The planar low temperature Coulomb gas: separation and equidistribution

TL;DR

This paper studies large planar Coulomb gases at low temperatures, proving they are uniformly separated and equidistributed under certain conditions, generalizing results from Fekete configurations to finite temperature regimes.

Contribution

It extends the understanding of Coulomb gases by establishing separation and equidistribution at low temperatures, generalizing previous zero-temperature results and introducing a versatile proof method.

Findings

Gas configurations are with high probability uniformly separated.

Configurations are equidistributed with respect to the equilibrium measure.

Method applies to general families satisfying separation and sampling conditions.

Abstract

We consider planar Coulomb systems consisting of a large number $n$ of repelling point charges in the low temperature regime, where the inverse temperature $\beta$ grows at least logarithmically in $n$ as $n \longrightarrow \infty$, i.e., $\beta\gtrsim \log n$. Under suitable conditions on an external potential we prove results to the effect that the gas is with high probability uniformly separated and equidistributed with respect to the corresponding equilibrium measure (in the given external field). Our results generalize earlier results about Fekete configurations, i.e., the case $\beta=\infty$. There are also several auxiliary results which could be of independent interest. For example, our method of proof of equidistribution (a variant of "Landau's method") works for general families of configurations which are uniformly separated and which satisfy certain sampling and interpolation inequalities.