Concentration for Coulomb gases and Coulomb transport inequalities

Djalil Chafaï (1, 2) , Adrien Hardy (3) , Mylène Maïda (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Université Paris Dauphine-PSL
3 LPP - Laboratoire Paul Painlevé - UMR 8524

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann–Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we obtain for the first time a concentration inequality for the empirical spectral distribution of Ginibre random matrices. Our approach relies crucially on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest.

Mots clés: Coulomb gas; Ginibre ensemble; Random matrix; Transport of measure; Transport inequality; Talagrand inequality; Wasserstein distance; Concentration of measure

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

Djalil Chafaï, Adrien Hardy, Mylène Maïda. Concentration for Coulomb gases and Coulomb transport inequalities. 2016. ⟨hal-01374624v1⟩