Dynamics of a planar Coulomb gas

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

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]

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.

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

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

François Bolley, Djalil Chafaï, Joaquin Fontbona. Dynamics of a planar Coulomb gas. The Annals of Applied Probability, 2018, 28 (5), pp.3152-3183. ⟨10.1214/18-AAP1386⟩. ⟨hal-01546288v4⟩