Limit theorems for Coulomb gases on a Jordan curve in an external potential
Abstract
We consider a Coulomb gas on a Jordan curve γ in an external potential V at inverse temperature β>0 and obtain an asymptotic expansion of the free energy up to o(1) and a central limit theorem for linear statistics. We focus on the one-cut regime, where the density of the weighted equilibrium measure of γ in V is strictly positive on γ. The constant term in the (normalized) expansion consists of two parts: the Fredholm determinant of a generalized Grunsky operator and the Dirichlet energy of the logarithm of the density of the weighted equilibrium measure of γ. The coefficient of the latter vanishes for β=2. The variance of the fluctuations of the linear statistics only depends on the Dirichlet energy of the test function and is therefore independent of V. Essential in our approach is that the generalized Grunsky operator and the accompanying equilibrium parametrization allow us to transport the particles on the curve in the external potential to a reference object in a way that preserves the equilibrium measure. In our setting, the unit circle is the natural reference object.
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BibTeX
@misc{johansson2026limit,
title = {{Limit theorems for Coulomb gases on a Jordan curve in an external potential}},
author = {Kurt Johansson and Thomas Wolfs},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20675},
eprint = {2609.20675},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20675},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Limit theorems for Coulomb gases on a Jordan curve in an external potential
AU - Kurt Johansson
AU - Thomas Wolfs
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20675
UR - https://arxiv.org/abs/2609.20675
AB - We consider a Coulomb gas on a Jordan curve γ in an external potential V at inverse temperature β>0 and obtain an asymptotic expansion of the free energy up to o(1) and a central limit theorem for linear statistics. We focus on the one-cut regime, where the density of the weighted equilibrium measure of γ in V is strictly positive on γ. The constant term in the (normalized) expansion consists of two parts: the Fredholm determinant of a generalized Grunsky operator and the Dirichlet energy of the logarithm of the density of the weighted equilibrium measure of γ. The coefficient of the latter vanishes for β=2. The variance of the fluctuations of the linear statistics only depends on the Dirichlet energy of the test function and is therefore independent of V. Essential in our approach is that the generalized Grunsky operator and the accompanying equilibrium parametrization allow us to transport the particles on the curve in the external potential to a reference object in a way that preserves the equilibrium measure. In our setting, the unit circle is the natural reference object.
ER -