On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions
Florent Benaych-Georges
Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 3
(2010), 644-652.
Abstract
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.
Résumé
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.
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Keywords: Free probability; Random matrices; Free convolution; Infinitely divisible laws; Marchenko–Pastur law
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aihp/1281100393
Digital Object Identifier: doi:10.1214/09-AIHP324
Mathematical Reviews number (MathSciNet): MR2682261
Zentralblatt MATH identifier: 1206.46055
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Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
