Concentration inequality around the thermal equilibrium measure of Coulomb gases

Abstract: This article deals with Coulomb gases at an intermediate temperature regime, in which no structure is observed at the microscopic level, but the mass in confined to a compact set. Our main result is a concentration inequality around the thermal equilibrium measure, stating that with probability exponentially close to $1,$ the empirical measure is $\mathcal{O}\left( \frac{1}{N^{\frac{1}{d}}}\right)$ close to the thermal equilibrium measure. We also prove that this concentration inequality is optimal in some sense. The main new tool are functional inequalities that allow us to compare the bounded Lipschitz norm of a measure to its $H^{-1}$ norm in some cases when the measure does not have compact support.