Random matrices and complexity of spin glasses

TLDR: This study enables detailed information about the bottom of the energy landscape, including the absolute minimum, and the other local minima, and describes an interesting layered structure of the low critical values for the Hamiltonians of these models.

Abstract: CERN ´ Y Abstract. We give an asymptotic evaluation of the complexity of spherical p-spin spin- glass models via random matrix theory. This study enables us to obtain detailed infor- mation about the bottom of the energy landscape, including the absolute minimum (the ground state), the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our ap- proach allows us to compute the related TAP-complexity and extend the results known in the physics literature. As an independent tool, we prove a LDP for the k-th largest eigenvalue of the GOE, extending the results of (BDG01). How many critical values of given index and below a given level does a typical random Morse function have on a high dimensional manifold? Our work addresses this question in a very special case. We look at certain natural random Gaussian functions on the N- dimensional sphere known as p-spin spherical spin glass models. We cannot yet answer the question above about the typical number, but we can study thoroughly the mean number, which we show is exponentially large in N. We introduce a new identity, based on the classical Kac-Rice formula, relating random matrix theory and the problem of counting these critical values. Using this identity and tools from random matrix theory, we give an asymptotic evaluation of the complexity of these spherical spin-glass models. The complexity mentioned here is defined as the mean number of critical points of given index whose value is below (or above) a given level. This includes the important question of counting the mean number of local minima below a given level, and in particular the question of finding the ground state energy (the minimal value of the Hamiltonian). We show that this question is directly related to the study of the edge of the spectrum of the Gaussian Orthogonal Ensemble (GOE). The question of computing the complexity of mean-field spin glass models has recently been thoroughly studied in the physics literature (see for example (CLR03) and the refer- ences therein), mainly for a different measure of the complexity, i.e. the mean number of solutions to the Thouless-Anderson-Palmer equations, or TAP-complexity. Our approach to the complexity enables us to recover known results in the physics literature about TAP-complexity, to compute the ground state energy (when p is even), and to describe an interesting layered structure of the low energy levels of the Hamiltonians of these models, which might prove useful for the study of the metastability of Langevin dynamics for these models (in longer time scales than those studied in (BDG01)). The paper is organised as follows. In Section 2, we give our main results. In Section 3, we prove two main formulas (Theorem 2.1 and 2.2), relating random matrix theory (specif- ically the GOE) and spherical spin glasses. These formulas are direct consequences of the Kac-Rice formula (we learned the version needed here in the book (AT07), for another modern account see (AW09)). The main new ingredient is the fact that, for spherical spin- glass models, the Hessian of the Hamiltonian at a critical point, conditioned on the value of the Hamiltonian, is a symmetric Gaussian random matrix with independent entries (up to symmetry) plus a diagonal matrix. This implies, in particular, that it is possible to