分析和微分方程讨论班——The Brown measure of a sum of two free random variables, one of which is R-diagonal

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.

报告人简介：

钟平，现任美国怀俄明大学数学与统计系助理教授；于2014年在印第安纳大学取得博士学位。其研究方向是自由概率论，随机矩阵，算子代数，及量子信息；研究论文发表于JEMS, Trans. AMS, JFA, Math.Z等期刊。