## Concentration for Coulomb gases on compact manifolds.(English)Zbl 1412.60011

Summary: 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 D. Chafaï et al. [J. Funct. Anal. 275, No. 6, 1447–1483 (2018; Zbl 1407.82045)] 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.

### MSC:

 60B05 Probability measures on topological spaces 82D05 Statistical mechanics of gases 26D10 Inequalities involving derivatives and differential and integral operators 35K05 Heat equation

Zbl 1407.82045
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### References:

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