A Spectral Condition for Spectral Gap: Fast Mixing in High-Temperature Ising Models

TL;DR

This paper establishes a spectral condition ensuring rapid mixing of Glauber dynamics in high-temperature Ising models on the hypercube, by proving a Poincaré inequality under a norm constraint on the interaction matrix.

Contribution

It introduces a spectral condition based on the operator norm of the interaction matrix that guarantees fast mixing in high-temperature Ising models, along with a novel localization technique.

Findings

Proves Poincaré inequality for Ising models with interaction matrix norm < 1.

Shows the inequality leads to bounds on Glauber dynamics mixing time.

Develops a structural decomposition of Ising measures into rank-one quadratic potentials.

Abstract

We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincar\'{e} inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than $1$. The inequality implies a control on the mixing time of the Glauber dynamics. Our techniques rely on a localization procedure which establishes a structural result, stating that Ising measures may be decomposed into a mixture of measures with quadratic potentials of rank one, and provides a framework for proving concentration bounds for high temperature Ising models.