On a surprising relation between rectangular and square free convolutions

TL;DR

This paper explores a new relationship between rectangular and square free convolutions, providing a simplified proof of a recent limit distribution result for large random matrices without requiring compact support.

Contribution

It reformulates a recent limit distribution result in terms of rectangular free convolution and offers a shorter proof under weaker assumptions.

Findings

Established a connection between rectangular and square free convolutions.

Provided a simplified proof of the limit empirical singular distribution result.

Extended the applicability of the result to non-compactly supported measures.

Abstract

Debbah and Ryan have recently proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free convolution introduced in a previous paper and then we give a new, shorter, proof of this result under weaker hypothesis: we do not suppose the \pro measure in question in this result to be compactly supported anymore. At last, we discuss the inclusion of this result in the family of relations between rectangular and square random matrices.