Fast mixing in Ising models with a negative spectral outlier via Gaussian approximation

TL;DR

This paper introduces a Gaussian approximation method to analyze the rapid mixing of Glauber dynamics in Ising models with a negative spectral outlier, overcoming limitations of previous approaches.

Contribution

It develops a new covariance approximation technique using Stein's method, enabling analysis of models where spectral width bounds are ineffective.

Findings

Provides near-optimal mixing time bounds for certain anti-ferromagnetic Ising models.

Establishes exponential lower bounds on mixing times for low temperature regimes on sparse graphs.

Introduces a novel iterative Stein's method approach for quadratic tilts of sums of bounded variables.

Abstract

We study the mixing time of Glauber dynamics for Ising models in which the interaction matrix contains a single negative spectral outlier. This class includes the anti-ferromagnetic Curie-Weiss model, the anti-ferromagnetic Ising model on expander graphs, and the Sherrington-Kirkpatrick model with disorder of negative mean. Existing approaches to rapid mixing rely crucially on log-concavity or spectral width bounds and therefore can break down in the presence of a negative outlier. To address this difficulty, we develop a new covariance approximation method based on Gaussian approximation. This method is implemented via an iterative application of Stein's method to quadratic tilts of sums of bounded random variables, which may be of independent interest. The resulting analysis provides an operator-norm control of the full correlation structure under arbitrary external fields. Combined with the localization schemes of Eldan and Chen, these estimates lead to a modified logarithmic Sobolev inequality and near-optimal mixing time bounds in regimes where spectral width bounds fail. We complement these results by proving exponential lower bounds on the mixing time for low temperature anti-ferromagnetic Ising models on sparse random regular graphs and Erd\"os-R\'enyi graphs, based on the existence of gapped states as in the recent work of Sellke.