CLT FOR FLUCTUATIONS OF BETA-ENSEMBLES WITH GENERAL POTENTIAL

Florent Bekerman (1) , Thomas Leblé (2) , Sylvia Serfaty (2)

1 MIT - Massachusetts Institute of Technology
2 CIMS - Courant Institute of Mathematical Sciences [New York]

We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or β-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.

Mots clés: β-ensembles; Log Gas; Central Limit Theorem; Linear statistics. MSC classification: 60F05; 60K35; 60B10; 60B20; 82B05; 60G15

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

Florent Bekerman, Thomas Leblé, Sylvia Serfaty. CLT FOR FLUCTUATIONS OF BETA-ENSEMBLES WITH GENERAL POTENTIAL. 2020. ⟨hal-03040294⟩