Decomposition of mean-field Gibbs distributions into product measures

TL;DR

This paper demonstrates that under certain low complexity conditions, Gibbs distributions on the Boolean hypercube can be approximated as mixtures of product measures, aiding in the analysis of Ising models and nonlinear large deviations.

Contribution

It extends previous work by showing how Gibbs distributions decompose into mixtures of product measures under low complexity conditions.

Findings

Gibbs distributions are approximate mixtures of product measures.

Application to characterize Ising models with mean-field conditions.

Application to analyze conditional distributions in nonlinear large deviations.

Abstract

We show that under a low complexity condition on the gradient of a Hamiltonian, Gibbs distributions on the Boolean hypercube are approximate mixtures of product measures whose probability vectors are critical points of an associated mean-field functional. This extends a previous work by the first author. As an application, we demonstrate how this framework helps characterize both Ising models satisfying a mean-field condition and the conditional distributions which arise in the emerging theory of nonlinear large deviations, both in the dense case and in the polynomially-sparse case.