Title: The Brown measure of the sum of a free random variable and an elliptic deformation of Voiculescu's circular element
Speaker: Ping Zhong (University of Wyoming)
Date: June 15, 2022
Abstract: The circular element is the most important example of non-normal random variable used in free probability, and its Brown measure is the uniform measure in the unit disk. The circular element has connection to asymptotics of non-normal random matrices with i.i.d. entries. We obtain a formula for the Brown measure of the addition $x_0+c$ of an arbitrary free random variable $x_0$ and circular element $c$, which is known to be the limit empirical spectral distribution of deformed i.i.d. random matrices.
Generalizing the case of circular and semi-circular elements, we also consider $g$, a family of elliptic deformations of $c$, that is $*$-free from $x_0$. Possible degeneracy then prevents a direct calculation of the Brown measure of $x_0+g$. We instead show that the whole family of Brown measures of operators $x_0+g$ are the push-forward measures of the Brown measure of $x_0+c$ under a family of self-maps of the complex plane, which could possibly be singular. The main results offer potential applications to various deformed random matrix models. This work generalizes earlier results of Bordenave-Caputo-Chafai, Hall-Ho, and a joint work with Ho.
