Limit theorems for Coulomb gases on a Jordan curve in an external potential

Abstract

We consider a Coulomb gas on a Jordan curve $\gamma$ in an external potential $V$ at inverse temperature $\beta >0$ 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 $\gamma$ in $V$ is strictly positive on $\gamma$. 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 $\gamma$. The coefficient of the latter vanishes for $\beta =2$. 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.

45 pages