Jiaming Xu: CLT of Beta ensembles in fixed and high temperature

Abstract:

 It is a classical result that the global fluctuation of a large class of Beta ensembles (a.k.a log gases) is given by Gaussian vectors, for any fixed \(\beta>0\). Moreover, such a result can be extended to the so-called high temperature regime, i.e, when \(\beta\) goes to \(0\) as the number of particles tends to infinity. We give an if and only if condition of the LLN and CLT of an arbitrary sequence of \(N\)-particle systems on the real line, in both fixed and high temperature regime, so that it can be applied to the ensembles beyond log gases. Our conditions and toolbox are in terms of certain symmetric functions and differential operators, and using such conditions, we manage to prove the global CLT for beta additions and single level beta corners processes. As a by-product, we introduce a class of second order free cumulants which linearize high temperature beta additions, and extend the results in Colins-Mingo-Sniady-Speicher and Benaych Georges-Cuenca-Gorin. This is an ongoing joint work with Cesar Cuenca.