The two-dimensional Coulomb gas: fluctuations through a spectral gap

TL;DR

This paper analyzes the fluctuations and asymptotic behavior of a radially symmetric Coulomb gas with gaps, revealing oscillatory particle distributions near the gap edges and providing explicit formulas involving special functions.

Contribution

It introduces a detailed asymptotic analysis of Coulomb gas ensembles with gaps, including explicit formulas for fluctuations and oscillations near the gap edges.

Findings

Oscillatory behavior in particle distribution near the gap edge

Explicit formulas involving discrete Gaussian, Szegő kernels, and Jacobi theta functions

Fine asymptotics for edge density, correlation kernel, and fluctuation cumulants

Abstract

We study a class of radially symmetric Coulomb gas ensembles at inverse temperature $\beta=2$, for which the droplet consists of a number of concentric annuli, having at least one bounded ``gap'' $G$, i.e., a connected component of the complement of the droplet, which disconnects the droplet. Let $n$ be the total number of particles. Among other things, we deduce fine asymptotics as $n \to \infty$ for the edge density and the correlation kernel near the gap, as well as for the cumulant generating function of fluctuations of smooth linear statistics. We typically find an oscillatory behaviour in the distribution of particles which fall near the edge of the gap. These oscillations are given explicitly in terms of a discrete Gaussian distribution, weighted Szeg\H{o} kernels, and the Jacobi theta function, which depend on the parameter $n$.