Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Florent Benaych-Georges

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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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 3 (2010), 644-652.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1281100393

Digital Object Identifier
doi:10.1214/09-AIHP324

Mathematical Reviews number (MathSciNet)
MR2682261

Zentralblatt MATH identifier
1206.46055

Subjects
Primary: 46L54: Free probability and free operator algebras 15A52

Keywords
Free probability Random matrices Free convolution Infinitely divisible laws Marchenko–Pastur law

Citation

Benaych-Georges, Florent. On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 3, 644--652. doi:10.1214/09-AIHP324. https://projecteuclid.org/euclid.aihp/1281100393


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