Optimal local law and central limit theorem for beta-ensembles

Abstract: In this talk, I will present a joint work with Paul Bourgade and Krishnan Mody. We consider beta-ensembles with general potentials (or equivalently a log-gas in dimension 1), which are a generalization of Gaussian beta-ensembles and of classical invariant ensembles of random matrices. We prove a multivariate central limit theorem for the logarithm of the characteristic polynomial, showing that it behaves as a log-correlated field. A key ingredient is an optimally sharp local law for the the Stieljes transform of the empirical measure which can be of independent interest. Both the proofs of the CLT and the local law are based essentially on loop equations techniques.

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

Video recordings of talks are posted online at www.youtube.com/channel/UC0CXjG-ZSIZHy0S40Px2FEQ .