Mixing of Glauber Dynamics on High Overlap Gibbs Measures

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 $β$, there exists a strength of external field $θ$, 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).

Disclosure

“thank Yuansi Chen for numerous insightful discussions on the topic in this paper. AR would like to thank Roland Bauerschmidt for an instructive discussion on this topic, and for pointing out the connections with [Per75]. We used modern AI tools in this work, primarily ChatGPT Pro 5.5. In fact, what are (in our view) the two key ingredients in our argument were found by ChatGPT: One is the realization that it is not too lossy to attempt to control correlations matrices for all pos”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-24T00:34:12.480Z

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