Wasserstein-infinity stability and mean field limit of discrete interaction energy minimizers

TL;DR

This paper establishes a quantitative stability and mean field limit for discrete interaction energy minimizers on the torus with Riesz potentials, showing that low-energy configurations approximate the uniform distribution as particle number grows.

Contribution

It provides the first quantitative stability result in Wasserstein-infinity distance for discrete interaction energies with singular potentials, extending previous continuous stability results.

Findings

Discrete energy minimizers are close to uniform distribution for large N.

Quantitative bounds on the Wasserstein-infinity distance are derived.

The results apply to singular Riesz potentials on the torus.

Abstract

In this paper we give a quantitative stability result for the discrete interaction energy on the multi-dimensional torus, for the periodic Riesz potential. It states that if the number of particles $N$ is large and the discrete interaction energy is low, then the particle distribution is necessarily close to the uniform distribution (i.e., the continuous energy minimizer) in the Wasserstein-infinity distance. As a consequence, we obtain a quantitative mean field limit of interaction energy minimizers in the Wasserstein-infinity distance. The proof is based on the application of the author's previous joint work with J. Wang on the stability of continuous energy minimizer, together with a new mollification trick for the empirical measure in the case of singular interaction potentials.