# Analysis Seminar

Wednesday, February 9, 2022 — 4:00 PM EST

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.

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