Analysis Seminar: Upgrading free convolution to non-normal random variables

Abstract: The free probability theory is a probability theory of noncommutative random variables, where usual independence is replaced by free independence. It was initially designed to study longstanding problems about von Neumann algebras of free groups. It turns out to be an extremely powerful framework to study the universality laws in random matrix theory due to the groundbreaking work of Voiculescu. These limiting laws are encoded in abstract operators, called free random variables.

Brown measure is a sort of spectral measure for free random variables, not necessarily normal. I will report some recent progress on the Brown measure of the sum X+Y of two free random variables X and Y, where Y has certain symmetry or explicit R-transform. The procedure relies on Hermitian reduction and subordination functions. The Brown measure results can predict the limit eigenvalue distribution of various full rank perturbations of random matrix models. The talk is based on my work on Brown measure of elliptic operators and joint works with Hari Bercovici, Serban Belinschi and Zhi Yin.

Host: Xiang Tang