Journal of Applied Mathematics and Stochastic Analysis, 13:2 (2000), 147-160. RIGOROUS SOLUTION OF A MEAN FIELD SPIN GLASS MODEL

using large deviation theory [14] is rigorously treated. The almost sure convergence criteria associated with the cumulant generating function C(t) with respect to the quenched random variables ( is carefully investigated, and it is proved that the related excluded null set N is independent of t. The free energy and hence the other thermodynamic quantities are rederived using Varadhan’s Large Deviation Theorem. A simulation is also presented for the entropy when ( assumes a Gaussian distribution.


Introduction
In this paper, an exactly solvable model of a spin glass which was introduced by van Hemmen et al. [12, 13] is studied.The Hamiltonian function is given by 3N({S}' {J}) E Jij w(i)w(j) Jo E w(i)w(j) h E w(i), (1.1) (,J) (,J) where co(i)-+1 are N Ising spins interacting with each other in pairs (i, j) and with an external magnetic field h.J0 is a direct ferromagnetic coupling and 1Supported by the Commonwealth Scholarship Commission, United Kingdom.
Printed in the U.S.A. ()2000 by North Atlantic Science Publishing Company 147 Jij JNir/j + ji] (1.2) where the i and r/j are i.i.d, random variables with a symmetric distribution with zero mean and variance 1.This model resembles a similar model proposed by Pastur and Figotin in [8-10].However, it was van Hemmen et al. who first used a large de- viation (LD) argument to successfully solve this model.In the absence of ferromagnetic coupling and an external magnetic field h, the Hamiltonian can also be written as

JN({S}, {J})
NJ N-1 N 1 (1.3) j=l To compute, using LD theory, the free energy defined at inverse temperature fl by -/3f(/3) =Nli_,m-log E {w(i)}/N--1 (1.4)   it must be shown that the so-called cumulant generating function exists, defined by C(t) lim -logV[exp(t, WN> (1.5) where E denotes a configuration average over the Ising spins w(i), (-,-), represents Euclidean inner product in R 2 and W N are the 2-dimensional random variables an WN E iw(i)'" i w(i) The focus of this paper is to rigorously investigate the a.s.convergence criteria of C(t) with respect to the common distribution of and r/.The main result is Theorem 3.1 in which the t independency of the excluded null set associated with the above mentioned convergence result is proved.Furthermore, we show in Proposition S-"*e(l + 2 0)In 2 as T---,0, when has a Gaussian distribution, s--.0 as T--0. (1.6)

Definitions
Consider the configuration space h-XA (i.e., Ft A --{w:A--.X}), where X is a compact metric space, and A {1,...,N} where N is any non-negative integer.In this particular case, we take X {-1, 1}.Define the single spin distribution # by P 1/2((1 + 5 1) (2.1) so that #({1})-#({-1})-1 / 2 .Here 5(_)is the Dirac-point measure.Also denote the infinite product measure on %(A) (the Borel subsets of A) with identical one- dimensional marginals # by u(w) 2-AI for each w E A" Define a random inter- action (-{(I) A" Here, w(i) E X, and h represents an external magnetic field.In the following, we index all quantities by N instead of A.
Let where i (il'2"'"id) are i.i.d, random variables each with a common symmetric distribution denoted by .Denote by _, the sequence (i)/N= 1" We assume that E([i a [) < oc; 1 _< a _< d and that ia are i.i.d, random variables.Q is a symmetric function of the i's which means :N -E Q((i; j)w(i)w(j) h E w(i) (,j) where (i, j) represents an index pair.The partition function is defined by where/3 is the inverse temperature.
The range of the interactions tends to c as Ncx.defined as (2.6) The specific free energy is We shall prove that the above limit exists FCa.s. in the following case. (2.7)

Separable Interactions
Let h 0, d 2, and consider a separable quadratic interaction of the form Q(i; j) J[SilSj2 + jli2]" (3.1) Define the observables and define their distribution by the image measure F N [2, 3] where FN(A)-P,(q I(A)), (3.3)   for all A e %(R2), under the measurable map qN-(qlN, q2N):N --+2" =N depends on the random variables i" The partition function (2.6) becomes N(/, 0) 2 N / eNKqlq2:N(dq) where K flJ and the specific free energy can be written as Notice that (3.4)

2
Let {Yn E Rd;n-1,2,...}, where d E +, be a sequence of random variables defined on some probability space (, ,).We define the cumulant generating function by C(t) =nli_,ml-ln E[e (t' Yn)] t Rd, (3.6)   where F denotes the expectation with respect to P, and (-, -) denotes the Euclidean inner produce in d.Our aim is to show the existence of C(t) -a.e., k/t and then to use Cramr's Theorem [14, 4] to find a candidate for the rate function I(ql,q2 which would be the Legendre-Fenchel transform [4] of C(t).Then we discuss the PCa.s.
convergence of (3.6) considering the bounded and unbounded distributions for , and also show that the null set, where this convergence is not valid, is independent of t.
3.1 Evaluation of the Free Energy  C(s) < c by the assumption.
bound that For A n closed, it follows by the large deviation upper limsupln[P(An) < -inf I(x) This implies that 'e > 0, Sn 0 E N such that n >_ n0:=lln P(An) <_ I M + =vP(An) <_ e-n(IM-e).(3.14)Let d 2, {i be i.i.d, random variables with E(e I] Q ][) < oo for all Then for every t (tl, t2) E 2 and _-a.e. the function exists and is independent of .
For _-a.e., the distributions :N (LDP) with rate function given by satisfy the Large Deviation Principle I(q) sup {(t, q} C(t)}.
N gN(t'--E In cosh((t, i)) ,=1 (3.18) and (3.19) g(t) [ln cosh((t, so that g(t) is independent of i.We show that there exists a null set h c such that /t 2 and V .N', gN(t, )g(t), i.e., C(t) :[ln cosh((t, i))] _ a.e. and (3.20) Notice that at a fixed t, X -lncosh((t,i)) are independent random variables with identical distributions and hence it follows from the strong law of large numbers that gN(t, _ )-g(t) a.s.This means that for each t E D, there is a null set t such that, if _ 6 Let > 0 be given and set lit-t' I I < e/3M for some M > IV[ I I I I ].Then gm(t,_ )--gN(t',_ )1 E [ln cosh((t, i))-In cosh((t', i))] i=1 N (3.23) Now suppose that has bounded distributions.Then, we can find an M such that I I ia I I -< M with probability 1 (for example, i,-:t:l with probability 1/2).For unbounded distributions we shall assume :(e II II < oc (notice that this condition is satisfied for Gaussian distributions).By Lemma 3.1, we have P(lim supAn) O.
This implies that if A" limsupn__,An, then eventually n I I I I < M a.s.
(3.24) i=1 So, take N t_l A when has an unbounded distribution and get, for those e with probability 1.
(3.27) (b) Since C(t) exists -a.e. for all t and is a convex function of t, we apply Cramr's Theorem [14, 4].Since (qlN, q2N) is a pair of independent random variables with common distribution function and is in the form of the averages described in Cramfir's Theorem, we have I(ql, q2) sup {;tlq -t-t2q 2 C(t 1, t2)) -a.e.tl,t 2 (c) Now []:N satisfies the LDP with rate function I(ql,q2), and qlNq2N is a continuous function from N2N.In order to apply Varadhan's Theorem [14] to evaluate the limit (3.4), we need KqlNq2N t be bounded above.This follows from Lemma for N>_N O Hence we get for N>_N0, qlNq2N <--M2 a.s.In the following, we determine the maximizers of the free energy functional (3.17), the critical temperature of the spin glass phase transition, and also find expressions for some thermodynamic quantities.Definition 3.1: Define the specific entropy as a function of the specific internal energy u by (3.30)   Proposition 3.1: Let ('1,2) be a maximizer of the free energy functional (3.17) with (1,2) and 1 1 + 2. Also take J 1 so that K ft.Then (i) Yi -'2 -' and " satisfies 2 E,[tanh ].Furthermore, y has non- ff > Zc (ii (iv) The specific inernal energy and the specific entropy are given by and (3.31) s(#) En[1 n cosh ] 22 + In 2 kB respectively.(k B is the Boltzmann constant.)Proof: Since C(t) is differentiable for all t, the maximization conditions of (3.16)If ('1,2) is an interior point of the essential domain of I, which we shall denote by essdomI, then I is differentiable [4].For ('1,2) int(essdom I) we have by (3.29) 2 =[2 tanh/(1"2 + 21)]" (3.34) We now prove (i).
3. The solutions of for fl > tic have to be obtained by numerically solving the implicit equation (3.38).In the following, we consider the solutions with specific distributions in more detail.Proof: (a) By (3.45) with k B -1 we have s(Z)-g0[lncosh/ r/]-2/2+1n2.
The entropy, computed using (3.45) at several values of L for the above discrete distribution as well as the case where ('s have a Gaussian distribution, is plotted as a function of the temperature T in Figure 3.1.In the next section, we compare this entropy with a simulation of this model using the method of coincidence counting [7].We consider the symmetric discrete distribution which was introduced in Example 1 for L-3.The method of coincidence counting is used for computing the entropy in a Monte-Carlo simulation [1] introduced by S.K. Ma [7].The method involves count- ing the number of states coinciding with a given state where a state is a configuration of the Ising spins.Two states are "coincident" if they do not differ by more than m spins (m 0,1,...), (when m = 0 the states are identical), and their energies are approximately the same.A detailed discussion about the algorithm can be found in We compute the analytical result for 12 Ising spins for this distribution by generat- ing all possible 212 configurations, and compare this with our simulation (see Figure 4.1).We also plot the entropy in the thermodynamic limit as was done in Figure 3.1 It becomes clear that 12 spins are still far removed from th thermodynamic limit.Indeed, 12 Ising spins are not sufficient to obtain a non-random limit for the entropy.
With L 3, we have 7 possible random values for to appear in a sequence of 22 elements (each spin is connected to 11 others, and each bond has two -values).In fact, there are fluctuations in the simulation as well as the analytical result (for 12 spins) between different runs of the program.These random fluctuations cause the slight discrepancy between the simulated and the analytical graphs.Comparison of entropy simulated using the method of coincidence counting for 12 Ising spins with the analytical results for N-12 and in the limit N---,c.

Lemma 3 . 1 "
Assume E(e s II II) < oo for all s > 0 and M > [ I I I I ].Define closed sets Then P( li_,supA n) 0.
I I I are independent random variables with the same distribution.Cramr's Theorem, we have a large deviation property for X n with rate function By where I(x) sup {sx-C(s)}, (3.9) s>0 C(s)-lnE(e s ]1Q [[ ).
is closed an E[ I I 113 An, we have I M > I (El I[ { I I 3) 0, Choose e < I M and we get -n(IM-e E e(An)-< E e < c. (3.13) n _> n o n >_ n oNow it follows from Borel-Cantelli's Lemma[6] that Theorem 3.1" t>0.

Figure 4 . 1 :
Figure 4.1: Comparison of entropy simulated using the method of coincidence counting for 12 Ising spins with the analytical results for N-12 and in the