Large Deviation Principle for local empirical measure of Coulomb gases at intermediate temperature regime

TL;DR

This paper investigates the large deviation principles for the local empirical measure of Coulomb gases at various temperature scalings, revealing different rate functions and a critical regime where entropy and energy compete.

Contribution

It introduces a comprehensive analysis of the local empirical field for Coulomb gases across subcritical, critical, and supercritical temperature regimes, highlighting the interplay of entropy and energy in large deviations.

Findings

Supercritical regime: LDP with entropy-based rate function

Subcritical regime: LDP with energy-based rate function

Critical regime: LDP with combined entropy and energy terms

Abstract

This paper deals with Coulomb gases at an intermediate temperature regime. We define a local empirical field and identify a critical temperature scaling. We show that if the scaling of the temperature is supercritical, the local empirical field satisfies an LDP with an entropy-based rate function. We also show that if the scaling of the temperature is subcritical, the local empirical field satisfies an LDP with an energy-based rate function. In the critical temperature scaling regime, we derive an LDP-type result in which the "rate function" features the competition of an entropy and energy terms. An important idea in this work is to exploit the different scaling relations satisfied by the Coulomb energy and the entropy.