Concentration for Coulomb gases on compact manifolds

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafa\"i, Hardy and Ma\"ida on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.