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 of $Z_N$ such that $A_N$ converges in distribution to a self-adjoint random variable $x_0$ in a $W^{\ast }$-probability space $(\mathcal{A},\tau )$. 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 a circular variable of variance $t$, freely independent of $x_0$. 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 a semicircular variable of variance $t$, freely independent of $x_0$. 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 of $G_N(t)$ such that $U_N$ converges in distribution to a unitary random variable $u$ in $(\mathcal{A},\tau )$. The random matrix $U_N{G}_N(t)$ converges in $\ast$-distribution to $u{b}_t$ where $b_t$ is the free multiplicative Brownian motion, freely independent of $u$. We compute the Brown measure ${\mu }_t$ of $u{b}_t$, 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 ${\mu }_t$ is related to the subordination function of the free multiplicative convolution of $u{u}_t$ where $u_t$ is the free unitary Brownian motion, freely independent of $u$. The push-forward of ${\mu }_t$ by a natural map is the law of $u{u}_t$.

In the special case that $u$ is Haar unitary, the Brown measure ${\mu }_t$ follows the annulus law . The support of the Brown measure of $u{b}_t$ 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 $\frac{1}{2\pi t}\frac{1}{r^2}.$