First order global asymptotics for confined particles with singular pair repulsion

TL;DR

This paper establishes a large deviations principle for a system of particles with singular pairwise repulsion under confinement, characterizing the equilibrium measure and its support in various interaction and external field scenarios.

Contribution

It provides the first rigorous derivation of the large deviations principle for confined particles with Riesz and Coulomb interactions in arbitrary dimensions.

Findings

Empirical distribution converges to the equilibrium measure as particle number grows.

Equilibrium measure is uniquely characterized by its potential.

Support of the equilibrium measure varies with external field shape.

Abstract

We study a physical system of $N$ interacting particles in $\mathbb{R}^d$, $d\geq1$, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as $N$ tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension $d>2$, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as $N$ tends to infinity. In the more specific case of Coulomb interaction in dimension $d>2$, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.