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arXiv · 1809.04231

Concentration for Coulomb gases on compact manifolds

David García-Zelada
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Abstract

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.

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David García-Zelada. 2018-09-12. Concentration for Coulomb gases on compact manifolds. https://doi.org/10.1214/19-ecp211

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