A note on the second order universality at the edge of Coulomb gases on the plane

Djalil Chafaï (1) , Sandrine Péché (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan and Rider, boils down to identities in law and to an instance of the Laplace method.

Mots clés: System of particles; Coulomb gases; Extreme values; Gumbel law

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

Djalil Chafaï, Sandrine Péché. A note on the second order universality at the edge of Coulomb gases on the plane. 2013. ⟨hal-00868922v1⟩