## Electronic Communications in Probability

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

### Concentration for Coulomb gases on compact manifolds

#### 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 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

**Source**

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

**Dates**

Received: 12 September 2018

Accepted: 16 January 2019

First available in Project Euclid: 21 March 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1553133701

**Digital Object Identifier**

doi:10.1214/19-ECP211

**Mathematical Reviews number (MathSciNet)**

MR3933036

**Zentralblatt MATH identifier**

1412.60011

**Subjects**

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

**Keywords**

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

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

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