Transport methods and universality for [beta]-ensembles

Abstract

In this thesis, we investigate the local and global properties of the eigenvalues of [beta]-ensembles. A lot of attention has been drawn recently on the universal properties of [beta]-ensembles, and how their local statistics relate to those of Gaussian ensembles. We use transport methods to prove universality of the eigenvalue gaps in the bulk and at the edge, in the single cut and multicut regimes. In a different direction, we also prove Central Limit Theorems for the linear statistics of [beta]-ensembles at the macroscopic and mesoscopic scales.