The high temperature crossover for general 2D Coulomb gases

TL;DR

This paper investigates the behavior of 2D Coulomb gases at different temperatures, deriving a PDE for the crossover density and analyzing asymptotics, extending previous 1D results to more general potentials.

Contribution

It introduces a PDE for the crossover density in 2D Coulomb gases and extends analysis to radially symmetric potentials, generalizing prior 1D findings.

Findings

Derived a PDE of generalized Liouville type for the crossover density.

Provided asymptotic results for radially symmetric potentials.

Presented numerical solutions and analytic density for specific potentials.

Abstract

We consider $N$ particles in the plane influenced by a general external potential that are subject to the Coulomb interaction in two dimensions at inverse temperature $\beta$. At large temperature, when scaling $\beta=2c/N$ with some fixed constant $c>0$, in the large-$N$ limit we observe a crossover from Ginibre's circular law or its generalization to the density of non-interacting particles at $\beta=0$. Using several different methods we derive a partial differential equation of generalized Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results and give examples for the numerical solution of the crossover density. These findings generalise previous results when the interacting particles are confined to the real line. In that situation we derive an integral equation for the resolvent valid for a general potential and present the analytic solution for the density in case of a Gaussian plus logarithmic potential.