On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions

TL;DR

This paper reveals a surprising connection between square and rectangular free convolutions through the Marchenko-Pastur law, impacting random matrix theory and free probability operations.

Contribution

It establishes a novel relation between square and rectangular R-transforms and free convolutions, with implications for random matrices and infinite divisibility.

Findings

Link between square and rectangular R-transforms

Relation involving the Marchenko-Pastur law

Implications for free additive and multiplicative convolutions

Abstract

n this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko-Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.