Low Temperature Asymptotics in Spherical Mean Field Spin Glasses

TL;DR

This paper rigorously analyzes the low temperature limit of spherical spin glasses, identifying the ground state energy via a variational approach, and clarifies the conditions for the 1RSB phase in these models.

Contribution

It establishes the $ ext{Gamma}$-limit of the Crisanti-Sommers functional, introduces a dual obstacle problem framework, and characterizes the 1RSB phase conditions in spherical mixed p-spin glasses.

Findings

Identified the $ ext{Gamma}$-limit for the variational problem.

Developed a dual obstacle problem approach for analysis.

Characterized the 1RSB phase conditions in $2+p$-spin models.

Abstract

In this paper, we study the low temperature limit of the spherical Crisanti-Sommers variational problem. We identify the $\Gamma$-limit of the Crisanti-Sommers functionals, thereby establishing a rigorous variational problem for the ground state energy of spherical mixed $p$-spin glasses. As an application, we compute moderate deviations of the corresponding minimizers in the low temperature limit. In particular, for a large class of models this yields moderate deviations for the overlap distribution. We then analyze the ground state energy problem. We show that this variational problem is dual to an obstacle-type problem. This duality is at the heart of our analysis. We present the regularity theory of the optimizers of the primal and dual problems. This culminates in a simple method for constructing a finite dimensional space in which these optimizers live for any model. As a consequence of these results, we unify independent predictions of Crisanti-Leuzzi and Auffinger-Ben Arous regarding the 1RSB phase in this limit. We find that the "positive replicon eigenvalue" and "pure-like" conditions are together necessary for optimality, but that neither are themselves sufficient, answering a question of Auffinger and Ben Arous in the negative. We end by proving that these conditions completely characterize the 1RSB phase in $2+p$-spin models.