Free energy of spherical Coulomb gases with point charges

TL;DR

This paper derives precise asymptotic expansions of the free energy for two-dimensional Coulomb gases on the Riemann sphere with point charges at the poles, revealing detailed energetic properties of these models.

Contribution

It provides the first detailed asymptotic expansion of free energy including constant terms for Coulomb gases with point charges on the sphere.

Findings

Asymptotic free energy expansions are obtained.

Results apply to models with determinantal or Pfaffian structures.

The droplet is the entire complex plane in these models.

Abstract

We consider two-dimensional Coulomb gases on the Riemann sphere with determinantal or Pfaffian structures, under external potentials that are invariant under rotations around the axis connecting the north and south poles, and with microscopic point charges inserted at the poles. These models can be interpreted as Coulomb gases on the complex plane with weakly confining potentials, where the associated droplet is the entire complex plane. For these models, we derive precise asymptotic expansions of the free energies, including the constant terms.