Electronic Communications in Probability

Concentration for Coulomb gases on compact manifolds

David García-Zelada

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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 Chafaï, Hardy and Maïda 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.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 12, 18 pp.

Received: 12 September 2018
Accepted: 16 January 2019
First available in Project Euclid: 21 March 2019

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Zentralblatt MATH identifier

Primary: 60B05: Probability measures on topological spaces 26D10: Inequalities involving derivatives and differential and integral operators 35K05: Heat equation

Gibbs measure Green function Coulomb gas empirical measure concentration of measure interacting particle system singular potential heat kernel

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García-Zelada, David. Concentration for Coulomb gases on compact manifolds. Electron. Commun. Probab. 24 (2019), paper no. 12, 18 pp. doi:10.1214/19-ECP211. https://projecteuclid.org/euclid.ecp/1553133701

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