The Brown measure of a sum of two free random variables, one of which is R-diagonal

Speaker: Prof. ZHONG Ping (University of Wyoming)

Venue: Online (Tencent Meeting ID: 974-516-332)

Abstract:

In the 1980s, L. Brown introduced an analogue of eigenvalue distribution of a square matrix in the framework of operator algebras, now called Brown measure. The Brown measure is a fascinating object that has successful applications in operator algebras and non-Hermitian random matrix theory. The Brown measures of random variables in free probability theory can often predict the limiting eigenvalue distributions of non-Hermitian random matrices.

I will speak on joint work with Hari Bercovici on the Brown measure of a sum of two free random variables, one of which is R-diagonal. This answers an open question of Biane and Lehner posed in early 2000s. It is shown that subordination functions that appear in the study of free additive convolution can detect some information about the Brown measure. In many cases, this leads to an explicit calculation of Brown measures. Some ideas were used implicitly in earlier works of Dykema, Haagerup and Schultz. In another joint work with Ching-Wei Ho, we prove that the Brown measure is the limiting eigenvalue distribution of a full rank perturbation of the single ring random matrix model under some mild assumptions.

Contact person: Prof. LIU Weihua (lwh.math@zju.edu.cn)