Free compressions of R-diagonal random variables and the semigroup of Brown measures

TL;DR

This paper studies how the Brown measures of R-diagonal random variables behave under compression, showing convergence to uniform distribution in finite variance cases and stability properties in infinite variance cases.

Contribution

It extends previous results to unbounded R-diagonal variables and characterizes stable Brown measures under compression.

Findings

Brown measures of finite variance R-diagonal variables converge to uniform on the unit disc.

In infinite variance cases, certain Brown measures remain stable under compression.

The paper provides detailed properties of stable Brown measures in the infinite variance setting.

Abstract

We investigate the Brown measures of compressions of $R$-diagonal random variables, extending previous results to include unbounded cases. For random variables with finite variance, we demonstrate that the Brown measures of their compressions converge to the uniform distribution on the unit disc. In the case of infinite variance, we characterize the Brown measures that remain stable under the compression operation and explore their properties in detail.