Size of chaos for Gibbs measures of mean field interacting diffusions

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.