Density of Brown measure of free circular Brownian motion

Abstract

We consider the Brown measure of the free circular Brownian motion, $\mathbf{a}+\sqrt{t}\mathbf{x}$, with an arbitrary initial condition $\mathbf{a}$, i.e. $\mathbf{a}$ is a general non-normal operator and $\mathbf{x}$ is a circular element $\ast$-free from $\mathbf{a}$. We prove that, under a mild assumption on $\mathbf{a}$, the density of the Brown measure has one of the following two types of behavior around each point on the boundary of its support – either (i) sharp cut, i.e. a jump discontinuity along the boundary, or (ii) quadratic decay at certain critical points on the boundary. Our result is in direct analogy with the previously known phenomenon for the spectral density of free semicircular Brownian motion, whose singularities are either a square-root edge or a cubic cusp. We also provide several examples and counterexamples, one of which shows that our assumption on $\mathbf{a}$ is necessary.