Fluctuations of Two-Dimensional Coulomb Gases

TL;DR

This paper establishes a Central Limit Theorem for linear statistics of two-dimensional Coulomb gases across various temperatures and scales, demonstrating convergence to a Gaussian Free Field and extending previous results.

Contribution

It provides the first CLT at arbitrary temperature and mesoscopic scales for Coulomb gases, including boundary cases, with new stability estimates and deviation bounds.

Findings

Proves CLT for Coulomb gases at all temperatures and scales.

Shows convergence of electrostatic potential to Gaussian Free Field.

Establishes rigidity and deviation bounds for linear statistics.

Abstract

We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the support of the equilibrium measure. This can be stated in terms of convergence of the random electrostatic potential to a Gaussian Free Field. Our result is the first to be valid at arbitrary temperature and at the mesoscopic scales, and we recover previous results of Ameur-Hendenmalm-Makarov and Rider-Vir\'ag concerning the determinantal case, with weaker assumptions near the boundary. We also prove moderate deviations upper bounds, or rigidity estimates, for the linear statistics and a convergence result for those corresponding to energy-minimizers. The method relies on a change of variables, a perturbative expansion of the energy, and the comparison of partition functions deduced from our previous work. Near the boundary, we use recent quantitative stability estimates on the solutions to the obstacle problem obtained by Serra and the second author.