Partition function for the 2d Coulomb gas on a Jordan curve

TL;DR

This paper derives an asymptotic formula for the partition function of a 2D Coulomb gas constrained to a Jordan curve, establishing a central limit theorem for particle statistics and linking results to conformal mappings and the Grunsky operator.

Contribution

It provides the first asymptotic analysis of the partition function for Coulomb gases on Jordan curves, connecting geometric conformal mappings with statistical properties.

Findings

Asymptotic formula for the partition function derived

Central limit theorem established for linear statistics

Expressions involve exterior conformal mapping and Grunsky operator

Abstract

We prove an asymptotic formula for the partition function of a 2d Coulomb gas at inverse temperature $\beta>0$ confined to lie on a Jordan curve. This also gives a central limit theorem for a linear statistic of the particles in the gas. We obtain different expressions for the asymptotic mean and variance which involve either the exterior conformal mapping of the curve or the Grunsky operator.