# Spectral gap estimates in mean field spin glasses

## TL;DR

This paper demonstrates that in mean field spin glasses, local reversible dynamics mix exponentially slowly at low temperatures due to free energy barriers, with conditions identified for their existence across various models.

## Contribution

It introduces a new framework linking free energy barriers to slow mixing, providing conditions that apply to a broad class of spin glass models and dynamics.

## Key findings

- Spectral gap is exponentially small at low temperatures.
- Conditions for free energy barriers are established using replicon eigenvalues.
- A wider criterion for slow mixing is derived from Panchenko's free energy calculations.

## Abstract

We show that mixing for local, reversible dynamics of mean field spin glasses is exponentially slow in the low temperature regime. We introduce a notion of free energy barriers for the overlap, and prove that their existence imply that the spectral gap is exponentially small, and thus that mixing is exponentially slow. We then exhibit sufficient conditions on the equilibrium Gibbs measure which guarantee the existence of these barriers, using the notion of replicon eigenvalue and 2D Guerra Talagrand bounds. We show how these sufficient conditions cover large classes of Ising spin models for reversible nearest-neighbor dynamics and spherical models for Langevin dynamics. Finally, in the case of Ising spins, Panchenko's recent rigorous calculation [79] of the free energy for a system of "two real replica" enables us to prove a quenched LDP for the overlap distribution, which gives us a wider criterion for slow mixing directly related to the Franz-Parisi-Virasoro approach [43,60]. This condition holds in a wider range of temperatures.