Size of chaos for Gibbs measures of mean field interacting diffusions

TL;DR

This paper provides a quantitative analysis of chaos size in Gibbs measures for mean field diffusions, using gradient structures and concentration inequalities, applicable to various interaction regimes and models.

Contribution

It introduces a unified framework to bound chaos in Gibbs measures, including unbounded interactions, and extends results to the quartic Curie-Weiss model.

Findings

Derived sharp bounds on chaos size for Gibbs measures.

Established a defective Talagrand inequality for concentration analysis.

Applied framework to the quartic Curie-Weiss model in the sub-critical regime.

Abstract

We investigate Gibbs measures for diffusive particles interacting through a two-body mean field energy. By identifying a gradient structure for the conditional law, we derive sharp bounds on the size of chaos, providing a quantitative characterization of particle independence. To handle interaction forces that are unbounded at infinity, we study the concentration of measure phenomenon for Gibbs measures via a defective Talagrand inequality, which may hold independent interest. Our approach provides a unified framework for both the flat semi-convex and displacement convex cases. Additionally, we establish sharp chaos bounds for the quartic Curie-Weiss model in the sub-critical regime, demonstrating the generality of this method.