The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element

TL;DR

This paper calculates the Brown measure of the sum of a self-adjoint element and an imaginary semicircular element, revealing its support, density, and connections to other free probability distributions, refining previous physics results.

Contribution

It provides a rigorous computation of the Brown measure for a specific free sum, including support, density, and transformations relating to other free distributions.

Findings

Brown measure supported in a specific bounded region

Density is constant in the vertical direction within the support

Pushforward of the measure yields the distribution of related sums

Abstract

We compute the Brown measure of $x_{0}+i\sigma_{t}$, where $\sigma_{t}$ is a free semicircular Brownian motion and $x_{0}$ is a freely independent self-adjoint element that is not a multiple of the identity. The Brown measure is supported in the closure of a certain bounded region $\Omega_{t}$ in the plane. In $\Omega_{t},$ the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature. We also show that pushing forward the Brown measure of $x_{0}+i\sigma_{t}$ by a certain map $Q_{t}:\Omega_{t}\rightarrow\mathbb{R}$ gives the distribution of $x_{0}+\sigma_{t}.$ We also establish a similar result relating the Brown measure of $x_{0}+i\sigma_{t}$ to the Brown measure of $x_{0}+c_{t}$, where $c_{t}$ is the free circular Brownian motion.