Brown measures of deformed $L^\infty$-valued circular elements

TL;DR

This paper characterizes the Brown measure of deformed circular elements in operator algebras, detailing its density, boundary behavior, and singularities, with implications for large random matrices.

Contribution

It provides a comprehensive classification of the edge singularities and interior zeros of the Brown measure for deformed circular elements, including explicit examples.

Findings

Brown measure has a real analytic density except at boundary singularities.

All types of edge singularities occur for suitable choices of the operator.

The Brown measure describes the spectral distribution of large random matrices with diagonal deformations.

Abstract

We consider the Brown measure of $a+\mathfrak{c}$, where $a$ lies in a commutative tracial von Neumann algebra $\mathcal{B}$ and $\mathfrak{c}$ is a $\mathcal{B}$-valued circular element. Under certain regularity conditions on $a$ and the covariance of $\mathfrak{c}$ this Brown measure has a density with respect to the Lebesgue measure on the complex plane which is real analytic apart from jump discontinuities at the boundary of its support. With the exception of finitely many singularities this one-dimensional spectral edge is real analytic. We provide a full description of all possible edge singularities as well as all points in the interior, where the density vanishes. The edge singularities are classified in terms of their local edge shape while internal zeros of the density are classified in terms of the shape of the density locally around these points. We also show that each of these countably infinitely many singularity types occurs for an appropriate choice of $a$ when $\mathfrak{c}$ is a standard circular element. The Brown measure of $a+\mathfrak{c}$ arises as the empirical spectral distribution of a diagonally deformed non-Hermitian random matrix with independent entries when its dimension tends to infinity.