Dynamics of a planar Coulomb gas

François Bolley (1) , Djalil Chafaï (2, 3, 4) , Joaquín Fontbona (5)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Université Paris Dauphine-PSL
3 IUF - Institut universitaire de France
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
5 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]

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.

Mots clés: Coulomb gas; Ginibre Ensemble; Interacting particle system; Poincaré inequality; Lyapunov function; McKean-Vlasov equation

Domaines: Physique mathématique [math-ph]; Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

François Bolley, Djalil Chafaï, Joaquín Fontbona. Dynamics of a planar Coulomb gas. 2017. ⟨hal-01546288v1⟩