Large deviations for empirical measures of mean field Gibbs measures

TL;DR

This paper establishes large deviation principles for empirical measures of mean-field Gibbs models under minimal conditions, extending previous results by removing continuity and boundedness constraints on interaction potentials.

Contribution

It proves large deviation principles for mean-field Gibbs measures without requiring continuity or boundedness of interaction potentials, using novel probabilistic techniques.

Findings

Large deviation principle holds under weak topology and Wasserstein metric.

No continuity or boundedness assumptions needed on interaction potentials.

Proof utilizes law of large numbers and exponential decoupling inequalities.

Abstract

In this paper, we show that the empirical measure of mean-field model satisfies the large deviation principle with respect to the weak convergence topology or the stronger Wasserstein metric, under the strong exponential integrability condition on the negative part of the interaction potentials. In contrast to the known results we prove this without any continuity or boundedness condition on the interaction potentials. The proof relies mainly on the law of large numbers and the exponential decoupling inequality of de la Pena for U-statistics.