Critical temperatures and collapsing of two-dimensional Log gases

TL;DR

This paper analyzes the critical behavior of two-dimensional log gases on a sphere, deriving explicit formulas for the critical temperature and studying the system's collapse phenomena using advanced geometric and analytic methods.

Contribution

It introduces a novel approach using Fulton--MacPherson compactification and complex analysis to relate critical temperatures to a discrete optimization problem in log gases.

Findings

Explicit formulas for critical temperature and partition function divergence.

Description of dipole formation near the critical temperature for two-component plasma.

Application of the theory to models like the Onsager turbulence model.

Abstract

We consider the canonical ensemble of a system of point particles on the sphere interacting via a logarithmic pair potential. In this setting, we study the associated Gibbs measure and partition function, and we derive explicit formulas relating the critical temperature, at which the partition function diverges, to a certain discrete optimization problem. We further show that the asymptotic behavior of both the partition function and the Gibbs measure near the critical temperature is governed by the same optimization problem. Our approach relies on the Fulton--MacPherson compactification of configuration spaces and analytic continuation of complex powers. To illustrate the results, we apply them to well-studied systems, including the two-component plasma and the Onsager model of turbulence. In particular, for the two-component plasma with general charges, we describe the formation of dipoles close to the critical temperature, which we determine explicitly.