Density of Brown measure of free circular Brownian motion

TL;DR

This paper analyzes the density behavior of the Brown measure for free circular Brownian motion with arbitrary initial conditions, revealing boundary behaviors analogous to known spectral phenomena in free probability.

Contribution

It establishes the boundary behavior of the Brown measure density under mild assumptions, identifying sharp cutoffs or quadratic decay at boundary points, extending known spectral edge phenomena.

Findings

Density exhibits sharp cut or quadratic decay at boundary points

Boundary behaviors mirror those in free semicircular Brownian motion

Assumption on initial operator is necessary for the results

Abstract

We consider the Brown measure of the free circular Brownian motion, $\boldsymbol{a}+\sqrt{t}\boldsymbol{x}$, with an arbitrary initial condition $\boldsymbol{a}$, i.e. $\boldsymbol{a}$ is a general non-normal operator and $\boldsymbol{x}$ is a circular element $*$-free from $\boldsymbol{a}$. We prove that, under a mild assumption on $\boldsymbol{a}$, 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 $\boldsymbol{a}$ is necessary.