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2019 Concentration for Coulomb gases on compact manifolds
David García-Zelada
Electron. Commun. Probab. 24: 1-18 (2019). DOI: 10.1214/19-ECP211


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.


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David García-Zelada. "Concentration for Coulomb gases on compact manifolds." Electron. Commun. Probab. 24 1 - 18, 2019.


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

zbMATH: 1412.60011
MathSciNet: MR3933036
Digital Object Identifier: 10.1214/19-ECP211

Primary: 26D10 , 35K05 , 60B05

Keywords: concentration of measure , Coulomb gas , empirical measure , Gibbs measure , Green function , heat kernel , Interacting particle system , Singular potential


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