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October 2018 Dynamics of a planar Coulomb gas
François Bolley, Djalil Chafaï, Joaquín Fontbona
Ann. Appl. Probab. 28(5): 3152-3183 (October 2018). DOI: 10.1214/18-AAP1386


We study the long-time behavior of the dynamics of interacting planar Brownian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann–Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincaré inequalities are available. Moreover, the second moment dynamics turns out to be a nice Cox–Ingersoll–Ross process, in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.


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François Bolley. Djalil Chafaï. Joaquín Fontbona. "Dynamics of a planar Coulomb gas." Ann. Appl. Probab. 28 (5) 3152 - 3183, October 2018.


Received: 1 July 2017; Revised: 1 February 2018; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974776
MathSciNet: MR3847984
Digital Object Identifier: 10.1214/18-AAP1386

Primary: 60K35, 65C35, 82C22
Secondary: 60B20

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.28 • No. 5 • October 2018
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