A Note on Large Deviations for 2D Coulomb Gas with Weakly Confining Potential

TL;DR

This paper establishes a large deviation principle for a 2D Coulomb gas under weaker potential growth conditions, showing convergence of empirical measures to a non-compact limiting measure via potential theory.

Contribution

It introduces a new approach to large deviations for Coulomb gases with less restrictive potential growth assumptions, including a novel compactification technique.

Findings

Empirical measures converge to a non-compact limiting measure.

Large deviation upper bound proved using a new compactification method.

The limiting measure is characterized by a variational principle from potential theory.

Abstract

We investigate a Coulomb gas in a potential satisfying a weaker growth assumption than usual and establish a large deviation principle for its empirical measure. As a consequence the empirical measure is seen to converge towards a non-random limiting measure, characterized by a variational principle from logarithmic potential theory, which may not have compact support. The proof of the large deviation upper bound is based on a compactification procedure which may be of help for further large deviation principles.