# Equilibrium large deviations for mean-field systems with translation invariance

## TL;DR

This paper establishes large deviation principles for mean-field particle systems with translation invariance, covering McKean-Vlasov and rank-based diffusions, with applications to capital distribution analysis.

## Contribution

It introduces a framework for large deviations in translation-invariant mean-field systems, including new results for systems without external potential and in orbit spaces.

## Key findings

- Large deviation principles are proved for equilibrium empirical measures.
- Results apply to systems with and without external potential.
- Application to atypical capital distribution is demonstrated.

## Abstract

We consider particle systems with mean-field interactions whose distribution is invariant by translations. Under the assumption that the system seen from its centre of mass be reversible with respect to a Gibbs measure, we establish large deviation principles for its empirical measure at equilibrium. Our study covers the cases of McKean-Vlasov particle systems without external potential, and systems of rank-based interacting diffusions. Depending on the strength of the interaction, the large deviation principles are stated in the space of centered probability measures endowed with the Wasserstein topology of appropriate order, or in the orbit space of the action of translations on probability measures. An application to the study of atypical capital distribution is detailed.