First order global asymptotics for Calogero-Sutherland gases

Djalil Chafaï (1) , Nathael Gozlan (1) , Pierre-André Zitt (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We study a physical system of N interacting particles in Rd subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical measure as N tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension d > 2, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as N tends to infinity. In the more specific case of Coulomb interaction in dimension d > 2, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.

Mots clés: Coulomb interaction; Interacting particle systems; Potential theory; Large deviations; Riesz kernel; Coulomb interaction.

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

Djalil Chafaï, Nathael Gozlan, Pierre-André Zitt. First order global asymptotics for Calogero-Sutherland gases. 2013. ⟨hal-00818472v1⟩