# Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions

### Ching-Wei Ho

Academia Sinica, Taipei, Taiwan; University of Notre Dame, USA### Ping Zhong

University of Wyoming, Laramie, USA

## Abstract

Let $Z_{N}$ be a Ginibre ensemble and let $A_{N}$ be a Hermitian random matrix independent of $Z_{N}$ such that $A_{N}$ converges in distribution to a self-adjoint random variable $x_{0}$ in a $W_{∗}$-probability space $(A,τ)$. For each $t>0$, the random matrix $A_{N}+t Z_{N}$ converges in $∗$-distribution to $x_{0}+c_{t}$, where $c_{t}$ is a circular variable of variance $t$, freely independent of $x_{0}$. We use the Hamilton–Jacobi method to compute the Brown measure $ρ_{t}$ of $x_{0}+c_{t}$. The Brown measure has a density that is *constant along the vertical direction* inside the support. The support of the Brown measure of $x_{0}+c_{t}$ is related to the subordination function of the free additive convolution of $x_{0}+s_{t}$, where $s_{t}$ is a semicircular variable of variance $t$, freely independent of $x_{0}$. Furthermore, the push-forward of $ρ_{t}$ by a natural map is the law of $x_{0}+s_{t}$.

Let $G_{N}(t)$ be the Brownian motion on the general linear group and let $U_{N}$ be a unitary random matrix independent of $G_{N}(t)$ such that $U_{N}$ converges in distribution to a unitary random variable $u$ in $(A,τ)$. The random matrix $U_{N}G_{N}(t)$ converges in $∗$-distribution to $ub_{t}$ where $b_{t}$ is the free multiplicative Brownian motion, freely independent of $u$. We compute the Brown measure $μ_{t}$ of $ub_{t}$, extending the recent work by Driver–Hall–Kemp, which corresponds to the case $u=I$. The measure has a density of the special form

in polar coordinates in its support. The support of $μ_{t}$ is related to the subordination function of the free multiplicative convolution of $uu_{t}$ where $u_{t}$ is the free unitary Brownian motion, freely independent of $u$. The push-forward of $μ_{t}$ by a natural map is the law of $uu_{t}$.

In the special case that $u$ is Haar unitary, the Brown measure $μ_{t}$ follows the *annulus law* . The support of the Brown measure of $ub_{t}$ is an annulus with inner radius $e_{−t/2}$ and outer radius $e_{t/2}$. In its support, the density in polar coordinates is given by $2πt1 r_{2}1 .$

## Cite this article

Ching-Wei Ho, Ping Zhong, Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions. J. Eur. Math. Soc. 25 (2023), no. 6, pp. 2163–2227

DOI 10.4171/JEMS/1233