The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal

TL;DR

This paper develops a method to compute the Brown measure of the sum of two free nonselfadjoint random variables when one is R-diagonal, extending previous approaches and emphasizing subordination functions for effective calculations.

Contribution

It introduces a new method leveraging classical free additive convolutions and subordination functions to determine the Brown measure of sums involving R-diagonal variables.

Findings

Effective calculation of Brown measures in key cases

Extension of previous methods to broader classes of variables

Use of subordination functions for explicit density determination

Abstract

Suppose that $X_{1}$ and $X_{2}$ are two $*$-free (generally unbounded) random variables with Brown measures $\mu_{X_{1}}$ and $\mu_{X_{2}}$, respectively. Using properties of classical free additive convolutions, we develop a method for calculating $\mu_{X_{1}+X_{2}}$when $X_{2}$ is $R$-diagonal. This method determines a density relative to Lebesgue measure on an open set whose closure contains the support of $\mu_{X_{1}+X_{2}}$. Effective calculations are possible in important cases. Biane and Lehner were the first to make significant progress on the problem we consider, even in some cases in which neither $X_{1}$ nor $X_{2}$ is $R$-diagonal. Our examples overlap with theirs, but we emphasize the use of subordination functions. When $X_{2}$ is circular, $\mu_{X_{1}+X_{2}}$ was studied earlier using two different approaches, one involving Hamilton-Jacobi equations, and another using standard free probability techniques. Our work extends the second approach.