Quantitative approximate independence for continuous mean field Gibbs measures

TL;DR

This paper provides a quantitative measure of approximate independence in continuous mean field Gibbs measures, showing improved bounds at high temperature and establishing the optimality of the quadratic rate.

Contribution

It introduces non-asymptotic bounds on the distance between finite particle marginals and their product measures for continuous Gibbs measures, improving previous results.

Findings

Distance between marginals and product measure is $O((k/n)^{c \,\wedge\, 2})$

High temperature bounds improve upon previous entropy-based results

Quadratic rate $O((k/n)^2)$ is shown to be optimal with a Gaussian example.

Abstract

Many Gibbs measures with mean field interactions are known to be chaotic, in the sense that any collection of $k$ particles in the $n$-particle system are asymptotically independent, as $n\to\infty$ with $k$ fixed or perhaps $k=o(n)$. This paper quantifies this notion for a class of continuous Gibbs measures on Euclidean space with pairwise interactions, with main examples being systems governed by convex interactions and uniformly convex confinement potentials. The distance between the marginal law of $k$ particles and its limiting product measure is shown to be $O((k/n)^{c \wedge 2})$, with $c$ proportional to the squared temperature. In the high temperature case, this improves upon prior results based on subadditivity of entropy, which yield $O(k/n)$ at best. The bound $O((k/n)^2)$ cannot be improved, as a Gaussian example demonstrates. The results are non-asymptotic, and distance is quantified via relative Fisher information, relative entropy, or the squared quadratic Wasserstein metric. The method relies on an a priori functional inequality for the limiting measure, used to derive an estimate for the $k$-particle distance in terms of the $(k+1)$-particle distance.