Dynamics of a planar Coulomb gas

Abstract : 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. The invariant law in our model is neither product nor log-concave. Nevertheless 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, and its 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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https://hal.archives-ouvertes.fr/hal-01546288
Contributeur : Djalil Chafaï <>
Soumis le : vendredi 23 juin 2017 - 15:20:05
Dernière modification le : jeudi 29 juin 2017 - 01:12:44

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  • HAL Id : hal-01546288, version 1
  • ARXIV : 1706.08776

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François Bolley, Djalil Chafaï, Joaquín Fontbona. Dynamics of a planar Coulomb gas. 2017. <hal-01546288>

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