Content-Type: multipart/mixed; boundary="-------------9901121103408" This is a multi-part message in MIME format. ---------------9901121103408 Content-Type: text/plain; name="99-14.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-14.comments" 1991 Mathematics Subject Classification: 60F05, 60K35, 82C32 e-mail: gentz@wias-berlin.de, lowe@eurandom.tue.nl ---------------9901121103408 Content-Type: text/plain; name="99-14.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-14.keywords" Hopfield model, spin glasses, neural networks, random disorder, limit theorems, non-Gaussian fluctuations, critical temperature ---------------9901121103408 Content-Type: application/postscript; name="critical_fluctuations.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="critical_fluctuations.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: kopf.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o kopf.ps kopf %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.12.17:1507 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N 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y(1)p 2484 1416 49 4 v 2484 1508 a(2)118 1640 y Ft(for)39 b(all)f Fr(i)i Ft(and)g Fr(\026)p Ft(.)64 b(Exp)s(ectations)40 b(with)g(resp)s(ect)h(to)e Fp(P)i Ft(will)d(b)s(e)i(denoted)g(b)m(y)h Fp(E)12 b Ft(.)71 b(Whenev)m(er)118 1757 y(con)m(v)m(enien)m(t,)35 b(w)m(e)f(shall)d(write)i Fr(\030)k Ft(for)32 b(the)h Fo(\()p Fr(N)g Fq(\002)23 b Fr(M)10 b Fo(\))p Ft(-matrix)31 b(consisting)i(of)f(the)h Fo(\()p Fr(\030)3205 1709 y Fl(\026)3200 1782 y(i)3251 1757 y Fo(\))3289 1772 y Fl(i;\026)3379 1757 y Ft(,)g(while)118 1874 y Fr(\030)161 1889 y Fl(i)217 1874 y Fo(=)27 b(\()p Fr(\030)406 1838 y Fj(1)401 1899 y Fl(i)445 1874 y Fr(;)17 b(:)g(:)g(:)32 b(;)17 b(\030)728 1838 y Fl(M)723 1899 y(i)806 1874 y Fo(\))32 b Ft(and)g Fr(\030)1113 1838 y Fl(\026)1187 1874 y Fo(=)27 b(\()p Fr(\030)1376 1827 y Fl(\026)1371 1899 y Fj(1)1422 1874 y Fr(;)17 b(:)g(:)g(:)32 b(;)17 b(\030)1705 1827 y Fl(\026)1700 1901 y(N)1767 1874 y Fo(\))p Ft(,)32 b(resp)s(ectiv)m(ely)-8 b(,)33 b(stand)f(for)f(the)h Fr(i)p Ft(th)g(ro)m(w)g(and)g(the)118 1991 y Fr(\026)p Ft(th)g(column)g(of)g(this)g(matrix,)f(resp)s(ectiv)m (ely)-8 b(.)218 2107 y(The)34 b(spin)g(v)-5 b(ariables)32 b(are)i(assumed)g(to)f(b)s(e)h(indep)s(enden)m(t)g(with)g(an)f(un)m (biased)h(a)g(priori)d(dis-)118 2223 y(tribution)g Fo(P)p Ft(,)i(i.)16 b(e.,)1242 2397 y Fo(P\()p Fr(\033)1401 2412 y Fl(i)1457 2397 y Fo(=)28 b(+1\))f(=)h(P\()p Fr(\033)2014 2412 y Fl(i)2070 2397 y Fo(=)g Fq(\000)p Fo(1\))g(=)2479 2330 y(1)p 2479 2375 V 2479 2466 a(2)118 2578 y Ft(for)38 b(all)d Fr(i)j Fq(2)f Fp(N)9 b Ft(.)66 b(In)38 b(addition,)f(w)m(e)i (shall)e(assume)h(throughout)g(this)g(pap)s(er)g(that)g(the)g(family) 118 2694 y Fq(f)17 b Fr(\030)233 2647 y Fl(\026)228 2720 y(i)326 2694 y Fo(:)48 b Fr(i)g Fq(2)g(f)p Fo(1)p Fr(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fq(g)p Fr(;)17 b(\026)47 b Fq(2)h(f)p Fo(1)p Fr(;)17 b(:)g(:)g(:)e(;)i(M)10 b Fq(g)17 b(g)44 b Ft(is)g(indep)s(enden)m(t)i(of)d(the)i(family)d(of)i(the)g(spin)118 2811 y(v)-5 b(ariables)31 b Fq(f)17 b 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Fk(X)1576 3730 y Fl(\033)r Fi(2f\000)p Fj(1)p Fl(;)p Fj(+1)p Fi(g)1935 3711 y Fh(N)2010 3606 y Fo(exp)q Fq(f\000)p Fr(\014)6 b(H)2428 3621 y Fl(N)2495 3606 y Fo(\()p Fr(\033)t Fo(\))p Fq(g)781 b Ft(\(1.3\))118 3889 y(is)32 b(the)h(normalization)c(whic)m(h)k(mak)m (es)g Fr(\045)1624 3904 y Fl(N)s(;\014)1783 3889 y Ft(a)f(probabilit)m (y)e(measure.)218 4006 y(In)39 b(order)g(to)f(understand)j(the)e(in)m (tro)s(duction)e(of)i(the)g(order)g(parameter)f(in)g(the)i(Hop\034eld) 118 4122 y(mo)s(del)27 b(note)i(that)f(the)i(Hamiltonian)24 b(\(1.1\))29 b(ma)m(y)f(b)s(e)h(rewritten)g(in)f(the)h(follo)m(wing)d (con)m(v)m(enien)m(t)118 4238 y(form)31 b(as)i(a)f(quadratic)h (functional)d(of)i(the)h(so-called)e(o)m(v)m(erlap)i Fr(m)2542 4253 y Fl(N)2610 4238 y Ft(:)1355 4457 y Fr(H)1436 4472 y Fl(N)1503 4457 y Fo(\()p Fr(\033)t Fo(\))27 b(=)h Fq(\000)1856 4390 y Fr(N)p 1856 4434 89 4 v 1876 4525 a Fo(2)1955 4457 y Fq(k)p Fr(m)2090 4472 y Fl(N)2157 4457 y Fo(\()p Fr(\033)t Fo(\))p Fq(k)2342 4416 y Fj(2)2342 4482 y(2)2381 4457 y Fr(;)1053 b Ft(\(1.4\))118 4658 y(where)656 4902 y Fr(m)741 4917 y Fl(N)809 4902 y Fo(\()p Fr(\033)t Fo(\))27 b(=)h(\()p Fr(m)1198 4855 y Fl(\026)1198 4929 y(N)1265 4902 y Fo(\()p Fr(\033)t Fo(\)\))1438 4917 y Fl(\026)p Fj(=1)p Fl(;:::)11 b(;M)1955 4902 y Ft(with)194 b Fr(m)2424 4855 y Fl(\026)2424 4929 y(N)2492 4902 y Fo(\()p Fr(\033)t Fo(\))28 b(=)2798 4778 y Fl(N)2758 4808 y Fk(X)2773 5018 y Fl(i)p Fj(=1)2919 4902 y Fr(\030)2967 4855 y Fl(\026)2962 4928 y(i)3013 4902 y Fr(\033)3068 4917 y Fl(i)3096 4902 y Fr(:)338 b Ft(\(1.5\))118 5173 y(Here)32 b(and)g(b)s(elo)m(w,)f Fq(k)19 b(\001)h(k)1005 5188 y Fj(2)1075 5173 y Ft(denotes)33 b(the)e(Euclidean)g(norm)g(in)f Fp(R)2475 5137 y Fl(M)2560 5173 y Ft(.)43 b(The)33 b Fr(\026)p Ft(th)e(comp)s(onen)m(t)g Fr(m)3594 5126 y Fl(\026)3594 5200 y(N)118 5289 y Ft(of)j(the)g(o)m(v)m(erlap)g Fr(m)825 5304 y Fl(N)926 5289 y Ft(compares)g(the)h(spin)e (con\034guration)g(to)h(the)g Fr(\026)p Ft(th)g(pattern)g Fr(\030)3196 5253 y Fl(\026)3276 5289 y Ft(in)f(suc)m(h)j(a)118 5406 y(w)m(a)m(y)26 b(that)f(a)g(large)f(absolute)h(v)-5 b(alue)24 b(of)h Fr(m)1628 5358 y Fl(\026)1628 5433 y(N)1695 5406 y Fo(\()p Fr(\033)t Fo(\))g Ft(means)g(that)g(the)h(spin)f (con\034guration)f Fr(\033)29 b Ft(largely)118 5522 y(agrees)38 b(with)f Fr(\030)691 5486 y Fl(\026)775 5522 y Ft(\(or)g(its)g(negativ) m(e\).)58 b(These)39 b(con\034gurations)e(are)h(of)f(lo)m(w)g(energy)h (according)118 5638 y(to)29 b(\(1.4\))o(.)43 b(Therefore,)31 b(the)e(o)m(v)m(erlap)g(is)g(an)g(imp)s(ortan)m(t)e(quan)m(tit)m(y)j (for)e(the)i(in)m(v)m(estigation)d(of)i(the)p eop %%Page: 3 3 3 2 bop 317 -67 a Fm(FLUCTUA)-6 b(TIONS)32 b(IN)g(THE)i(HOPFIELD)e (MODEL)i(A)-6 b(T)33 b(THE)g(CRITICAL)g(TEMPERA)-6 b(TURE)184 b(3)94 123 y Ft(Hop\034eld)38 b(mo)s(del,)g(a)f(so-called)g(order)h (parameter.)59 b(Its)38 b(distribution)e(under)j Fr(\045)3097 138 y Fl(N)s(;\014)3261 123 y Ft(has)f(b)s(een)94 239 y(of)33 b(ma)5 b(jor)31 b(in)m(terest)i(in)f(the)h(study)h(of)e(the)h (mo)s(del)d(and)j(also)f(will)e(b)s(e)j(cen)m(tral)f(in)g(this)g(pap)s (er.)194 355 y(In)46 b([7],)i(Bo)m(vier,)h(Ga)m(yrard,)g(and)c(Picco)g (established)h(a)f(la)m(w)g(of)g(large)f(n)m(um)m(b)s(ers)i(for)f(the) 94 471 y(distribution)22 b(of)h(the)h(o)m(v)m(erlap)g(under)g(the)g (Gibbs)f(measure)h Fr(\045)2329 486 y Fl(N)s(;\014)2479 471 y Ft(whic)m(h)g(holds)g(for)f Fp(P)p Ft(-almost)g(all)94 588 y(realizations)31 b(of)h(the)h(random)f(patterns)h Fr(\030)5 b Ft(.)43 b(They)34 b(sho)m(w)m(ed)h(that,)d(whenev)m(er)k Fr(M)10 b Fo(\()p Fr(N)g Fo(\))p Fr(=)-5 b(N)38 b Fq(!)28 b Fo(0)p Ft(,)94 704 y(for)g Fp(P)p Ft(-almost)h(all)d Fr(\030)5 b Ft(,)29 b(the)g(distribution)d(of)i(the)h(o)m(v)m(erlap)g Fr(m)2261 719 y Fl(N)2357 704 y Ft(under)g(the)g(Gibbs)f(measure)h (with)94 820 y(external)34 b(magnetic)f(\034eld)g(of)h(strength)g Fr(h)c Fq(6)p Fo(=)g(0)j Ft(in)g(the)i(direction)d(of)i(the)g(\034rst)g (unit)f(v)m(ector)i Fr(e)3598 835 y Fj(1)94 936 y Ft(of)j(the)g (canonical)e(basis)i(in)f Fp(R)1247 900 y Fl(M)1369 936 y Ft(con)m(v)m(erges)j(w)m(eakly)f(to)m(w)m(ards)f(the)g(Dirac)g (measure)g Fr(\016)3378 952 y Fi(\006)p Fl(z)s Fj(\()p Fl(\014)s Fj(\))p Fl(e)3599 961 y Fg(1)94 1053 y Ft(concen)m(trated)45 b(in)e Fq(\006)p Fr(z)t Fo(\()p Fr(\014)6 b Fo(\))p Fr(e)1115 1068 y Fj(1)1199 1053 y Ft(as)43 b(\034rst)h(the)g(system)g(size)g Fr(N)56 b Fq(!)46 b(1)d Ft(and)g(then)h(the)g(strength)94 1169 y Fr(h)28 b Fq(!)g Fo(0)p Fq(\006)p Ft(.)43 b(Here)34 b Fr(z)t Fo(\()p Fr(\014)6 b Fo(\))32 b Ft(denotes)i(the)f(largest)f (ro)s(ot)f Fr(z)i Fq(2)28 b Fo([0)p Fr(;)17 b Fo(1\))32 b Ft(of)g(the)h(Curie)8 b(\025)-8 b(W)g(eiss)32 b(equation)1571 1345 y Fr(z)h Fo(=)27 b(tanh)q(\()p Fr(\014)6 b(z)t Fo(\))p Fr(:)94 1522 y Ft(Note)33 b(that)g Fr(z)t Fo(\()p Fr(\014)6 b Fo(\))28 b(=)g(0)k Ft(for)g Fr(\014)i Fq(\024)28 b Fr(\014)1339 1537 y Fj(c)1403 1522 y Fo(=)g(1)p Ft(,)k(so)h(that)g Fr(\016)1990 1537 y Fj(0)2062 1522 y Ft(is)f(the)h(unique)g(limiting)28 b(measure)33 b(in)f(the)94 1638 y(high-temp)s(erature)j(region)h Fr(\014)k Fq(\024)35 b Fr(\014)1429 1653 y Fj(c)1498 1638 y Fo(=)g(1)p Ft(,)i(whereas)h Fr(z)t Fo(\()p Fr(\014)6 b Fo(\))34 b Fr(>)g Fo(0)j Ft(for)f Fr(\014)j(>)c(\014)2924 1653 y Fj(c)2959 1638 y Ft(,)j(so)e(that)g(in)g(this)94 1755 y(regime)c(there)h(is)f(no)g(unique)h(limiting)28 b(p)s(oin)m(t.)194 1871 y(Note)40 b(that)f(this)g(result)g(strongly)g (resem)m(bles)h(the)g(la)m(w)f(of)g(large)f(n)m(um)m(b)s(ers)j(for)d (the)i(mean)94 1987 y(magnetization)31 b(in)h(the)i(Curie)8 b(\025)-8 b(W)g(eiss)33 b(mo)s(del,)f(see)i([14,)f(Theorem)h (IV.4.1\(a\)],)f(for)f(example.)94 2103 y(As)50 b(already)e(explained)g (at)g(the)h(b)s(eginning)e(this)h(is,)k(of)c(course,)53 b(not)c(acciden)m(tal,)j(as)c(the)94 2220 y(Hop\034eld)31 b(mo)s(del)f(can)h(b)s(e)h(considered)g(as)f(a)g(disordered)g(v)m (ersion)h(of)f(the)g(Curie)8 b(\025)-8 b(W)g(eiss)31 b(mo)s(del)94 2336 y(and,)36 b(indeed,)g(for)e Fr(M)43 b Fo(=)31 b(1)k Ft(the)g(Hop\034eld)g(mo)s(del)e(and)i(the)g(Curie)8 b(\025)-8 b(W)g(eiss)35 b(mo)s(del)e(agree)i(b)m(y)h(a)94 2452 y(simple)31 b(\020gauge)h(transformation\021)38 b(\(i.)16 b(e.,)33 b(replacing)e Fr(\033)2123 2467 y Fl(i)2184 2452 y Ft(b)m(y)i Fr(\033)2374 2467 y Fl(i)2403 2452 y Fr(\030)2451 2416 y Fj(1)2446 2477 y Fl(i)2490 2452 y Ft(\).)194 2578 y(On)25 b(the)g(scale)f(of)g(\035uctuations,)i (when)g(analyzing)d(the)i(distribution)e(of)2851 2493 y Fq(p)p 2934 2493 89 4 v 85 x Fr(N)10 b Fo(\()p Fr(m)3145 2593 y Fl(N)3219 2578 y Fq(\000)c Fr(z)t Fo(\()p Fr(\014)g Fo(\))p Fr(e)3533 2593 y Fj(1)3573 2578 y Fo(\))p Ft(,)94 2694 y(the)38 b(c)m(haracter)f(of)g(the)g(disorder)g(b)s(ecomes)g (visible.)55 b(Indeed,)40 b(for)c Fr(M)5 b(=)-5 b(N)45 b Fq(!)35 b Fo(0)i Ft(and)g Fo(\()p Fr(\014)6 b(;)17 b(h)p Fo(\))34 b Fq(6)p Fo(=)94 2810 y(\(1)p Fr(;)17 b Fo(0\))p Ft(,)34 b(the)h(o)m(v)m(erlap)f(satis\034es)h Fp(P)p Ft(-almost)f(surely)g(a)g(cen)m(tral)g(limit)d(theorem)j(with)f (the)i(co)m(v)-5 b(ari-)94 2926 y(ance)31 b(matrix)d(whic)m(h)i(could)f (b)s(e)h(exp)s(ected)i(from)c(the)j(analogy)d(with)h(the)i(Curie)8 b(\025)-8 b(W)g(eiss)29 b(mo)s(del)94 3042 y(and)35 b(a)f(cen)m(tring)g (whic)m(h)g(di\033ers)g(in)g(the)g(cases)i Fr(\014)g(>)30 b Fo(1)k Ft(or)g Fr(h)c Fq(6)p Fo(=)h(0)j Ft(from)f(the)h(naiv)m(ely)g (exp)s(ected)94 3159 y(one)f(b)m(y)h(a)e Fr(\030)5 b Ft(-dep)s(enden)m(t)33 b(adjustmen)m(t,)g(see)h([16],)e([17],)h([19)o (])g(and)f(Bo)m(vier)h(and)g(Ga)m(yrard)f([4].)194 3275 y(As)k(sho)m(wn)h(in)d(a)h(previous)h(pap)s(er)g([20)o(],)h(the)e (in\035uence)i(of)e(the)g(disorder)h(is)e(ev)m(en)j(stronger)94 3391 y(when)31 b(in)m(v)m(estigating)e(the)h(\035uctuations)g(of)f(the) h(o)m(v)m(erlap)g(at)g(the)g(critical)d(temp)s(erature)j Fo(1)p Fr(=\014)j Fo(=)94 3507 y(1)p Fr(=\014)247 3522 y Fj(c)335 3507 y Fo(=)53 b(1)p Ft(,)d(ev)m(en)f(when)f Fr(M)10 b Fo(\()p Fr(N)g Fo(\))49 b Ft(remains)d(b)s(ounded.)88 b(Recall)46 b(that)h(in)f(the)i(Curie)8 b(\025)-8 b(W)g(eiss)94 3624 y(mo)s(del)40 b(the)i(criticalit)m(y)c(at)j(temp)s(erature)g Fo(1)p Fr(=\014)48 b Fo(=)42 b(1)f Ft(can)h(also)e(b)s(e)i(seen)g(as)g (the)f(breakdo)m(wn)94 3740 y(of)g(the)h(cen)m(tral)f(limit)d(theorem.) 69 b(As)42 b(a)f(matter)g(of)g(fact)g(at)g(the)g(critical)e(temp)s (erature)i(the)94 3856 y(magnetization)21 b(in)h(the)h(Curie)8 b(\025)-8 b(W)g(eiss)23 b(mo)s(del\026scaled)e(b)m(y)j(a)e(factor)h Fr(N)2713 3820 y Fj(1)p Fl(=)p Fj(4)2823 3856 y Ft(\026con)m(v)m(erges) i(w)m(eakly)94 3972 y(to)m(w)m(ards)40 b(a)e(random)f(v)-5 b(ariable)36 b(giv)m(en)j(b)m(y)g(its)f(densit)m(y)h(with)f(resp)s(ect) h(to)f(Leb)s(esgue)h(measure)94 4089 y(whic)m(h)g(is)e(prop)s(ortional) f(to)h Fo(exp)q(\()p Fq(\000)p Fr(x)1492 4052 y Fj(4)1533 4089 y Fr(=)p Fo(12\))p Ft(,)h(cf.)g([14,)h(Theorem)f(V.9.5].)60 b(In)38 b([20])g(w)m(e)h(sho)m(w)m(ed)94 4205 y(that)30 b(in)f(the)h(Hop\034eld)g(mo)s(del)e(with)i(\034nitely)f(man)m(y)g (patterns)i(\(i.)16 b(e.,)30 b(with)f Fr(M)41 b Ft(not)30 b(dep)s(ending)94 4324 y(on)41 b Fr(N)10 b Ft(\))41 b(the)g (distribution)d(of)i(the)h(o)m(v)m(erlap\026scaled)g(b)m(y)g(the)g (same)f(factor)g Fr(N)3061 4288 y Fj(1)p Fl(=)p Fj(4)3172 4324 y Ft(\026regarded)94 4440 y(as)35 b(a)f(random)f(v)-5 b(ariable)32 b Fr(Q)1102 4455 y Fl(N)1204 4440 y Ft(taking)h(v)-5 b(alues)34 b(in)g(the)g(P)m(olish)g(space)h Fq(M)2759 4455 y Fj(1)2798 4440 y Fo(\()p Fp(R)2902 4404 y Fl(M)2987 4440 y Fo(\))f Ft(of)g(probabilit)m(y)94 4556 y(measures)29 b(on)e Fp(R)704 4520 y Fl(M)817 4556 y Ft(con)m(v)m(erges)i(w)m(eakly)f (\(with)g(resp)s(ect)g(to)f Fp(P)p Ft(\))j(to)d(a)g(limiting)c(random)j (measure)p 94 4598 78 4 v 94 4678 a Fr(Q)172 4701 y Fl(M)251 4678 y Ft(.)40 b(This)24 b(limiting)19 b(random)j(measure)p 1605 4598 V 24 w Fr(Q)1682 4701 y Fl(M)1784 4678 y Ft(is)h(giv)m(en)h (b)m(y)g(its)f(\(random\))f(densit)m(y)j(with)e(resp)s(ect)94 4794 y(to)33 b(the)g Fr(M)10 b Ft(-dimensional)30 b(Leb)s(esgue)k (measure)e(whic)m(h)i(is)e(prop)s(ortional)d(to)646 5062 y Fo(exp)795 4921 y Fk(\022)868 5062 y Fq(\000)980 4995 y Fo(1)p 955 5039 98 4 v 955 5130 a(12)1114 4937 y Fl(M)1079 4967 y Fk(X)1085 5176 y Fl(\026)p Fj(=1)1240 5062 y Fr(x)1295 5021 y Fj(4)1295 5087 y Fl(\026)1364 5062 y Fq(\000)1474 4995 y Fo(1)p 1474 5039 49 4 v 1474 5130 a(2)1655 4967 y Fk(X)1549 5179 y Fj(1)p Fi(\024)p Fl(\026<\027)t Fi(\024)p Fl(M)1922 5062 y Fr(x)1977 5021 y Fj(2)1977 5087 y Fl(\026)2024 5062 y Fr(x)2079 5021 y Fj(2)2079 5087 y Fl(\027)2145 5062 y Fo(+)2349 4967 y Fk(X)2243 5179 y Fj(1)p Fi(\024)p Fl(\026<\027)t Fi(\024)p Fl(M)2616 5062 y Fr(\021)2664 5077 y Fl(\026;\027)2769 5062 y Fr(x)2824 5077 y Fl(\026)2871 5062 y Fr(x)2926 5077 y Fl(\027)2970 4921 y Fk(\023)3043 5062 y Fr(;)367 b Ft(\(1.6\))94 5345 y(where)36 b Fr(\021)i Ft(is)d(an)f Fr(M)10 b Fo(\()p Fr(M)35 b Fq(\000)24 b Fo(1\))p Fr(=)p Fo(2)p Ft(-dimensional)31 b(Gaussian)j(random)f(v)-5 b(ariable)33 b(with)h(mean)g(zero)94 5461 y(and)f(the)g(co)m(v)-5 b(ariance)32 b(matrix)f(b)s(eing)h(the)h(iden)m(tit)m(y)f(matrix,)g (namely)-8 b(,)1312 5638 y Fo(\006)28 b(=)f(\(\006)1621 5654 y Fj(\()p Fl(\026;\027)t Fj(\))p Fl(;)p Fj(\()p Fl(\026)1865 5635 y Ff(0)1890 5654 y Fl(;\027)1949 5635 y Ff(0)1971 5654 y Fj(\))2002 5638 y Fo(\))2040 5654 y Fj(\()p Fl(\026;\027)t Fj(\))p Fl(;)p Fj(\()p Fl(\026)2284 5635 y Ff(0)2308 5654 y Fl(;\027)2367 5635 y Ff(0)2389 5654 y Fj(\))p eop %%Page: 4 4 4 3 bop 118 -51 a Fm(4)1242 b(B.)34 b(GENTZ)g(AND)e(M.)i(L\326)n(WE)118 123 y Ft(and)1056 288 y Fo(\006)1126 304 y Fj(\()p Fl(\026;\027)t Fj(\))p Fl(;)p Fj(\()p Fl(\026)1370 285 y Ff(0)1394 304 y Fl(;\027)1453 285 y Ff(0)1475 304 y Fj(\))1534 288 y Fo(=)1638 148 y Fk(\032)1754 228 y Fo(1)p Fr(;)83 b Ft(if)31 b Fo(\()p Fr(\026;)17 b(\027)6 b Fo(\))28 b(=)f(\()p Fr(\026)2463 192 y Fi(0)2486 228 y Fr(;)17 b(\027)2584 192 y Fi(0)2608 228 y Fo(\))p Fr(;)1754 345 y Fo(0)p Fr(;)83 b Ft(otherwise,)118 497 y(for)32 b Fo(1)c Fq(\024)g Fr(\026)f(<)h(\027)34 b Fq(\024)28 b Fr(M)43 b Ft(and)33 b Fo(1)27 b Fq(\024)h Fr(\026)1393 461 y Fi(0)1444 497 y Fr(<)g(\027)1602 461 y Fi(0)1653 497 y Fq(\024)g Fr(M)10 b Ft(.)218 613 y(This)26 b(sho)m(ws)h(that)f(ev)m(en)h(for)e(\034nite)h Fr(M)36 b Ft(at)26 b(the)g(critical)d(temp)s(erature)j Fo(1)p Fr(=\014)33 b Fo(=)27 b(1)p Ft(,)g(the)f(\035uctua-)118 730 y(tions)g(of)h(the)g(o)m(v)m(erlap)g(dep)s(end)h(strongly)f(on)f (the)i(random)e(disorder)g(as)i(ev)m(en)g(the)f(distribution)118 846 y(of)35 b(the)h(limiting)31 b(\035uctuations)36 b(is)f(random.)52 b(Ev)m(en)37 b(to)e(form)m(ulate)f(the)i(corresp)s(onding)f(result)118 962 y(for)27 b(the)i(case)g(where)g(the)f(n)m(um)m(b)s(er)g(of)g (patterns)g Fr(M)10 b Fo(\()p Fr(N)g Fo(\))29 b Ft(is)f(actually)e(gro) m(wing)i(with)f Fr(N)38 b Ft(seemed)118 1078 y(to)33 b(b)s(e)g(di\036cult,)g(since,)g(on)h(one)f(hand,)h(w)m(e)g(don't)f(ha) m(v)m(e)i(an)e(\020in\034nite-dimensional)28 b(Leb)s(esgue)118 1195 y(measure\021)40 b(as)33 b(reference)g(measure)g(and,)f(on)g(the)h (other)f(hand,)h(w)m(e)g(cannot)f(w)m(ork)h(with)f(\034nite-)118 1311 y(dimensional)e(pro)5 b(jections)33 b(\(as)f(in)g(the)h(Cen)m (tral)f(Limit)e(Theorem\))i(either,)h(since)g(the)g(\020mixed)118 1427 y(terms\021)430 1352 y Fk(P)535 1456 y Fj(1)p Fi(\024)p Fl(\026<\027)t Fi(\024)p Fl(M)912 1427 y Fr(\021)960 1442 y Fl(\026;\027)1066 1427 y Fr(x)1121 1442 y Fl(\026)1168 1427 y Fr(x)1223 1442 y Fl(\027)1299 1427 y Ft(tend)g(to)f (\020glue\021)39 b(together)33 b(the)g(co)s(ordinates.)218 1546 y(In)25 b(this)f(pap)s(er)h(w)m(e)h(circum)m(v)m(en)m(t)f(these)h (di\036culties)e(b)m(y)h(not)g(stating)f(a)g(limit)d(theorem)k(but)f(b) m(y)118 1662 y(sho)m(wing)29 b(instead)f(that)g(the)h(distance)g(b)s (et)m(w)m(een)h(the)f(distribution)d Fr(Q)2705 1677 y Fl(N)2801 1662 y Ft(of)i(the)h(scaled)f(o)m(v)m(erlap)118 1778 y(and)e(the)g(random)e(measure)p 1185 1698 78 4 v 26 w Fr(Q)1262 1802 y Fl(M)1367 1778 y Ft(b)s(ecomes)i(small)d(with)i (high)g(probabilit)m(y)e(for)i(large)f Fr(N)10 b Ft(.)42 b(More)118 1895 y(precisely)-8 b(,)43 b(w)m(e)e(shall)e(sho)m(w,)44 b(under)d(the)f(constrain)m(t)h Fr(M)2242 1858 y Fj(15)2317 1895 y Fr(=)-5 b(N)51 b Fq(!)41 b Fo(0)f Ft(on)g(the)h(gro)m(wth)g (rate)f(of)118 2011 y Fr(M)10 b Fo(\()p Fr(N)g Fo(\))p Ft(,)40 b(that)e(for)f(eac)m(h)h(large)f(enough)h Fr(N)48 b Ft(there)38 b(exists)g(a)g(set)g(of)f Fr(\030)5 b Ft('s)38 b(of)f(probabilit)m(y)f(larger)118 2127 y(than)g Fo(1)24 b Fq(\000)h Fo(exp)q Fq(f\000)p Fr(M)5 b(=L)p Fq(g)37 b Ft(\(with)f(some)f(constan)m(t)i Fr(L)d(>)g Fo(0)p Ft(\))h(on)h(whic)m(h)h(the)f(distance)g(b)s(et)m(w)m(een)118 2248 y Fr(Q)195 2263 y Fl(N)295 2248 y Ft(and)p 485 2168 V 33 w Fr(Q)562 2272 y Fl(M)674 2248 y Ft(is)c(smaller)e(than)j Fr(")1376 2263 y Fl(N)1470 2248 y Fq(&)28 b Fo(0)p Ft(.)218 2365 y(This)35 b(pap)s(er)f(has)h(three)h(more)e(sections.)50 b(Section)34 b(2)h(con)m(tains)g(the)g(explicit)e(statemen)m(t)i(of)118 2481 y(the)f(result)f(concerning)h(the)g(non-Gaussian)e(\035uctuations) i(of)f(the)g(o)m(v)m(erlap)h(at)f Fr(\014)i Fo(=)29 b(1)k Ft(for)g(the)118 2597 y(Hop\034eld)43 b(mo)s(del)e(with)i(a)g(gro)m (wing)f(n)m(um)m(b)s(er)i(of)e(patterns.)76 b(Section)43 b(3)g(is)g(dev)m(oted)h(to)f(one)118 2713 y(of)c(our)h(basic)f(to)s (ols,)h(a)f(m)m(ultidimensional)c(v)m(ersion)40 b(of)f(a)h(strong)f (appro)m(ximation)e(result)j(of)118 2829 y(K)m(oml\363s,)f(Ma)5 b(jor)40 b(and)f(T)-8 b(usn\341dy)41 b([22)o(],)g(whic)m(h)f(allo)m(ws) e(to)h(con)m(trol)f(the)i(di\033erence)g(of)f(a)f(sum)118 2946 y(of)i(i.i.d.)f(random)g(v)-5 b(ariables)39 b(and)h(a)g(sum)g(of)g (i.i.d.)e(Gaussian)i(random)f(v)-5 b(ariables)39 b(with)h(the)118 3062 y(same)f(co)m(v)-5 b(ariance)39 b(matrix.)63 b(These)41 b(results)f(go)e(bac)m(k)j(to)e(Zaitsev)g([32)o(],)j([33)o(],)f (Einmahl)d([12)o(])118 3178 y(and)43 b(Einmahl)e(and)h(Mason)i([13)o (].)74 b(They)44 b(also)e(pro)m(v)m(ed)i(useful)f(in)f([10].)74 b(Section)42 b(4)h(\034nally)118 3294 y(is)38 b(dev)m(oted)h(to)f(the)g (pro)s(of)f(whic)m(h)i(is)e(based)i(on)f(the)h(Hubbard\025Stratono)m (vic)m(h)f(transform)f(of)118 3411 y(the)30 b(measures)h(of)e(in)m (terest)h(together)g(with)f(a)h(T)-8 b(a)m(ylor)29 b(expansion)h(of)g (the)g(resulting)e(densit)m(y)-8 b(,)32 b(a)118 3527 y(saddle)25 b(p)s(oin)m(t)f(appro)m(ximation)f(as)i(w)m(ell)g(as)g(the) h(strong)f(Gaussian)f(appro)m(ximation)f(men)m(tioned)118 3643 y(b)s(efore.)118 3813 y FA(A)m(c)m(kno)m(wledgemen)m(t)p Ft(.)94 b(W)-8 b(e)50 b(are)g(grateful)e(to)i(An)m(ton)g(Bo)m(vier)g (for)f(bringing)f(the)i(strong)118 3929 y(Gaussian)45 b(appro)m(ximation)f(to)h(our)g(atten)m(tion,)k(and,)g(in)c (particular,)i(for)e(sharing)h(the)g(re-)118 4045 y(sults)33 b(of)g([10)o(])g(with)g(us)h(prior)d(to)i(publication.)43 b(W)-8 b(e)33 b(b)s(ene\034ted)h(from)e(in)m(teresting)h(discussions) 118 4161 y(with)39 b(him.)63 b(The)40 b(results)g(presen)m(ted)i(here)e (w)m(ere)h(obtained)e(while)g(the)h(second)g(author)f(w)m(as)118 4278 y(visiting)31 b(at)h(the)h Fe(Wias)p Ft(.)43 b(He)33 b(thanks)h(the)f Fe(Wias)f Ft(for)g(its)g(hospitalit)m(y)-8 b(.)1269 4512 y(2.)53 b Fs(St)-8 b(a)g(tement)34 b(of)i(Resul)-8 b(ts)218 4686 y Ft(This)52 b(section)g(con)m(tains)g(the)h (mathematically)48 b(precise)53 b(statemen)m(t)f(of)g(the)g(result)g (an-)118 4802 y(nounced)h(in)d(the)i(in)m(tro)s(duction.)99 b(W)-8 b(e)51 b(shall)f(state)i(the)g(theorem)f(only)g(for)g(the)h (case)g(of)118 4918 y Fr(\014)34 b Fo(=)28 b Fr(\014)366 4933 y Fj(c)430 4918 y Fo(=)h(1)k Ft(b)s(eing)f(\034xed.)46 b(In)33 b([20],)g(where)h(w)m(e)g(considered)g Fr(M)44 b Ft(indep)s(enden)m(t)34 b(of)e Fr(N)44 b Ft(only)-8 b(,)32 b(w)m(e)118 5035 y(also)j(treated)g(the)h(case)g(of)f(v)-5 b(ariable)34 b(temp)s(erature)h Fr(\014)2131 5050 y Fl(N)2234 5035 y Ft(con)m(v)m(erging)g(to)g Fr(\014)2898 5050 y Fj(c)2967 5035 y Fo(=)d(1)j Ft(as)h Fr(N)43 b Fq(!)32 b(1)p Ft(.)118 5155 y(It)40 b(turned)g(out)g(that)g(for)f Fr(\014)1163 5170 y Fl(N)1270 5155 y Ft(con)m(v)m(erging)h(to)f Fr(\014)1943 5170 y Fj(c)2019 5155 y Ft(faster)h(than)f Fo(1)p Fr(=)2627 5070 y Fq(p)p 2710 5070 89 4 v 85 x Fr(N)50 b Ft(\(recall)38 b(that)i Fr(M)50 b Ft(w)m(as)118 5271 y(c)m(hosen)39 b(as)e(a)f(constan)m(t\),)j(the)e(limiting)c (distribution)h(is)j(the)g(same,)h(while)e(for)g Fr(\014)3202 5286 y Fl(N)3307 5271 y Ft(con)m(v)m(erg-)118 5396 y(ing)f(to)h Fr(\014)462 5411 y Fj(c)533 5396 y Ft(slo)m(w)m(er)g(than)g Fo(1)p Fr(=)1158 5312 y Fq(p)p 1241 5312 V 84 x Fr(N)10 b Ft(,)37 b(w)m(e)g(ha)m(v)m(e)g(a)f(Cen)m(tral-Limit-Theorem)c(t)m(yp) s(e)37 b(result)f(and)g(at)118 5522 y(\020the)e(b)s(orderline\021,)e (i.)16 b(e.,)34 b(when)h Fr(\014)1357 5537 y Fl(N)1448 5522 y Fq(\000)23 b Fr(\014)1603 5537 y Fj(c)1673 5522 y Ft(is)33 b(of)g(the)i(same)e(order)h(as)g Fo(1)p 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y(Lipschitz)36 b(c)-5 b(onstant)36 b(of)g Fr(f)47 b Fc(and)36 b Fq(k)p Fr(f)11 b Fq(k)1700 1990 y Fi(1)1804 1975 y Fo(=)31 b(sup)2057 1998 y Fl(x)p Fi(2)p Fd(R)2192 1979 y Fh(M)2280 1975 y Fq(j)p Fr(f)11 b Fo(\()p Fr(x)p Fo(\))p Fq(j)p Fc(.)49 b(This)35 b(implies)h(the)g(the)-5 b(or)g(em)335 2091 y(by)44 b Ft(\(4.2\))34 b Fc(b)-5 b(elow.)1008 2302 y Ft(3.)53 b Fs(Str)m(ong)35 b(Ga)m(ussian)h(Appr)m (o)m(xima)-8 b(tion)218 2477 y Ft(In)26 b(this)g(section)g(w)m(e)i(are) e(going)f(to)h(collect)f(some)h(facts)g(ab)s(out)g(the)h(so-called)d (strong)j(Gauss-)118 2593 y(ian)39 b(appro)m(ximation)g(and)h(apply)g (them)g(to)g(the)h(situation)e(of)h(our)g(in)m(terest.)67 b(The)42 b(problem)118 2709 y(of)34 b(the)g(Gaussian)g(appro)m (ximation)d(is)j(quic)m(kly)g(stated.)49 b(Giv)m(en)34 b(a)f(sequence)k Fo(\()p Fr(X)3136 2724 y Fl(i)3164 2709 y Fo(\))3202 2724 y Fl(i)p Fi(2)p Fd(N)3359 2709 y Ft(of)d(i.i.d.)118 2825 y(random)i(v)m(ectors)i(in)e Fp(R)999 2789 y Fl(d)1045 2825 y 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Fr(s;)k(X)2807 5585 y Fl(i)2840 5570 y Fq(i)2879 5529 y Fj(2)3461 5570 y Ft(\(3.4\))p eop %%Page: 7 7 7 6 bop 317 -67 a Fm(FLUCTUA)-6 b(TIONS)32 b(IN)g(THE)i(HOPFIELD)e (MODEL)i(A)-6 b(T)33 b(THE)g(CRITICAL)g(TEMPERA)-6 b(TURE)184 b(7)94 123 y Ft(with)33 b(some)f Fr(\034)44 b Ft(for)32 b(all)e Fr(m)e Fq(\025)h Fo(3)j Ft(and)h(all)d Fr(s;)17 b(t)28 b Fq(2)g Fp(R)1869 87 y Fl(d)1915 123 y Ft(.)194 239 y(Under)e(the)h(condition)d(\(3.4\))o(,)j(Zaitsev)e(pro)m(v)m(ed)j (in)c([32,)j(Theorem1.1])f(the)g(follo)m(wing)d(b)s(ound)94 355 y(on)40 b Fr(\025)p Fo(\()p Fr(P)395 370 y Fj(1)p Fl(;n)497 355 y Fr(;)17 b(P)604 370 y Fj(2)p Fl(;n)705 355 y Fr(;)g(\016)t Fo(\))p Ft(,)41 b(where)g Fr(P)1254 370 y Fj(1)p Fl(;n)1396 355 y Ft(is)e(the)h(distribution)e(of)h Fr(X)2414 370 y Fj(1)2481 355 y Fo(+)27 b Fr(:)17 b(:)g(:)26 b Fo(+)h Fr(X)2909 370 y Fl(n)2996 355 y Ft(and)40 b Fr(P)3256 370 y Fj(2)p Fl(;n)3397 355 y Ft(is)f(the)94 471 y Fr(d)p Ft(-dimensional)26 b(normal)h(distribution)f(with)j(mean)f 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y(of)45 b(notation,)h(w)m(e)g (denote)g(the)f(iden)m(tit)m(y)f(matrix)f(b)m(y)j Fo(Id)f Ft(whatev)m(er)i(the)e(dimension)e(of)h(the)94 4867 y(underlying)i (space)g Fp(R)932 4831 y Fl(d)1024 4867 y Ft(is.\))83 b(In)46 b(order)g(to)f(apply)h(Corollary)e(3.3,)k(w)m(e)f(ha)m(v)m(e)g (to)e(c)m(hec)m(k)j(the)94 4983 y(Bernstein-t)m(yp)s(e)34 b(condition)d(\(3.4\))o(.)44 b(This)33 b(is)f(done)h(in)e(the)i(follo)m (wing)d(lemma.)94 5166 y FA(Lemma)38 b(3.4.)49 b Fc(In)d(the)i(ab)-5 b(ove)46 b(setting)i Fr(X)1713 5181 y Fj(1)1752 5166 y Fr(;)17 b(:)g(:)g(:)f(;)h(X)2052 5181 y Fl(n)2146 5166 y Fc(ful\034l)5 b(l)47 b(the)g(Bernstein-typ)-5 b(e)47 b(c)-5 b(ondition)94 5282 y(\(3.4\))35 b(with)f Fr(\034)39 b Fo(=)28 b Fr(M)10 b Fc(.)94 5466 y(Pr)-5 b(o)g(of.)42 b Ft(By)33 b(Sc)m(h)m(w)m(arz')h(inequalit)m(y)-8 b(,)1211 5638 y Fq(jh)p Fr(t;)17 b(X)1438 5653 y Fl(i)1466 5638 y Fq(ij)27 b(\024)h(k)p Fr(t)p Fq(k)1800 5653 y Fj(2)1839 5638 y Fq(k)p Fr(X)1970 5653 y Fl(i)1998 5638 y Fq(k)2048 5653 y Fj(2)2115 5638 y Fq(\024)g(k)p Fr(t)p Fq(k)2355 5653 y Fj(2)2395 5638 y Fr(M)5 b(:)p eop %%Page: 8 8 8 7 bop 118 -51 a Fm(8)1242 b(B.)34 b(GENTZ)g(AND)e(M.)i(L\326)n(WE)118 123 y Ft(Th)m(us,)g(for)e(an)m(y)h(c)m(hoice)g(of)f Fr(s;)17 b(t)28 b Fq(2)g Fp(R)1439 87 y Fl(M)1557 123 y Ft(and)k(all)f Fr(m)d Fq(\025)g Fo(3)333 330 y Fq(j)p Fp(E)12 b Fq(h)p Fr(s;)17 b(X)631 345 y Fl(i)665 330 y Fq(i)704 289 y Fj(2)743 330 y Fq(h)p Fr(t;)g(X)942 345 y Fl(i)970 330 y Fq(i)1009 289 y Fl(m)p Fi(\000)p Fj(2)1165 330 y Fq(j)28 b(\024)g Fr(\034)1379 289 y Fl(m)p Fi(\000)p Fj(2)1536 330 y Fq(k)p Fr(t)p Fq(k)1671 289 y Fl(m)p Fi(\000)p Fj(2)1671 354 y(2)1828 330 y Fr(E)6 b Fq(h)p Fr(s;)17 b(X)2116 345 y Fl(i)2143 330 y Fq(i)2182 289 y Fj(2)2249 330 y Fq(\024)2364 262 y Fo(1)p 2364 307 49 4 v 2364 398 a(2)2423 330 y Fr(m)p Fo(!)p Fr(\034)2588 289 y Fl(m)p Fi(\000)p Fj(2)2746 330 y Fq(k)p Fr(t)p Fq(k)2881 289 y Fl(m)p Fi(\000)p Fj(2)2881 354 y(2)3037 330 y Fp(E)c Fq(h)p Fr(s;)k(X)3308 345 y Fl(i)3341 330 y Fq(i)3380 289 y Fj(2)3419 330 y Fr(;)118 523 y Ft(where)34 b(w)m(e)f(ha)m(v)m(e)h (already)e(c)m(hosen)i Fr(\034)40 b Fo(=)27 b Fr(M)10 b Ft(.)p 3591 523 4 66 v 3595 460 59 4 v 3595 523 V 3653 523 4 66 v 218 697 a(No)m(w)33 b(w)m(e)g(are)g(ready)g(to)f(deduce)i (the)f(desired)g(appro)m(ximation.)118 876 y FA(Corollary)k(3.5.)49 b Fc(If)43 b Fo(\(\012)p Fr(;)17 b Fq(F)10 b Fr(;)17 b Fp(P)p Fo(\))45 b Fc(is)f(rich)f(enough,)i(for)e(e)-5 b(ach)43 b Fr(N)54 b Fc(and)43 b Fr(\016)48 b Fq(\025)c Fo(0)p Fc(,)i(ther)-5 b(e)43 b(exist)h(a)118 992 y(me)-5 b(an)44 b(zer)-5 b(o)45 b(Gaussian)g(r)-5 b(andom)44 b(variable)g Fr(\021)k Fc(with)d(c)-5 b(ovarianc)g(e)44 b(matrix)g Fo(Id)i Fc(and)e(numeric)-5 b(al)118 1108 y(c)g(onstants)34 b Fr(c)588 1123 y Fj(1)628 1108 y Fr(;)17 b(c)714 1123 y Fj(2)781 1108 y Fr(>)27 b Fo(0)p Fc(,)35 b(such)f(that)774 1363 y Fr(P)851 1223 y Fk(\032)926 1219 y(\015)926 1279 y(\015)926 1338 y(\015)926 1398 y(\015)1052 1296 y Fo(1)p 991 1340 172 4 v 991 1360 a Fq(p)p 1074 1360 89 4 v 85 x Fr(N)1229 1239 y Fl(N)1189 1269 y Fk(X)1204 1479 y Fl(i)p Fj(=1)1350 1363 y Fr(X)1431 1378 y Fl(i)1481 1363 y Fq(\000)22 b Fr(\021)1632 1219 y Fk(\015)1632 1279 y(\015)1632 1338 y(\015)1632 1398 y(\015)1687 1462 y Fj(2)1755 1363 y Fq(\025)28 b Fr(\016)1907 1223 y Fk(\033)2009 1363 y Fq(\024)g Fr(c)2156 1378 y Fj(1)2196 1363 y Fr(M)2300 1322 y Fj(5)2356 1363 y Fo(exp)2506 1223 y Fk(\032)2580 1363 y Fq(\000)2671 1296 y Fr(\016)2718 1212 y Fq(p)p 2801 1212 V 84 x Fr(N)p 2667 1340 226 4 v 2667 1432 a(c)2709 1447 y Fj(2)2749 1432 y Fr(M)2853 1403 y Fj(6)2903 1223 y Fk(\033)2978 1363 y Fr(:)456 b Ft(\(3.8\))118 1628 y Fc(Pr)-5 b(o)g(of.)41 b Ft(Apply)33 b(Lemma)e(3.4)h(and)h(Corollary)d(3.3)i(with)h Fr(\034)39 b Fo(=)27 b Fr(M)10 b Ft(.)p 3591 1628 4 66 v 3595 1565 59 4 v 3595 1628 V 3653 1628 4 66 v 118 1806 a FA(Remark)37 b(3.6.)49 b Fc(Observe)37 b(that)i Fr(\016)j Fc(in)c(\(3.8\))g (may\026and)f(wil)5 b(l)38 b(inde)-5 b(e)g(d)37 b(in)h(our)g(applic)-5 b(ations\026)118 1922 y(dep)g(end)34 b(on)g Fr(N)46 b Fc(and)34 b Fr(M)10 b Fc(.)1647 2140 y Ft(4.)53 b Fs(Pr)m(oofs)218 2314 y Ft(T)-8 b(o)29 b(pro)m(v)m(e)h(Theorem)g(2.1,)f(w)m(e)h(need)h (to)d(sho)m(w)j(that)e(for)f(large)g(system)i(size)g Fr(N)39 b Ft(the)30 b(distribu-)118 2431 y(tion)i Fr(Q)396 2446 y Fl(N)497 2431 y Ft(of)g(the)i(scaled)f(o)m(v)m(erlap)g(under)h (the)g(Gibbs)e(measure)i Fr(\045)2564 2446 y Fl(N)s(;)p Fj(1)2715 2431 y Ft(is)f(close)g(to)g(the)g(random)118 2555 y(measure)p 493 2475 78 4 v 28 w Fr(Q)570 2578 y Fl(M)677 2555 y Ft(with)27 b(resp)s(ect)i(to)e(the)h(metric)f Fr(d)g Ft(on)h(a)f(set)i(of)e(large)f Fp(P)p Ft(-measure.)44 b(First)27 b(w)m(e)h(sho)m(w)118 2671 y(that)35 b Fr(Q)409 2686 y Fl(N)512 2671 y Ft(and)g(its)g(smo)s(othed)f(v)m(ersion)i (obtained)f(b)m(y)h(a)f(Hubbard\025Stratono)m(vic)m(h)g(transform)118 2787 y(are)25 b(close,)i(so)f(that)f(w)m(e)h(ma)m(y)f(in)m(v)m (estigate)g(the)h(Hubbard\025Stratono)m(vic)m(h)f(transform)g(instead)g (of)118 2903 y(the)32 b(measure)f(itself.)41 b(W)-8 b(e)32 b(recall)d(the)j(Hubbard\025Stratono)m(vic)m(h)f(transform)f(of)g Fr(Q)3155 2918 y Fl(N)3254 2903 y Ft(from)g([20)o(].)118 3020 y(The)25 b(core)f(of)f(the)h(pro)s(of)f(is)g(the)h(in)m(v)m (estigation)e(of)h(the)h(densit)m(y)h(of)e(this)g(Hubbard\025Stratono)m (vic)m(h)118 3136 y(transform)32 b(b)m(y)h(an)g(adaptation)e(of)h (Laplace's)g(metho)s(d.)118 3314 y FA(Notation)k(4.1.)49 b Fc(W)-7 b(e)35 b(denote)f(by)h Fr(\026)22 b Fq(\003)g Fr(\027)41 b Fc(the)35 b(c)-5 b(onvolution)34 b(of)g(two)h(me)-5 b(asur)g(es)34 b Fr(\026)h Fc(and)f Fr(\027)6 b Fc(.)118 3495 y FA(Lemma)37 b(4.2.)49 b Fc(F)-7 b(or)33 b(al)5 b(l)34 b Fr(M)k Fq(\025)28 b Fo(8)p Fc(,)34 b(al)5 b(l)34 b Fr(f)k Fq(2)28 b Fo(BL\()p Fp(R)1961 3459 y Fl(M)2046 3495 y Fr(;)17 b Fp(R)5 b Fo(\))40 b Fc(and)33 b(al)5 b(l)34 b(pr)-5 b(ob)g(ability)33 b(me)-5 b(asur)g(es)3468 3470 y Fo(~)3446 3495 y Fr(Q)34 b Fc(on)118 3612 y Fp(R)184 3575 y Fl(M)269 3612 y Fc(,)406 3687 y Fk(\014)406 3747 y(\014)406 3807 y(\014)406 3866 y(\014)440 3696 y(Z)556 3831 y Fr(f)27 b(d)p Fo(\()742 3806 y(~)720 3831 y Fr(Q)22 b Fq(\003)g(N)15 b Fo(\(0)p Fr(;)i(N)1207 3790 y Fi(\000)p Fj(1)p Fl(=)p Fj(2)1371 3831 y Fo(Id)q(\)\))22 b Fq(\000)1658 3696 y Fk(Z)1775 3831 y Fr(f)27 b(d)1923 3806 y Fo(~)1901 3831 y Fr(Q)1978 3687 y Fk(\014)1978 3747 y(\014)1978 3807 y(\014)1978 3866 y(\014)2039 3831 y Fq(\024)h Fo(2)2193 3744 y Fq(p)p 2276 3744 49 4 v 87 x Fo(2)o Fr(K)2407 3846 y Fl(f)2453 3755 y Fq(p)p 2536 3755 63 4 v 76 x Fr(\013)23 b Fo(+)f Fq(k)p Fr(f)11 b Fq(k)2878 3846 y Fi(1)2952 3831 y Fo(e)2995 3790 y Fi(\000)p Fl(M)3129 3831 y Fr(;)305 b Ft(\(4.1\))118 4057 y Fc(wher)-5 b(e)36 b Fr(K)478 4072 y Fl(f)560 4057 y Fc(denotes)g(again)g(the)g(Lipschitz) g(c)-5 b(onstant)36 b(of)h Fr(f)47 b Fc(and)36 b Fq(k)p Fr(f)11 b Fq(k)2708 4072 y Fi(1)2813 4057 y Fo(=)31 b(sup)3067 4080 y Fl(x)p Fi(2)p Fd(R)3202 4061 y Fh(M)3290 4057 y Fq(j)p Fr(f)11 b Fo(\()p Fr(x)p Fo(\))p Fq(j)36 b Fc(as)118 4173 y(b)-5 b(efor)g(e.)218 4351 y Ft(No)m(w,)871 4571 y Fr(d)p Fo(\()p Fr(P)1023 4586 y Fj(1)1062 4571 y Fr(;)17 b(P)1169 4586 y Fj(2)1208 4571 y Fo(\))28 b(=)f(sup)1524 4431 y Fk(\032)1599 4427 y(\014)1599 4487 y(\014)1599 4546 y(\014)1599 4606 y(\014)1632 4436 y(Z)1748 4571 y Fr(f)11 b(dQ)1935 4586 y Fj(1)1997 4571 y Fq(\000)2096 4436 y Fk(Z)2213 4571 y Fr(f)g(dQ)2400 4586 y Fj(2)2439 4427 y Fk(\014)2439 4487 y(\014)2439 4546 y(\014)2439 4606 y(\014)2500 4571 y Fo(:)27 b Fr(f)39 b Fq(2)28 b(G)2794 4586 y Fj(0)2833 4431 y Fk(\033)3461 4571 y Ft(\(4.2\))118 4791 y(with)1385 4954 y Fq(G)1444 4969 y Fj(0)1511 4954 y Fo(=)g Fq(G)g(\\)23 b(f)p Fr(f)38 b Fo(:)28 b Fr(f)11 b Fo(\(0\))27 b(=)g(0)p Fq(g)1066 b Ft(\(4.3\))118 5131 y(and)43 b Fq(G)377 5146 y Fj(0)463 5131 y Fq(\032)j Fo(BL\()p Fp(R)820 5095 y Fl(M)905 5131 y Fr(;)17 b Fp(R)5 b Fo(\))p Ft(.)81 b(Therefore,)47 b(the)d(follo)m(wing)c(corollary)i (is)g(an)h(immediate)e(conse-)118 5247 y(quence)34 b(of)f(the)g (preceding)f(lemma.)118 5428 y FA(Corollary)37 b(4.3.)49 b Fc(F)-7 b(or)34 b(al)5 b(l)35 b Fr(M)j Fq(\025)28 b Fo(8)35 b Fc(and)f(al)5 b(l)35 b(pr)-5 b(ob)g(ability)34 b(me)-5 b(asur)g(es)2721 5403 y Fo(~)2698 5428 y Fr(Q)35 b Fc(on)g Fp(R)3016 5392 y Fl(M)3101 5428 y Fc(,)1026 5599 y Fr(d)p Fo(\()1137 5574 y(~)1115 5599 y Fr(Q)22 b Fq(\003)g(N)15 b Fo(\(0)p Fr(;)i(N)1602 5558 y Fi(\000)p Fj(1)p Fl(=)p Fj(2)1766 5599 y Fo(Id)q(\))p Fr(;)1960 5574 y Fo(~)1938 5599 y Fr(Q)p Fo(\))27 b Fq(\024)i Fo(2)2235 5512 y Fq(p)p 2317 5512 112 4 v 2317 5599 a Fo(2)p Fr(\013)23 b Fo(+)f(e)2592 5558 y Fi(\000)p Fl(M)2726 5599 y Fr(:)708 b Ft(\(4.4\))p eop %%Page: 9 9 9 8 bop 317 -67 a Fm(FLUCTUA)-6 b(TIONS)32 b(IN)g(THE)i(HOPFIELD)e (MODEL)i(A)-6 b(T)33 b(THE)g(CRITICAL)g(TEMPERA)-6 b(TURE)184 b(9)94 131 y Fc(Pr)-5 b(o)g(of)35 b(of)g(L)-5 b(emma)34 b(4.2.)40 b Ft(Let)48 b Fr(f)64 b Fq(2)54 b Fo(BL\()p Fp(R)1672 95 y Fl(M)1757 131 y Fr(;)17 b Fp(R)5 b Fo(\))53 b Ft(and)48 b(let)2341 106 y Fo(~)2319 131 y Fr(Q)g Ft(b)s(e)g(an)f (arbitrary)g(probabilit)m(y)94 247 y(measure)33 b(on)g Fp(R)676 211 y Fl(M)761 247 y Ft(.)43 b(Then,)34 b(for)e Fr(\016)g(>)27 b Fo(0)p Ft(,)517 336 y Fk(\014)517 396 y(\014)517 456 y(\014)517 516 y(\014)550 345 y(Z)667 481 y Fr(f)g(d)p Fo(\()853 456 y(~)831 481 y Fr(Q)22 b Fq(\003)g(N)15 b Fo(\(0)p Fr(;)i(N)1318 440 y Fi(\000)p Fj(1)p Fl(=)p Fj(2)1482 481 y Fo(Id)q(\)\))22 b Fq(\000)1769 345 y Fk(Z)1885 481 y Fr(f)28 b(d)2034 456 y Fo(~)2012 481 y Fr(Q)2089 336 y Fk(\014)2089 396 y(\014)2089 456 y(\014)2089 516 y(\014)907 748 y Fq(\024)1012 613 y Fk(Z)12 b(Z)1195 748 y Fo(1)1244 764 y Fl(B)s Fj(\(0)p Fl(;\016)r Fj(\))1448 748 y Fo(\()p Fr(x)p Fo(\))p Fq(j)p Fr(f)f Fo(\()p Fr(x)22 b Fo(+)g Fr(y)t Fo(\))f Fq(\000)i Fr(f)11 b Fo(\()p Fr(y)t Fo(\))p Fq(j)2342 723 y Fo(~)2322 748 y Fr(Q)n Fo(\()p Fr(dy)t Fo(\))17 b Fq(N)e Fo(\(0)p Fr(;)i(N)2909 707 y Fi(\000)p Fj(1)p Fl(=)p Fj(2)3072 748 y Fo(Id\)\()p Fr(dx)p Fo(\))1062 1039 y(+)22 b(2)p Fq(k)p Fr(f)11 b Fq(k)1368 1054 y Fi(1)1442 899 y Fk(\022)1525 888 y Fq(p)p 1608 888 89 4 v 84 x Fr(N)p 1525 1017 172 4 v 1557 1108 a Fo(2)p Fr(\031)1707 899 y Fk(\023)1780 921 y Fl(M)s(=)p Fj(2)1942 904 y Fk(Z)2059 1039 y Fo(1)2108 1055 y Fl(B)s Fj(\(0)p Fl(;\016)r Fj(\))2306 1036 y Fg(c)2343 1039 y Fo(\()p Fr(x)p Fo(\))17 b(exp)2640 929 y Fk(n)2706 1039 y Fq(\000)2793 972 y Fr(N)p 2793 1017 89 4 v 2814 1108 a Fo(2)2892 1039 y Fq(k)p Fr(x)p Fq(k)3047 998 y Fj(2)3047 1064 y(2)3087 929 y Fk(o)3170 1039 y Fr(dx)907 1262 y Fq(\024)28 b Fr(K)1095 1277 y Fl(f)1141 1262 y Fr(\016)e Fo(+)c(2)p Fq(k)p Fr(f)11 b Fq(k)1516 1277 y Fi(1)1590 1262 y Fr(\015)1641 1277 y Fl(M)1719 1262 y Fo(\()p Fr(B)5 b Fo(\(0)p Fr(;)17 b(\016)t(N)2102 1221 y Fj(1)p Fl(=)p Fj(2)2212 1262 y Fo(\))2250 1221 y Fj(c)2286 1262 y Fo(\))p Fr(;)1086 b Ft(\(4.5\))94 1440 y(where)29 b Fr(\015)422 1455 y Fl(M)528 1440 y Ft(denotes)f(the)g Fr(M)10 b Ft(-dimensional)24 b(Gaussian)j(measure)g(with)g(mean)g(zero) g(and)h(the)f(co-)94 1556 y(v)-5 b(ariance)26 b(matrix)e(b)s(eing)h (the)i(iden)m(tit)m(y)f(matrix.)39 b(The)27 b(radius)f Fr(\034)2427 1571 y Fl(M)2532 1556 y Ft(satisfying)f Fr(\015)3011 1571 y Fl(M)3090 1556 y Fo(\()p Fr(B)5 b Fo(\(0)p Fr(;)17 b(\034)3380 1571 y Fl(M)3458 1556 y Fo(\)\))28 b(=)94 1682 y(1)p Fr(=)p Fo(2)k Ft(is)g(b)s(ounded)i(b)m(y) 905 1598 y Fq(p)p 988 1598 154 4 v 84 x Fo(2)p Fr(M)43 b Ft(for)32 b Fr(M)39 b Fq(\025)28 b Fo(8)p Ft(,)k(cf.)h([18,)f (Equation)g(\(4.4\)].)43 b(Cho)s(osing)32 b Fr(\016)g Fo(=)27 b(2)3358 1599 y Fq(p)p 3441 1599 112 4 v 83 x Fo(2)p Fr(\013)q Ft(,)681 1908 y Fr(\015)732 1923 y Fl(M)811 1908 y Fo(\()p Fr(B)5 b Fo(\(0)p Fr(;)17 b(\016)t(N)1194 1867 y Fj(1)p Fl(=)p Fj(2)1304 1908 y Fo(\))1342 1867 y Fj(c)1377 1908 y Fo(\))28 b Fq(\024)1558 1841 y Fo(1)p 1558 1885 49 4 v 1558 1976 a(2)1633 1908 y(exp)1782 1798 y Fk(n)1849 1908 y Fq(\000)1936 1841 y Fo(1)p 1936 1885 V 1936 1976 a(2)1995 1827 y Fk(\002)2036 1908 y Fr(N)2124 1867 y Fj(1)p Fl(=)p Fj(2)2235 1908 y Fr(\016)e Fq(\000)d Fr(\034)2446 1923 y Fl(M)2525 1827 y Fk(\003)2566 1850 y Fj(2)2606 1798 y Fk(o)2700 1908 y Fq(\024)2815 1841 y Fo(1)p 2815 1885 V 2815 1976 a(2)2874 1908 y(e)2917 1867 y Fi(\000)p Fl(M)3437 1908 y Ft(\(4.6\))94 2121 y(follo)m(ws)32 b(b)m(y)h([24,)f(Theorem)h(1.2].)43 b(This)33 b(concludes)g(the)g(pro)s(of.)p 3567 2121 4 66 v 3571 2058 59 4 v 3571 2121 V 3629 2121 4 66 v 194 2312 a(The)i (Hubbard\025Stratono)m(vic)m(h)g(transform)e(of)h(the)h(distribution)d (of)i(the)g(scaled)h(o)m(v)m(erlap)f(is)94 2429 y(giv)m(en)f(b)m(y)h (its)e(densit)m(y)h(with)f(resp)s(ect)i(to)e(Leb)s(esgue)i(measure.)94 2615 y FA(Lemma)k(4.4.)49 b Fc(L)-5 b(et)35 b Fo(0)27 b Fr(<)h(\014)33 b(<)28 b Fq(1)34 b Fc(and)g Fr(a)28 b(>)g Fo(0)p Fc(.)44 b(Then)34 b(the)h(c)-5 b(onvolution)1290 2814 y Fr(\037)1351 2829 y Fl(N)s(;\014)s(;a)1562 2814 y Fo(=)27 b Fr(Q)1742 2829 y Fl(N)1832 2814 y Fq(\003)22 b(N)2001 2734 y Fk(\000)2046 2814 y Fo(0)p Fr(;)2197 2747 y(a)p 2148 2791 150 4 v 2148 2883 a(N)10 b(\014)2307 2814 y Fo(Id)2397 2734 y Fk(\001)3437 2814 y Ft(\(4.7\))94 3052 y Fc(of)39 b Fr(Q)290 3067 y Fl(N)392 3052 y Fo(=)c Fr(\045)553 3067 y Fl(N)s(;\014)679 3052 y Fo(\()717 2980 y Fq(p)p 800 2980 52 4 v 72 x Fr(am)936 3067 y Fl(N)1004 3052 y Fo(\))1042 3015 y Fi(\000)p Fj(1)1175 3052 y Fc(with)j(the)h Fr(M)10 b Fc(-dimensional)37 b(Gaussian)i(distribution)f(with)h(me)-5 b(an)94 3169 y(zer)g(o)39 b(and)f(c)-5 b(ovarianc)g(e)38 b(matrix)1342 3130 y Fl(a)p 1308 3146 107 4 v 1308 3204 a(N)7 b(\014)1424 3169 y Fo(Id)39 b Fc(is)g(the)g(r)-5 b(andom)38 b(me)-5 b(asur)g(e)38 b(on)h Fp(R)2777 3133 y Fl(M)2901 3169 y Fc(which)f(is)h(given)f(by)94 3299 y(the)d(\(r)-5 b(andom\))34 b(density)652 3541 y Fr(f)700 3556 y Fl(N)s(;\014)s(;a)883 3541 y Fo(\()p Fr(x)p Fo(\))28 b(=)1308 3474 y(exp)q Fq(f\000)p Fr(N)10 b(\014)c Fo(\010)1803 3489 y Fl(N)s(;\014)1930 3474 y Fo(\()p Fr(x=)2072 3402 y Fq(p)p 2155 3402 52 4 v 72 x Fr(a)q Fo(\))p Fq(g)p 1156 3518 1292 4 v 1156 3530 a Fk(R)1203 3645 y Fd(R)1251 3626 y Fh(M)1339 3611 y Fo(exp)q Fq(f\000)p Fr(N)k(\014)c Fo(\010)1834 3626 y Fl(N)s(;\014)1960 3611 y Fo(\()p Fr(x=)2102 3539 y Fq(p)p 2186 3539 52 4 v 2186 3611 a Fr(a)p Fo(\))p Fq(g)17 b Fr(dx)2466 3541 y(;)216 b(x)28 b Fq(2)g Fp(R)2952 3500 y Fl(M)3037 3541 y Fr(;)373 b Ft(\(4.8\))94 3784 y Fc(with)35 b(r)-5 b(esp)g(e)g(ct)35 b(to)g(the)g Fr(M)10 b Fc(-dimensional)33 b(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e,)34 b(wher)-5 b(e)608 4053 y Fo(\010)678 4068 y Fl(N)s(;\014)804 4053 y Fo(\()p Fr(x)p Fo(\))28 b(=)1077 3985 y(1)p 1077 4030 49 4 v 1077 4121 a(2)1136 4053 y Fq(k)p Fr(x)p Fq(k)1291 4011 y Fj(2)1291 4077 y(2)1352 4053 y Fq(\000)1512 3985 y Fo(1)p 1462 4030 150 4 v 1462 4121 a Fr(\014)6 b(N)1678 3928 y Fl(N)1638 3958 y Fk(X)1652 4168 y Fl(i)p Fj(=1)1798 4053 y Fo(log)17 b(cosh)q(\()p Fr(\014)6 b Fq(h)p Fr(x;)17 b(\030)2406 4068 y Fl(i)2433 4053 y Fq(i)p Fo(\))p Fr(;)215 b(x)28 b Fq(2)h Fp(R)2995 4011 y Fl(M)3080 4053 y Fr(;)330 b Ft(\(4.9\))94 4329 y Fc(dep)-5 b(ends)34 b(on)g(the)h(r)-5 b(andom)34 b(p)-5 b(atterns.)45 b(Her)-5 b(e)35 b Fq(h\001)p Fr(;)17 b Fq(\001i)33 b Fc(stands)h(for)h(the)g(inner)f(pr)-5 b(o)g(duct)35 b(in)g Fp(R)3382 4293 y Fl(M)3467 4329 y Fc(.)194 4515 y Ft(W)-8 b(e)40 b(omit)d(the)i(pro)s(of)f(as)i(it)e (follo)m(ws)f(b)m(y)j(a)f(straigh)m(t-forw)m(ard)f(calculation)f (similar)f(to)i(the)94 4632 y(ones)c(giv)m(en)e(in)g([7,)h(Lemma)e (2.2])h(or)g([15,)g(Lemma)f(3.3].)194 4748 y(Before)h(turning)f(to)h (the)g(pro)s(of)f(of)h(Theorem)g(2.1,)g(w)m(e)h(gather)e(some)h (estimates)g(whic)m(h)g(will)94 4864 y(pro)m(v)m(e)38 b(useful)d(in)g(the)i(sequel.)54 b(The)37 b(\034rst)f(of)f(these)i (estimates)f(is)f(a)h(b)s(ound)g(on)f(the)h(op)s(erator)94 4980 y(norm)c(of)g(the)h(random)f(matrix)f(arising)g(from)g(the)i (patterns.)94 5166 y FA(Lemma)38 b(4.5)f Ft(\([6,)32 b(Theorem)h(4.1]\))p FA(.)49 b Fc(Ther)-5 b(e)29 b(exist)i(a)f(c)-5 b(onstant)30 b Fr(K)35 b(>)27 b Fo(0)j Fc(and)g(an)g Fr(N)3187 5181 y Fj(1)3255 5166 y Fq(2)e Fp(N)45 b Fc(such)94 5283 y(that)866 5460 y Fp(P)925 5380 y Fk(\010)985 5376 y(\014)985 5435 y(\014)1018 5460 y Fq(k)1092 5421 y Fj(1)p 1078 5437 64 4 v 1078 5495 a Fl(N)1151 5460 y Fr(\030)1199 5419 y Fj(T)1254 5460 y Fr(\030)5 b Fq(k)1352 5475 y Fj(Op)1471 5460 y Fq(\000)23 b Fo(\(1)f(+)1778 5384 y Fq(p)p 1861 5384 63 4 v 76 x Fr(\013)p Fo(\))1961 5419 y Fj(2)2001 5376 y Fk(\014)2001 5435 y(\014)2062 5460 y Fq(\025)2167 5384 y(p)p 2250 5384 V 76 x Fr(\013)2312 5380 y Fk(\011)2398 5460 y Fq(\024)28 b Fr(K)7 b Fo(e)2636 5419 y Fi(\000)p Fl(M)s(=K)3389 5460 y Ft(\(4.10\))94 5638 y Fc(for)35 b(al)5 b(l)35 b Fr(N)j Fq(\025)28 b Fr(N)689 5653 y Fj(1)728 5638 y Fc(.)p eop %%Page: 10 10 10 9 bop 118 -51 a Fm(10)1204 b(B.)34 b(GENTZ)g(AND)e(M.)i(L\326)n(WE) 218 123 y Ft(F)-8 b(or)31 b(later)h(use,)i(w)m(e)f(de\034ne)887 284 y Fo(\012)957 299 y Fj(1)997 284 y Fo(\()p Fr(N)10 b Fo(\))28 b(=)1293 203 y Fk(\010)1351 284 y Fr(\030)k Fo(:)1481 199 y Fk(\014)1481 259 y(\014)1514 284 y Fq(k)1588 244 y Fj(1)p 1574 261 64 4 v 1574 318 a Fl(N)1647 284 y Fr(\030)1695 243 y Fj(T)1750 284 y Fr(\030)5 b Fq(k)1848 299 y Fj(Op)1967 284 y Fq(\000)23 b Fo(\(1)f(+)2274 207 y Fq(p)p 2357 207 63 4 v 77 x Fr(\013)p Fo(\))2457 243 y Fj(2)2497 199 y Fk(\014)2497 259 y(\014)2558 284 y Fr(<)2661 207 y Fq(p)p 2744 207 V 77 x Fr(\013)2807 203 y Fk(\011)2865 284 y Fr(:)520 b Ft(\(4.11\))118 457 y(In)33 b(particular,)e(w)m(e)i(kno)m(w)h(that)e(for)g Fr(N)39 b Fq(\025)28 b Fr(N)1775 472 y Fj(1)1814 457 y Ft(,)33 b Fr(\030)f Fq(2)c Fo(\012)2113 472 y Fj(1)2153 457 y Fo(\()p Fr(N)10 b Fo(\))33 b Ft(and)f(all)f Fr(x;)17 b(y)31 b Fq(2)d Fp(R)3013 421 y Fl(M)3098 457 y Ft(,)933 571 y Fk(\014)933 631 y(\014)933 690 y(\014)933 750 y(\014)996 648 y Fo(1)p 976 692 89 4 v 976 784 a Fr(N)1131 591 y Fl(N)1091 621 y Fk(X)1106 831 y Fl(i)p Fj(=1)1235 715 y Fq(h)p Fr(x;)17 b(\030)1416 730 y Fl(i)1443 715 y Fq(ih)p Fr(y)t(;)g(\030)1660 730 y Fl(i)1686 715 y Fq(i)22 b(\000)h(h)p Fr(x;)17 b(y)t Fq(i)2076 571 y Fk(\014)2076 631 y(\014)2076 690 y(\014)2076 750 y(\014)2136 715 y Fq(\024)28 b Fo(4)2290 639 y Fq(p)p 2373 639 63 4 v 76 x Fr(\013)p Fq(k)p Fr(x)p Fq(k)2590 730 y Fj(2)2630 715 y Fq(k)p Fr(y)t Fq(k)2782 730 y Fj(2)2820 715 y Fr(:)565 b Ft(\(4.12\))218 962 y(W)-8 b(e)43 b(also)f(need)i(the)f(follo)m(wing)e(estimates)h(to)h (treat)g(terms)g(whic)m(h)g(in)m(v)m(olv)m(e)g(pro)s(ducts)h(of)118 1078 y(comp)s(onen)m(ts)j Fr(\030)717 1031 y Fl(\026)712 1104 y(i)810 1078 y Ft(for)f(four)g(or)h(six)g(di\033eren)m(t)f(v)-5 b(alues)47 b(of)f Fr(\026)p Ft(.)86 b(These)48 b(are)f(pro)m(vided)g(b) m(y)h(the)118 1195 y(follo)m(wing)30 b(lemma.)218 1311 y(F)-8 b(or)31 b Fr(\016)h(>)c Fo(0)k Ft(let)505 1547 y Fo(\012)575 1562 y Fj(2)614 1547 y Fo(\()p Fr(N)5 b(;)17 b(\016)t Fo(\))83 b(=)1106 1406 y Fk(\022)1250 1452 y([)1179 1661 y Fl(\026)1221 1670 y Fg(1)1256 1661 y Fl(;:::)o(;\026)1397 1670 y Fg(4)1431 1406 y Fk(\032)1506 1402 y(\014)1506 1462 y(\014)1506 1522 y(\014)1506 1581 y(\014)1569 1479 y Fo(1)p 1549 1524 89 4 v 1549 1615 a Fr(N)1705 1422 y Fl(N)1664 1452 y Fk(X)1679 1662 y Fl(i)p Fj(=1)1825 1547 y Fr(\030)1873 1499 y Fl(\026)1915 1508 y Fg(1)1868 1572 y Fl(i)1953 1547 y Fr(\030)2001 1499 y Fl(\026)2043 1508 y Fg(2)1996 1572 y Fl(i)2082 1547 y Fr(\030)2130 1499 y Fl(\026)2172 1508 y Fg(3)2125 1572 y Fl(i)2211 1547 y Fr(\030)2259 1499 y Fl(\026)2301 1508 y Fg(4)2254 1572 y Fl(i)2339 1402 y Fk(\014)2339 1462 y(\014)2339 1522 y(\014)2339 1581 y(\014)2400 1547 y Fr(>)28 b(\016)2551 1470 y Fq(p)p 2634 1470 63 4 v 77 x Fr(\013)2696 1406 y Fk(\033)1201 1901 y Fq([)1361 1807 y Fk([)1290 2016 y Fl(\026)1332 2025 y Fg(1)1367 2016 y Fl(;:::)n(;\026)1507 2025 y Fg(6)1542 1761 y Fk(\032)1617 1757 y(\014)1617 1817 y(\014)1617 1876 y(\014)1617 1936 y(\014)1680 1834 y Fo(1)p 1660 1878 89 4 v 1660 1970 a Fr(N)1815 1777 y Fl(N)1775 1807 y Fk(X)1790 2017 y Fl(i)p Fj(=1)1936 1901 y Fr(\030)1984 1854 y Fl(\026)2026 1863 y Fg(1)1979 1927 y Fl(i)2064 1901 y Fr(\030)2112 1854 y Fl(\026)2154 1863 y Fg(2)2107 1927 y Fl(i)2193 1901 y Fr(\030)2241 1854 y Fl(\026)2283 1863 y Fg(3)2236 1927 y Fl(i)2321 1901 y Fr(\030)2369 1854 y Fl(\026)2411 1863 y Fg(4)2364 1927 y Fl(i)2450 1901 y Fr(\030)2498 1854 y Fl(\026)2540 1863 y Fg(5)2493 1927 y Fl(i)2579 1901 y Fr(\030)2627 1854 y Fl(\026)2669 1863 y Fg(6)2622 1927 y Fl(i)2707 1757 y Fk(\014)2707 1817 y(\014)2707 1876 y(\014)2707 1936 y(\014)2768 1901 y Fr(>)g(\016)2919 1825 y Fq(p)p 3002 1825 63 4 v 76 x Fr(\013)3064 1761 y Fk(\033\023)3212 1783 y Fj(c)3248 1901 y Fr(;)137 b Ft(\(4.13\))118 2160 y(where)54 b(eac)m(h)f(of)f(the)h(unions)f(is)g(tak)m(en)h(o)m(v)m(er)h (all)c(sets)k(of)e(pairwise)f(di\033eren)m(t)i(indices)f(in)118 2277 y Fq(f)p Fo(1)p Fr(;)17 b(:)g(:)g(:)f(;)h(M)10 b Fq(g)p Ft(.)118 2453 y FA(Lemma)37 b(4.6.)49 b Fc(F)-7 b(or)34 b(every)h Fr(\016)c(>)d Fo(0)p Fc(,)34 b(ther)-5 b(e)35 b(exists)g(an)f Fr(N)2196 2468 y Fj(2)2236 2453 y Fo(\()p Fr(\016)t Fo(\))g Fc(such)h(that)g(for)g(al)5 b(l)34 b Fr(N)k Fq(\025)29 b Fr(N)3409 2468 y Fj(2)3448 2453 y Fo(\()p Fr(\016)t Fo(\))1227 2615 y Fp(P)p Fq(f)p Fo(\012)1406 2630 y Fj(2)1448 2615 y Fo(\()p Fr(N)5 b(;)17 b(\016)t Fo(\))1698 2574 y Fj(c)1733 2615 y Fq(g)28 b(\024)g Fo(exp)q Fq(f\000)p Fr(\016)2239 2574 y Fj(2)2278 2615 y Fr(M)5 b(=)p Fo(4)p Fq(g)p Fr(:)860 b Ft(\(4.14\))118 2792 y Fc(Pr)-5 b(o)g(of.)41 b Ft(Let)814 3044 y Fr(B)888 3059 y Fl(N)s(;\016)1004 3044 y Fo(\()p Fr(\026)1101 3059 y Fj(1)1140 3044 y Fr(;)17 b(:)g(:)g(:)f(;)h(\026)1418 3059 y Fj(4)1457 3044 y Fo(\))28 b(=)1626 2904 y Fk(\032)1701 2900 y(\014)1701 2960 y(\014)1701 3019 y(\014)1701 3079 y(\014)1764 2977 y Fo(1)p 1744 3021 89 4 v 1744 3113 a Fr(N)1900 2920 y Fl(N)1859 2950 y Fk(X)1874 3160 y Fl(i)p Fj(=1)2020 3044 y Fr(\030)2068 2997 y Fl(\026)2110 3006 y Fg(1)2063 3070 y Fl(i)2148 3044 y Fr(\030)2196 2997 y Fl(\026)2238 3006 y Fg(2)2191 3070 y Fl(i)2277 3044 y Fr(\030)2325 2997 y Fl(\026)2367 3006 y Fg(3)2320 3070 y Fl(i)2406 3044 y Fr(\030)2454 2997 y Fl(\026)2496 3006 y Fg(4)2449 3070 y Fl(i)2534 2900 y Fk(\014)2534 2960 y(\014)2534 3019 y(\014)2534 3079 y(\014)2595 3044 y Fr(>)g(\016)2746 2968 y Fq(p)p 2829 2968 63 4 v 76 x Fr(\013)2891 2904 y Fk(\033)3412 3044 y Ft(\(4.15\))118 3291 y(and)674 3524 y Fr(C)744 3539 y Fl(N)s(;\016)860 3524 y Fo(\()p Fr(\026)957 3539 y Fj(1)996 3524 y Fr(;)17 b(:)g(:)g(:)f(;)h(\026)1274 3539 y Fj(6)1313 3524 y Fo(\))27 b(=)1482 3384 y Fk(\032)1557 3380 y(\014)1557 3440 y(\014)1557 3499 y(\014)1557 3559 y(\014)1620 3457 y Fo(1)p 1600 3501 89 4 v 1600 3593 a Fr(N)1755 3400 y Fl(N)1715 3430 y Fk(X)1730 3640 y Fl(i)p Fj(=1)1875 3524 y Fr(\030)1923 3477 y Fl(\026)1965 3486 y Fg(1)1918 3550 y Fl(i)2004 3524 y Fr(\030)2052 3477 y Fl(\026)2094 3486 y Fg(2)2047 3550 y Fl(i)2133 3524 y Fr(\030)2181 3477 y Fl(\026)2223 3486 y Fg(3)2176 3550 y Fl(i)2261 3524 y Fr(\030)2309 3477 y Fl(\026)2351 3486 y Fg(4)2304 3550 y Fl(i)2390 3524 y Fr(\030)2438 3477 y Fl(\026)2480 3486 y Fg(5)2433 3550 y Fl(i)2518 3524 y Fr(\030)2566 3477 y Fl(\026)2608 3486 y Fg(6)2561 3550 y Fl(i)2647 3380 y Fk(\014)2647 3440 y(\014)2647 3499 y(\014)2647 3559 y(\014)2708 3524 y Fr(>)g(\016)2858 3448 y Fq(p)p 2941 3448 63 4 v 76 x Fr(\013)3004 3384 y Fk(\033)3079 3524 y Fr(:)306 b Ft(\(4.16\))118 3777 y(F)-8 b(or)33 b(pairwise)h(di\033eren)m(t)g (indices)f Fr(\026)1442 3792 y Fj(1)1481 3777 y Fr(;)17 b(:)g(:)g(:)f(;)h(\026)1759 3792 y Fj(6)1828 3777 y Fq(2)31 b(f)p Fo(1)p Fr(;)17 b(:)g(:)g(:)e(;)i(M)10 b Fq(g)p Ft(,)35 b(Cheb)m(yc)m(hev's)j(inequalit)m(y)33 b(with)118 3894 y Fr(t)28 b Fo(=)g Fr(\016)332 3822 y Fq(p)p 415 3822 V 72 x Fr(\013)33 b Ft(implies)460 4055 y Fp(P)p Fo(\()p Fr(B)631 4070 y Fl(N)s(;\016)749 4055 y Fo(\()p Fr(\026)846 4070 y Fj(1)885 4055 y Fr(;)17 b(:)g(:)g(:)f(;)h(\026)1163 4070 y Fj(4)1202 4055 y Fo(\)\))27 b Fq(\024)i Fo(exp)q Fq(f\000)p Fr(t\016)1769 3979 y Fq(p)p 1852 3979 V 76 x Fr(\013)q(N)10 b Fq(g)17 b Fo(exp)q Fq(f)p Fr(N)10 b(t)2392 4014 y Fj(2)2432 4055 y Fr(=)p Fo(2)p Fq(g)27 b Fo(=)g(exp)q Fq(f\000)p Fr(\016)3033 4014 y Fj(2)3073 4055 y Fr(M)5 b(=)p Fo(2)p Fq(g)118 4216 y Ft(and,)33 b(similarly)-8 b(,)1098 4337 y Fp(P)p Fo(\()p Fr(C)1265 4352 y Fl(N)s(;\016)1383 4337 y Fo(\()p Fr(\026)1480 4352 y Fj(1)1519 4337 y Fr(;)17 b(:)g(:)g(:)f(;)h(\026)1797 4352 y Fj(6)1836 4337 y Fo(\)\))28 b Fq(\024)g Fo(exp)q Fq(f\000)p Fr(\016)2368 4295 y Fj(2)2408 4337 y Fr(M)5 b(=)p Fo(2)p Fq(g)p Fr(:)118 4477 y Ft(Therefore,)222 4681 y Fp(P)p Fo(\(\012)389 4696 y Fj(2)431 4681 y Fo(\()p Fr(N)g(;)17 b(\016)t Fo(\))681 4640 y Fj(c)716 4681 y Fo(\))27 b Fq(\024)887 4571 y Fk(\020)956 4614 y Fo(1)p 956 4658 49 4 v 956 4750 a(2)1015 4681 y Fr(M)10 b Fo(\()p Fr(M)33 b Fq(\000)23 b Fo(1\))f(+)1614 4614 y(1)p 1601 4658 76 4 v 1601 4750 a(4!)1687 4681 y Fr(M)10 b Fo(\()p Fr(M)33 b Fq(\000)23 b Fo(1\)\()p Fr(M)32 b Fq(\000)23 b Fo(2\)\()p Fr(M)33 b Fq(\000)22 b Fo(3\))2845 4571 y Fk(\021)2921 4681 y Fo(exp)q Fq(f\000)p Fr(\016)3244 4640 y Fj(2)3284 4681 y Fr(M)5 b(=)p Fo(2)p Fq(g)p Fr(:)3412 4833 y Ft(\(4.17\))118 4993 y(Cho)s(osing)32 b Fr(M)43 b Ft(large)32 b(concludes)h(the)g(pro)s(of.)p 3591 4993 4 66 v 3595 4931 59 4 v 3595 4993 V 3653 4993 4 66 v 218 5165 a(The)28 b(next)f(lemma)e(pro)m(vides)j(a)e(b)s(ound)h (similar)d(to)i(\(4.12\))g(for)h(terms)g(in)m(v)m(olving)e(the)j (Gauss-)118 5284 y(ian)k Fr(\021)k Ft(instead)c(of)g Fr(N)900 5248 y Fi(\000)p Fj(1)p Fl(=)p Fj(2)1066 5284 y Fr(\030)1114 5248 y Fj(T)1168 5284 y Fr(\030)5 b Ft(.)43 b(Let)368 5501 y Fo(\012)438 5516 y Fj(3)478 5501 y Fo(\()p Fr(N)5 b(;)17 b(R)q(;)g(\024)p Fo(\))27 b(=)987 5360 y Fk(\032)1061 5501 y Fr(\030)32 b Fo(:)1191 5356 y Fk(\014)1191 5416 y(\014)1191 5476 y(\014)1191 5535 y(\014)1224 5406 y(X)1228 5615 y Fl(\026<\027)1385 5501 y Fr(\021)1433 5516 y Fl(\026;\027)1538 5501 y Fo(\()p Fr(\030)5 b Fo(\))p Fr(x)1717 5516 y Fl(\026)1764 5501 y Fr(x)1819 5516 y Fl(\027)1862 5356 y Fk(\014)1862 5416 y(\014)1862 5476 y(\014)1862 5535 y(\014)1923 5501 y Fr(<)28 b(\024R)2158 5459 y Fj(2)2197 5411 y Fq(p)p 2280 5411 105 4 v 90 x Fr(M)11 b Fq(k)p Fr(x)p Fq(k)2540 5459 y Fj(2)2540 5525 y(2)2677 5501 y Fq(8)p Fr(x)29 b Fq(2)f Fp(R)2976 5459 y Fl(M)3061 5360 y Fk(\033)3135 5501 y Fr(:)250 b Ft(\(4.18\))p eop %%Page: 11 11 11 10 bop 317 -67 a Fm(FLUCTUA)-6 b(TIONS)32 b(IN)g(THE)i(HOPFIELD)e (MODEL)i(A)-6 b(T)33 b(THE)g(CRITICAL)g(TEMPERA)-6 b(TURE)146 b(11)94 123 y FA(Lemma)38 b(4.7.)964 243 y Fp(P)p Fq(f)p Fo(\012)1143 258 y Fj(3)1184 243 y Fo(\()p Fr(N)5 b(;)17 b(R)q(;)g(\024)p Fo(\))1562 202 y Fj(c)1597 243 y Fq(g)28 b(\024)g Fo(5)1829 202 y Fj(2)p Fl(M)1960 243 y Fo(exp)q Fq(f\000)p Fr(\024)2292 202 y Fj(2)2332 243 y Fr(R)2407 202 y Fj(4)2446 243 y Fr(M)5 b(=)p Fo(16)p Fq(g)p Fr(:)94 420 y Fc(Pr)-5 b(o)g(of.)42 b Ft(Let)d Fr(x;)17 b(y)42 b Fq(2)d Fp(R)934 384 y Fl(M)1019 420 y Ft(.)63 b(First)38 b(note)h(that)1794 345 y Fk(P)1899 449 y Fl(\026<\027)2057 420 y Fr(\021)2105 435 y Fl(\026;\027)2210 420 y Fr(x)2265 435 y Fl(\026)2312 420 y Fr(y)2360 435 y Fl(\027)2442 420 y Ft(can)g(b)s(e)g(view)m(ed)i(as)e(the)g(scalar)94 536 y(pro)s(duct)34 b(of)g Fr(\021)j Ft(and)d(the)g(v)m(ector)h Fo(\()p Fr(x)1406 551 y Fl(\026)1453 536 y Fr(y)1501 551 y Fl(\027)1544 536 y Fo(\))1582 551 y Fl(\026<\027)1756 536 y Ft(and)f(that)f Fq(k)p Fo(\()p Fr(x)2302 551 y Fl(\026)2349 536 y Fr(y)2397 551 y Fl(\027)2440 536 y Fo(\))2478 551 y Fl(\026<\027)2618 536 y Fq(k)2668 551 y Fj(2)2737 536 y Fq(\024)e Fo(2)2894 500 y Fi(\000)p Fj(1)p Fl(=)p Fj(2)3058 536 y Fq(k)p Fr(x)p Fq(k)3213 551 y Fj(2)3253 536 y Fq(k)p Fr(y)t Fq(k)3405 551 y Fj(2)3443 536 y Ft(.)47 b(By)94 652 y(Cheb)m(yc)m(hev's)37 b(inequalit)m(y)-8 b(,)589 871 y Fp(P)648 731 y Fk(\032)724 776 y(X)728 986 y Fl(\026<\027)885 871 y Fr(\021)933 886 y Fl(\026;\027)1038 871 y Fr(x)1093 886 y Fl(\026)1140 871 y Fr(y)1188 886 y Fl(\027)1259 871 y Fq(\025)28 b Fr(\024)1420 830 y Fi(0)1443 731 y Fk(\033)1601 871 y Fq(\024)83 b Fo(exp)r Fq(f\000)p Fr(t\024)2129 830 y Fi(0)2152 871 y Fq(g)17 b Fo(exp)2368 760 y Fk(n)2444 804 y Fr(t)2479 767 y Fj(2)p 2444 848 75 4 v 2457 939 a Fo(2)2529 871 y Fq(k)p Fo(\()p Fr(x)2672 886 y Fl(\026)2719 871 y Fr(y)2767 886 y Fl(\027)2809 871 y Fo(\))2847 886 y Fl(\026<\027)2988 871 y Fq(k)3038 830 y Fj(2)3038 896 y(2)3077 760 y Fk(o)1601 1192 y Fq(\024)83 b Fo(exp)r Fq(f\000)p Fr(t\024)2129 1151 y Fi(0)2152 1192 y Fq(g)17 b Fo(exp)2368 1082 y Fk(n)2444 1125 y Fr(t)2479 1089 y Fj(2)p 2444 1169 V 2457 1261 a Fo(4)2529 1192 y Fq(k)p Fr(x)p Fq(k)2684 1151 y Fj(2)2684 1217 y(2)2723 1192 y Fq(k)p Fr(y)t Fq(k)2875 1151 y Fj(2)2875 1217 y(2)2914 1082 y Fk(o)3389 1192 y Ft(\(4.19\))94 1393 y(for)32 b Fr(t)c(>)g Fo(0)p Ft(.)43 b(Cho)s(osing)32 b Fr(t)c Fo(=)f(2)p Fr(\024)1226 1356 y Fi(0)1250 1393 y Fr(=)p Fo(\()p Fq(k)p Fr(x)p Fq(k)1492 1356 y Fj(2)1492 1417 y(2)1531 1393 y Fq(k)p Fr(y)t Fq(k)1683 1356 y Fj(2)1683 1417 y(2)1721 1393 y Fo(\))p Ft(,)953 1628 y Fp(P)1012 1487 y Fk(\032)1089 1533 y(X)1092 1742 y Fl(\026<\027)1249 1628 y Fr(\021)1297 1643 y Fl(\026;\027)1403 1628 y Fr(x)1458 1643 y Fl(\026)1505 1628 y Fr(y)1553 1643 y Fl(\027)1623 1628 y Fq(\025)h Fr(\024)1784 1586 y Fi(0)1808 1487 y Fk(\033)1910 1628 y Fq(\024)g Fo(exp)2164 1517 y Fk(n)2231 1628 y Fq(\000)2451 1560 y Fr(\024)2507 1524 y Fi(0)2530 1513 y Fj(2)p 2318 1605 385 4 v 2318 1696 a Fq(k)p Fr(x)p Fq(k)2473 1662 y Fj(2)2473 1720 y(2)2513 1696 y Fq(k)p Fr(y)t Fq(k)2665 1662 y Fj(2)2665 1720 y(2)2713 1517 y Fk(o)3389 1628 y Ft(\(4.20\))94 1887 y(follo)m(ws.)43 b(T)-8 b(o)32 b(obtain)g(a)g(uniform)f(b)s(ound,)h(note)h(that)114 2104 y Fp(P)173 1963 y Fk(\032)250 2104 y Fq(9)p Fr(x)28 b Fq(2)h Fp(R)548 2063 y Fl(M)661 2104 y Fo(:)716 2009 y Fk(X)720 2218 y Fl(\026<\027)876 2104 y Fr(\021)924 2119 y Fl(\026;\027)1030 2104 y Fr(x)1085 2119 y Fl(\026)1132 2104 y Fr(x)1187 2119 y Fl(\027)1258 2104 y Fq(\025)f Fr(\024)1419 2063 y Fi(0)1443 2104 y Fq(k)p Fr(x)p Fq(k)1598 2063 y Fj(2)1598 2128 y(2)1637 1963 y Fk(\033)1796 2104 y Fo(=)83 b Fp(P)2014 1963 y Fk(\032)2091 2104 y Fq(9)p Fr(x)29 b Fq(2)f Fr(B)5 b Fo(\(0)p Fr(;)17 b Fo(1\))27 b(:)2702 2009 y Fk(X)2706 2218 y Fl(\026<\027)2863 2104 y Fr(\021)2911 2119 y Fl(\026;\027)3016 2104 y Fr(x)3071 2119 y Fl(\026)3118 2104 y Fr(x)3173 2119 y Fl(\027)3245 2104 y Fq(\025)h Fr(\024)3406 2063 y Fi(0)3429 1963 y Fk(\033)1795 2423 y Fq(\024)83 b Fp(P)2014 2282 y Fk(\032)2091 2423 y Fq(9)p Fr(x;)17 b(y)31 b Fq(2)d Fr(B)5 b Fo(\(0)p Fr(;)17 b Fo(1\))27 b(:)2797 2328 y Fk(X)2801 2538 y Fl(\026<\027)2958 2423 y Fr(\021)3006 2438 y Fl(\026;\027)3111 2423 y Fr(x)3166 2438 y Fl(\026)3213 2423 y Fr(y)3261 2438 y Fl(\027)3331 2423 y Fq(\025)i Fr(\024)3493 2382 y Fi(0)3516 2282 y Fk(\033)3591 2423 y Fr(:)94 2695 y(B)5 b Fo(\(0)p Fr(;)17 b Fo(1\))37 b Ft(b)s(eing)g(a)h(\(b)s(ounded\))g (con)m(v)m(ex,)j(balanced)d(set)g(in)f Fp(R)2369 2659 y Fl(M)2454 2695 y Ft(,)i(there)g(exists)f(a)g(subset)h Fr(D)g Fq(\032)94 2811 y Fr(B)5 b Fo(\(0)p Fr(;)17 b Fo(2\))28 b Ft(suc)m(h)i(that)e Fr(B)5 b Fo(\(0)p Fr(;)17 b Fo(1\))27 b Ft(is)h(con)m(tained)h(in)e(the)i(con)m(v)m(ex)h(h)m(ull) d(of)h Fr(D)j Ft(and)e Fr(D)i Ft(has)d(at)g(most)g Fo(5)3559 2775 y Fl(M)94 2928 y Ft(elemen)m(ts)g(\(see)g(for)e(example)h([31,)h (Lemma)d(10.2)i(in)f(the)i(App)s(endix]\).)42 b(No)m(w,)28 b(b)m(y)g(our)f(previous)94 3044 y(b)s(ound)33 b(and)g(the)g (de\034nition)e(of)h(the)h(set)h Fr(D)s Ft(,)261 3261 y Fp(P)320 3120 y Fk(\032)396 3261 y Fq(9)p Fr(x)29 b Fq(2)f Fp(R)695 3220 y Fl(M)807 3261 y Fo(:)862 3166 y Fk(X)866 3375 y Fl(\026<\027)1023 3261 y Fr(\021)1071 3276 y Fl(\026;\027)1176 3261 y Fr(x)1231 3276 y Fl(\026)1278 3261 y Fr(x)1333 3276 y Fl(\027)1404 3261 y Fq(\025)g Fr(\024)1565 3220 y Fi(0)1589 3261 y Fq(k)p Fr(x)p Fq(k)1744 3220 y Fj(2)1744 3285 y(2)1783 3120 y Fk(\033)678 3580 y Fq(\024)g Fp(P)842 3440 y Fk(\032)919 3580 y Fq(9)p Fr(x;)17 b(y)31 b Fq(2)d Fr(D)j Fo(:)1413 3485 y Fk(X)1416 3695 y Fl(\026<\027)1573 3580 y Fr(\021)1621 3595 y Fl(\026;\027)1726 3580 y Fr(x)1781 3595 y Fl(\026)1829 3580 y Fr(y)1877 3595 y Fl(\027)1947 3580 y Fq(\025)d Fr(\024)2108 3539 y Fi(0)2132 3440 y Fk(\033)2234 3580 y Fq(\024)g Fo(5)2388 3539 y Fj(2)p Fl(M)2547 3580 y Fo(sup)2519 3661 y Fl(x;y)r Fi(2)p Fl(D)2739 3580 y Fp(P)2798 3440 y Fk(\032)2874 3485 y(X)2878 3695 y Fl(\026<\027)3035 3580 y Fr(\021)3083 3595 y Fl(\026;\027)3188 3580 y Fr(x)3243 3595 y Fl(\026)3290 3580 y Fr(y)3338 3595 y Fl(\027)3409 3580 y Fq(\025)g Fr(\024)3570 3539 y Fi(0)3593 3440 y Fk(\033)678 3912 y Fq(\024)g Fo(5)832 3871 y Fj(2)p Fl(M)991 3912 y Fo(sup)963 3993 y Fl(x;y)r Fi(2)p Fl(D)1183 3912 y Fo(exp)1332 3801 y Fk(n)1399 3912 y Fq(\000)1619 3844 y Fr(\024)1675 3808 y Fi(0)1698 3798 y Fj(2)p 1486 3889 V 1486 3980 a Fq(k)p Fr(x)p Fq(k)1641 3946 y Fj(2)1641 4005 y(2)1681 3980 y Fq(k)p Fr(y)t Fq(k)1833 3946 y Fj(2)1833 4005 y(2)1881 3801 y Fk(o)1975 3912 y Fq(\024)g Fo(5)2129 3871 y Fj(2)p Fl(M)2260 3912 y Fo(exp)2409 3801 y Fk(n)2475 3912 y Fq(\000)2562 3844 y Fr(\024)2618 3808 y Fi(0)2642 3798 y Fj(2)p 2562 3889 119 4 v 2573 3980 a Fo(16)2691 3801 y Fk(o)2758 3912 y Fr(:)634 b Ft(\(4.21\))94 4161 y(Cho)s(osing)32 b Fr(\024)576 4125 y Fi(0)627 4161 y Fo(=)c Fr(\024R)862 4125 y Fj(2)902 4077 y Fq(p)p 985 4077 105 4 v 84 x Fr(M)43 b Ft(with)32 b Fr(\024)c(>)g Fo(0)k Ft(concludes)h(the)g(pro)s(of.)p 3567 4161 4 66 v 3571 4099 59 4 v 3571 4161 V 3629 4161 4 66 v 194 4333 a(With)f(these)i(preparations)e(w)m(e)h(are)g(able)f (to)g(pro)m(v)m(e)i(Theorem)f(2.1.)94 4504 y Fc(Pr)-5 b(o)g(of)35 b(of)g(The)-5 b(or)g(em)33 b(2.1.)41 b Ft(By)49 b(\(4.2\),)h(Theorem)e(2.1)f(follo)m(ws,)i(once)f(w)m(e)g(ha)m(v)m(e)g (sho)m(wn)h(that,)94 4620 y(under)34 b(the)f(conditions)e(of)h(the)h (theorem,)660 4692 y Fk(\014)660 4752 y(\014)660 4812 y(\014)660 4872 y(\014)693 4701 y(Z)748 4927 y Fd(R)796 4908 y Fh(M)884 4837 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))p Fr(Q)1151 4852 y Fl(N)1219 4837 y Fo(\()p Fr(dx)p Fo(\))22 b Fq(\000)1522 4701 y Fk(Z)1578 4927 y Fd(R)1626 4908 y Fh(M)1714 4837 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))p 1904 4757 78 4 v Fr(Q)1981 4860 y Fl(M)2060 4837 y Fo(\()p Fr(dx)p Fo(\))2242 4692 y Fk(\014)2242 4752 y(\014)2242 4812 y(\014)2242 4872 y(\014)2303 4837 y Fq(\024)28 b Fr(")2448 4852 y Fl(N)2515 4837 y Fo(\()p Fr(K)2636 4852 y Fl(f)2704 4837 y Fo(+)22 b Fq(k)p Fr(f)11 b Fq(k)2961 4852 y Fi(1)3035 4837 y Fo(\))316 b Ft(\(4.22\))94 5073 y(holds)33 b(for)f(all)f Fr(\030)h Fq(2)p 804 4993 71 4 v 28 w Fo(\012)q(\()p Fr(N)10 b Fo(\))33 b Ft(and)g(all)d Fr(f)39 b Fq(2)28 b Fo(BL\()p Fp(R)1812 5036 y Fl(M)1897 5073 y Fr(;)17 b Fp(R)5 b Fo(\))p Ft(.)50 b(By)33 b(Lemma)e(4.2,)h(w)m (e)i(ma)m(y)f(replace)f Fr(Q)3570 5088 y Fl(N)94 5189 y Ft(b)m(y)i(its)e(Hubbard\025Stratono)m(vic)m(h)h(transform.)194 5305 y(So)g(let)e Fr(f)39 b Fq(2)28 b Fo(BL\()p Fp(R)885 5269 y Fl(M)970 5305 y Fr(;)17 b Fp(R)t Fo(\))p Ft(.)50 b(W)-8 b(e)33 b(need)g(to)g(in)m(v)m(estigate)1201 5389 y Fk(R)1284 5469 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)p Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)2149 5433 y Fj(1)p Fl(=)p Fj(4)2261 5469 y Fo(\))p Fq(g)17 b Fr(dx)p 1201 5515 1271 4 v 1304 5527 a Fk(R)1387 5608 y Fo(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)2046 5579 y Fj(1)p Fl(=)p Fj(4)2158 5608 y Fo(\))p Fq(g)17 b Fr(dx)2498 5538 y(;)864 b Ft(\(4.23\))p eop %%Page: 12 12 12 11 bop 118 -51 a Fm(12)1204 b(B.)34 b(GENTZ)g(AND)e(M.)i(L\326)n(WE) 118 123 y Ft(where)426 354 y Fo(\010\()p Fr(y)t Fo(\))27 b(=)h(\010)825 369 y Fl(N)s(;)p Fj(1)944 354 y Fo(\()p Fr(y)t Fo(\))e(=)1212 286 y(1)p 1212 331 49 4 v 1212 422 a(2)1270 354 y Fq(k)p Fr(y)t Fq(k)1422 312 y Fj(2)1422 378 y(2)1483 354 y Fq(\000)1612 286 y Fo(1)p 1592 331 89 4 v 1592 422 a Fr(N)1748 229 y Fl(N)1707 259 y Fk(X)1722 469 y Fl(i)p Fj(=1)1868 354 y Fo(log)17 b(cosh\()p Fq(h)p Fr(y)t(;)g(\030)2411 369 y Fl(i)2438 354 y Fq(i)p Fo(\))p Fr(;)211 b(y)31 b Fq(2)d Fp(R)2992 312 y Fl(M)3077 354 y Fr(:)308 b Ft(\(4.24\))118 598 y(Consider)32 b(the)h(nominator)c (\034rst)k(as)f(the)g(denominator)e(is)i(a)f(sp)s(ecial)g(case)i(of)e (the)h(nominator.)118 715 y(The)e(main)e(con)m(tribution)g(to)h(the)h (in)m(tegral)e(arises)h(from)f(the)i Fc(inner)h(r)-5 b(e)g(gion)29 b Fr(B)5 b Fo(\(0)p Fr(;)17 b(R)q(M)3327 679 y Fj(1)p Fl(=)p Fj(4)3437 715 y Fo(\))29 b Ft(and)118 831 y(w)m(e)44 b(shall)c(c)m(ho)s(ose)k(a)e(suitable)f Fr(R)46 b(>)e Fo(0)e Ft(later)g(on.)73 b(In)42 b(the)h(inner)f(region)g (as)g(w)m(ell)g(as)g(in)g(the)118 947 y Fc(interme)-5 b(diate)42 b(r)-5 b(e)g(gion)41 b Fr(B)5 b Fo(\(0)p Fr(;)17 b(r)s(N)1332 911 y Fj(1)p Fl(=)p Fj(4)1442 947 y Fo(\))28 b Fq(n)g Fr(B)5 b Fo(\(0)p Fr(;)17 b(R)q(M)1975 911 y Fj(1)p Fl(=)p Fj(4)2085 947 y Fo(\))42 b Ft(with)f Fr(r)k(>)e Fo(0)e Ft(to)g(b)s(e)h(c)m(hosen)h(later,)f(w)m(e)118 1063 y(in)m(v)m(estigate)31 b(the)h(b)s(eha)m(viour)f(of)g(the)h(in)m (tegral)e(in)g(the)i(nominator)d(with)i(the)h(help)f(of)g(a)g(T)-8 b(a)m(ylor)118 1180 y(expansion)33 b(of)f Fo(\010)p Ft(.)44 b(The)34 b Fc(outer)h(r)-5 b(e)g(gion)31 b Fr(B)5 b Fo(\(0)p Fr(;)17 b(r)s(N)1911 1143 y Fj(1)p Fl(=)p Fj(4)2021 1180 y 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y Fo(\()p Fr(x;)17 b(\030)5 b Fo(\))p Fr(;)288 b Ft(\(4.25\))118 1975 y(with)1015 2206 y Fr(R)1089 2221 y Fl(N)1156 2206 y Fo(\()p Fr(x;)17 b(\030)5 b Fo(\))27 b(=)h Fq(\000)1617 2138 y Fo(1)p 1597 2183 89 4 v 1597 2274 a Fr(N)1753 2081 y Fl(N)1712 2111 y Fk(X)1727 2321 y Fl(i)p Fj(=1)1907 2138 y Fo(1)p 1883 2183 98 4 v 1883 2274 a(15)2007 2206 y Fr(h)p Fo(\()p Fq(h)p Fr(\022)s(x;)17 b(\030)2330 2221 y Fl(i)2358 2206 y Fq(i)p Fo(\))p Fq(h)p Fr(x;)g(\030)2616 2221 y Fl(i)2643 2206 y Fq(i)2682 2165 y Fj(5)2721 2206 y Fr(;)664 b Ft(\(4.26\))118 2451 y(where)1041 2644 y Fr(h)p Fo(\()p Fr(t)p Fo(\))28 b(=)1364 2576 y(tanh\()p Fr(t)p Fo(\))p 1349 2621 336 4 v 1349 2719 a(cosh)1534 2676 y Fj(4)1574 2719 y Fo(\()p Fr(t)p Fo(\))1695 2644 y([2)22 b Fq(\000)g Fo(sinh)2066 2601 y Fj(2)2106 2644 y Fo(\()p Fr(t)p Fo(\)])p Fr(;)212 b(t)28 b Fq(2)g Fp(R)5 b Fr(:)679 b Ft(\(4.27\))118 2863 y(Regrouping)32 b(the)h(terms)f(of)g (the)h(T)-8 b(a)m(ylor)32 b(expansion)h(of)f Fo(\010)p 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y Fr(x)2448 3636 y Fl(\026)2490 3645 y Fg(2)2529 3621 y Fr(x)2584 3580 y Fj(2)2584 3646 y Fl(\026)2626 3655 y Fg(3)2696 3554 y Fo(1)p 2676 3598 89 4 v 2676 3690 a Fr(N)2831 3497 y Fl(N)2791 3527 y Fk(X)2806 3737 y Fl(i)p Fj(=1)2952 3621 y Fr(\030)3000 3574 y Fl(\026)3042 3583 y Fg(1)2995 3647 y Fl(i)3080 3621 y Fr(\030)3128 3574 y Fl(\026)3170 3583 y Fg(2)3123 3647 y Fl(i)3412 3621 y Ft(\(4.28\))561 3976 y Fq(\000)695 3909 y Fo(1)p 671 3953 98 4 v 671 4044 a(12)887 3881 y Fk(X)1031 3898 y Fi(\003)795 4090 y Fl(\026)837 4099 y Fg(1)872 4090 y Fl(;\026)934 4099 y Fg(2)969 4090 y Fl(;\026)1031 4099 y Fg(3)1065 4090 y Fl(;\026)1127 4099 y Fg(4)1178 3976 y Fr(x)1233 3991 y Fl(\026)1275 4000 y Fg(1)1315 3976 y Fr(x)1370 3991 y Fl(\026)1412 4000 y Fg(2)1452 3976 y Fr(x)1507 3991 y Fl(\026)1549 4000 y Fg(3)1588 3976 y Fr(x)1643 3991 y Fl(\026)1685 4000 y Fg(4)1755 3909 y Fo(1)p 1735 3953 89 4 v 1735 4044 a Fr(N)1890 3851 y Fl(N)1850 3881 y Fk(X)1865 4091 y Fl(i)p Fj(=1)2011 3976 y Fr(\030)2059 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b(:)g(:)g(:)f(;)h(\026)1963 4398 y Fl(k)2005 4383 y Fo(\))48 b Fq(2)g(f)p Fo(1)p Fr(;)17 b(:)g(:)g(:)e(;)i(M)10 b Fq(g)45 b Ft(with)f(pairwise)g(disjoin)m(t)118 4499 y(comp)s(onen)m(ts.)218 4616 y(Let)29 b(us)g(consider)g(the)g (di\033eren)m(t)g Fr(\030)5 b Ft(-dep)s(enden)m(t)29 b(terms.)42 b(By)29 b(the)g(strong)g(Gaussian)f(appro)m(xi-)118 4732 y(mation)22 b(Corollary)h(3.3,)i(there)g(exist)f(a)g(constan)m(t)h Fr(N)2039 4747 y Fj(0)2107 4732 y Fq(2)j Fp(N)39 b Ft(and)24 b(an)g Fr(M)10 b Fo(\()p Fr(M)15 b Fq(\000)5 b Fo(1\))p Fr(=)p Fo(2)p Ft(-dimensional)118 4848 y(Gaussian)33 b(v)m(ector)h Fr(\021)i Ft(with)d(mean)g(zero)g(and)g(co)m(v)-5 b(ariance)33 b(matrix)e(b)s(eing)i(the)g(iden)m(tit)m(y)g(matrix)118 4964 y(suc)m(h)h(that)879 5195 y Fo(\012)949 5210 y Fj(0)989 5195 y Fo(\()p Fr(N)5 b(;)17 b(\016)1197 5210 y Fl(N)1264 5195 y Fo(\))27 b(=)1433 5055 y Fk(\032)1508 5051 y(\015)1508 5111 y(\015)1508 5170 y(\015)1508 5230 y(\015)1634 5128 y Fo(1)p 1573 5172 172 4 v 1573 5192 a Fq(p)p 1656 5192 89 4 v 85 x Fr(N)1812 5071 y Fl(N)1771 5101 y Fk(X)1786 5311 y Fl(i)p Fj(=1)1915 5195 y Fo(\()p Fr(\030)2001 5148 y Fl(\026)1996 5221 y(i)2047 5195 y Fr(\030)2095 5154 y Fl(\027)2090 5220 y(i)2138 5195 y Fo(\))2176 5210 y Fl(\026<\027)2338 5195 y Fq(\000)c Fr(\021)2490 5051 y Fk(\015)2490 5111 y(\015)2490 5170 y(\015)2490 5230 y(\015)2545 5294 y Fj(2)2612 5195 y Fr(<)28 b(\016)2759 5210 y Fl(N)2826 5055 y Fk(\033)3412 5195 y Ft(\(4.29\))118 5440 y(with)1542 5599 y Fr(\016)1585 5614 y Fl(N)1680 5599 y Fo(=)f Fr(K)7 b(M)1977 5558 y Fj(7)2018 5599 y Fr(=)2067 5510 y Fq(p)p 2149 5510 V 2149 5599 a Fr(N)1185 b Ft(\(4.30\))p eop %%Page: 13 13 13 12 bop 317 -67 a Fm(FLUCTUA)-6 b(TIONS)32 b(IN)g(THE)i(HOPFIELD)e (MODEL)i(A)-6 b(T)33 b(THE)g(CRITICAL)g(TEMPERA)-6 b(TURE)146 b(13)94 123 y Ft(for)32 b(some)h Fr(K)i(>)27 b Fo(0)32 b Ft(satis\034es)1117 299 y Fp(P)p Fo(\(\012)1284 314 y Fj(0)1326 299 y Fo(\()p Fr(N)5 b(;)17 b(\016)1534 314 y Fl(N)1601 299 y Fo(\))1639 258 y Fj(c)1674 299 y Fo(\))28 b Fq(\024)g Fo(exp)q Fq(f\000)p Fr(K)7 b(M)e(=)p Fo(\(2)p Fr(c)2488 314 y Fj(2)2528 299 y Fo(\))p Fq(g)773 b Ft(\(4.31\))94 475 y(for)32 b(all)f Fr(N)38 b Fq(\025)28 b Fr(N)678 490 y Fj(0)750 475 y Ft(and)100 594 y Fk(\014)100 654 y(\014)100 714 y(\014)100 774 y(\014)143 671 y Fo(1)p 143 716 49 4 v 143 807 a(2)218 644 y Fk(X)362 660 y Fi(\003)223 853 y Fl(\026)265 862 y Fg(1)300 853 y Fl(;\026)362 862 y Fg(2)418 739 y Fr(x)473 754 y Fl(\026)515 763 y Fg(1)555 739 y Fr(x)610 754 y Fl(\026)652 763 y Fg(2)763 671 y Fo(1)p 701 716 172 4 v 701 736 a Fq(p)p 784 736 89 4 v 84 x Fr(N)940 614 y Fl(N)900 644 y Fk(X)914 854 y Fl(i)p Fj(=1)1060 739 y Fr(\030)1108 692 y Fl(\026)1150 701 y Fg(1)1103 764 y Fl(i)1189 739 y Fr(\030)1237 692 y Fl(\026)1279 701 y Fg(2)1232 764 y Fl(i)1339 739 y Fq(\000)1472 644 y Fk(X)1439 853 y Fl(\026)1481 862 y Fg(1)1516 853 y Fl(<\026)1613 862 y Fg(2)1665 739 y Fr(\021)1713 754 y Fl(\026)1755 763 y Fg(1)1790 754 y Fl(;\026)1852 763 y 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Fr(N)394 1389 y Fj(1)466 1374 y Ft(and)33 b Fr(\030)f Fq(2)c Fo(\012)895 1389 y Fj(1)935 1374 y Fo(\()p Fr(N)10 b Fo(\))p Ft(,)33 b(the)g(b)s(ound)f(\(4.12\))g(on)g(the)h(random)f (matrix)f(yields)1011 1503 y Fk(\014)1011 1562 y(\014)1011 1622 y(\014)1011 1682 y(\014)1055 1580 y Fo(1)p 1055 1624 V 1055 1716 a(3)1130 1553 y Fk(X)1274 1569 y Fi(\003)1135 1762 y Fl(\026)1177 1771 y Fg(1)1212 1762 y Fl(;\026)1274 1771 y Fg(2)1330 1647 y Fr(x)1385 1662 y Fl(\026)1427 1671 y Fg(1)1467 1647 y Fr(x)1522 1606 y Fj(3)1522 1672 y Fl(\026)1564 1681 y Fg(2)1633 1580 y Fo(1)p 1613 1624 89 4 v 1613 1716 a Fr(N)1769 1523 y Fl(N)1728 1553 y Fk(X)1743 1762 y Fl(i)p Fj(=1)1889 1647 y Fr(\030)1937 1600 y Fl(\026)1979 1609 y Fg(1)1932 1673 y Fl(i)2017 1647 y Fr(\030)2065 1600 y Fl(\026)2107 1609 y Fg(2)2060 1673 y Fl(i)2146 1503 y Fk(\014)2146 1562 y(\014)2146 1622 y(\014)2146 1682 y(\014)2207 1647 y Fq(\024)2322 1580 y Fo(4)p 2322 1624 49 4 v 2322 1716 a(3)2381 1571 y Fq(p)p 2464 1571 63 4 v 76 x 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2502 49 4 v 631 2593 a(2)706 2430 y Fk(X)850 2446 y Fi(\003)711 2639 y Fl(\026)753 2648 y Fg(1)788 2639 y Fl(;\026)850 2648 y Fg(2)906 2525 y Fr(x)961 2540 y Fl(\026)1003 2549 y Fg(1)1043 2525 y Fr(x)1098 2540 y Fl(\026)1140 2549 y Fg(2)1180 2525 y Fq(k)p Fr(x)p Fq(k)1335 2484 y Fj(2)1335 2549 y(2)1404 2457 y Fo(1)p 1384 2502 89 4 v 1384 2593 a Fr(N)1540 2400 y Fl(N)1499 2430 y Fk(X)1514 2640 y Fl(i)p Fj(=1)1660 2525 y Fr(\030)1708 2478 y Fl(\026)1750 2487 y Fg(1)1703 2550 y Fl(i)1788 2525 y Fr(\030)1836 2478 y Fl(\026)1878 2487 y Fg(2)1831 2550 y Fl(i)1939 2525 y Fq(\000)2039 2430 y Fk(X)2183 2446 y Fi(\003)2043 2639 y Fl(\026)2085 2648 y Fg(1)2120 2639 y Fl(;\026)2182 2648 y Fg(2)2239 2525 y Fr(x)2294 2540 y Fl(\026)2336 2549 y Fg(1)2375 2525 y Fr(x)2430 2484 y Fj(3)2430 2549 y Fl(\026)2472 2558 y Fg(2)2542 2457 y Fo(1)p 2522 2502 V 2522 2593 a Fr(N)2677 2400 y Fl(N)2637 2430 y Fk(X)2652 2640 y Fl(i)p Fj(=1)2797 2525 y Fr(\030)2845 2478 y Fl(\026)2887 2487 y Fg(1)2840 2550 y 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Fg(1)1092 3079 y Fr(x)1147 3094 y Fl(\026)1189 3103 y Fg(2)1228 3079 y Fr(x)1283 3094 y Fl(\026)1325 3103 y Fg(3)1365 3079 y Fr(x)1420 3094 y Fl(\026)1462 3103 y Fg(4)1531 3011 y Fo(1)p 1511 3056 89 4 v 1511 3147 a Fr(N)1667 2954 y Fl(N)1627 2984 y Fk(X)1641 3194 y Fl(i)p Fj(=1)1787 3079 y Fr(\030)1835 3031 y Fl(\026)1877 3040 y Fg(1)1830 3104 y Fl(i)1916 3079 y Fr(\030)1964 3031 y Fl(\026)2006 3040 y Fg(2)1959 3104 y Fl(i)2044 3079 y Fr(\030)2092 3031 y Fl(\026)2134 3040 y Fg(3)2087 3104 y Fl(i)2173 3079 y Fr(\030)2221 3031 y Fl(\026)2263 3040 y Fg(4)2216 3104 y Fl(i)2302 2934 y Fk(\014)2302 2994 y(\014)2302 3054 y(\014)2302 3114 y(\014)3389 3079 y Ft(\(4.35\))627 3401 y Fq(\024)742 3333 y Fr(\016)789 3261 y Fq(p)p 872 3261 63 4 v 72 x Fr(\013)p 742 3378 193 4 v 790 3469 a Fo(12)1073 3306 y Fk(X)961 3515 y Fl(\026)1003 3524 y Fg(1)1038 3515 y Fl(;\026)1100 3524 y Fg(2)1135 3515 y Fl(;\026)1197 3524 y Fg(3)1232 3515 y Fl(;\026)1294 3524 y Fg(4)1345 3401 y Fq(j)p Fr(x)1428 3416 y Fl(\026)1470 3425 y Fg(1)1509 3401 y Fr(x)1564 3416 y Fl(\026)1606 3425 y Fg(2)1646 3401 y Fr(x)1701 3416 y Fl(\026)1743 3425 y Fg(3)1782 3401 y Fr(x)1837 3416 y Fl(\026)1879 3425 y Fg(4)1919 3401 y Fq(j)27 b(\024)2089 3333 y Fr(\016)2136 3261 y Fq(p)p 2219 3261 63 4 v 72 x Fr(\013)q(M)2386 3297 y Fj(2)p 2089 3378 337 4 v 2209 3469 a Fo(12)2436 3401 y Fq(k)p Fr(x)p Fq(k)2591 3359 y Fj(4)2591 3425 y(2)2658 3401 y Fo(=)2797 3333 y Fr(\016)p 2772 3378 98 4 v 2772 3469 a Fo(12)2879 3290 y Fk(\020)2949 3333 y Fr(M)3053 3297 y Fj(5)p 2949 3378 145 4 v 2977 3469 a Fr(N)3103 3290 y Fk(\021)3163 3312 y Fj(1)p Fl(=)p Fj(2)3273 3401 y Fq(k)p Fr(x)p Fq(k)3428 3359 y Fj(4)3428 3425 y(2)3467 3401 y Fr(:)194 3681 y Ft(It)34 b(remains)g(to)f(consider)i(the)f (remainder)g(of)f(the)i(T)-8 b(a)m(ylor)34 b(expansion.)48 b(No)m(w,)35 b Fq(j)p Fr(h)p Fo(\()p Fr(t)p Fo(\))p Fq(j)c(\024)f Fo(2)p Fq(j)p Fr(t)p Fq(j)94 3797 y Ft(and)j Fo(0)28 b Fr(<)f(\022)k(<)d Fo(1)k Ft(together)g(with)h(Sc)m(h)m(w)m(arz')h 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3921 y Fk(\014)3170 3981 y(\014)3170 4040 y(\014)3170 4100 y(\014)3203 4065 y Fr(:)159 b Ft(\(4.36\))94 4340 y(The)39 b(righ)m(t-hand)e(side)h(is)g (b)s(ounded)h(ab)s(o)m(v)m(e)f(b)m(y)h(a)f(com)m(binatorial)c(factor)k (times)f(the)h(sum)g(of)94 4456 y(terms)33 b(similar)c(to)j(the)h(ones) h(treated)f(ab)s(o)m(v)m(e)g(\(with)f(t)m(w)m(o,)i(four)e(or)g(six)h (di\033eren)m(t)f Fr(\030)3179 4409 y Fl(\026)3174 4482 y(i)3225 4456 y Ft(\))h(plus)f(the)94 4572 y(term)g(arising)f(from)h Fr(\026)932 4587 y Fj(1)998 4572 y Fo(=)c Fq(\001)17 b(\001)g(\001)26 b Fo(=)h Fr(\026)1408 4587 y Fj(6)1447 4572 y Ft(.)44 b(This)33 b(yields)303 4806 y Fq(j)p Fr(R)405 4821 y Fl(N)472 4806 y Fo(\()p Fr(y)t(;)17 b(\030)5 b Fo(\))p Fq(j)26 b(\024)i Fr(C)928 4666 y Fk(\024)980 4730 y Fq(p)p 1063 4730 63 4 v 76 x Fr(\013)q Fq(k)p Fr(y)t Fq(k)1278 4765 y Fj(6)1278 4831 y(2)1338 4806 y Fo(+)22 b Fr(\016)1483 4696 y Fk(\020)1553 4739 y Fr(M)1657 4703 y Fj(5)p 1553 4783 145 4 v 1581 4875 a Fr(N)1707 4696 y Fk(\021)1766 4718 y Fj(1)p Fl(=)p Fj(2)1876 4806 y Fq(k)p Fr(y)t Fq(k)2028 4765 y Fj(6)2028 4831 y(2)2089 4806 y Fo(+)g Fr(\016)2234 4696 y Fk(\020)2303 4739 y Fr(M)2407 4703 y Fj(7)p 2303 4783 V 2331 4875 a Fr(N)2457 4696 y Fk(\021)2517 4718 y Fj(1)p Fl(=)p Fj(2)2627 4806 y Fq(k)p Fr(y)t Fq(k)2779 4765 y Fj(6)2779 4831 y(2)2839 4806 y Fo(+)g Fq(k)p Fr(y)t Fq(k)3089 4765 y Fj(6)3089 4831 y(6)3127 4666 y Fk(\025)3389 4806 y Ft(\(4.37\))94 5044 y(for)32 b Fr(N)39 b Fq(\025)28 b Fo(max)o Fq(f)p Fr(N)774 5059 y Fj(1)814 5044 y Fr(;)17 b(N)936 5059 y Fj(2)975 5044 y Fo(\()p Fr(\016)t Fo(\))p Fq(g)32 b Ft(and)h Fr(\030)f Fq(2)c Fo(\012)1609 5059 y Fj(1)1649 5044 y Fo(\()p Fr(N)10 b Fo(\))22 b Fq(\\)h Fo(\012)1994 5059 y Fj(2)2034 5044 y Fo(\()p Fr(N)5 b(;)17 b(\016)t Fo(\))p Ft(,)32 b(so)h(that)409 5284 y Fr(N)10 b Fq(j)p Fr(R)599 5299 y Fl(N)667 5284 y Fo(\()p Fr(x=)-5 b(N)892 5243 y Fj(1)p Fl(=)p Fj(4)1002 5284 y Fr(;)17 b(\030)5 b Fo(\))p Fq(j)27 b(\024)1349 5217 y Fr(C)p 1302 5261 172 4 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Fo(\()p Fr(N)10 b Fo(\))30 b Fq(\\)f Fo(\012)1315 5653 y Fj(2)1355 5638 y Fo(\()p Fr(N)5 b(;)17 b(\016)t Fo(\))p Ft(.)74 b(W)-8 b(e)43 b(ha)m(v)m(e)h(already)f(seen)h(that)e(this)h(implies)d(that)p eop %%Page: 14 14 14 13 bop 118 -51 a Fm(14)1204 b(B.)34 b(GENTZ)g(AND)e(M.)i(L\326)n(WE) 118 126 y Fq(\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)578 89 y Fj(1)p Fl(=)p Fj(4)690 126 y Fo(\))32 b Ft(di\033ers)h(from)924 357 y Fo(\011\()p Fr(x)p Fo(\))28 b(=)g Fq(\000)1375 289 y Fo(1)p 1350 334 98 4 v 1350 425 a(12)1458 357 y Fq(k)p Fr(x)p Fq(k)1613 316 y Fj(4)1613 381 y(4)1674 357 y Fq(\000)1784 289 y Fo(1)p 1784 334 49 4 v 1784 425 a(2)1859 262 y Fk(X)1863 471 y Fl(\026<\027)2020 357 y Fr(x)2075 316 y Fj(2)2075 381 y Fl(\026)2122 357 y Fr(x)2177 316 y Fj(2)2177 381 y Fl(\027)2243 357 y Fo(+)2341 262 y Fk(X)2344 471 y Fl(\026<\027)2501 357 y Fr(\021)2549 372 y Fl(\026;\027)2654 357 y Fr(x)2709 372 y Fl(\026)2757 357 y Fr(x)2812 372 y Fl(\027)3412 357 y Ft(\(4.39\))118 637 y(b)m(y)34 b(at)e(most)g(a)g(constan)m(t)h (times)141 878 y Fr(g)188 893 y Fl(N)255 878 y Fo(\()p Fr(x)p Fo(\))28 b(=)f Fr(\016)560 893 y Fl(M)640 878 y Fq(k)p Fr(x)p Fq(k)795 836 y Fj(2)795 902 y(2)856 878 y Fo(+)954 737 y Fk(\024)1007 801 y Fq(p)p 1090 801 63 4 v 77 x Fr(\013)c Fo(+)f Fr(\016)1320 767 y Fk(\020)1389 810 y Fr(M)1493 774 y Fj(5)p 1389 855 145 4 v 1417 946 a Fr(N)1543 767 y Fk(\021)1603 789 y Fj(1)p Fl(=)p Fj(2)1713 737 y Fk(\025)1765 878 y Fq(k)p Fr(x)p Fq(k)1920 836 y Fj(4)1920 902 y(2)1982 878 y Fo(+)2151 810 y(1)p 2090 855 172 4 v 2090 875 a Fq(p)p 2173 875 89 4 v 84 x Fr(N)2271 737 y Fk(\024)2324 801 y Fq(p)p 2407 801 63 4 v 77 x Fr(\013)h Fo(+)f Fr(\016)2637 767 y Fk(\020)2706 810 y Fr(M)2810 774 y Fj(7)p 2706 855 145 4 v 2734 946 a Fr(N)2860 767 y Fk(\021)2920 789 y Fj(1)p Fl(=)p Fj(2)3030 737 y Fk(\025)3083 878 y Fq(k)p Fr(x)p Fq(k)3238 836 y Fj(6)3238 902 y(2)3299 878 y Fo(+)3407 810 y Fq(k)p Fr(x)p Fq(k)3562 774 y Fj(6)3562 835 y(6)p 3407 855 195 4 v 3419 875 a Fq(p)p 3502 875 89 4 v 84 x Fr(N)3612 878 y(:)3412 1055 y Ft(\(4.40\))118 1298 y FA(The)34 b(inner)e(region.)48 b Ft(F)-8 b(or)28 b Fq(k)p Fr(x)p Fq(k)1336 1313 y Fj(2)1404 1298 y Fq(\024)g Fr(R)q(M)1688 1262 y Fj(1)p Fl(=)p Fj(4)1799 1298 y Ft(,)h(the)g(main)f(con)m (tribution)f(to)i Fr(g)2974 1313 y Fl(N)3041 1298 y Fo(\()p Fr(x)p Fo(\))g Ft(arises)g(from)118 1414 y(the)k(\034rst)g(summand.)43 b(Therefore,)34 b(w)m(e)f(shall)e(use)j(the)f(estimate)894 1654 y Fr(g)941 1669 y Fl(N)1008 1654 y Fo(\()p Fr(x)p Fo(\))28 b Fq(\024)g Fr(h)1328 1669 y Fl(N)1395 1654 y Fo(\()p Fr(\016)n(;)17 b(R)q Fo(\))28 b(=)1762 1544 y Fk(\020)1832 1587 y Fr(M)1936 1551 y Fj(15)p 1832 1632 180 4 v 1877 1723 a Fr(N)2021 1544 y Fk(\021)2081 1566 y Fj(1)p Fl(=)p Fj(2)2191 1654 y Fo(\()p Fr(K)h Fo(+)22 b Fr(\016)t Fo(\))p Fr(R)2599 1613 y Fj(6)2666 1654 y Fq(!)27 b Fo(0)p Fr(;)543 b Ft(\(4.41\))118 1890 y(pro)m(vided)36 b Fr(M)626 1854 y Fj(15)701 1890 y Fr(=)-5 b(N)43 b Fq(!)32 b Fo(0)p Ft(.)51 b(\(Recall)34 b(that)h Fr(\016)1717 1905 y Fl(M)1829 1890 y Fo(=)d Fr(K)7 b(M)2131 1854 y Fj(7)2171 1890 y Fr(=)2220 1805 y Fq(p)p 2303 1805 89 4 v 85 x Fr(N)k Ft(.\))52 b(Therefore,)37 b(the)e(estimate)g(for)118 2006 y(the)e(inner)f(region)g(is)g(immediate:)41 b(F)-8 b(or)31 b Fr(f)39 b Fq(2)28 b Fo(BL\()p Fp(R)2030 1970 y Fl(M)2115 2006 y Fr(;)17 b Fp(R)t Fo(\))p Ft(,)602 2105 y Fk(Z)658 2331 y Fl(B)s Fj(\(0)p Fl(;RM)924 2311 y Fg(1)p Fh(=)p Fg(4)1022 2331 y Fj(\))1070 2241 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1936 2200 y Fj(1)p Fl(=)p Fj(4)2047 2241 y Fo(\))p Fq(g)17 b Fr(dx)1215 2520 y Fo(=)28 b(exp)q Fq(fO)s Fo(\()p Fr(h)1694 2535 y Fl(N)1761 2520 y Fo(\()p Fr(\016)n(;)17 b(R)q Fo(\))p Fq(g)2064 2385 y Fk(Z)2119 2610 y Fl(B)s Fj(\(0)p Fl(;RM)2385 2591 y Fg(1)p Fh(=)p Fg(4)2483 2610 y Fj(\))2531 2520 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx:)68 b Ft(\(4.42\))118 2831 y FA(The)38 b(in)m(termediate)d(region.)48 b Ft(F)-8 b(or)32 b Fr(R)q(M)1738 2794 y Fj(1)p Fl(=)p Fj(4)1876 2831 y Fq(\024)d(k)p Fr(x)p Fq(k)2137 2846 y Fj(2)2204 2831 y Fq(\024)f Fr(r)s(N)2444 2794 y Fj(1)p Fl(=)p Fj(4)2554 2831 y Ft(,)399 3076 y Fr(g)446 3091 y Fl(N)513 3076 y Fo(\()p Fr(x)p Fo(\))g Fq(\024)g Fr(\016)820 3091 y Fl(M)899 3076 y Fq(k)p Fr(x)p Fq(k)1054 3035 y Fj(2)1054 3101 y(2)1116 3076 y Fo(+)1214 2936 y Fk(\024)1267 3076 y Fo(\(1)21 b(+)h Fr(r)1520 3035 y Fj(2)1559 3076 y Fo(\))1597 3000 y Fq(p)p 1680 3000 63 4 v 76 x Fr(\013)h Fo(+)f(\(1)g(+)g Fr(r)2117 3035 y Fj(2)2156 3076 y Fo(\))p Fr(\016)2241 2966 y Fk(\020)2311 3009 y Fr(M)2415 2973 y Fj(7)p 2311 3054 145 4 v 2338 3145 a Fr(N)2465 2966 y Fk(\021)2524 2988 y Fj(1)p Fl(=)p Fj(2)2656 3076 y Fo(+)g Fr(r)2801 3035 y Fj(2)2841 2936 y Fk(\025)2893 3076 y Fq(k)p Fr(x)p Fq(k)3048 3035 y Fj(4)3048 3101 y(2)3088 3076 y Fr(;)297 b Ft(\(4.43\))118 3320 y(whic)m(h)33 b(implies,)d(that)j(there)g (exists)g(an)g Fr(N)1699 3335 y Fj(3)1738 3320 y Fo(\()p Fr(\016)n(;)17 b(r)s Fo(\))28 b Fq(2)g Fp(N)47 b Ft(suc)m(h)34 b(that)1318 3504 y Fr(g)1365 3519 y Fl(N)1432 3504 y Fo(\()p Fr(x)p Fo(\))28 b Fq(\024)g Fr(\016)1739 3519 y Fl(M)1818 3504 y Fq(k)p Fr(x)p Fq(k)1973 3463 y Fj(2)1973 3528 y(2)2035 3504 y Fo(+)22 b(2)p Fr(r)2229 3463 y Fj(2)2267 3504 y Fq(k)p Fr(x)p Fq(k)2422 3463 y Fj(4)2422 3528 y(2)3412 3504 y Ft(\(4.44\))118 3696 y(for)32 b(all)f Fr(N)38 b Fq(\025)28 b Fr(N)702 3711 y Fj(3)741 3696 y Fo(\()p Fr(\016)n(;)17 b(r)s Fo(\))p Ft(,)33 b(pro)m(vided)g(pro)m (vided)g Fr(M)1915 3660 y Fj(7)1955 3696 y Fr(=)-5 b(N)37 b Fq(!)28 b Fo(0)p Ft(.)218 3813 y(Assuming)e Fr(N)38 b Fq(\025)29 b Fo(max)o Fq(f)p Fr(N)1191 3828 y Fj(0)1230 3813 y Fr(;)17 b(N)1352 3828 y Fj(1)1392 3813 y Fr(;)g(N)1514 3828 y Fj(2)1553 3813 y Fo(\()p Fr(\016)t Fo(\))p Fr(;)g(N)1798 3828 y Fj(3)1837 3813 y Fo(\()p Fr(\016)t Fq(g)27 b Ft(and)g Fr(\030)32 b Fq(2)c Fo(\012)2422 3828 y Fj(0)2462 3813 y Fo(\()p Fr(N)5 b(;)17 b(\016)2670 3828 y Fl(N)2737 3813 y Fo(\))11 b Fq(\\)g Fo(\012)2933 3828 y Fj(1)2973 3813 y Fo(\()p Fr(N)f Fo(\))h Fq(\\)g Fo(\012)3295 3828 y Fj(2)3335 3813 y Fo(\()p Fr(N)5 b(;)17 b(\016)t Fo(\))11 b Fq(\\)118 3929 y Fo(\012)188 3944 y Fj(3)228 3929 y Fo(\()p Fr(N)5 b(;)17 b(R)q(;)g(\024)p Fo(\))45 b Ft(from)g(no)m(w)i (on,)i(our)d(previous)g(estimates)f(together)h(with)g(the)g (de\034nition)f(of)118 4045 y Fo(\012)188 4060 y Fj(3)228 4045 y Fo(\()p Fr(N)5 b(;)17 b(R)q(;)g(\024)p Fo(\))32 b Ft(yield)325 4236 y Fq(\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)785 4194 y Fj(1)p Fl(=)p Fj(4)897 4236 y Fo(\))2477 b Ft(\(4.45\))548 4381 y Fq(\024)28 b Fo(\011\()p Fr(x)p Fo(\))23 b(+)f Fq(O)s Fo(\()p Fr(g)1148 4396 y Fl(N)1215 4381 y Fo(\()p Fr(x)p Fo(\)\))548 4575 y Fq(\024)28 b(\000)765 4507 y Fo(1)p 740 4552 98 4 v 740 4643 a(12)848 4575 y Fq(k)p Fr(x)p Fq(k)1003 4534 y Fj(4)1003 4599 y(4)1065 4575 y Fq(\000)1174 4507 y Fo(1)p 1174 4552 49 4 v 1174 4643 a(2)1250 4480 y Fk(X)1254 4689 y Fl(\026<\027)1410 4575 y Fr(x)1465 4534 y Fj(2)1465 4599 y Fl(\026)1512 4575 y Fr(x)1567 4534 y Fj(2)1567 4599 y Fl(\027)1633 4575 y Fo(+)1731 4480 y Fk(X)1735 4689 y Fl(\026<\027)1892 4575 y Fr(\021)1940 4590 y Fl(\026;\027)2045 4575 y Fr(x)2100 4590 y Fl(\026)2147 4575 y Fr(x)2202 4590 y Fl(\027)2268 4575 y Fo(+)22 b Fq(O)s Fo(\()p Fr(\016)2529 4590 y Fl(M)2608 4575 y Fq(k)p Fr(x)p Fq(k)2763 4534 y Fj(2)2763 4599 y(2)2825 4575 y Fo(+)g(2)p Fr(r)3019 4534 y Fj(2)3057 4575 y Fq(k)p Fr(x)p Fq(k)3212 4534 y Fj(4)3212 4599 y(2)3252 4575 y Fo(\))548 4881 y Fq(\024)28 b(\000)765 4814 y Fo(1)p 740 4858 98 4 v 740 4949 a(12)848 4881 y Fq(k)p Fr(x)p Fq(k)1003 4840 y Fj(4)1003 4906 y(4)1065 4881 y Fq(\000)1199 4814 y Fo(1)p 1174 4858 V 1174 4949 a(12)1282 4800 y Fk(\002)1323 4881 y Fq(k)p Fr(x)p Fq(k)1478 4840 y Fj(4)1478 4906 y(2)1540 4881 y Fq(\000)23 b(k)p Fr(x)p Fq(k)1795 4840 y Fj(4)1795 4906 y(4)1834 4800 y Fk(\003)1898 4881 y Fo(+)f Fr(\024R)2127 4840 y Fj(2)2167 4792 y Fq(p)p 2250 4792 105 4 v 89 x Fr(M)10 b Fq(k)p Fr(x)p Fq(k)2509 4840 y Fj(2)2509 4906 y(2)2571 4881 y Fo(+)22 b Fq(O)s Fo(\()p Fr(\016)2832 4896 y Fl(M)2911 4881 y Fq(k)p Fr(x)p Fq(k)3066 4840 y Fj(2)3066 4906 y(2)3128 4881 y Fo(+)g(2)p Fr(r)3322 4840 y Fj(2)3361 4881 y Fq(k)p Fr(x)p Fq(k)3516 4840 y Fj(4)3516 4906 y(2)3555 4881 y Fo(\))p Fr(:)118 5116 y Ft(F)-8 b(or)33 b Fq(k)p Fr(x)p Fq(k)449 5131 y Fj(2)518 5116 y Fq(\025)d Fr(R)q(M)804 5080 y Fj(1)p Fl(=)p Fj(4)915 5116 y Ft(,)k Fq(k)p Fr(x)p Fq(k)1131 5080 y Fj(4)1131 5141 y(2)1200 5116 y Fq(\025)c Fr(R)1382 5080 y Fj(2)1421 5032 y Fq(p)p 1504 5032 V 84 x Fr(M)11 b Fq(k)p Fr(x)p Fq(k)1764 5080 y Fj(2)1764 5141 y(2)1837 5116 y Ft(is)33 b(trivial.)45 b(By)34 b(c)m(ho)s(osing)f Fr(r)j Ft(and)e Fo(0)29 b Fr(<)h(\024)f Fq(\024)h Fo(1)p Fr(=)p Fo(48)118 5233 y Ft(small)k(enough,)39 b(w)m(e)e(see)h(that)f(there)g(exists)g(an)g Fr(N)2015 5248 y Fj(4)2054 5233 y Fo(\()p Fr(R)q(;)17 b(K)7 b Fo(\))35 b Fq(2)g Fp(N)52 b Ft(suc)m(h)38 b(that)e Fr(\016)3066 5248 y Fl(M)3182 5233 y Ft(b)s(ecomes)h(so)118 5349 y(small)30 b(that)1221 5570 y Fq(\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1681 5529 y Fj(1)p Fl(=)p Fj(4)1793 5570 y Fo(\))28 b Fq(\024)g(\000)2051 5502 y Fr(R)2126 5466 y Fj(2)p 2051 5547 115 4 v 2059 5638 a Fo(24)2176 5481 y Fq(p)p 2259 5481 105 4 v 89 x Fr(M)11 b Fq(k)p Fr(x)p Fq(k)2519 5529 y Fj(2)2519 5594 y(2)3412 5570 y Ft(\(4.46\))p eop %%Page: 15 15 15 14 bop 317 -67 a Fm(FLUCTUA)-6 b(TIONS)32 b(IN)g(THE)i(HOPFIELD)e (MODEL)i(A)-6 b(T)33 b(THE)g(CRITICAL)g(TEMPERA)-6 b(TURE)146 b(15)94 123 y Ft(holds)36 b(for)f(all)e Fr(N)44 b Fq(\025)33 b Fr(N)953 138 y Fj(4)993 123 y Fo(\()p Fr(R)q(;)17 b(K)7 b Fo(\))35 b Ft(and)h(all)d Fr(x)j Ft(from)f(the)h(in)m(termediate)e (region.)52 b(Therefore,)37 b(for)94 239 y(all)31 b Fr(f)38 b Fq(2)28 b Fo(BL\()p Fp(R)644 203 y Fl(M)729 239 y Fr(;)17 b Fp(R)5 b Fo(\))38 b Ft(and)33 b Fr(N)43 b Ft(and)33 b Fr(\030)k Ft(c)m(hosen)d(as)f(b)s(efore,)616 336 y Fk(\014)616 396 y(\014)616 456 y(\014)616 516 y(\014)649 345 y(Z)705 571 y Fi(f)p Fl(RM)868 551 y Fg(1)p Fh(=)p Fg(4)965 571 y Fi(\024k)p Fl(x)p Fi(k)1130 580 y Fg(2)1165 571 y Fi(\024)p Fl(r)r(N)1317 551 y Fg(1)p Fh(=)p Fg(4)1414 571 y Fi(g)1470 481 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)p Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)2335 440 y Fj(1)p Fl(=)p Fj(4)2447 481 y Fo(\))p Fq(g)17 b Fr(dx)2658 336 y Fk(\014)2658 396 y(\014)2658 456 y(\014)2658 516 y(\014)1034 771 y Fq(\024)28 b(k)p Fr(f)11 b Fq(k)1298 786 y Fi(1)1389 636 y Fk(Z)1444 861 y Fi(fk)p Fl(x)p Fi(k)1589 870 y Fg(2)1624 861 y Fi(\025)p Fl(RM)1807 841 y Fg(1)p Fh(=)p Fg(4)1905 861 y Fi(g)1961 771 y Fo(exp)2110 661 y Fk(n)2176 771 y Fq(\000)2263 704 y Fr(R)2338 668 y Fj(2)p 2263 748 115 4 v 2271 839 a Fo(24)2388 682 y Fq(p)p 2471 682 105 4 v 89 x Fr(M)g Fq(k)p Fr(x)p Fq(k)2731 730 y Fj(2)2731 796 y(2)2770 661 y Fk(o)2853 771 y Fr(dx)1034 1061 y Fq(\024)28 b(k)p Fr(f)11 b Fq(k)1298 1076 y Fi(1)1389 1061 y Fo(exp)q Fq(f\000)p Fr(R)1740 1020 y Fj(4)1780 1061 y Fr(M)5 b(=)p Fo(48)p Fq(g)2093 926 y Fk(Z)2147 1151 y Fd(R)2195 1132 y Fh(M)2283 1061 y Fo(exp)2432 951 y Fk(n)2499 1061 y Fq(\000)2586 994 y Fr(R)2661 958 y Fj(2)p 2586 1038 115 4 v 2594 1130 a Fo(48)2711 972 y Fq(p)p 2794 972 105 4 v 89 x Fr(M)10 b Fq(k)p Fr(x)p Fq(k)3053 1020 y Fj(2)3053 1086 y(2)3093 951 y Fk(o)3176 1061 y Fr(dx)1034 1357 y Fo(=)28 b Fq(k)p Fr(f)11 b Fq(k)1297 1372 y Fi(1)1387 1357 y Fo(exp)q Fq(f\000)p Fr(R)1738 1316 y Fj(4)1778 1357 y Fr(M)5 b(=)p Fo(48)p Fq(g)2074 1216 y Fk(\022)2230 1289 y Fo(48)p Fr(\031)p 2157 1334 303 4 v 2157 1438 a(R)2232 1409 y Fj(2)2271 1354 y Fq(p)p 2354 1354 105 4 v 84 x Fr(M)2469 1216 y Fk(\023)2542 1238 y Fl(M)s(=)p Fj(2)2688 1357 y Fr(:)674 b Ft(\(4.47\))94 1593 y(This)32 b(b)s(ound)f(will)e(allo)m(w)g(us)j(to)f(deduce)h(that)f (the)g(in)m(tegral)f(o)m(v)m(er)i(the)f(in)m(termediate)f(region)g(is) 94 1709 y(negligible.)94 1930 y FA(The)52 b(outer)f(region.)e Ft(The)c(in)m(v)m(estigation)f(of)g(the)h(outer)g(region)f(consists)i (of)e(t)m(w)m(o)h(parts.)94 2047 y(First,)32 b(w)m(e)i(sho)m(w)g(that)f (there)g(exists)h(an)f Fr(r)1656 2062 y Fj(0)1723 2047 y Fr(>)28 b Fo(0)33 b Ft(suc)m(h)h(that)f(the)g(in)m(tegral)e(o)m(v)m (er)j Fr(B)5 b Fo(\(0)p Fr(;)17 b(r)3327 2062 y Fj(0)3366 2047 y Fr(N)3454 2010 y Fj(1)p Fl(=)p Fj(4)3564 2047 y Fo(\))3602 2010 y Fj(c)94 2163 y Ft(is)32 b(negligible)d(and)j(then,) h(in)e(a)h(second)i(step,)f(w)m(e)g(sho)m(w)g(that)f(this)g Fr(r)2634 2178 y Fj(0)2705 2163 y Ft(can)g(b)s(e)h(replaced)f(b)m(y)h (an)94 2279 y(arbitrarily)d(small)g Fr(r)h(>)c Fo(0)p Ft(.)194 2395 y(F)-8 b(or)33 b(con)m(v)m(enience,)i(w)m(e)g(denote)f(b) m(y)g Fr(f)1575 2410 y Fj(CW)1704 2395 y Fo(\()p Fr(\014)6 b Fo(\))33 b Ft(the)g(free)h(energy)h(in)d(the)i(Curie)8 b(\025)-8 b(W)g(eiss)33 b(mo)s(del)94 2512 y(at)g(temp)s(erature)f Fo(1)p Fr(=\014)6 b Ft(,)32 b(i.)16 b(e.,)1053 2746 y Fr(f)1101 2761 y Fj(CW)1230 2746 y Fo(\()p Fr(\014)6 b Fo(\))27 b(=)h Fq(\000)1585 2678 y Fr(\014)p 1585 2723 61 4 v 1591 2814 a Fo(2)1656 2746 y Fr(z)t Fo(\()p Fr(\014)6 b Fo(\))1842 2704 y Fj(2)1904 2746 y Fo(+)22 b(log)16 b(cosh)q(\()p Fr(\014)6 b(z)t Fo(\()p Fr(\014)g Fo(\)\))p Fr(:)710 b Ft(\(4.48\))94 2955 y(Then,)307 3192 y Fo(log)16 b(cosh)i Fr(x)28 b Fq(\024)879 3124 y Fo(1)p 849 3169 110 4 v 849 3260 a(4)p Fr(\014)968 3192 y(x)1023 3150 y Fj(2)1085 3192 y Fo(+)22 b(max)1214 3254 y Fl(t)p Fi(2)p Fd(R)1365 3051 y Fk(\032)1440 3192 y Fq(\000)1557 3124 y Fo(1)p 1527 3169 V 1527 3260 a(4)p Fr(\014)1646 3192 y(t)1681 3150 y Fj(2)1743 3192 y Fo(+)g(log)17 b(cosh)g Fr(t)2220 3051 y Fk(\033)2323 3192 y Fo(=)2467 3124 y(1)p 2436 3169 V 2436 3260 a(4)p Fr(\014)2556 3192 y(x)2611 3150 y Fj(2)2673 3192 y Fo(+)22 b Fr(f)2819 3207 y Fj(CW)2947 3192 y Fo(\(2)p Fr(\014)6 b Fo(\))p Fr(;)229 b Ft(\(4.49\))94 3427 y(whic)m(h)34 b(implies)c(in)h(particular)g(that)527 3699 y Fq(\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)987 3658 y Fj(1)p Fl(=)p Fj(4)1099 3699 y Fo(\))84 b(=)f Fq(\000)1467 3548 y(p)p 1551 3548 89 4 v 1551 3632 a Fr(N)p 1467 3676 172 4 v 1529 3768 a Fo(2)1649 3699 y Fq(k)p Fr(x)p Fq(k)1804 3658 y Fj(2)1804 3724 y(2)1866 3699 y Fo(+)2004 3575 y Fl(N)1964 3605 y Fk(X)1979 3815 y Fl(i)p Fj(=1)2124 3699 y Fo(log)17 b(cosh)q Fq(h)p Fr(x=)-5 b(N)2678 3658 y Fj(1)p Fl(=)p Fj(4)2788 3699 y Fr(;)17 b(\030)2875 3714 y Fl(i)2902 3699 y Fq(i)1220 4042 y(\024)83 b(\000)1467 3890 y(p)p 1551 3890 89 4 v 1551 3974 a Fr(N)p 1467 4019 172 4 v 1529 4110 a Fo(2)1649 4042 y Fq(k)p Fr(x)p Fq(k)1804 4001 y Fj(2)1804 4066 y(2)1866 4042 y Fo(+)2060 3974 y(1)p 1974 4019 221 4 v 1974 4123 a(4)2023 4039 y Fq(p)p 2106 4039 89 4 v 84 x Fr(N)2261 3917 y Fl(N)2221 3947 y Fk(X)2236 4157 y Fl(i)p Fj(=1)2365 4042 y Fq(h)p Fr(x;)17 b(\030)2546 4057 y Fl(i)2573 4042 y Fq(i)2612 4001 y Fj(2)2674 4042 y Fo(+)22 b Fr(N)27 b(f)2925 4057 y Fj(CW)3053 4042 y Fo(\(2\))p Fr(:)184 b Ft(\(4.50\))94 4322 y(Estimating)29 b(the)h(sum)g(with)g(the)h(help)f(of)g(the)g(b)s(ound)h(\(4.12\))e(on)h (the)h(random)e(matrix)3402 4282 y Fj(1)p 3388 4299 64 4 v 3388 4356 a Fl(N)3461 4322 y Fr(\030)3509 4286 y Fj(T)3563 4322 y Fr(\030)5 b Ft(,)94 4438 y(w)m(e)34 b(see)g(that)e(there)h(exist)g Fr(r)1131 4453 y Fj(0)1198 4438 y Fr(>)28 b Fo(0)k Ft(and)h Fr(N)1651 4453 y Fj(5)1718 4438 y Fq(\025)28 b Fr(N)1901 4453 y Fj(1)1973 4438 y Ft(suc)m(h)34 b(that)1263 4685 y Fq(\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1723 4644 y Fj(1)p Fl(=)p Fj(4)1835 4685 y Fo(\))27 b Fq(\024)i(\000)2093 4533 y(p)p 2176 4533 89 4 v 85 x Fr(N)p 2093 4662 172 4 v 2154 4754 a Fo(6)2275 4685 y Fq(k)p Fr(x)p Fq(k)2430 4644 y Fj(2)2430 4710 y(2)3389 4685 y Ft(\(4.51\))94 4912 y(holds)k(for)f(all)e Fr(x)j Ft(satisfying)f Fq(k)p Fr(x)p Fq(k)1312 4927 y Fj(2)1379 4912 y Fq(\025)c Fr(r)1528 4927 y Fj(0)1567 4912 y Fr(N)1655 4876 y Fj(1)p Fl(=)p Fj(4)1766 4912 y Ft(,)k(all)f Fr(N)38 b Fq(\025)28 b Fr(N)2260 4927 y Fj(5)2332 4912 y Ft(and)33 b(all)d Fr(\030)j Fq(2)28 b Fo(\012)2897 4927 y Fj(1)2937 4912 y Fo(\()p Fr(N)10 b Fo(\))p Ft(.)194 5031 y(Let)33 b(no)m(w)g Fr(r)s(N)707 4995 y Fj(1)p Fl(=)p Fj(4)845 5031 y Fq(\024)28 b(k)p Fr(x)p Fq(k)1105 5046 y Fj(2)1172 5031 y Fq(\024)g Fr(r)1321 5046 y Fj(0)1361 5031 y Fr(N)1449 4995 y Fj(1)p Fl(=)p Fj(4)1592 5031 y Ft(with)k(an)g(arbitrary)g Fr(r)e Fq(2)e Fo(\(0)p Fr(;)17 b(r)2707 5046 y Fj(0)2746 5031 y Fo(\))p Ft(.)44 b(First)31 b(note)i(that)216 5260 y Fo(\010\()p Fr(x=)-5 b(N)511 5219 y Fj(1)p Fl(=)p Fj(4)621 5260 y Fo(\))83 b Fq(\025)h Fp(E)963 5150 y Fk(n)1046 5193 y Fo(1)p 1046 5237 49 4 v 1046 5329 a(2)1104 5260 y Fq(h)p Fr(x=)-5 b(N)1330 5219 y Fj(1)p Fl(=)p Fj(4)1440 5260 y Fr(;)17 b(\030)1527 5275 y Fj(1)1566 5260 y Fq(i)1605 5219 y Fj(2)1666 5260 y Fq(\000)23 b Fo(log)17 b(cosh)p Fq(h)p Fr(x=)-5 b(N)2319 5219 y Fj(1)p Fl(=)p Fj(4)2429 5260 y Fr(;)17 b(\030)2516 5275 y Fj(1)2555 5260 y Fq(i)2594 5150 y Fk(o)3389 5260 y Ft(\(4.52\))925 5546 y Fq(\000)81 b Fo(sup)1024 5632 y Fi(k)p Fl(y)r Fi(k)1131 5641 y Fg(2)1167 5632 y Fi(\024)p Fl(r)1254 5641 y Fg(0)1288 5402 y Fk(\014)1288 5461 y(\014)1288 5521 y(\014)1288 5581 y(\014)1351 5479 y Fo(1)p 1331 5523 89 4 v 1331 5614 a Fr(N)1487 5421 y Fl(N)1446 5451 y Fk(X)1461 5661 y Fl(i)p Fj(=1)1607 5546 y Fo(log)17 b(cosh)p Fq(h)p Fr(x=)-5 b(N)2160 5505 y Fj(1)p Fl(=)p Fj(4)2270 5546 y Fr(;)17 b(\030)2357 5561 y Fl(i)2385 5546 y Fq(i)22 b(\000)g Fp(E)30 b Fo(log)22 b(cosh)q Fq(h)p Fr(x=)-5 b(N)3182 5505 y Fj(1)p Fl(=)p Fj(4)3292 5546 y Fr(;)17 b(\030)3379 5561 y Fj(1)3418 5546 y Fq(i)3457 5402 y Fk(\014)3457 5461 y(\014)3457 5521 y(\014)3457 5581 y(\014)3490 5546 y Fr(:)p eop %%Page: 16 16 16 15 bop 118 -51 a Fm(16)1204 b(B.)34 b(GENTZ)g(AND)e(M.)i(L\326)n(WE) 118 123 y Ft(The)g(\034rst)f(summand)e(on)i(the)g(righ)m(t-hand)e(side) i(is)f(b)s(ounded)h(b)s(elo)m(w)g(b)m(y)1261 286 y Fr(c)1303 301 y Fl(r)n(;r)1385 310 y Fg(0)1451 286 y Fo(=)185 b(inf)1554 353 y Fl(y)14 b Fj(:)e Fl(r)r Fi(\024k)p Fl(y)r Fi(k)1831 362 y Fg(2)1866 353 y Fi(\024)p Fl(r)1953 362 y Fg(0)2004 286 y Fp(E)29 b Fr(\036)p Fo(\()p Fq(h)p Fr(y)t(;)17 b(\030)2355 301 y Fj(1)2398 286 y Fq(i)p Fo(\))p Fr(;)910 b Ft(\(4.53\))118 504 y(where)1147 667 y Fr(\036)p Fo(\()p Fr(t)p Fo(\))28 b(=)f Fr(t)1482 626 y Fj(2)1522 667 y Fr(=)p Fo(2)22 b Fq(\000)g Fo(log)17 b(cosh)g Fr(t;)212 b(t)28 b Fq(2)g Fp(R)5 b Fr(;)803 b Ft(\(4.54\))118 837 y(attains)32 b(its)g(unique)h(minim)m(um)d(at)i Fr(t)c Fo(=)g(0)p Ft(.)44 b(The)34 b(fact)f(that)f Fq(h)p Fr(y)t(;)17 b(\030)2523 852 y Fj(1)2561 837 y Fq(i)32 b Ft(is)g(a)h(\(\034nite\))f (Rademac)m(her)118 953 y(a)m(v)m(erage)i(\(see)f([24,)65 b(Chapter)33 b(I.4],)g(for)f(instance\),)h(implies)d(that)1339 1117 y Fp(P)p Fo(\()p Fq(jh)p Fr(y)t(;)17 b(\030)1642 1132 y Fj(1)1681 1117 y Fq(ij)27 b(\025)1890 1077 y Fj(1)p 1890 1094 36 4 v 1890 1151 a(8)1935 1117 y Fq(k)p Fr(y)t Fq(k)2087 1132 y Fj(2)2125 1117 y Fo(\))h Fr(>)g Fo(1)p Fr(=)p Fo(3)970 b Ft(\(4.55\))118 1286 y(\(cf.)33 b([17)o(,)g(Lemma)e (4.3]\),)h(so)h(that)1171 1450 y Fr(c)1213 1465 y Fl(r)n(;r)1295 1474 y Fg(0)1361 1450 y Fo(=)185 b(inf)1464 1516 y Fl(y)14 b Fj(:)e Fl(r)r Fi(\024k)p Fl(y)r Fi(k)1741 1525 y Fg(2)1776 1516 y Fi(\024)p Fl(r)1863 1525 y Fg(0)1914 1450 y Fp(E)29 b Fr(\036)p Fo(\()p Fq(h)p Fr(y)t(;)17 b(\030)2265 1465 y Fj(1)2308 1450 y Fq(i)p Fo(\))27 b Fr(>)h Fo(0)p Fr(;)820 b Ft(\(4.56\))118 1668 y(b)s(ecause)37 b(there)g(is)e(a)h(set)g(of)g(p) s(ositiv)m(e)f Fp(P)p Ft(-measure,)j(on)e(whic)m(h)h Fr(\036)e Ft(is)h(b)s(ounded)g(a)m(w)m(a)m(y)h(from)e(its)118 1784 y(unique)e(minim)m(um)c(at)j(zero.)218 1900 y(The)k(second)h (summand)e(on)g(the)h(righ)m(t-hand)f(side)g(of)42 b(\(4.52\))35 b(b)s(ecomes)h(small)d(due)j(to)g(so-)118 2016 y(called)42 b(self-a)m(v)m(eraging.)76 b(Insp)s(ection)43 b(of)g(the)h(pro)s(of)f (of)g([17,)j(Lemma)c(4.2])h(sho)m(ws)i(that)f(not)118 2132 y(only)933 2387 y Fo(lim)899 2449 y Fl(N)7 b Fi(!1)1179 2387 y Fo(sup)1119 2473 y Fi(k)p Fl(x)p Fi(k)1229 2482 y Fg(2)1264 2473 y Fi(\024)p Fl(r)1351 2482 y Fg(0)1386 2243 y Fk(\014)1386 2303 y(\014)1386 2363 y(\014)1386 2422 y(\014)1449 2320 y Fo(1)p 1429 2364 89 4 v 1429 2456 a Fr(N)1584 2263 y Fl(N)1544 2293 y Fk(X)1559 2503 y Fl(i)p Fj(=1)1704 2387 y Fr(f)k Fo(\()p Fq(h)p Fr(x;)17 b(\030)1982 2402 y Fl(i)2010 2387 y Fq(i)p Fo(\))k Fq(\000)i Fp(E)13 b Fr(f)e Fo(\()p Fq(h)p Fr(x;)17 b(\030)2547 2402 y Fj(1)2591 2387 y Fq(i)p Fo(\))2668 2243 y Fk(\014)2668 2303 y(\014)2668 2363 y(\014)2668 2422 y(\014)2729 2387 y Fo(=)27 b(0)531 b Ft(\(4.57\))118 2638 y(holds)29 b Fp(P)p Ft(-almost)g(surely)h(for)f(Lipsc)m(hitz)g(con)m(tin)m(uous)h Fr(f)11 b Ft(,)29 b(but)h(w)m(e)g(obtained)f(also)f(b)s(ounds)i(v)-5 b(alid)118 2754 y(for)32 b(large)g(but)g(\034xed)i Fr(N)10 b Ft(:)118 2932 y FA(Lemma)37 b(4.8)h Ft(\([17)o(,)33 b(Lemma)e(4.2]\))p FA(.)48 b Fc(Ther)-5 b(e)31 b(exist)g(a)g(c)-5 b(onstant)31 b Fr(c)c(>)h Fo(0)j Fc(and)g(an)f Fr(N)3153 2947 y Fj(6)3221 2932 y Fq(\025)e Fr(N)3404 2947 y Fj(1)3475 2932 y Fc(such)118 3048 y(that)35 b(for)g(al)5 b(l)34 b Fr(")28 b(>)f Fo(0)35 b Fc(and)f(al)5 b(l)35 b Fr(N)j Fq(\025)28 b Fo(max)o Fq(f)p Fr(N)1732 3063 y Fj(6)1772 3048 y Fr(;)17 b Fo(2)p Fr(=")1960 3012 y Fj(2)1998 3048 y Fq(g)499 3309 y Fp(P)558 3168 y Fk(\032)693 3309 y Fo(sup)635 3395 y Fi(k)p Fl(y)r Fi(k)742 3404 y Fg(2)777 3395 y Fi(\024)p Fl(r)864 3404 y Fg(0)899 3164 y Fk(\014)899 3224 y(\014)899 3284 y(\014)899 3344 y(\014)962 3241 y Fo(1)p 942 3286 V 942 3377 a Fr(N)1097 3184 y Fl(N)1057 3214 y Fk(X)1072 3424 y Fl(i)p Fj(=1)1217 3309 y Fo(log)g(cosh)q Fq(h)p Fr(y)t(;)g(\030)1723 3324 y Fl(i)1749 3309 y Fq(i)22 b(\000)h Fp(E)29 b Fo(log)22 b(cosh)q Fq(h)p Fr(y)t(;)17 b(\030)2498 3324 y Fj(1)2536 3309 y Fq(i)2575 3164 y Fk(\014)2575 3224 y(\014)2575 3284 y(\014)2575 3344 y(\014)2635 3309 y Fq(\025)29 b Fo(\(3)21 b(+)h(2)p Fr(r)3040 3324 y Fj(0)3080 3309 y Fo(\))p Fr(")3164 3168 y Fk(\033)1097 3555 y Fq(\024)28 b Fo(2)17 b(exp)p Fq(f)p Fr(M)10 b Fo(\(log)q(\()p Fr(r)1817 3570 y Fj(0)1856 3555 y Fr(=")p Fo(\))22 b(+)g Fr(c)p Fo(\))p Fq(g)17 b Fo(exp)p Fq(f\000)p Fr(N)10 b(")2665 3514 y Fj(2)2705 3555 y Fr(=)p Fo(8)p Fq(g)22 b Fo(+)g Fp(P)p Fo(\(\012)3140 3570 y Fj(1)3181 3555 y Fo(\()p Fr(N)10 b Fo(\))3345 3514 y Fj(c)3381 3555 y Fo(\))p Fr(:)218 3733 y Ft(With)1578 3859 y Fr(")28 b Fo(=)1897 3791 y Fr(c)1939 3806 y Fl(r)n(;r)2021 3815 y Fg(0)p 1765 3836 426 4 v 1765 3927 a Fo(2\(3)22 b(+)g(2)p Fr(r)2114 3942 y Fj(0)2153 3927 y Fo(\))118 4056 y Ft(and)176 4292 y Fo(\012)246 4307 y Fj(4)286 4292 y Fo(\()p Fr(N)5 b(;)17 b(r)m(;)g(r)580 4307 y Fj(0)619 4292 y Fo(\))27 b(=)788 4151 y Fk(\032)862 4292 y Fr(\030)33 b Fo(:)86 b(sup)992 4377 y Fi(k)p Fl(y)r Fi(k)1099 4386 y Fg(2)1135 4377 y Fi(\024)p Fl(r)1222 4386 y Fg(0)1256 4147 y Fk(\014)1256 4207 y(\014)1256 4267 y(\014)1256 4327 y(\014)1319 4224 y Fo(1)p 1299 4269 89 4 v 1299 4360 a Fr(N)1455 4167 y Fl(N)1414 4197 y Fk(X)1429 4407 y Fl(i)p Fj(=1)1575 4292 y Fo(log)16 b(cosh)q Fq(h)p Fr(y)t(;)h(\030)2080 4307 y Fl(i)2107 4292 y Fq(i)22 b(\000)g Fp(E)30 b Fo(log)22 b(cosh)q Fq(h)p Fr(y)t(;)17 b(\030)2856 4307 y Fj(1)2893 4292 y Fq(i)2932 4147 y Fk(\014)2932 4207 y(\014)2932 4267 y(\014)2932 4327 y(\014)2993 4292 y Fq(\024)3108 4224 y Fr(c)3150 4239 y Fl(r)n(;r)3232 4248 y Fg(0)p 3108 4269 163 4 v 3165 4360 a Fo(2)3280 4151 y Fk(\033)3412 4292 y Ft(\(4.58\))118 4541 y(w)m(e)34 b(obtain)d(the)i(follo)m(wing)d (corollary)-8 b(.)118 4719 y FA(Corollary)37 b(4.9.)49 b Fc(Ther)-5 b(e)33 b(exist)g(a)g(c)-5 b(onstant)33 b Fr(K)7 b Fo(\()p Fr(r)m(;)17 b(r)2071 4734 y Fj(0)2111 4719 y Fo(\))28 b Fr(>)f Fo(0)33 b Fc(and)g(an)g Fr(N)2766 4734 y Fj(7)2806 4719 y Fo(\()p Fr(r)m(;)17 b(r)2973 4734 y Fj(0)3012 4719 y Fo(\))27 b Fq(2)i Fp(N)48 b Fc(such)33 b(that)118 4836 y(for)i(al)5 b(l)34 b Fr(N)k Fq(\025)29 b Fr(N)713 4851 y Fj(7)752 4836 y Fo(\()p Fr(r)m(;)17 b(r)919 4851 y Fj(0)958 4836 y Fo(\))825 4999 y Fp(P)p Fo(\(\012)992 5014 y Fj(4)1034 4999 y Fo(\()p Fr(N)5 b(;)17 b(r)m(;)g(r)1328 5014 y Fj(0)1367 4999 y Fo(\))1405 4958 y Fj(c)1441 4999 y Fo(\))27 b Fq(\024)h Fo(exp)q Fq(f\000)p Fr(K)7 b Fo(\()p Fr(r)m(;)17 b(r)2144 5014 y Fj(0)2184 4999 y Fo(\))p Fr(N)10 b Fq(g)23 b Fo(+)f Fp(P)p Fo(\(\012)2648 5014 y Fj(1)2689 4999 y Fo(\()p Fr(N)10 b Fo(\))2853 4958 y Fj(c)2889 4999 y Fo(\))p Fr(:)218 5177 y Ft(No)m(w,)31 b(b)m(y)g(our)f(estimates)g(on)g(the)h(t) m(w)m(o)g(summands)f(on)g(the)h(righ)m(t-hand)e(side)i(of)36 b(\(4.52\),)30 b(w)m(e)118 5294 y(\034nd)1306 5457 y Fq(\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1766 5416 y Fj(1)p Fl(=)p Fj(4)1877 5457 y Fo(\))28 b Fq(\024)g(\000)p Fr(N)10 b(c)2255 5472 y Fl(r)n(;r)2337 5481 y Fg(0)2376 5457 y Fr(=)p Fo(2)938 b Ft(\(4.59\))118 5638 y(for)26 b(all)e Fr(x)j Ft(suc)m(h)h(that)e Fr(r)s(N)1026 5602 y Fj(1)p Fl(=)p Fj(4)1164 5638 y Fq(\024)i(k)p Fr(x)p Fq(k)1424 5653 y Fj(2)1491 5638 y Fq(\024)g Fr(r)1640 5653 y Fj(0)1680 5638 y Fr(N)1768 5602 y Fj(1)p Fl(=)p Fj(4)1878 5638 y Ft(,)g(all)c Fr(N)38 b Fq(\025)28 b Fr(N)2361 5653 y Fj(7)2401 5638 y Fo(\()p Fr(r)m(;)17 b(r)2568 5653 y Fj(0)2607 5638 y Fo(\))27 b Ft(and)f(all)e Fr(\030)32 b Fq(2)c Fo(\012)3223 5653 y Fj(4)3263 5638 y Fo(\()p Fr(N)5 b(;)17 b(r)m(;)g(r)3557 5653 y Fj(0)3596 5638 y Fo(\))p Ft(.)p eop %%Page: 17 17 17 16 bop 317 -67 a Fm(FLUCTUA)-6 b(TIONS)32 b(IN)g(THE)i(HOPFIELD)e (MODEL)i(A)-6 b(T)33 b(THE)g(CRITICAL)g(TEMPERA)-6 b(TURE)146 b(17)194 123 y Ft(Gathering)31 b(our)i(estimates)f(on)g(the)h(outer)g (region)e(yields)707 236 y Fk(\014)707 296 y(\014)707 355 y(\014)707 415 y(\014)741 245 y(Z)796 470 y Fi(fk)p Fl(x)p Fi(k)941 479 y Fg(2)976 470 y Fi(\025)p Fl(r)r(N)1128 451 y Fg(1)p Fh(=)p Fg(4)1225 470 y Fi(g)1281 380 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)p Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)2146 339 y Fj(1)p Fl(=)p Fj(4)2258 380 y Fo(\))p Fq(g)17 b Fr(dx)2469 236 y Fk(\014)2469 296 y(\014)2469 355 y(\014)2469 415 y(\014)1125 684 y Fq(\024)1230 548 y Fk(Z)1286 774 y Fi(fk)p Fl(x)p Fi(k)1431 783 y Fg(2)1466 774 y Fi(\025)p Fl(r)1553 783 y Fg(0)1587 774 y Fl(N)1650 754 y Fg(1)p Fh(=)p Fg(4)1747 774 y Fi(g)1803 684 y Fq(k)p Fr(f)11 b Fq(k)1962 699 y Fi(1)2053 684 y Fo(exp)2202 543 y Fk(\032)2277 684 y Fq(\000)2364 532 y(p)p 2447 532 89 4 v 85 x Fr(N)p 2364 661 172 4 v 2425 752 a Fo(6)2546 684 y Fq(k)p Fr(x)p Fq(k)2701 643 y Fj(2)2701 709 y(2)2740 543 y Fk(\033)2831 684 y Fr(dx)1225 963 y Fo(+)1323 828 y Fk(Z)1378 1053 y Fi(f)p Fl(r)r(N)1510 1034 y Fg(1)p Fh(=)p Fg(4)1607 1053 y Fi(\024k)p Fl(x)p Fi(k)1772 1062 y Fg(2)1807 1053 y Fi(\024)p Fl(r)1894 1062 y Fg(0)1928 1053 y Fl(N)1991 1034 y Fg(1)p Fh(=)p Fg(4)2088 1053 y Fi(g)2144 963 y Fq(k)p Fr(f)g Fq(k)2303 978 y Fi(1)2394 963 y Fo(exp)q Fq(f\000)p Fr(N)f(c)2800 978 y Fl(r)n(;r)2882 987 y Fg(0)2921 963 y Fr(=)p Fo(2)p Fq(g)17 b Fr(dx)1125 1192 y Fq(\024)28 b(k)p Fr(f)11 b Fq(k)1389 1207 y Fi(1)1463 1111 y Fk(\002)1505 1192 y Fo(exp)q Fq(f\000)p Fr(N)f(r)1916 1151 y Fj(2)1913 1217 y(0)1956 1192 y Fr(=)p Fo(12)p Fq(g)21 b Fo(+)h(exp)q Fq(f\000)p Fr(N)10 b(c)2678 1207 y Fl(r)n(;r)2760 1216 y Fg(0)2799 1192 y Fr(=)p Fo(4)p Fq(g)2947 1111 y Fk(\003)3389 1192 y Ft(\(4.60\))94 1396 y(for)32 b(all)f Fr(N)38 b Fq(\025)28 b Fr(N)678 1411 y Fj(8)718 1396 y Fo(\()p Fr(r)m(;)17 b(r)885 1411 y Fj(0)924 1396 y Fo(\))33 b Ft(for)f(some)g Fr(N)1466 1411 y Fj(8)1505 1396 y Fo(\()p Fr(r)m(;)17 b(r)1672 1411 y Fj(0)1712 1396 y Fo(\))27 b Fq(2)h Fp(N)9 b Ft(.)94 1660 y FA(Completing)36 b(the)h(pro)s(of.)49 b Ft(F)-8 b(rom)31 b(no)m(w)i(on)g(w)m(e)g(shall)f(alw)m(a)m(ys)h (assume)g(that)212 1861 y Fr(\030)f Fq(2)p 381 1781 71 4 v 28 w Fo(\012\()p Fr(N)10 b Fo(\))84 b(=)p 858 1781 V 83 w(\012\()p Fr(N)5 b(;)17 b(R)q(;)g(r)m(;)g(r)1341 1876 y Fj(0)1380 1861 y Fr(;)g(\016)n(;)g(\024)p Fo(\))699 2007 y(=)83 b(\012)928 2022 y Fj(0)967 2007 y Fo(\()p Fr(N)5 b(;)17 b(\016)1175 2022 y Fl(N)1243 2007 y Fo(\))22 b Fq(\\)g Fo(\012)1461 2022 y Fj(1)1501 2007 y Fo(\()p Fr(N)10 b Fo(\))23 b Fq(\\)f Fo(\012)1846 2022 y Fj(2)1886 2007 y Fo(\()p Fr(N)5 b(;)17 b(\016)t Fo(\))22 b Fq(\\)g Fo(\012)2316 2022 y Fj(3)2356 2007 y Fo(\()p Fr(N)5 b(;)17 b(R)q(;)g(\024)p Fo(\))22 b Fq(\\)h Fo(\012)2915 2022 y Fj(4)2955 2007 y Fo(\()p Fr(N)5 b(;)17 b(r)m(;)g(r)3249 2022 y Fj(0)3288 2007 y Fo(\))63 b Ft(\(4.61\))94 2209 y(and)33 b(that)213 2410 y Fr(N)38 b Fq(\025)28 b Fo(max)616 2330 y Fk(\010)674 2410 y Fr(N)752 2425 y Fj(0)791 2410 y Fr(;)17 b(N)913 2425 y Fj(1)953 2410 y Fr(;)g(N)1075 2425 y Fj(2)1114 2410 y Fo(\()p Fr(\016)t Fo(\))p Fr(;)g(N)1359 2425 y Fj(3)1398 2410 y Fo(\()p Fr(\016)n(;)g(r)s Fo(\))p Fr(;)g(N)1728 2425 y Fj(4)1767 2410 y Fo(\()p Fr(R)q(;)g(K)7 b Fo(\))p Fr(;)17 b(N)2174 2425 y Fj(5)2213 2410 y Fr(;)g(N)2335 2425 y Fj(6)2374 2410 y Fr(;)g(N)2496 2425 y Fj(7)2535 2410 y Fo(\()p Fr(r)m(;)g(r)2702 2425 y Fj(0)2742 2410 y Fo(\))p Fr(;)g(N)2902 2425 y Fj(8)2941 2410 y Fo(\()p Fr(r)m(;)g(r)3108 2425 y Fj(0)3147 2410 y Fo(\))3185 2330 y Fk(\011)3243 2410 y Fr(:)119 b Ft(\(4.62\))94 2612 y(Note)33 b(that)f(there)i(exists)f(a)f(constan)m(t)i Fr(L)28 b(>)f Fo(0)32 b Ft(suc)m(h)i(that)1304 2814 y Fp(P)p Fo(\()p 1401 2734 V(\012)r(\()p Fr(N)10 b Fo(\))1637 2773 y Fj(c)1673 2814 y Fo(\))28 b Fq(\024)g Fo(exp)q Fq(f\000)p Fr(M)5 b(=L)p Fq(g)p Fr(;)978 b Ft(\(4.63\))94 3016 y(pro)m(vided)38 b Fr(R)h Ft(is)e(c)m(hosen)i(large)d(compared)h (to)h Fr(\024)f Ft(and)h Fr(M)48 b Ft(is)37 b(large)f(enough,)j(cf.)f (Lemma)e(4.7.)94 3132 y(Naturally)-8 b(,)31 b Fr(L)i Ft(dep)s(ends)h(on)f(our)f(c)m(hoice)h(of)f Fr(R)q Ft(,)h Fr(r)s Ft(,)f Fr(r)2023 3147 y Fj(0)2062 3132 y Ft(,)h Fr(\016)j Ft(and)d Fr(\024)p Ft(.)194 3248 y(Let)g Fr(f)38 b Fq(2)28 b Fo(BL\()p Fp(R)783 3212 y Fl(M)868 3248 y Fr(;)17 b Fp(R)5 b Fo(\))38 b Ft(b)s(e)33 b(arbitrary)-8 b(.)42 b(W)-8 b(e)33 b(ha)m(v)m(e)h(already)e(sho)m(wn)i(that)649 3367 y Fk(Z)765 3503 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1631 3461 y Fj(1)p Fl(=)p Fj(4)1742 3503 y Fo(\))p Fq(g)17 b Fr(dx)1067 3761 y Fo(=)27 b(exp)q Fq(fO)s Fo(\()p Fr(h)1545 3776 y Fl(N)1613 3761 y Fo(\()p Fr(\016)n(;)17 b(R)q Fo(\))p Fq(g)1916 3626 y Fk(Z)1971 3851 y Fl(B)s Fj(\(0)p Fl(;RM)2237 3832 y Fg(1)p Fh(=)p Fg(4)2335 3851 y Fj(\))2383 3761 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx)1165 4080 y Fo(+)22 b Fq(O)1345 3909 y Fk( )1424 4080 y Fq(k)p Fr(f)11 b Fq(k)1583 4095 y Fi(1)1674 4080 y Fo(exp)q Fq(f\000)p Fr(R)2025 4038 y Fj(4)2064 4080 y Fr(M)5 b(=)p Fo(48)p Fq(g)2360 3939 y Fk(\022)2516 4012 y Fo(48)p Fr(\031)p 2443 4057 303 4 v 2443 4161 a(R)2518 4132 y Fj(2)2558 4077 y Fq(p)p 2641 4077 105 4 v 84 x Fr(M)2755 3939 y Fk(\023)2829 3961 y Fl(M)s(=)p Fj(2)2974 3909 y Fk(!)1165 4386 y Fo(+)22 b Fq(O)1345 4245 y Fk(\022)1418 4386 y Fq(k)p Fr(f)11 b Fq(k)1577 4401 y Fi(1)1651 4305 y Fk(\002)1693 4386 y Fo(exp)q Fq(f\000)p Fr(N)f(r)2104 4345 y Fj(2)2101 4410 y(0)2144 4386 y Fr(=)p Fo(12)p Fq(g)21 b Fo(+)h(exp)q Fq(f\000)p Fr(N)10 b(c)2866 4401 y Fl(r)n(;r)2948 4410 y Fg(0)2987 4386 y Fr(=)p Fo(4)p Fq(g)3135 4305 y Fk(\003)3176 4245 y(\023)3389 4386 y Ft(\(4.64\))94 4649 y(with)33 b Fr(h)373 4664 y Fl(N)440 4649 y Fo(\()p Fr(\016)n(;)17 b(R)q Fo(\))28 b Fq(!)f Fo(0)p Ft(.)43 b(Next,)34 b(w)m(e)f(w)m(an)m(t) h(to)e(replace)h(the)f(in)m(tegral)1240 4768 y Fk(Z)1295 4993 y Fl(B)s Fj(\(0)p Fl(;RM)1561 4974 y Fg(1)p Fh(=)p Fg(4)1659 4993 y Fj(\))1707 4903 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx)896 b Ft(\(4.65\))94 5185 y(b)m(y)38 b(the)g(in)m(tegral)d (o)m(v)m(er)j Fp(R)1045 5149 y Fl(M)1130 5185 y Ft(.)57 b(First)36 b(note,)j(that)e(\(4.45\))f(already)g(pro)m(vides)i(an)f (upp)s(er)h(b)s(ound)94 5305 y(on)33 b Fo(\011\()p Fr(x)p Fo(\))p Ft(,)g(v)-5 b(alid)30 b(for)i(all)f Fr(x)i Ft(satisfying)e Fq(k)p Fr(x)p Fq(k)1694 5320 y Fj(2)1762 5305 y Fq(\025)d Fr(R)q(M)2046 5268 y Fj(1)p Fl(=)p Fj(4)2157 5305 y Ft(:)247 5570 y Fo(\011\()p Fr(x)p Fo(\))f Fq(\024)i(\000)698 5502 y Fo(1)p 674 5547 98 4 v 674 5638 a(12)782 5570 y Fq(k)p Fr(x)p Fq(k)937 5529 y Fj(4)937 5594 y(4)998 5570 y Fq(\000)1132 5502 y Fo(1)p 1108 5547 V 1108 5638 a(12)1215 5489 y Fk(\002)1257 5570 y Fq(k)p Fr(x)p Fq(k)1412 5529 y Fj(4)1412 5594 y(2)1474 5570 y Fq(\000)22 b(k)p Fr(x)p Fq(k)1728 5529 y Fj(4)1728 5594 y(4)1768 5489 y Fk(\003)1831 5570 y Fo(+)g Fr(\024R)2060 5529 y Fj(2)2100 5481 y Fq(p)p 2183 5481 105 4 v 89 x Fr(M)11 b Fq(k)p Fr(x)p Fq(k)2443 5529 y Fj(2)2443 5594 y(2)2510 5570 y Fq(\024)28 b(\000)2702 5502 y Fr(R)2777 5466 y Fj(2)p 2702 5547 115 4 v 2710 5638 a Fo(24)2827 5481 y Fq(p)p 2910 5481 105 4 v 89 x Fr(M)11 b Fq(k)p Fr(x)p Fq(k)3170 5529 y Fj(2)3170 5594 y(2)3209 5570 y Fr(:)153 b Ft(\(4.66\))p eop %%Page: 18 18 18 17 bop 118 -51 a Fm(18)1204 b(B.)34 b(GENTZ)g(AND)e(M.)i(L\326)n(WE) 118 123 y Ft(As)f(an)g(immediate)c(consequence,)801 227 y Fk(\014)801 287 y(\014)801 347 y(\014)801 407 y(\014)834 236 y(Z)890 462 y Fi(fk)p Fl(x)p Fi(k)1035 471 y Fg(2)1070 462 y Fi(\025)p Fl(RM)1253 442 y Fg(1)p Fh(=)p Fg(4)1350 462 y Fi(g)1406 372 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx)2192 227 y Fk(\014)2192 287 y(\014)2192 347 y(\014)2192 407 y(\014)1219 662 y Fq(\024)28 b(k)p Fr(f)11 b Fq(k)1483 677 y Fi(1)1574 527 y Fk(Z)1629 752 y Fi(fk)p Fl(x)p Fi(k)1774 761 y Fg(2)1809 752 y Fi(\025)p Fl(RM)1992 732 y Fg(1)p Fh(=)p Fg(4)2090 752 y Fi(g)2146 662 y Fo(exp)2295 552 y Fk(n)2361 662 y Fq(\000)2448 595 y Fr(R)2523 559 y Fj(2)p 2448 639 115 4 v 2456 730 a Fo(24)2573 573 y Fq(p)p 2656 573 105 4 v 89 x Fr(M)g Fq(k)p Fr(x)p Fq(k)2916 621 y Fj(2)2916 687 y(2)2955 552 y Fk(o)3038 662 y Fr(dx)1219 974 y Fq(\024)28 b(k)p Fr(f)11 b Fq(k)1483 989 y Fi(1)1574 974 y Fo(exp)q Fq(f\000)p Fr(R)1925 933 y Fj(4)1965 974 y Fr(M)5 b(=)p Fo(48)p Fq(g)2261 834 y Fk(\022)2416 907 y Fo(48)p Fr(\031)p 2343 951 303 4 v 2343 1055 a(R)2418 1027 y Fj(2)2458 971 y Fq(p)p 2541 971 105 4 v 84 x Fr(M)2656 834 y Fk(\023)2729 856 y Fl(M)s(=)p Fj(2)2875 974 y Fr(;)510 b Ft(\(4.67\))118 1229 y(whic)m(h)33 b(implies)d(b)m(y)35 b(\(4.64\))d(that)659 1339 y Fk(Z)775 1475 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1641 1433 y Fj(1)p Fl(=)p Fj(4)1753 1475 y Fo(\))p Fq(g)17 b Fr(dx)1077 1733 y Fo(=)27 b(exp)r Fq(fO)s Fo(\()p Fr(h)1556 1748 y Fl(N)1623 1733 y Fo(\()p Fr(\016)n(;)17 b(R)q Fo(\))p Fq(g)1926 1598 y Fk(Z)1981 1823 y Fd(R)2029 1804 y Fh(M)2117 1733 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)p Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx)1175 2035 y Fo(+)22 b Fq(O)1355 1865 y Fk( )1434 2035 y Fq(k)p Fr(f)11 b Fq(k)1593 2050 y Fi(1)1684 2035 y Fo(exp)q Fq(f\000)p Fr(R)2035 1994 y Fj(4)2074 2035 y Fr(M)5 b(=)p Fo(48)p Fq(g)2370 1895 y Fk(\022)2526 1968 y Fo(48)p Fr(\031)p 2453 2012 303 4 v 2453 2116 a(R)2528 2088 y Fj(2)2568 2032 y Fq(p)p 2651 2032 105 4 v 84 x Fr(M)2765 1895 y Fk(\023)2839 1917 y Fl(M)s(=)p Fj(2)2984 1865 y Fk(!)1175 2341 y Fo(+)22 b Fq(O)1355 2201 y Fk(\022)1428 2341 y Fq(k)p Fr(f)11 b Fq(k)1587 2356 y Fi(1)1661 2261 y Fk(\002)1703 2341 y Fo(exp)q Fq(f\000)p Fr(N)f(r)2114 2300 y Fj(2)2111 2366 y(0)2154 2341 y Fr(=)p Fo(12)p Fq(g)21 b Fo(+)h(exp)q Fq(f\000)p Fr(N)10 b(c)2876 2356 y Fl(r)n(;r)2958 2365 y Fg(0)2997 2341 y Fr(=)p Fo(4)p Fq(g)3145 2261 y Fk(\003)3186 2201 y(\023)3259 2341 y Fr(:)126 b Ft(\(4.68\))118 2590 y(In)33 b(order)g(to)f(compare)577 2700 y Fk(R)624 2815 y Fd(R)672 2796 y Fh(M)760 2781 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1626 2744 y Fj(1)p Fl(=)p Fj(4)1737 2781 y Fo(\))p Fq(g)p 577 2831 1249 4 v 619 2843 a Fk(R)666 2958 y Fd(R)714 2939 y Fh(M)802 2923 y Fo(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1461 2895 y Fj(1)p Fl(=)p Fj(4)1573 2923 y Fo(\))p Fq(g)17 b Fr(dx)2030 2854 y Ft(to)2322 2700 y Fk(R)2369 2815 y Fd(R)2417 2796 y Fh(M)2505 2781 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)p Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)p 2322 2831 846 4 v 2364 2843 a Fk(R)2411 2958 y Fd(R)2459 2939 y Fh(M)2547 2923 y Fo(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx)3186 2854 y(;)118 3124 y Ft(w)m(e)36 b(need)h(a)e(lo)m(w)m(er)g(b)s(ound)h(on)1278 3043 y Fk(R)1325 3158 y Fd(R)1373 3139 y Fh(M)1461 3124 y Fo(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)17 b Fr(dx)p Ft(.)52 b(T)-8 b(o)35 b(obtain)f(a)h(lo)m(w)m(er)h(b)s(ound)f (on)h Fo(\011)f Ft(\034rst,)118 3241 y(w)m(e)f(pro)s(ceed)f(as)g(in)f (\(4.45\))o(:)208 3483 y Fo(\011\()p Fr(x)p Fo(\))c Fq(\025)g(\000)660 3416 y Fo(1)p 635 3460 98 4 v 635 3551 a(12)743 3483 y Fq(k)p Fr(x)p Fq(k)898 3442 y Fj(4)898 3508 y(4)959 3483 y Fq(\000)1069 3416 y Fo(1)p 1069 3460 49 4 v 1069 3551 a(4)1128 3402 y Fk(\002)1169 3483 y Fq(k)p Fr(x)p Fq(k)1324 3442 y Fj(4)1324 3508 y(2)1386 3483 y Fq(\000)23 b(k)p Fr(x)p Fq(k)1641 3442 y Fj(4)1641 3508 y(4)1680 3402 y Fk(\003)1744 3483 y Fq(\000)f Fr(\024R)1974 3442 y Fj(2)2014 3394 y Fq(p)p 2097 3394 105 4 v 89 x Fr(M)11 b Fq(k)p Fr(x)p Fq(k)2357 3442 y Fj(2)2357 3508 y(2)2424 3483 y Fq(\025)28 b(\000)2616 3416 y Fo(1)p 2616 3460 49 4 v 2616 3551 a(4)2675 3483 y Fq(k)p Fr(x)p Fq(k)2830 3442 y Fj(4)2830 3508 y(2)2892 3483 y Fq(\000)23 b Fr(\024R)3123 3442 y Fj(2)3162 3394 y Fq(p)p 3245 3394 105 4 v 89 x Fr(M)11 b Fq(k)p Fr(x)p Fq(k)3505 3442 y Fj(2)3505 3508 y(2)3545 3483 y Fr(:)3412 3634 y Ft(\(4.69\))118 3845 y(F)-8 b(or)32 b Fq(k)p Fr(x)p Fq(k)448 3860 y Fj(2)515 3845 y Fq(\024)c Fr(R)q(M)799 3809 y Fj(1)p Fl(=)p Fj(4)910 3845 y Ft(,)1423 4101 y Fo(\011\()p Fr(x)p Fo(\))f Fq(\025)i(\000)1850 4034 y Fr(R)1925 3998 y Fj(2)p 1850 4078 115 4 v 1883 4170 a Fo(3)1975 4012 y Fq(p)p 2058 4012 105 4 v 89 x Fr(M)10 b Fq(k)p Fr(x)p Fq(k)2317 4060 y Fj(2)2317 4126 y(2)3412 4101 y Ft(\(4.70\))118 4330 y(follo)m(ws.)42 b(\(Recall,)31 b(that)h Fr(\024)c Fq(\024)g Fo(1)p Fr(=)p Fo(48)p Ft(.\))43 b(No)m(w,)172 4472 y Fk(Z)228 4698 y Fd(R)276 4679 y Fh(M)364 4608 y Fo(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)17 b Fr(dx)27 b Fq(\025)1075 4472 y Fk(Z)1131 4698 y Fl(B)s Fj(\(0)p Fl(;RM)1397 4678 y Fg(1)p Fh(=)p Fg(4)1495 4698 y Fj(\))1543 4608 y Fo(exp)1692 4497 y Fk(n)1758 4608 y Fq(\000)1845 4541 y Fr(R)1920 4505 y Fj(2)p 1845 4585 115 4 v 1879 4676 a Fo(3)1970 4519 y Fq(p)p 2053 4519 105 4 v 89 x Fr(M)11 b Fq(k)p Fr(x)p Fq(k)2313 4567 y Fj(2)2313 4633 y(2)2353 4497 y Fk(o)2436 4608 y Fr(dx)27 b Fq(\025)2685 4541 y Fo(1)p 2685 4585 49 4 v 2685 4676 a(2)2743 4468 y Fk(\022)2924 4541 y Fo(3)p Fr(\031)p 2827 4585 303 4 v 2827 4689 a(R)2902 4661 y Fj(2)2941 4605 y Fq(p)p 3024 4605 105 4 v 84 x Fr(M)3139 4468 y Fk(\023)3212 4490 y Fl(M)s(=)p Fj(2)3412 4608 y Ft(\(4.71\))118 4875 y(for)32 b Fr(M)43 b Ft(large)32 b(enough,)h(i.)16 b(e.,)32 b Fr(N)38 b Fq(\025)29 b Fr(N)1506 4890 y Fj(9)1545 4875 y Fo(\()p Fr(R)q Fo(\))k Ft(for)f(some)g Fr(N)2200 4890 y Fj(9)2240 4875 y Fo(\()p Fr(R)q Fo(\))27 b Fq(2)h Fp(N)9 b Ft(.)218 4991 y(With)32 b(these)h(preparations,)g(it) e(is)h(easy)i(to)e(see)i(that)812 5096 y Fk(\014)812 5155 y(\014)812 5215 y(\014)812 5275 y(\014)812 5335 y(\014)855 5116 y(R)902 5231 y Fd(R)950 5212 y Fh(M)1038 5197 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1904 5160 y Fj(1)p Fl(=)p Fj(4)2015 5197 y Fo(\))p Fq(g)p 855 5247 1249 4 v 897 5259 a Fk(R)944 5374 y Fd(R)992 5355 y Fh(M)1080 5339 y Fo(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1739 5311 y Fj(1)p Fl(=)p Fj(4)1851 5339 y Fo(\))p Fq(g)17 b Fr(dx)2135 5270 y Fq(\000)2245 5116 y Fk(R)2292 5231 y Fd(R)2340 5212 y Fh(M)2428 5197 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)p 2245 5247 846 4 v 2287 5259 a Fk(R)2334 5374 y Fd(R)2382 5355 y Fh(M)2470 5339 y Fo(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx)3100 5096 y Fk(\014)3100 5155 y(\014)3100 5215 y(\014)3100 5275 y(\014)3100 5335 y(\014)937 5567 y Fq(\024)28 b(k)p Fr(f)11 b Fq(k)1201 5582 y Fi(1)1726 5500 y Fq(O)p 1285 5544 964 4 v 1285 5556 a Fk(R)1332 5671 y Fd(R)1380 5652 y Fh(M)1468 5637 y Fo(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)17 b Fr(dx)22 b Fo(+)g Fq(O)2268 5567 y Fr(;)1117 b Ft(\(4.72\))p eop %%Page: 19 19 19 18 bop 317 -67 a Fm(FLUCTUA)-6 b(TIONS)32 b(IN)g(THE)i(HOPFIELD)e (MODEL)i(A)-6 b(T)33 b(THE)g(CRITICAL)g(TEMPERA)-6 b(TURE)146 b(19)94 123 y Ft(where)34 b(w)m(e)g(use)f Fq(O)j Ft(as)d(an)f (abbreviation)f(for)431 356 y Fq(O)513 215 y Fk(\022)587 356 y Fr(h)643 371 y Fl(N)710 356 y Fo(\()p Fr(\016)n(;)17 b(R)q Fo(\))963 220 y Fk(Z)1018 446 y Fd(R)1066 427 y Fh(M)1154 356 y Fo(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx)1733 215 y Fk(\023)1828 356 y Fo(+)22 b Fq(O)2008 185 y Fk( )2087 356 y Fo(exp)q Fq(f\000)p Fr(R)2438 315 y Fj(4)2478 356 y Fr(M)5 b(=)p Fo(48)p Fq(g)2774 215 y Fk(\022)2930 288 y Fo(48)p Fr(\031)p 2857 333 303 4 v 2857 437 a(R)2932 408 y Fj(2)2971 353 y Fq(p)p 3054 353 105 4 v 84 x Fr(M)3169 215 y Fk(\023)3243 237 y Fl(M)s(=)p Fj(2)3388 185 y Fk(!)648 662 y Fo(+)22 b Fq(O)828 522 y Fk(\022)902 662 y Fo(exp)q Fq(f\000)p Fr(N)10 b(r)1313 621 y Fj(2)1310 687 y(0)1353 662 y Fr(=)p Fo(12)p Fq(g)21 b Fo(+)h(exp)q Fq(f\000)p Fr(N)10 b(c)2075 677 y Fl(r)n(;r)2157 686 y Fg(0)2196 662 y Fr(=)p Fo(4)p Fq(g)2344 522 y Fk(\023)2417 662 y Fr(:)94 896 y Ft(By)28 b(our)g(lo)m(w)m(er)f(b)s(ound)h(on)1086 815 y Fk(R)1133 930 y Fd(R)1181 911 y Fh(M)1269 896 y Fo(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)17 b Fr(dx)p Ft(,)28 b(w)m(e)h(see)f(that)f Fr(R)i Ft(can)f(b)s(e)f(c)m (hosen)i(so)f(large)e(that)94 1026 y(there)34 b(exist)f(a)f(constan)m (t)p 1049 946 91 4 v 33 w Fr(K)j(>)28 b Fo(0)k Ft(and)g(an)h Fr(N)1755 1041 y Fj(10)1830 1026 y Fo(\()p Fr(R)q(;)17 b(r)m(;)g(r)2116 1041 y Fj(0)2155 1026 y Fr(;)g(\016)n(;)g(\024)p Fo(\))27 b Fq(2)i Fp(N)47 b Ft(suc)m(h)34 b(that)788 1110 y Fk(\014)788 1170 y(\014)788 1230 y(\014)788 1289 y(\014)788 1349 y(\014)831 1131 y(R)879 1245 y Fd(R)927 1227 y Fh(M)1015 1211 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)p Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1880 1175 y Fj(1)p Fl(=)p Fj(4)1992 1211 y Fo(\))p Fq(g)p 831 1261 1249 4 v 873 1273 a Fk(R)920 1388 y Fd(R)968 1369 y Fh(M)1057 1354 y Fo(exp)q Fq(f\000)p Fr(N)10 b Fo(\010\()p Fr(x=)-5 b(N)1716 1325 y Fj(1)p Fl(=)p Fj(4)1827 1354 y Fo(\))p Fq(g)17 b Fr(dx)2112 1284 y Fq(\000)2221 1131 y Fk(R)2268 1245 y Fd(R)2316 1227 y Fh(M)2404 1211 y Fr(f)11 b Fo(\()p Fr(x)p Fo(\))17 b(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)p 2221 1261 846 4 v 2263 1273 a Fk(R)2310 1388 y Fd(R)2358 1369 y Fh(M)2446 1354 y Fo(exp)q Fq(f)p Fo(\011\()p Fr(x)p Fo(\))p Fq(g)g Fr(dx)3077 1110 y Fk(\014)3077 1170 y(\014)3077 1230 y(\014)3077 1289 y(\014)3077 1349 y(\014)913 1534 y Fq(\024)29 b(k)p Fr(f)11 b Fq(k)1178 1549 y Fi(1)1252 1454 y Fk(\002)1293 1534 y Fq(O)s Fo(\()p Fr(h)1469 1549 y Fl(N)1537 1534 y Fo(\()p Fr(\016)n(;)17 b(R)q Fo(\)\))22 b(+)g Fq(O)s Fo(\(exp)q Fq(f\000)p Fr(R)2402 1493 y Fj(4)2442 1534 y Fr(M)5 b(=)p 2590 1454 91 4 v(K)i Fq(g)p Fo(\))2768 1454 y Fk(\003)3389 1534 y Ft(\(4.73\))94 1702 y(for)51 b(all)e Fr(N)69 b Fq(\025)60 b Fr(N)778 1717 y Fj(10)853 1702 y Fo(\()p Fr(R)q(;)17 b(r)m(;)g(r)1139 1717 y Fj(0)1178 1702 y Fr(;)g(\016)n(;)g(\024)p Fo(\))p Ft(.)99 b(No)m(w)51 b(the)h(theorem)e(follo)m(ws)g(from)g(Lemma)f(4.2)i (and)94 1824 y(Lemma)32 b(4.4)g(with)p 822 1744 71 4 v 32 w Fo(\012\()p Fr(N)10 b Fo(\))33 b Ft(as)g(de\034ned)h(in)e(the)h (b)s(eginning)e(of)h(this)g(subsection)h(and)193 1995 y Fr(N)38 b Fq(\025)p 414 1915 89 4 v 28 w Fr(N)94 b Fo(=)p 745 1915 V 83 w Fr(N)10 b Fo(\()p Fr(R)q(;)17 b(r)m(;)g(r)1119 2010 y Fj(0)1158 1995 y Fr(;)g(\016)n(;)g(\024)p Fo(\))586 2142 y(=)83 b(max)926 2061 y Fk(\010)984 2142 y Fr(N)1062 2157 y Fj(0)1102 2142 y Fr(;)17 b(N)1224 2157 y Fj(1)1263 2142 y Fr(;)g(N)1385 2157 y Fj(2)1424 2142 y Fo(\()p Fr(\016)t Fo(\))p Fr(;)g(N)1669 2157 y Fj(3)1708 2142 y Fo(\()p Fr(\016)n(;)g(r)s Fo(\))p Fr(;)g(N)2038 2157 y Fj(4)2077 2142 y Fo(\()p Fr(R)q(;)g(K)7 b Fo(\))p Fr(;)17 b(N)2484 2157 y Fj(5)2524 2142 y Fr(;)g(N)2646 2157 y Fj(6)2685 2142 y Fr(;)g(N)2807 2157 y Fj(7)2846 2142 y Fo(\()p Fr(r)m(;)g(r)3013 2157 y Fj(0)3052 2142 y Fo(\))p Fr(;)g(N)3212 2157 y Fj(8)3251 2142 y Fo(\()p Fr(r)m(;)g(r)3418 2157 y Fj(0)3458 2142 y Fo(\))p Fr(;)984 2299 y(N)1062 2314 y Fj(9)1102 2299 y Fo(\()p Fr(R)q Fo(\))p Fr(;)g(N)1375 2314 y Fj(10)1449 2299 y Fo(\()p Fr(R)q(;)g(r)m(;)g(r)1735 2314 y Fj(0)1774 2299 y Fr(;)g(\016)n(;)g (\024)p Fo(\))1997 2218 y Fk(\011)2056 2299 y Fr(:)1306 b Ft(\(4.74\))p 3567 2466 4 66 v 3571 2403 59 4 v 3571 2466 V 3629 2466 4 66 v 1574 2691 a Fs(References)136 2849 y Fz(1.)41 b(D.)14 b(J.)28 b(Amit,)i(H.)e(Gutfreund,)i(and)e(H.)g (Somp)r(olinsky)-7 b(,)28 b Fn(Statistic)l(al)j(me)l(chanics)g(of)g (neur)l(al)f(networks)h(ne)l(ar)242 2948 y(satur)l(ation)p Fz(,)d(Ann.)g(Ph)n(ys.)f Fb(173)g Fz(\(1987\),)f(30)h(\025)g(67.)136 3048 y(2.)41 b(R.)14 b(N.)26 b(Bhattac)n(hary)n(a)c(and)j(R.)g(Ranga)f (Rao,)h Fn(Normal)j(appr)l(oximation)h(and)f(asymptotic)h(exp)l(ansion) p Fz(,)d(Wi-)242 3147 y(ley)-7 b(,)28 b(New)f(Y)-7 b(ork,)27 b(1976.)136 3247 y(3.)41 b(A.)34 b(Bo)n(vier)f(and)h(V.)g(Ga)n(yrard,)g Fn(A)n(n)g(almost)j(sur)l(e)e(lar)l(ge)h(deviation)h(principle)h(for)e (the)g(Hop\034eld)h(mo)l(del)p Fz(,)242 3347 y(Ann.)28 b(Probab.)f Fb(24)g Fz(\(1996\),)g(1444\0251475.)136 3446 y(4.)p 242 3446 250 4 v 304 w(,)f Fn(A)n(n)i(almost)g(sur)l(e)g(c) l(entr)l(al)g(limit)g(the)l(or)l(em)h(for)g(the)f(Hop\034eld)h(mo)l (del)p Fz(,)f(Mark)n(o)n(v)23 b(Pro)r(cesses)i(Relat.)242 3546 y(Fields)j Fb(3)f Fz(\(1997\),)f(151\025173.)136 3646 y(5.)p 242 3646 V 304 w(,)g Fn(The)i(r)l(etrieval)g(phase)h(of)f (the)f(Hop\034eld)i(mo)l(del:)38 b(A)27 b(rigor)l(ous)h(analysis)h(of)f (the)f(overlap)j(distribu-)242 3745 y(tion)p Fz(,)e(Probab.)f(Theory)f (Related)i(Fields)f Fb(107)g Fz(\(1997\),)g(61\02598.)136 3845 y(6.)p 242 3845 V 304 w(,)35 b Fn(Hop\034eld)g(mo)l(dels)h(as)f (gener)l(alize)l(d)h(r)l(andom)g(me)l(an)e(\034eld)i(mo)l(dels)p Fz(,)f(Mathematical)e(Asp)r(ects)g(of)242 3944 y(Spin)38 b(Glasses)d(and)i(Neural)g(Net)n(w)n(orks)e(\(A.)j(Bo)n(vier)d(and)i(P) -7 b(.)37 b(Picco,)j(eds.\),)f(Progress)c(in)i(Probabilit)n(y)-7 b(,)242 4044 y(Birkh\344user,)26 b(Boston,)g(1998,)g(pp.)i(3\02589.)136 4144 y(7.)41 b(A.)23 b(Bo)n(vier,)g(V.)g(Ga)n(yrard,)f(and)g(P)-7 b(.)24 b(Picco,)g Fn(Gibbs)i(states)f(of)i(the)e(Hop\034eld)i(mo)l(del) f(in)g(the)f(r)l(e)l(gime)h(of)g(p)l(erfe)l(ct)242 4243 y(memory)p Fz(,)i(Probab.)f(Theory)g(Related)g(Fields)h Fb(100)f Fz(\(1994\),)f(329\025363.)136 4343 y(8.)p 242 4343 V 304 w(,)f Fn(Gibbs)i(states)e(of)i(the)g(Hop\034eld)g(mo)l(del)g (with)g(extensively)f(many)h(p)l(atterns)p Fz(,)d(J.)f(Statist.)i(Ph)n (ys.)e Fb(79)242 4443 y Fz(\(1995\),)j(395\025414.)136 4542 y(9.)p 242 4542 V 304 w(,)c Fn(L)l(ar)l(ge)i(deviation)i (principles)f(for)g(the)f(Hop\034eld)g(mo)l(del)h(and)f(the)g (Kac\025Hop\034eld)h(mo)l(del)p Fz(,)f(Probab.)242 4642 y(Theory)i(Related)i(Fields)g Fb(101)f Fz(\(1995\),)f(511\025546.)94 4741 y(10.)41 b(A.)25 b(Bo)n(vier)d(and)i(D.)g(Mason,)g Fn(Extr)l(eme)i(value)h(b)l(ehaviour)h(in)f(the)g(Hop\034eld)g(mo)l (del)p Fz(,)f(preprin)n(t,)e(preliminary)242 4841 y(v)n(ersion,)i (1998.)94 4941 y(11.)41 b(A.)i(Bo)n(vier)d(and)j(P)-7 b(.)43 b(Picco)f(\(eds.\),)k Fn(Mathematic)l(al)f(asp)l(e)l(cts)f(of)g (spin)g(glasses)g(and)g(neur)l(al)f(networks)p Fz(,)242 5040 y(Progress)26 b(in)h(Probabilit)n(y)-7 b(,)27 b(Boston,)g (Birkh\344user,)f(1998.)94 5140 y(12.)41 b(U.)32 b(Einmahl,)h Fn(Extensions)g(of)h(a)g(r)l(esult)f(by)g(Koml\363s,)j(Major,)g(and)e (T)-6 b(usn\341dy)34 b(to)f(the)g(multidimensional)242 5240 y(c)l(ase)p Fz(,)28 b(J.)g(Multiv)-5 b(ariate)27 b(Anal.)g Fb(28)h Fz(\(1989\),)e(20\02568.)94 5339 y(13.)41 b(U.)c(Einmahl)f(and)g(D.)h(Mason,)h Fn(Gaussian)g(appr)l(oximation)h (of)g(lo)l(c)l(al)f(empiric)l(al)i(pr)l(o)l(c)l(esses)e(indexe)l(d)g (by)242 5439 y(functions)p Fz(,)28 b(Probab.)f(Theory)f(Related)i (Fields)f Fb(107)g Fz(\(1997\),)f(283\025311.)94 5538 y(14.)41 b(R.)14 b(S.)31 b(Ellis,)h Fn(Entr)l(opy,)i(lar)l(ge)f (deviations,)i(and)e(statistic)l(al)g(me)l(chanics)p Fz(,)g(Grundlehren)d(der)g(mathemati-)242 5638 y(sc)n(hen)d(Wissensc)n (haften,)g(v)n(ol.)g(271,)f(Springer,)h(New)h(Y)-7 b(ork,)26 b(1985.)p eop %%Page: 20 20 20 19 bop 118 -51 a Fm(20)1204 b(B.)34 b(GENTZ)g(AND)e(M.)i(L\326)n(WE) 118 123 y Fz(15.)41 b(R.)14 b(S.)27 b(Ellis)g(and)g(C.)14 b(M.)27 b(Newman,)h Fn(Limit)h(the)l(or)l(ems)h(for)g(sums)e(of)i(dep)l (endent)g(r)l(andom)g(variables)h(o)l(c)l(cur-)266 222 y(ring)e(in)g(statistic)l(al)h(me)l(chanics)p Fz(,)e(Z.)f(W)-7 b(ahrsc)n(heinlic)n(hk)n(eitstheorie)24 b(v)n(erw.)i(Gebiete)i Fb(44)e Fz(\(1978\),)g(117\025139.)118 322 y(16.)41 b(B.)d(Gen)n(tz,)j Fn(A)n(n)e(almost)h(sur)l(e)f(c)l(entr)l(al)g(limit)h(the)l(or)l(em)g (for)h(the)e(overlap)j(p)l(ar)l(ameters)e(in)g(the)f(Hop\034eld)266 422 y(mo)l(del)p Fz(,)28 b(Sto)r(c)n(hastic)f(Pro)r(cess.)g(Appl.)h Fb(62)f Fz(\(1996\),)g(243\025262.)118 521 y(17.)p 266 521 250 4 v 304 w(,)c Fn(A)i(c)l(entr)l(al)g(limit)g(the)l(or)l(em)h (for)g(the)f(overlap)i(in)e(the)g(Hop\034eld)h(mo)l(del)p Fz(,)f(Ann.)e(Probab.)f Fb(24)g Fz(\(1996\),)266 621 y(1809\0251841.)118 720 y(18.)p 266 720 V 304 w(,)h Fn(A)h(c)l(entr)l (al)g(limit)h(the)l(or)l(em)g(for)g(the)g(overlap)h(in)f(the)g (Hop\034eld)g(mo)l(del)p Fz(,)f(Ph.D.)f(thesis,)g(Univ)n(ersit\344t)266 820 y(Z\374ric)n(h,)j(Switzerland,)i(1996.)118 920 y(19.)p 266 920 V 304 w(,)41 b Fn(On)e(the)h(c)l(entr)l(al)g(limit)h(the)l(or)l (em)f(for)h(the)f(overlap)h(in)g(the)f(Hop\034eld)h(mo)l(del)p Fz(,)h(Mathematical)266 1019 y(Asp)r(ects)e(of)f(Spin)h(Glasses)e(and)i (Neural)f(Net)n(w)n(orks)f(\(A.)i(Bo)n(vier)e(and)h(P)-7 b(.)40 b(Picco,)j(eds.\),)g(Progress)37 b(in)266 1119 y(Probabilit)n(y)-7 b(,)27 b(Birkh\344user,)e(Boston,)i(1998,)f(pp.)i (115\025149.)118 1219 y(20.)41 b(B.)22 b(Gen)n(tz)g(and)g(M.)g(L\366)n (w)n(e,)g Fn(The)j(\035uctuations)f(of)i(the)e(overlap)j(in)d(the)h (Hop\034eld)h(mo)l(del)f(with)h(\034nitely)e(many)266 1318 y(p)l(atterns)29 b(at)h(the)f(critic)l(al)i(temp)l(er)l(atur)l(e)p Fz(,)c(preprin)n(t,)g(submitted,)i(1998.)118 1418 y(21.)41 b(J.)14 b(J.)35 b(Hop\034eld,)j Fn(Neur)l(al)f(networks)h(and)f(physic) l(al)j(systems)d(with)h(emer)l(gent)f(c)l(ol)t(le)l(ctive)h(c)l (omputational)266 1517 y(abilities)p Fz(,)29 b(Pro)r(c.)f(Natl.)f(A)n (cad.)h(Sci.)f(U.S.A.)i Fb(79)e Fz(\(1982\),)f(2554\0252558.)118 1617 y(22.)41 b(J.)27 b(K)n(oml\363s,)e(P)-7 b(.)28 b(Ma)5 b(jor,)26 b(and)h(G.)h(T)-7 b(usn\341dy)g(,)27 b Fn(A)n(n)h(appr)l (oximation)k(of)e(p)l(artial)g(sums)f(of)h(indep)l(endent)g(R)-8 b(V's)266 1717 y(and)30 b(the)g(sample)h(DF.)e(I)p Fz(,)f(Z.)g(W)-7 b(ahrsc)n(heinlic)n(hk)n(eitstheorie)25 b(v)n(erw.)h(Gebiete)i Fb(32)f Fz(\(1975\),)g(111\025131.)118 1816 y(23.)41 b(J.)18 b(K)n(oml\363s)f(and)h(R.)h(P)n(aturi,)h Fn(Conver)l(genc)l(e)i (r)l(esults)e(in)h(an)h(asso)l(ciative)h(memory)f(mo)l(del)p Fz(,)f(Neural)d(Net)n(w)n(orks)266 1916 y Fb(1)27 b Fz(\(1988\),)f (239\025250.)118 2016 y(24.)41 b(M.)29 b(Ledoux)f(and)h(M.)g(T)-7 b(alagrand,)28 b Fn(Pr)l(ob)l(ability)33 b(in)e(Banach)h(sp)l(ac)l(es)p Fz(,)f(Ergebnisse)d(der)g(Mathematik)h(und)266 2115 y(ihrer)d (Grenzgebiete,)h(Springer,)g(Berlin,)g(1991.)118 2215 y(25.)41 b(D.)30 b(Loukiano)n(v)-5 b(a,)28 b Fn(L)l(ower)33 b(b)l(ounds)f(on)f(the)h(r)l(estitution)f(err)l(or)h(in)g(the)g (Hop\034eld)h(mo)l(del)p Fz(,)e(Probab.)f(Theory)266 2314 y(Related)d(Fields)h Fb(107)f Fz(\(1997\),)f(161\025176.)118 2414 y(26.)41 b(M.)24 b(L\366)n(w)n(e,)f Fn(On)j(the)g(stor)l(age)h(c)l (ap)l(acity)h(of)f(Hop\034eld)g(mo)l(dels)g(with)g(we)l(akly)g(c)l(orr) l(elate)l(d)h(p)l(atterns)p Fz(,)c(to)f(app)r(ear)266 2514 y(in)28 b(Ann.)g(Appl.)g(Probab.)118 2613 y(27.)41 b(R.)14 b(J.)22 b(McEliece,)h(E.)14 b(C.)23 b(P)n(osner,)f(E.)14 b(R.)23 b(Ro)r(demic)n(h,)g(and)f(S.)14 b(S.)22 b(V)-7 b(enk)i(atesh,)23 b Fn(The)j(c)l(ap)l(acity)g(of)g(the)e(Hop\034eld)266 2713 y(asso)l(ciative)31 b(memory)p Fz(,)e(IEEE)g(T)-7 b(rans.)26 b(Inform.)i(Theory)f Fb(33)g Fz(\(1987\),)g(461\025482.)118 2813 y(28.)41 b(C.)14 b(M.)34 b(Newman,)i Fn(Memory)i(c)l(ap)l(acity)f (in)f(neur)l(al)g(network)g(mo)l(dels:)53 b(R)n(igor)l(ous)36 b(lower)h(b)l(ounds)p Fz(,)f(Neural)266 2912 y(Net)n(w)n(orks)26 b Fb(1)h Fz(\(1988\),)f(223\025238.)118 3012 y(29.)41 b(L.)14 b(A.)26 b(P)n(astur)f(and)h(A.)14 b(L.)26 b(Figotin,)g Fn(Exactly)i(soluble)h(mo)l(del)g(of)g(a)g(spin)f(glass)p Fz(,)f(So)n(v.)f(J.)f(Lo)n(w)g(T)-7 b(emp.)27 b(Ph)n(ys.)266 3111 y Fb(3)g Fz(\(1977\),)f(no.)i(6,)f(378\025383.)118 3211 y(30.)p 266 3211 V 304 w(,)g Fn(On)i(the)h(the)l(ory)g(of)h(disor) l(der)l(e)l(d)g(spin)g(systems)p Fz(,)c(Theor.)g(Math.)g(Ph)n(ys.)h Fb(35)f Fz(\(1977\),)f(403\025414.)118 3311 y(31.)41 b(M.)35 b(T)-7 b(alagrand,)35 b Fn(R)n(igor)l(ous)i(r)l(esults)f(for)h (the)g(Hop\034eld)h(mo)l(del)f(with)g(many)g(p)l(atterns)p Fz(,)g(Probab.)e(Theory)266 3410 y(Related)27 b(Fields)h Fb(110)f Fz(\(1998\),)f(177\025276.)118 3510 y(32.)41 b(A.)29 b(Y)-7 b(u.)29 b(Zaitsev,)f Fn(On)i(the)h(Gaussian)g(appr)l (oximation)i(of)e(c)l(onvolutions)h(under)e(multidimensional)i(ana-)266 3610 y(lo)l(gues)26 b(of)h(S.)13 b(N.)26 b(Bernstein)g(ine)l(quality)h (c)l(onditions)p Fz(,)f(Probab.)d(Theory)g(Related)g(Fields)h Fb(74)f Fz(\(1987\),)g(535\025)266 3709 y(566.)118 3809 y(33.)41 b(A.)34 b(Y)-7 b(u.)35 b(Zaitsev,)f Fn(Multidimensional)k (version)e(of)g(the)g(r)l(esults)f(of)h(Koml\363s,)i(Major,)i(and)c(T) -6 b(usn\341dy)36 b(for)266 3908 y(ve)l(ctors)30 b(with)g(\034nite)f (exp)l(onential)h(moments)p Fz(,)e(T)-7 b(ec)n(h.)27 b(Rep)r(ort)g(95-055,)e(SFB)j(343,)e(Bielefeld,)i(1995.)218 4095 y(\(Barbara)36 b(Gen)n(tz\))j Fy(Weierstra\377-Institut)44 b(f\374r)f(Angew)-10 b(andte)44 b(Anal)-7 b(ysis)42 b(und)h (Stochastik,)118 4195 y(Mohrenstr.)31 b(39,)g(D-10117)g(Berlin,)f (Germany)218 4295 y Fn(E-mail)g(addr)l(ess)p Fz(,)f(Barbara)c(Gen)n (tz:)37 b Fa(gentz@wias-berli)o(n.d)o(e)218 4464 y Fz(\(Matthias)26 b(L\366)n(w)n(e\))f Fy(Eurandom,)j(PO)i(Bo)n(x)f(513,)g(NL-5600)g(MB)h (Eindho)n(ven,)f(The)h(Netherlands)218 4563 y Fn(E-mail)g(addr)l(ess)p Fz(,)f(Matthias)e(L\366)n(w)n(e:)36 b Fa(lowe@eurandom.tu)o(e.)o(nl)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9901121103408--