# Critical points and the Gibbs measure of a spherical spin glass model

# Critical points and the Gibbs measure of a spherical spin glass model

For integers N let H_N(x) be an isotropic Gaussian field on the N-dimensional unit sphere, meaning that Cov(H_N(x),H_N(y)) is a function, f_N, of the inner product of . The spherical spin glass models of statistical mechanics are such random fields, with f_N = N f with the function independent of the dimension N. The intricate landscape of the graph of H_N(x) may be studies through its critical points and the corresponding values. Focusing on the pure p-spin models, I will review recent developments concerning the distribution of the number of critical values at a given height and the associated extremal point process. Combining these results with a local investigation of the behaviour of H_N(x) in neighborhoods around the critical points, we obtain a detailed geometric picture for the Gibbs measure at low enough temperature. The measure concentrates on spherical "bands" around the deepest critical points. The main focus of the talk will be on the structure of such states, and its consequences. The talk is based on a joint work with Ofer Zeitouni.