Remarks on absolute continuity in the context of free probability and random matrices

TL;DR

This paper establishes conditions under which the spectral distribution of certain symmetric random matrices and their free convolutions with the semicircle law are absolutely continuous, advancing understanding in free probability and random matrix theory.

Contribution

It provides new sufficient conditions for absolute continuity of spectral distributions in random matrices and free convolutions, linking stationary entries to spectral properties.

Findings

Spectral distribution of symmetric stationary matrices is absolutely continuous under certain conditions.

Free multiplicative convolution with the semicircle law can be absolutely continuous given specific measure conditions.

Results extend the understanding of spectral measures in free probability contexts.

Abstract

In this note, we show that the limiting spectral distribution of symmetric random matrices with stationary entries is absolutely continuous under some sufficient conditions. This result is applied to obtain sufficient conditions on a probability measure for its free multiplicative convolution with the semicircle law to be absolutely continuous.