Concentration inequality around the thermal equilibrium measure of Coulomb gases

Abstract

This article deals with Coulomb gases at an intermediate temperature regime, which are governed by a Gibb's measure in which the inverse temperature is much larger than 1N, where N is the number of particles. Our main result is a concentration inequality around the thermal equilibrium measure, stating that with probability exponentially close to 1, the empirical measure is O(1N1d) close to the thermal equilibrium measure. We also prove that this concentration inequality is optimal in some sense. The main new tool is functional inequalities that allow us to compare the bounded Lipschitz norm of a measure to its H−1 norm in some cases in which the measure does not have compact support.