(J)9 b(B)33 b Fr(=)27 b Fo(D)e Fr(with)c Fo(D)31 b Fr(=)c(diag)17 b(\()p Fo(d)3254 698 y Fw(1)3293 683 y Fo(;)g(:)g(:)g(:)f(;)h(d)3563 698 y Fx(N)3629 683 y Fr(\).)-24 833 y(Since)36 b(det)17 b Fo(J)42 b Fp(6)p Fr(=)32 b(0)j(w)m(e)h(ha)m(v)m(e)h Fo(d)1099 848 y Fx(i)1159 833 y Fo(>)c Fr(0,)j Fo(i)c Fr(=)h(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(N)10 b Fr(,)36 b(and)g(det)17 b Fo(J)2380 797 y Fj(\000)p Fw(1)2507 833 y Fr(=)2615 758 y Fn(Q)2709 862 y Fx(i)2754 833 y Fo(d)2805 791 y Fj(\000)p Fw(1)2805 858 y Fx(i)2899 833 y Fr(.)52 b(Let)35 b Fo(u)d Fp(2)h Fo(I)-25 b(R)3435 797 y Fx(N)3538 833 y Fr(b)s(e)-24 982 y(suc)m(h)34 b(that)f Fo(\033)e Fr(=)d Fo(B)5 b(u)p Fr(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h Fo(<)28 b(J)9 b(\033)n(;)17 b(\033)32 b(>)p Fr(=)p Fo(<)27 b(B)5 b(u;)17 b(J)9 b(B)c(u)27 b(>)p Fr(=)p Fo(<)g(u;)17 b(D)s(u)26 b(>)p Fr(,)33 b(and)f(th)m(us)29 1289 y(exp)q(\()266 1222 y Fo(\025)p 226 1266 138 4 v 226 1358 a Fr(2)p Fo(N)401 1289 y(<)27 b(J)9 b(\033)n(;)17 b(\033)33 b(>)p Fr(\))82 b(=)1139 1165 y Fx(N)1107 1195 y Fn(Y)1114 1405 y Fx(i)p Fw(=1)1251 1289 y Fr(exp)q(\()1488 1222 y Fo(\025)p 1448 1266 V 1448 1358 a Fr(2)p Fo(N)1595 1289 y(d)1646 1304 y Fx(i)1674 1289 y Fo(u)1730 1248 y Fw(2)1730 1314 y Fx(i)1769 1289 y Fr(\))948 1628 y(=)1139 1503 y Fx(N)1107 1533 y Fn(Y)1114 1743 y Fx(i)p Fw(=1)1332 1560 y Fr(1)p 1261 1605 191 4 v 1261 1625 a Fp(p)p 1344 1625 108 4 v 82 x Fr(2)p Fo(\031)1478 1492 y Fn(Z)1578 1518 y Fj(1)1534 1718 y(\0001)1680 1628 y Fr(exp)1846 1457 y Fn( )1925 1628 y Fp(\000)2012 1560 y Fo(x)2067 1524 y Fw(2)2067 1585 y Fx(i)p 2012 1605 95 4 v 2035 1696 a Fr(2)2139 1628 y(+)2237 1469 y Fn(r)p 2337 1469 156 4 v 2347 1560 a Fo(\025d)2455 1575 y Fx(i)p 2347 1605 136 4 v 2370 1696 a Fo(N)2525 1628 y(u)2581 1643 y Fx(i)2609 1628 y Fo(x)2664 1643 y Fx(i)2693 1457 y Fn(!)2788 1628 y Fo(dx)2894 1643 y Fx(i)948 1960 y Fr(=)1251 1893 y(1)p 1117 1938 316 4 v 1117 2029 a(\(2)p Fo(\031)t Fr(\))1301 2000 y Fx(N)q(=)p Fw(2)1459 1825 y Fn(Z)1515 2050 y Fx(I)-18 b(R)1581 2032 y Fl(N)1659 1960 y Fr(exp)1825 1790 y Fn( )1903 1960 y Fp(\000)1990 1893 y Fr(1)p 1990 1938 49 4 v 1990 2029 a(2)2077 1960 y Fo(<)28 b(x;)17 b(x)28 b(>)g Fr(+)2543 1801 y Fn(r)p 2642 1801 109 4 v 2668 1893 a Fo(\025)p 2652 1938 89 4 v 2652 2029 a(N)2811 1960 y(<)g(u;)17 b(D)3099 1919 y Fw(1)p Fx(=)p Fw(2)3208 1960 y Fo(x)28 b(>)3367 1790 y Fn(!)3479 1960 y Fo(dx)948 2282 y Fr(=)1141 2214 y(det)17 b Fo(J)1356 2177 y Fj(\000)1421 2150 y Fi(1)p 1421 2162 31 3 v 1421 2204 a(2)p 1117 2259 373 4 v 1117 2350 a Fr(\(2)p Fo(\031)t(\025)p Fr(\))1358 2321 y Fx(N)q(=)p Fw(2)1516 2146 y Fn(Z)1572 2372 y Fx(I)-18 b(R)1638 2353 y Fl(N)1716 2282 y Fr(exp)1881 2141 y Fn(\022)1955 2282 y Fp(\000)2071 2214 y Fr(1)p 2042 2259 106 4 v 2042 2350 a(2)p Fo(\025)2186 2282 y(<)27 b(y)t(;)17 b(J)2448 2241 y Fj(\000)p Fw(1)2541 2282 y Fo(y)31 b(>)d Fr(+)44 b Fo(<)27 b(\033)n(;)3115 2214 y(y)p 3055 2259 172 4 v 3055 2279 a Fp(p)p 3138 2279 89 4 v 84 x Fo(N)3264 2282 y(>)3340 2141 y Fn(\023)3446 2282 y Fo(dy)t(:)-24 2559 y Fr(By)33 b(\(2.6\))f(this)h(yields)96 2855 y Fo(Z)163 2870 y Fx(N)230 2855 y Fr(\()p Fo(\014)6 b Fr(\))27 b(=)g(2)546 2814 y Fx(N)647 2788 y Fr(det)17 b Fo(J)862 2751 y Fj(\000)927 2724 y Fi(1)p 927 2736 31 3 v 927 2777 a(2)p 623 2832 373 4 v 623 2924 a Fr(\(2)p Fo(\031)t(\025)p Fr(\))864 2895 y Fx(N)q(=)p Fw(2)1022 2720 y Fn(Z)1078 2945 y Fx(I)-18 b(R)1144 2926 y Fl(N)1222 2855 y Fr(exp)1371 2745 y Fn(\020)1430 2855 y Fp(\000)1546 2788 y Fr(1)p 1517 2832 106 4 v 1517 2924 a(2)p Fo(\025)1661 2855 y(<)28 b(y)t(;)17 b(J)1924 2814 y Fj(\000)p Fw(1)2017 2855 y Fo(y)31 b(>)c Fr(+)22 b(+)2395 2761 y Fn(X)2455 2971 y Fx(i)2556 2855 y Fr(log)16 b(cosh)2958 2788 y Fo(y)3006 2803 y Fx(i)p 2910 2832 172 4 v 2910 2852 a Fp(p)p 2993 2852 89 4 v 85 x Fo(N)3091 2745 y Fn(\021)3167 2855 y Fo(dy)123 b Fr(\(3.53\))-24 3163 y(to)27 b(b)s(e)g(compared)f(with)g(\(2.17\).)41 b(W)-8 b(e)27 b(p)s(oin)m(t)f(out)h(that)f(the)h(square)h(ro)s(ots)e (app)s(earing)g(in)g(the)h(ab)s(o)m(v)m(e)-24 3313 y(form)m(ula)39 b(are)h(only)f(apparen)m(tly)h(ill)e(de\014ned.)67 b(Indeed)41 b(they)g(disapp)s(ear)f(as)g(so)s(on)g(as)g(one)g(tak)m(es)-24 3462 y(its)f(dev)m(elopmen)m(t)h(in)e(p)s(o)m(w)m(ers)i(of)f Fo(\014)6 b Fr(,)40 b(b)s(ecause)g(the)g(latter)e(con)m(tains)h(only)f (ev)m(en)j(terms.)62 b Fo(Z)3434 3477 y Fx(N)3501 3462 y Fr(\()p Fo(\014)6 b Fr(\))-24 3611 y(has)45 b(b)s(een)g(computed)g(b) m(y)h(P)m(arisi)d(and)i(P)m(otters)g(in)f([PP)q(])g(using)g(standard)h (high-temp)s(erature)-24 3761 y(tec)m(hniques.)j(Relying)32 b(on)h(their)g(computation)e(w)m(e)k(are)e(no)m(w)h(in)e(the)i(p)s (osition)d(to)i(state)h(a)f(result)-24 3910 y(analogous)f(to)g(Prop)s (osition)f(2.3)h(for)g(this)g(class)h(of)f(mo)s(dels.)-24 4143 y Fm(Prop)s(osition)k(3.7)49 b Fq(F)-7 b(or)34 b Fr(0)27 b Fp(\024)h Fo(\014)34 b(<)27 b Fr(1)35 b Fq(and)f(for)h(any)g (ortho)-5 b(gonal)34 b(inter)-5 b(action)34 b(we)g(have)864 4392 y Fp(\000)p Fo(\014)g(F)14 b Fr(\()p Fo(\014)6 b Fr(\))27 b Fp(\021)h Fr(log)17 b Fo(Z)1586 4407 y Fx(N)1653 4392 y Fr(\()p Fo(\014)6 b Fr(\))27 b(=)g Fo(N)g Fr(log)17 b(2)39 b(+)22 b Fo(N)10 b(G)2519 4407 y Fx(N)2586 4392 y Fr(\()p Fo(\014)c Fr(\))p Fo(;)639 b Fr(\(3.54\))-24 4641 y Fq(wher)-5 b(e)46 4923 y Fo(G)123 4938 y Fx(N)190 4923 y Fr(\()p Fo(\014)6 b Fr(\))27 b Fp(\045)g Fo(G)p Fr(\()p Fo(\014)6 b Fr(\))27 b(=)836 4856 y(1)p 836 4900 49 4 v 836 4992 a(4)911 4753 y Fn(")969 4832 y(p)p 1069 4832 318 4 v 91 x Fr(1)22 b(+)g(4)p Fo(\014)1348 4894 y Fw(2)1409 4923 y Fp(\000)g Fr(log)1651 4753 y Fn( )1740 4855 y Fr(1)g(+)1909 4769 y Fn(p)p 2008 4769 V 2008 4855 a Fr(1)g(+)g(4)p Fo(\014)2287 4826 y Fw(2)p 1740 4900 587 4 v 2009 4992 a Fr(2)2336 4753 y Fn(!)2437 4923 y Fp(\000)h Fr(1)2586 4753 y Fn(#)2760 4923 y Fq(as)99 b Fo(N)38 b Fp(!)28 b(1)p Fo(:)69 b Fr(\(3.55\))1758 5682 y(13)p eop %%Page: 14 14 14 13 bop -24 171 a Fd(3.3)136 b(Limiting)70 b(b)t(eha)l(viour)g(and)f (connected)h(correlations)h(at)f(high)283 320 y(temp)t(erature)46 b(for)f(the)g(orthogonal)h(mo)t(del)-24 534 y Fr(W)-8 b(e)35 b(start)g(noticing)e(that)h(the)h(function)f Fo(G)p Fr(\()p Fo(\014)6 b Fr(\))34 b(de\014ned)i(in)e(Prop)s(osition)e(3.7)j (has)f(the)h(follo)m(wing)-24 683 y(expansion)e(in)f(the)h(vicinit)m(y) f(of)g Fo(\014)h Fr(=)28 b(0:)1386 952 y Fo(G)p Fr(\()p Fo(\014)6 b Fr(\))27 b(=)1740 885 y Fo(\014)1801 848 y Fw(2)p 1740 929 101 4 v 1766 1020 a Fr(4)1872 952 y(+)22 b Fp(O)s Fr(\()p Fo(\014)2151 911 y Fw(3)2191 952 y Fr(\))1160 b(\(3.56\))-24 1196 y(whic)m(h,)30 b(b)m(y)f(the)g(w)m(a)m(y)-8 b(,)30 b(coincides)e(with)f(what)i(one)f(obtains)g(for)f(the)i(SK)f(mo) s(del)e(if)h(truncated)i(after)-24 1345 y(the)36 b(\014rst)f(term.)49 b(Moreo)m(v)m(er,)37 b(according)d(to)h(Prop)s(osition)e(3.7)h(and)h (with)f(the)h(same)g(notation)e(of)-24 1495 y(the)g(previous)g (section,)g(w)m(e)h(ha)m(v)m(e)884 1776 y Fo(<)27 b(e)1032 1735 y Fj(\000)p Fx(\025h)1169 1746 y Fl(N)1259 1776 y Fo(>)p Fr(=)1448 1703 y Fo(Z)1515 1718 y Fx(N)1599 1622 y Fn(\000)1645 1703 y Fo(\014)g Fr(+)1846 1663 y Fx(\025)p 1835 1680 64 4 v 1835 1737 a(N)1908 1622 y Fn(\001)p 1448 1753 506 4 v 1566 1845 a Fo(Z)1633 1860 y Fx(N)1700 1845 y Fr(\()p Fo(\014)6 b Fr(\))1992 1776 y(=)27 b(exp)18 b(\()p Fo(\025)f(G)2450 1735 y Fj(0)2450 1801 y Fx(N)2516 1776 y Fr(\()p Fo(\014)2615 1735 y Fj(\003)2654 1776 y Fr(\)\))659 b(\(3.57\))-24 2047 y(for)30 b(some)h Fo(\014)427 2011 y Fj(\003)496 2047 y Fr(suc)m(h)h(that)f(0)c Fp(\024)h Fo(\014)1166 2011 y Fj(\003)1223 2047 y Fp(\000)19 b Fo(\014)33 b Fp(\024)28 b Fo(\025=)-5 b(N)10 b Fr(.)43 b(W)-8 b(e)31 b(no)m(w)g(observ)m(e)h(that)f(according)f(to)g(\(3.54\)) g(w)m(e)-24 2197 y(ha)m(v)m(e)986 2346 y Fo(<)d(h)1145 2361 y Fx(N)1240 2346 y Fo(>)p Fr(=)h Fp(\000)1527 2279 y Fr(1)p 1507 2323 89 4 v 1507 2414 a Fo(N)1622 2206 y Fn(\022)1706 2279 y Fo(@)22 b Fr(log)17 b Fo(Z)1989 2294 y Fx(N)p 1706 2323 351 4 v 1822 2414 a Fo(@)5 b(\014)2066 2206 y Fn(\023)2167 2346 y Fr(=)27 b Fp(\000)p Fo(G)2424 2305 y Fj(0)2424 2371 y Fx(N)2492 2346 y Fr(\()p Fo(\014)6 b Fr(\))760 b(\(3.58\))-24 2576 y(and)33 b(since)g Fo(<)28 b(h)565 2591 y Fx(N)660 2576 y Fo(>)k Fr(is)g(b)s(ounded)h(uniformly)e (in)h Fo(N)43 b Fr(prop)s(ert)m(y)33 b(\(3.55\))f(implies)551 2840 y Fo(<)c(h)711 2855 y Fx(N)806 2840 y Fo(>)f Fp(!)g Fo(<)h(h)g(>)17 b Fr(=)43 b Fo(G)1513 2799 y Fj(0)1537 2840 y Fr(\()p Fo(\014)6 b Fr(\))27 b(=)2077 2773 y Fo(\014)p 1814 2817 587 4 v 1814 2923 a Fr(1)22 b(+)1983 2837 y Fn(p)p 2083 2837 318 4 v 86 x Fr(1)g(+)g(4)p Fo(\014)2362 2895 y Fw(2)2508 2840 y Fr(as)98 b Fo(N)38 b Fp(!)27 b(1)p Fo(:)326 b Fr(\(3.59\))-24 3130 y(Therefore,)36 b(if)e(w)m(e)h(\014x)g Fo(\014)h Fp(2)31 b Fr([0)p Fo(;)17 b Fr(1\))34 b(and)h(expand)g(the)g(r.h.s.)49 b(of)34 b(\(3.57\))g(in)g(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)h Fo(\014)3599 3094 y Fj(\003)-24 3280 y Fr(w)m(e)g(obtain)d(for)i Fo(N)43 b Fr(large)31 b(enough)1022 3548 y Fo(<)d(e)1171 3507 y Fj(\000)p Fx(\025h)1308 3518 y Fl(N)1397 3548 y Fo(>)p Fr(=)g Fo(e)1622 3507 y Fx(\025)12 b(G)1730 3484 y Fc(0)1752 3507 y Fw(\()p Fx(\014)s Fw(\))1871 3408 y Fn(\022)1944 3548 y Fr(1)22 b(+)g Fp(O)2212 3408 y Fn(\022)2295 3481 y Fo(\025)2352 3445 y Fw(2)p 2295 3526 97 4 v 2299 3617 a Fo(N)2401 3408 y Fn(\023)q(\023)2565 3548 y Fo(:)797 b Fr(\(3.60\))-24 3821 y(Note)28 b(that)f Fo(G)490 3785 y Fj(0)514 3821 y Fr(\(0\))g(=)h(0,)g(so)g(that)f(at)g (in\014nite)g(temp)s(erature)g(\()p Fo(\014)34 b Fr(=)27 b(0\))g(w)m(e)i(reco)m(v)m(er)g(the)f(same)g(result)-24 3970 y(as)45 b(in)e(Prop)s(osition)f(2.4)i(for)f(this)h(class)g(of)g 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Fi(1)2853 5262 y Fp(\001)j(\001)g(\001)e Fo(\033)3041 5277 y Fx(j)3070 5285 y Fl(n)3144 5262 y Fo(>)1758 5682 y Fr(14)p eop %%Page: 15 15 15 14 bop -24 171 a Fr(and)33 b(the)g Fo(n)p Fr(-th)f(cum)m(ulan)m(t) 360 420 y Fo(<)27 b(h)519 379 y Fx(n)519 444 y(N)614 420 y Fo(>)690 435 y Fx(c)808 420 y Fr(=)977 352 y(\()p Fp(\000)p Fr(1\))1179 316 y Fx(n)p 977 397 250 4 v 1042 488 a Fo(N)1130 459 y Fx(n)1252 279 y Fn(\022)1336 352 y Fo(@)1392 316 y Fx(n)1456 352 y Fr(log)17 b Fo(Z)1666 367 y Fx(N)p 1336 397 398 4 v 1452 488 a Fo(@)5 b(\014)1569 459 y Fx(n)1743 279 y Fn(\023)3389 420 y Fr(\(3.62\))808 690 y(=)982 622 y(\()p Fp(\000)p Fr(1\))1184 586 y Fx(n)p 977 667 261 4 v 977 758 a Fr(\(2)p Fo(N)10 b Fr(\))1190 729 y Fx(n)1383 595 y Fn(X)1277 800 y Fl(i)1299 815 y Fi(1)1333 800 y Fl(;)p Fc(\001\001\001)p Fl(;i)1450 808 y(n)1493 800 y Fc(2)p Fb(Z)1577 782 y Fl(n)1577 826 y(N)1273 874 y(j)1298 889 y Fi(1)1333 874 y Fl(;)p Fc(\001\001\001)p Fl(;j)1453 882 y(n)1496 874 y Fc(2)p Fb(Z)1580 857 y Fl(n)1580 900 y(N)1664 690 y Fo(J)1718 705 y Fx(i)1742 714 y Fi(1)1776 705 y Fx(j)1805 714 y Fi(1)1860 690 y Fp(\001)17 b(\001)g(\001)e Fo(J)2047 705 y Fx(i)2071 713 y Fl(n)2113 705 y Fx(j)2142 713 y Fl(n)2216 690 y Fo(<)28 b(\033)2375 705 y Fx(i)2399 714 y Fi(1)2455 690 y Fp(\001)17 b(\001)g(\001)d Fo(\033)2642 705 y Fx(i)2666 713 y Fl(n)2714 690 y Fo(\033)2769 705 y Fx(j)2798 714 y Fi(1)2853 690 y Fp(\001)j(\001)g(\001)e Fo(\033)3041 705 y Fx(j)3070 713 y Fl(n)3144 690 y Fo(>)3220 705 y Fx(c)-24 1093 y Fr(where)39 b Fo(<)d(\033)430 1108 y Fx(i)454 1117 y Fi(1)510 1093 y Fp(\001)17 b(\001)g(\001)e Fo(\033)698 1108 y Fx(i)722 1116 y Fl(n)769 1093 y Fo(\033)824 1108 y Fx(j)853 1117 y Fi(1)908 1093 y Fp(\001)i(\001)g(\001)e Fo(\033)1096 1108 y Fx(j)1125 1116 y Fl(n)1208 1093 y Fo(>)1284 1108 y Fx(c)1356 1093 y Fr(is)37 b(de\014ned)h(as)g(ab)s(o)m (v)m(e)g(\(see)h(2.34)e(and)g(the)h(subsequen)m(t)j(for-)-24 1242 y(m)m(ula\).)75 b(W)-8 b(e)44 b(ma)m(y)f(no)m(w)h(use)g(Prop)s 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Fm(Prop)s(osition)36 b(3.9)49 b Fq(F)-7 b(or)30 b(any)g(ortho)-5 b(gonal)30 b(inter)-5 b(action)30 b(and)g Fr(0)e Fp(\024)g Fo(\014)33 b(<)28 b Fr(1)i Fq(we)h(c)-5 b(an)30 b(\014nd)g(a)g(p)-5 b(ositive)-24 2396 y(c)g(onstant)35 b Fo(C)g Fr(=)27 b Fo(C)7 b Fr(\()p Fo(n)p Fr(\))35 b Fq(so)g(that)g(as)f Fo(N)39 b Fp(!)27 b(1)390 2606 y Fr(1)p 346 2651 136 4 v 346 2742 a Fo(N)434 2713 y Fx(n)628 2579 y Fn(X)522 2784 y Fl(i)544 2799 y Fi(1)578 2784 y Fl(;)p Fc(\001\001\001)p Fl(;i)695 2792 y(n)738 2784 y Fc(2)p Fb(Z)822 2767 y Fl(n)822 2810 y(N)518 2858 y(j)543 2873 y Fi(1)578 2858 y Fl(;)p Fc(\001\001\001)p Fl(;j)698 2866 y(n)741 2858 y Fc(2)p Fb(Z)825 2841 y Fl(n)825 2884 y(N)909 2674 y Fo(J)963 2689 y Fx(i)987 2698 y Fi(1)1021 2689 y Fx(j)1050 2698 y Fi(1)1105 2674 y Fp(\001)17 b(\001)g(\001)e Fo(J)1292 2689 y Fx(i)1316 2697 y Fl(n)1359 2689 y Fx(j)1388 2697 y Fl(n)1461 2674 y Fo(<)28 b(\033)1620 2689 y Fx(i)1644 2698 y Fi(1)1700 2674 y Fp(\001)17 b(\001)g(\001)e Fo(\033)1888 2689 y Fx(i)1912 2697 y Fl(n)1959 2674 y Fo(\033)2014 2689 y Fx(j)2043 2698 y Fi(1)2098 2674 y Fp(\001)i(\001)g(\001)e Fo(\033)2286 2689 y Fx(j)2315 2697 y Fl(n)2389 2674 y Fo(>)2465 2689 y Fx(c)2527 2674 y Fr(=)28 b Fp(O)2730 2533 y Fn(\022)2819 2606 y Fo(C)p 2813 2651 89 4 v 2813 2742 a(N)2911 2533 y Fn(\023)3001 2674 y Fo(:)361 b Fr(\(3.63\))123 3080 y(W)-8 b(e)38 b(are)f(in)g(a)h(p)s(osition)e(to)h(sp)s(ecify)h (the)g(previous)g(prop)s(osition)e(in)m(to)h(a)h(Wic)m(k)g(lik)m(e)f (form)m(ula)-24 3229 y(b)m(y)42 b(sho)m(wing)e(that)g(for)g(eac)m(h)h (exp)s(ectation)g(only)f(the)g(terms)h(outside)f(the)h Fq(fat)h(diagonal)d Fr(giv)m(e)h(a)-24 3379 y(con)m(tribution)32 b(in)m(to)g(the)h(sums.)44 b(F)-8 b(or)31 b(this)i(purp)s(ose)g(w)m(e)h (can)e(pro)m(v)m(e)i(the)f(follo)m(wing:)-24 3636 y Fm(Prop)s(osition)j (3.10)49 b Fq(F)-7 b(or)34 b(the)h(sine)f(inter)-5 b(action)35 b Fr(0)27 b Fp(\024)h Fo(\014)34 b(<)27 b Fr(1)35 b Fq(we)f(c)-5 b(an)35 b(\014nd)f(a)h(p)-5 b(ositive)34 b(c)-5 b(onstant)-24 3785 y Fo(C)35 b Fr(=)28 b Fo(C)7 b Fr(\()p Fo(n)p Fr(\))34 b Fq(so)h(that)g(as)g Fo(N)j Fp(!)27 b(1)325 3995 y Fr(1)p 281 4040 136 4 v 281 4131 a Fo(N)369 4102 y Fx(n)775 3968 y Fn(X)443 4202 y Fx(i)467 4211 y Fi(1)502 4202 y Fx(;)p Fj(\001\001\001)n Fx(;i)624 4210 y Fl(n)667 4202 y Fx(;j)716 4211 y Fi(1)749 4202 y Fx(;)p Fj(\001\001\001)o Fx(;j)877 4210 y Fl(n)918 4202 y Fj(2)p 965 4143 249 3 v Fx(D)1023 4211 y Fi(2)p Fl(n)1096 4202 y Fw(\()p Fx(N)7 b Fw(\))1214 4157 y Fc(\003)1267 4063 y Fo(J)1321 4078 y Fx(i)1345 4087 y Fi(1)1379 4078 y Fx(j)1408 4087 y Fi(1)1463 4063 y Fp(\001)17 b(\001)g(\001)e Fo(J)1650 4078 y Fx(i)1674 4086 y Fl(n)1716 4078 y Fx(j)1745 4086 y Fl(n)1819 4063 y Fo(<)28 b(\033)1978 4078 y Fx(i)2002 4087 y Fi(1)2058 4063 y Fp(\001)17 b(\001)g(\001)e Fo(\033)2246 4078 y Fx(i)2270 4086 y Fl(n)2317 4063 y Fo(\033)2372 4078 y Fx(j)2401 4087 y Fi(1)2456 4063 y Fp(\001)i(\001)g(\001)e Fo(\033)2644 4078 y Fx(j)2673 4086 y Fl(n)2747 4063 y Fo(>)2823 4078 y Fx(c)2885 4063 y Fr(=)28 b Fp(O)3088 3922 y Fn(\022)3177 3995 y Fo(C)p 3171 4040 89 4 v 3171 4131 a(N)3269 3922 y Fn(\023)-24 4405 y Fq(wher)-5 b(e)p 1307 4468 328 4 v 1307 4555 a Fo(D)1388 4570 y Fw(2)p Fx(n)1470 4555 y Fr(\()p Fo(N)10 b Fr(\))1634 4486 y Fj(\003)1701 4555 y Fr(=)28 b Fp(\012)1882 4514 y Fx(n)1882 4579 y(k)r Fw(=1)p 2015 4468 293 4 v 2015 4555 a Fo(D)2096 4570 y Fx(n)2143 4555 y Fr(\()p Fo(N)10 b Fr(\))-24 4812 y Fk(Pro)s(of.)57 b Fr(By)37 b(induction)f(o)m(v)m(er)i Fo(n)e Fp(\025)g Fr(2.)56 b(F)-8 b(or)37 b Fo(n)e Fr(=)g(2)i(the)h(pro) s(of)e(is)h(come)f(from)g(the)i(lemmata)c(3.3)-24 4961 y(and)f(3.4.)43 b(The)34 b(inductiv)m(e)e(argumen)m(ts)h(follo)m(w)e (from)g(the)i(follo)m(wing:)-24 5219 y Fm(Lemma)k(3.5)49 b Fq(F)-7 b(or)34 b(the)h(sine)g(inter)-5 b(action)34 b Fo(J)44 b Fq(the)35 b(fol)5 b(lowing)33 b(r)-5 b(esult)36 b(holds:)1758 5682 y Fr(15)p eop %%Page: 16 16 16 15 bop 352 305 a Fn(\014)352 365 y(\014)352 425 y(\014)621 352 y Fr(1)p 578 397 136 4 v 578 488 a Fo(N)666 459 y Fx(n)919 325 y Fn(X)813 530 y Fl(i)835 545 y Fi(1)869 530 y Fl(;)p Fc(\001\001\001)p Fl(;i)986 538 y(n)1029 530 y Fc(2)p Fb(Z)1113 513 y Fl(n)1113 556 y(N)750 604 y(j)775 619 y Fi(1)810 604 y Fl(;)p Fc(\001\001\001)p Fl(;j)930 612 y(n)973 604 y Fc(2)p Fb(Z)1057 587 y Fl(n)1057 630 y(N)1114 604 y(;s)p Fi(=)p Fl(t)1259 420 y Fo(J)1313 435 y Fx(i)1337 444 y Fi(1)1371 435 y Fx(j)1400 444 y Fi(1)1455 420 y Fp(\001)17 b(\001)g(\001)e Fo(J)1642 435 y Fx(i)1666 443 y Fl(n)1709 435 y Fx(j)1738 443 y Fl(n)1811 420 y Fo(<)28 b(\033)1970 435 y Fx(i)1994 444 y Fi(1)2050 420 y Fp(\001)17 b(\001)g(\001)e Fo(\033)2238 435 y Fx(i)2262 443 y Fl(n)2309 420 y Fo(\033)2364 435 y Fx(j)2393 444 y Fi(1)2448 420 y Fp(\001)i(\001)g(\001)e Fo(\033)2636 435 y Fx(j)2665 443 y Fl(n)2739 420 y Fo(>)2815 435 y Fx(c)2850 305 y Fn(\014)2850 365 y(\014)2850 425 y(\014)2966 420 y Fp(\024)290 738 y Fr(1)p 270 782 89 4 v 270 873 a Fo(N)369 691 y Fn(\014)369 750 y(\014)369 810 y(\014)667 738 y Fr(1)p 578 782 226 4 v 578 873 a Fo(N)666 845 y Fx(n)p Fj(\000)p Fw(1)1021 710 y Fn(X)844 933 y Fl(i)866 948 y Fi(1)900 933 y Fl(;)p Fc(\001\001\001)p Fl(;i)1017 948 y(n)p Fc(\000)p Fi(1)1138 933 y Fc(2)p Fb(Z)1222 909 y Fl(n)p Fc(\000)p Fi(1)1222 959 y Fl(N)840 1024 y(j)865 1039 y Fi(1)900 1024 y Fl(;)p Fc(\001\001\001)p Fl(;j)1020 1039 y(n)p Fc(\000)p Fi(1)1141 1024 y Fc(2)p Fb(Z)1225 1000 y Fl(n)p Fc(\000)p Fi(1)1225 1051 y Fl(N)1372 805 y Fo(J)1426 820 y Fx(i)1450 829 y Fi(1)1485 820 y Fx(j)1514 829 y Fi(1)1569 805 y Fp(\001)i(\001)g(\001)e Fo(J)1756 820 y Fx(i)1780 829 y Fl(n)p Fc(\000)p Fi(1)1900 820 y Fx(j)1929 829 y Fl(n)p Fc(\000)p Fi(1)2082 805 y Fo(<)27 b(\033)2240 820 y Fx(i)2264 829 y Fi(1)2320 805 y Fp(\001)17 b(\001)g(\001)e Fo(\033)2508 820 y Fx(i)2532 829 y Fl(n)p Fc(\000)p Fi(1)2657 805 y Fo(\033)2712 820 y Fx(j)2741 829 y Fi(1)2797 805 y Fp(\001)i(\001)g(\001)e Fo(\033)2985 820 y Fx(j)3014 829 y Fl(n)p Fc(\000)p Fi(1)3166 805 y Fo(>)3242 820 y Fx(c)3277 691 y Fn(\014)3277 750 y(\014)3277 810 y(\014)3327 805 y Fo(:)35 b Fr(\(3.64\))-24 1246 y Fk(Pro)s(of)p Fr(.)46 b(Case)34 b Fo(s)29 b Fr(=)g Fo(i)731 1261 y Fx(l)790 1246 y Fr(and)34 b Fo(t)29 b Fr(=)g Fo(j)1190 1261 y Fx(l)1216 1246 y Fr(:)45 b(lik)m(e)33 b(in)f(the)i(second)h(order)e(case)h(the)g(inequalit)m(y)e(comes)i (from)-24 1396 y(the)k(orthogonalit)m(y)c(prop)s(ert)m(y)1157 1321 y Fn(P)1262 1425 y Fx(i)1286 1437 y Fl(l)1331 1396 y Fo(J)1385 1411 y Fx(i)1409 1423 y Fl(l)1433 1411 y Fx(i)1457 1423 y Fl(l)1520 1396 y Fr(=)h(1)h(and)h(from)e Fo(<)2255 1321 y Fn(Q)2349 1425 y Fx(A)2423 1396 y Fo(\033)k(>)p Fp(\024)p Fo(<)2781 1321 y Fn(Q)2875 1425 y Fx(B)2952 1396 y Fo(\033)g(>)d Fr(for)g Fo(B)k Fp(\032)c Fo(A)p Fr(.)-24 1545 y(Case)42 b Fo(i)251 1560 y Fx(l)318 1545 y Fr(=)e Fo(j)474 1560 y Fx(s)551 1545 y Fr(\(or)g Fo(i)749 1560 y Fx(l)816 1545 y Fr(=)h Fo(i)966 1560 y Fx(s)1043 1545 y Fr(or)f Fo(j)1210 1560 y Fx(l)1277 1545 y Fr(=)g Fo(j)1433 1560 y Fx(s)1470 1545 y Fr(\):)59 b(the)41 b(sum)f(on)g Fo(i)2160 1560 y Fx(l)2226 1545 y Fr(is)g(p)s(erformed)g (using)g(orthogonalit)m(y)-24 1620 y Fn(P)82 1724 y Fx(i)106 1736 y Fl(l)150 1695 y Fo(J)204 1710 y Fx(si)261 1722 y Fl(l)289 1695 y Fo(J)343 1710 y Fx(i)367 1722 y Fl(l)391 1710 y Fx(t)449 1695 y Fr(=)27 b Fo(\016)595 1710 y Fx(s;t)677 1695 y Fr(,)33 b(the)g(extra)g Fo(\033)k Fr(is)32 b(b)s(ounded)h(as)g (in)f(the)h(ab)s(o)m(v)m(e)g(lemma.)123 1844 y(The)24 b(previous)f(lemma)e(together)i(with)g(the)h(prop)s(osition)d(\(3.8\))h (giv)m(es)i(the)f(prop)s(osition)e(\(3.10\).)-24 2064 y Fm(Ac)m(kno)m(wledgmen)m(ts)35 b Fr(One)i(of)e(us)i(\(PC\))g(thanks)g (F)-8 b(rancesco)37 b(Guerra)f(for)f(man)m(y)h(enligh)m(tening)-24 2214 y(discussions.)-24 2633 y Fs(References)-24 2881 y Fr([A)m(C])261 b Fa(M.Aizenman,)54 b(P.Contucci)p Fr(,)48 b(On)d(the)g(Stabilit)m(y)e(of)i(the)g(Quenc)m(hed)i(state)f(in)431 3030 y(Mean)33 b(Field)e(Spin)h(Glass)g(Mo)s(dels,)h Fk(J.)f(Stat.)h(Ph)m(ys.)h Fr(,)f Fm(92)p Fr(,)g(\(1998\),)e(765-783.) -24 3263 y([DEGGI])71 b Fa(M)27 b(Degli)h(Esposti,)h(C)f(Giardin)1802 3255 y(\022)1799 3263 y(a,)i(S)e(Graffi,)j(S)d(Isola)p Fr(,)e(Statistics)d(of)g(Energy)431 3412 y(Lev)m(els)37 b(at)g(Zero)f(T)-8 b(emp)s(erature)37 b(Dynamics)e(for)h(Deterministic) e(Spin)j(Mo)s(dels)f(with)431 3562 y(Glassy)c(Beha)m(viour,)h Fk(J.)g(Stat.)f(Ph)m(ys.)67 b Fm(102)33 b Fr(\(2001\),)e(1285-1313.)-24 3794 y([G])324 b Fa(F.Guerra)p Fr(,)29 b(Ab)s(out)h(the)g(o)m(v)m (erlap)f(distibution)f(in)g(a)i(mean)f(\014eld)g(spin)g(glass)g(mo)s (del,)431 3943 y Fk(In)m(t.)k(J.)g(Mo)s(d.)f(Ph)m(ys.)j Fm(B10)d Fr(\(1996\))g(1675-1681.)-24 4176 y([MPR])174 b Fa(M.Marinari,)37 b(G.P)-9 b(arisi)37 b(and)h(F.Ritor)-7 b(t)p Fr(,)49 b(Replica)31 b(Field)g(Theory)j(for)e(Deter-)431 4325 y(ministic)21 b(Mo)s(dels)j(\(I)g(and)g(I)s(I\):)g(A)g(Non-Random) f(Spin)g(Glass)g(with)h(Glassy)g(Beha)m(vior.)431 4475 y Fk(J.)34 b(Ph)m(ys.)i(A:)f(Math)f(and)h(Gen)f Fm(27)g Fr(\(1994\),)g(7615;)g Fk(J.)h(Ph)m(ys.)h(A:)e(Math)h(and)f(Gen)h Fm(27)431 4624 y Fr(\(1994\),)c(7647.)-24 4857 y([MPV])173 b Fa(M)28 b(M)657 4849 y(\023)657 4857 y(ezard,)i(G)f(P)-9 b(arisi)29 b(and)g(M)f(Virasor)n(o)p Fr(,)43 b Fq(Spin)27 b(Glass)g(The)-5 b(ory)27 b(and)g(Beyond)p Fr(,)431 5006 y(W)-8 b(orld)31 b(Scien)m(ti\014c)i(1995.)-24 5239 y([PP])269 b Fa(G)43 b(P)-9 b(arisi,)45 b(M)e(Potters)p Fr(,)c(Mean-Field)d (Equations)j(for)e(Spin)h(Mo)s(dels)g(with)f(Or-)431 5388 y(thogonal)23 b(In)m(teraction)i(Matrices.)g Fk(J.)g(Ph)m(ys.)h (A:)f(Math.)g(Gen.)g Fm(28)g Fr(\(1995\),)g(5267-5285.)1758 5682 y(16)p eop %%Page: 17 17 17 16 bop -24 171 a Fr([R])329 b Fa(D)37 b(R)m(uelle)p Fr(,)32 b Fq(Statistic)-5 b(al)35 b(Me)-5 b(chanics)p Fr(,)31 b(W)i(A)g(Benjamin,)e(Inc.)j(1969.)-24 403 y([S])347 b Fa(A)38 b(N)f(Shir)-7 b(y)g(aev)p Fr(,)34 b Fq(Pr)-5 b(ob)g(ability)p Fr(,)32 b(Springer-V)-8 b(erlag)31 b(1995.)-24 636 y([Th])277 b Fa(C.J.Thompson)p Fr(,)34 b Fq(Equilibrium)g (Statistic)-5 b(al)35 b(Me)-5 b(chanics)p Fr(,)32 b(Springer-V)-8 b(erlag)30 b(1995.)1758 5682 y(17)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0106210857815--