On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 3, p. 644-652

In 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.

Dans cet article, on prouve un résultat reliant les versions carré et rectangulaire de la R-transformée, qui a pour conséquence une relation surprenante entre les versions carré et rectangulaire de la convolution libre additive, impliquant la loi de Marchenko-Pastur. On donne des conséquences de ce résultat portant sur les matrices aléatoires, sur l'infinie divisibilité et sur l'arithmétique des versions carré des convolutions additives et multiplicatives.

DOI : https://doi.org/10.1214/09-AIHP324
Classification:  46L54,  15A52
Keywords: free probability, random matrices, free convolution, infinitely divisible laws, Marchenko-Pastur law
@article{AIHPB_2010__46_3_644_0,
author = {Benaych-Georges, Florent},
title = {On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {46},
number = {3},
year = {2010},
pages = {644-652},
doi = {10.1214/09-AIHP324},
zbl = {1206.46055},
mrnumber = {2682261},
language = {en},
url = {http://www.numdam.org/item/AIHPB_2010__46_3_644_0}
}

Benaych-Georges, Florent. On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 3, pp. 644-652. doi : 10.1214/09-AIHP324. http://www.numdam.org/item/AIHPB_2010__46_3_644_0/

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