Fluctuations of Two-Dimensional Coulomb Gases

Thomas Leblé, Sylvia Serfaty

Published 2016-09-26Version 1

We study two-dimensional Coulomb gases with arbitrary inverse temperature $\beta$ and general confining potential (these are also called $\beta$-ensembles). The points of the ensemble are known to follow a certain equilibrium distribution characterized by an obstacle problem, and we consider the fluctuations of the random field, at both the macroscopic scale and mesoscopic scales, possibly near the boundary of the support of the equilibrium measure. We prove that the linear statistics have a Gaussian limit whose mean and variance is characterized, which can be stated as the convergence of the fluctuations 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-Virag valid in the determinantal case $\beta=2$, with weaker assumptions near the boundary. We also prove moderate deviations upper bounds for the linear statistics. Our proof relies on the energy approach and our previous LDP results, combined with a change of variables constructed with the help of quantitative stability estimates on the solutions to the obstacle problem.