Spectral Gap Estimates for Mixed $p$-Spin Models at High Temperature

TL;DR

This paper introduces a method to estimate the spectral gap of mixed p-spin models at high temperature, showing it remains bounded away from zero, which implies rapid mixing and stability of the system.

Contribution

The authors develop an iterative approach to relate spectral gaps of large systems to those of conditioned smaller subsystems in mixed p-spin models.

Findings

Spectral gap remains of order one at high temperature.

Method applies to general mixed p-spin models.

Ensures rapid convergence to equilibrium.

Abstract

We consider general mixed $p$-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the $N$-spin system to that of suitably conditioned subsystems.