On the Spectral Gap of Spherical Spin Glass Dynamics

TL;DR

This paper analyzes the spectral gap of spherical p-spin glass dynamics, revealing a phase transition from order 1 at high temperature to exponential decay at low temperature, indicating different dynamical phases.

Contribution

It establishes the temperature-dependent behavior of the spectral gap in spherical p-spin models, connecting spectral properties to dynamical phases.

Findings

Spectral gap is order 1 at high temperature.

Spectral gap decays exponentially at low temperature.

Confirms existence of dynamical high and glass phases.

Abstract

We consider the time to equilibrium for the Langevin dynamics of the spherical $p$-spin glass model of system size $N$. We show that the log-Sobolev constant and spectral gap are order $1$ in $N$ at sufficiently high temperature whereas the spectral gap decays exponentially in $N$ at sufficiently low temperatures. These verify the existence of a dynamical high temperature phase and a dynamical glass phase at the level of the spectral gap. Key to these results are the understanding of the extremal process and restricted free energy of Subag--Zeitouni and Subag.