### Optimal local law and central limit theorem for beta-ensembles

Probability Seminar

9th December 2022, 3:30 pm – 4:30 pm

Fry Building, 2.04

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). 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 Stieltjes transform of the empirical measure. Both the proofs of the CLT and the local law are based essentially on loop equations techniques.

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