Size of chaos for Gibbs measures of mean field interacting diffusions
Zhenjie Ren, Songbo Wang
TL;DR
The paper develops a quantitative theory of chaos for Gibbs measures of mean-field interacting diffusions by introducing a hierarchical framework of conditional entropies and a defective but non-perturbative concentration mechanism. By coupling a non-linear log-Sobolev landscape with a dimension-free defective $ extnormal{T}_2$ inequality and a $ extnormal{T}_1$ tightening, the authors obtain sharp chaos bounds that hold beyond small interaction strength, applicable to flat semi-convex and displacement-convex settings. They derive explicit entropy bounds for the conditional laws and for the $k$-particle marginals, yielding $H(m^{N,k}_*|m_*^{ ensor k}) = O(k^2/N^2)$ in many cases and $O(k/N^2)$ for the conditional levels, and illustrate the method on the quartic Curie–Weiss model up to the critical regime. The results unify static and dynamic chaos analyses, provide a broadly applicable framework for unbounded interactions, and offer new tools such as a dimension-free defective $ extnormal{T}_2$ inequality with large-deviation consequences.
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.