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

Florent Benaych-Georges (1, 2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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.

Mots clés: free probability; random matrices; free convolution; infinitely divisible laws; Marchenko-Pastur law

Domaines: Probabilités [math.PR]

Florent Benaych-Georges. On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions. 2008. ⟨hal-00315507v5⟩