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

Klara Courteaut, Kurt Johansson

TL;DR

The paper analyzes a planar $2d$ Coulomb gas confined to a Jordan curve and derives sharp asymptotics for the partition function and the Laplace transform of linear statistics. It develops a relative Szegő framework that reduces the problem to exterior conformal maps and the Grunsky operator $K$, yielding explicit mean and variance expressions in terms of $\phi$, $K$, and the associated vectors $\mathbf{g}$ and $\mathbf{d}$. The main contributions are a rigorous asymptotic formula for $D_n^{\beta}[e^g]$ (and hence $Z_{n,\beta}(\gamma)$) for general $\beta>0$, and a central limit theorem for linear statistics with a nontrivial limiting variance that depends on the exterior map and Grunsky data. The results connect the partition function asymptotics to the geometry of $\gamma$ via the exterior mapping and the Neumann-Poincaré/Grunsky structure, with implications for Weil-Petersson quasicircles and Loewner energy through the determinant $\det(I+K)$.

Abstract

We prove an asymptotic formula for the partition function of a 2d Coulomb gas at inverse temperature $β>0$ 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.

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

TL;DR

The paper analyzes a planar Coulomb gas confined to a Jordan curve and derives sharp asymptotics for the partition function and the Laplace transform of linear statistics. It develops a relative Szegő framework that reduces the problem to exterior conformal maps and the Grunsky operator , yielding explicit mean and variance expressions in terms of , , and the associated vectors and . The main contributions are a rigorous asymptotic formula for (and hence ) for general , and a central limit theorem for linear statistics with a nontrivial limiting variance that depends on the exterior map and Grunsky data. The results connect the partition function asymptotics to the geometry of via the exterior mapping and the Neumann-Poincaré/Grunsky structure, with implications for Weil-Petersson quasicircles and Loewner energy through the determinant .

Abstract

We prove an asymptotic formula for the partition function of a 2d Coulomb gas at inverse temperature 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.
Paper Structure (10 sections, 20 theorems, 254 equations)

This paper contains 10 sections, 20 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and an integral equation
  3. Preliminaries
  4. The integral equation
  5. A relative Szegő theorem
  6. Deforming the curve $\gamma$
  7. A smaller domain of integration
  8. The change of variables
  9. Computing the limit
  10. The partition function

Key Result

Theorem 1.1

Assume that $\gamma$ is a $C^{12+\alpha}$ Jordan curve and $g\in C^{4+\epsilon}(\gamma)$, for some $\alpha>0$, $\epsilon>0$. Then, as $n\to\infty$,

Theorems & Definitions (40)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1: The strengthened Grunsky inequality
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 30 more