Random Matrices and Complexity of Spin Glasses

Abstract

Abstract. We give an asymptotic evaluation of the complexity of spherical p-spin spinglass models via random matrix theory. This study enables us to obtain detailed information 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 approach 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].

Sum rules via large deviations

“…We may also refer to Auffinger et al (2013) for a general result in the same vein. For the sake of completeness let us shortly discuss the three requirements.…”

Large time zero temperature dynamics of the spherical p = 2-spin glass model of finite size

“…(27), to obtain the result given in introduction in Eq. (13) with the function e 1 (t) simply given by…”

Complexity of random smooth functions on compact manifolds

“…The distribution of critical points (or energy landscape) of isotropic random functions on R m was investigated by Fyodorov [15,16] who also relates this problem to the staistics of the eigenvalues in the ensemble GOE m+1 . Recently A. Auffinger [3,4] has investigated the distributions of critical values of certain isotropic random fields on a round sphere S m , where m → ∞, and described a connection with the distribution of eigenvalues of symmetric matrices in the ensemble GOE m+1 .…”

“…We have the following technical result whose proof is contained in Appendix A. Following the terminology on [3,4] we will refer to σ u as the variational complexity of u. Observe that supp µ u = Cr(u), supp σ u = D(u).…”