Concentration inequality around the thermal equilibrium measure of Coulomb gases

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