Propagation of chaos and residual dependence in Gibbs measures on finite sets

TL;DR

This paper investigates how mean-field Gibbs measures on finite sets relate to mixtures of product measures, extending previous work on propagation of chaos by analyzing residual dependencies beyond the classical asymptotic regime.

Contribution

It introduces a comparison framework between mean-field Gibbs measures and explicit mixtures, highlighting residual dependencies beyond traditional propagation of chaos assumptions.

Findings

Comparison between Gibbs measures and mixture models elucidates residual dependencies.

Extends classical propagation of chaos results to finite, non-asymptotic settings.

Provides insights into the structure of finite-volume Gibbs measures.

Abstract

We compare a mean-field Gibbs distribution on a finite state space on $N$ spins to that of an explicit simple mixture of product measures. This illustrates the situation beyond the so-called increasing propagation of chaos introduced by Ben Arous and Zeitouni 1999, where marginal distributions of size $k=o(N)$ are compared to product measures.