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Dynamics of a planar Coulomb gas

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Date
2018
Link to item file
https://hal.archives-ouvertes.fr/hal-01546288
Dewey
Probabilités et mathématiques appliquées
Sujet
Coulomb gas; Ginibre Ensemble; Interacting particle system; Poincaré inequality; Lyapunov function; McKean-Vlasov equation; Cox-Ingersoll-Ross process
Journal issue
The Annals of Applied Probability
Volume
28
Number
5
Publication date
2018
Article pages
3152-3183
Publisher
IMS
DOI
http://dx.doi.org/10.1214/18-AAP1386
URI
https://basepub.dauphine.fr/handle/123456789/17204
Collections
  • CEREMADE : Publications
Metadata
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Author
Bolley, François
Chafaï, Djalil
Fontbona, Joaquín
Type
Article accepté pour publication ou publié
Abstract (EN)
We study the long-time behavior of the dynamics of interacting planar Brow-nian 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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