Analysis Seminar

Ping Zhong, University of Wyoming

"A surprising connection between Brown measures of Voiculescu's circular element and its elliptic deformations"

The circular element is the most important example of non-normal random variable used in free probability, and its Brown measure is theuniform 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 $c$. This answers a question of Biane-Lehner. 

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 natural maps of the complex plane assumed to be non-singular. We calculate density formulas for various interesting examples.  

The talk is based on arXiv:2108.09844 and a joint work with Ching-Wei Ho.

Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09