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.

statistical mechanicsmathematical physicsprobability

Audience: researchers in the topic

Séminaire MEGA

Series comments: Description: Monthly seminar on random matrices and random graphs