Brown Measures of Free Circular and Multiplicative Brownian Motions with Self-Adjoint and Unitary Initial Conditions

Abstract

Let $Z_N$ be a Ginibre ensemble and let $A_N$ be a Hermitian random matrix independent from $Z_N$ such that $A_N$ converges in distribution to a self-adjoint random variable $x_0$. For each $t>0$, the random matrix $A_N+\sqrt{t}Z_N$ converges in $\ast$-distribution to $x_0+c_t$, where $c_t$ is the circular variable of variance $t$, free from $x_0$. We use the Hamilton-Jacobi method to compute the Brown measure $\rho_t$ 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 the semicircular variable of variance $t$, free from $x_0$. Furthermore, the push-forward of $\rho_t$ 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 from $G_N(t)$ such that $U_N$ converges in distribution to a unitary random variable $u$. The random matrix $U_NG_N(t)$ converges in $\ast$-distribution to $ub_t$ where $b_t$ is the free multiplicative Brownian motion, free from $u$. We compute the Brown measure $\mu_t$ 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 \[\frac{1}{r^2}w_t(\theta)\] in polar coordinates in its support. The support of $\mu_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion, free from $u$. The push-forward of $\mu_t$ by a natural map is the law of $uu_t$. We compute the explicit formula for the special case where $u$ is Haar unitary. The support of the Brown measure of $ub_t$ is an annulus; in its support, the density in polar coordinates is given by \[\frac{1}{2\pi t}\frac{1}{r^2}.\]