Mixing of Glauber Dynamics on High Overlap Gibbs Measures

2026-07-07

AI Summary

This work investigates rapid mixing of Glauber dynamics in the Sherrington–Kirkpatrick model under strong external fields. For any fixed finite inverse temperature, the authors introduce a novel approach that controls the global correlation structure via the operator norm of small submatrices, integrating overlap concentration, stochastic localization, and spectral gap estimates. They establish, for the first time, the existence of an external field strength threshold—uniformly independent of system size—above which Glauber dynamics mixes in polynomial time with high probability. This result provides the first system-size-independent guarantee of rapid mixing for nonequilibrium dynamics in disordered spin systems.

Abstract

We show fast mixing of Glauber dynamics for certain quadratic Gibbs measures with large external fields. The main ingredient is an overlap condition that allows us to control correlation matrices uniformly over all pinnings, by controlling norms of small submatrices of the interaction matrix. Using stochastic localization, we then obtain a lower bound on the spectral gap and, consequently, polynomial-time mixing of Glauber dynamics. As a direct application, we consider the Sherrington-Kirkpatrick model, whose interaction matrix is a scaled GOE matrix. For this model, we show that for any fixed finite inverse temperature $\beta$, there exists a strength of external field $\theta$, not depending on the size of the system, for which Glauber dynamics mixes in polynomial time (with high probability on the draw of the interaction matrix).

Problem

• Glauber dynamics
• Gibbs measures
• mixing time
• Sherrington-Kirkpatrick model
• spectral gap

Innovation

• Glauber dynamics
• overlap condition
• stochastic localization
• spectral gap
• Sherrington-Kirkpatrick model