Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model

Abstract

We consider the free energy of the bipartite spherical Sherrington-Kirkpatrick model. We find the critical temperature and prove the limiting free energy for all non-critical temperature. We also show that the law of the fluctuation of the free energy converges to the Gaussian distribution when the temperature is above the critical temperature, and to the GOE Tracy-Widom distribution when the temperature is below the critical temperature. The result is universal, and the analysis is applicable to a more general setting including the case where the disorders are non-identically distributed.We use a special structure of the Hamiltonian (1.3) to study its limiting free energy and the fluctuations. Setting the matrix J = (J ij ) and considering σ and τ as column vectors, the critical points of the function f (σ, τ ) = σ T Jτ (which is a constant multiple of the Hamiltonian) subject to the constraints σ 2 = N 1 and τ 2 = N 2 satisfy the equationswhere λ 1 and λ 2 are Lagrange multipliers. These equations imply that σ is an eigenvector of the matrix JJ T , τ is an eigenvector of J T J, and λ 1 λ 2 is an eigenvalue of J T J (and also JJ T ). The matrix J is a random matrix with independent and identically distributed entries. The matrix J T J is called a (constant multiple of) sample covariance matrix (with null covariance) in statistics and also is said to belong to the Laguerre orthogonal ensemble in random matrix theory [31,22,5]. It is one of the fundamental matrices in random matrix theory. The behavior of the eigenvalues of J T J (the squared singular values of J) in the large dimension limit is well-studied.There is a more direct connection between the random matrices and the free energy. In [27,7], it was shown that the partition function of the usual SSK model can be expressed as a random single integral. In this paper, we obtain a similar result for the bipartite SSK model, but this time the random integral is a double integral; see Lemma 2.5. This random double integral involves the eigenvalues of J T J. We analyze the double integral asymptotically using the method of steepestdescent. The reason that we can apply the method of steepest-descent to the random integral is that even though the eigenvalues are random, their fluctuations about their classical locations are small. Precise estimates for the locations of the eigenvalues were obtained recently in random matrix theory. The "rigidity" estimates for the eigenvalues of J T J were proved by Pillai and Yin in 2014 [35] for the case N 1 = N 2 . The estimates for the case N 1 = N 2 follow from [2]. The rigidity estimates are also crucial in proving the universality in random matrix. They are proved for several other random matrix ensembles [21,20,13]. Our analysis is applicable to a large class of random double integrals under certain general conditions (including the rigidity condition) on a sequence of random variables. We obtain the results for the bipartite SSK model as a special case of a more general asymptotic result for random double integrals.The s...

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“…Baik and Lee's result is based on a contour integral representation for the partition function, which reduces the evaluation of F N (β) to a (delicate) saddle point analysis involving quantities studied in random matrix theory. In a subsequent series of work, these authors exploit similar representations to derive the thermodynamic limit of variants of the SSK model, including a model with an additional ferromagnetic (Curie-Weiss) interaction in the Hamiltonian [3], the bipartite SSK model [4], for which the computation of the first order behavior at all temperatures was open, and in a recent joint work with H. Wu, the SSK/Curie-Weiss model at the critical coupling strength [5].…”

“…One such computation is the remarkable result by J. Baik and J.O. Lee [2] concerning the fluctuations of the logarithm partition function Z N (β), defined by…”

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“…As the minimum of the quadratic form (1) on a sphere is obviously given by (minus one half of) the largest eigenvalue λ max of the matrix J, the statistics of the ground state are governed by the Tracy-Widom distribution for GOE, which describes the fluctuations of the largest/smallest eigenvalue in that ensemble [7]. Besides, it was recently proved that for T < T c , the fluctuations of the free energy are also given by the Tracy-Widom distribution, see [6]. Throughout this paper, we will be interested in the limiting case T = 0, which already contains many interesting aspects.…”

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“…This completes the proof of the asymptotic normality of log dQ n dP n |P n . Proof of part (2) of Theorem 2.1: Before proving part (1) of Theorem 2.1, we prove part (2). Since…”

“…The case low temperature case (β > 1 2 ) is also well-known in this case where the free energy has a limiting GOE Tracy-Widom distribution with O n − 2 3 fluctuations. One might look at Baik and Lee [2] for a reference.…”