Large Deviation Principle for Empirical Fields of Log and Riesz Gases

TL;DR

This paper establishes a large deviation principle for the empirical fields of particles interacting via logarithmic, Coulomb, or Riesz potentials, revealing their probabilistic behavior and thermodynamic properties.

Contribution

It introduces a large deviation principle for the tagged empirical field of particle systems with specific interactions, connecting entropy, energy, and random matrix processes.

Findings

Proves a large deviation principle at speed N for the empirical field.

Derives a variational characterization of sine-beta processes.

Provides a next-to-leading order expansion of the free energy and confirms the thermodynamic limit.

Abstract

We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the system, proving the existence of the thermodynamic limit.