The Annals of Applied Probability

Dynamics of a planar Coulomb gas

François Bolley, Djalil Chafaï, and Joaquín Fontbona

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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.

Article information

Ann. Appl. Probab., Volume 28, Number 5 (2018), 3152-3183.

Received: July 2017
Revised: February 2018
First available in Project Euclid: 28 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C22: Interacting particle systems [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 65C35: Stochastic particle methods [See also 82C80]
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Coulomb gas Ginibre ensemble interacting particle system Poincaré inequality Lyapunov function Markov diffusion process McKean–Vlasov equation Cox–Ingersoll–Ross process


Bolley, François; Chafaï, Djalil; Fontbona, Joaquín. Dynamics of a planar Coulomb gas. Ann. Appl. Probab. 28 (2018), no. 5, 3152--3183. doi:10.1214/18-AAP1386.

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