报 告 人: 许嘉铭 (俄亥俄州立大学)

时间地点: 五教5106，6月10日10:00

题      目: CLT of Beta ensembles in fixed and high temperature

摘     要: 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 \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.

报告人简介: Jiaming Xu is a visiting assistant professor at The Ohio State University. He obtained his PhD in 2024, supervised by Vadim Gorin and Benedek Valko. He works in integrable probability and random matrices.