DYNAMICS OF A PLANAR COULOMB GAS

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 c...

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Bibliographic Details
Published in:The Annals of applied probability 2018-10, Vol.28 (5), p.3152-3183
Main Authors: Bolley, François, Chafaï, Djalil, Fontbona, Joaquín
Format: Article
Language:English
Subjects:
Brownian motion
Dependence
Dynamics
Functional Analysis
Invariants
Mathematical Physics
Mathematics
Probability
Well posed problems
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Summary: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.
ISSN:1050-5164
2168-8737
DOI:10.1214/18-AAP1386