The strong asymptotic freeness of Haar  and deterministic matrices

Collins, Benoît; Male, Camille

doi:10.24033/asens.2211

The strong asymptotic freeness of Haar and deterministic matrices
[Liberté asymptotique forte pour les matrices de Haar et les matrices déterministes]

Collins, Benoît ; Male, Camille

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 1, pp. 147-163.

Résumé
Abstract

Dans cet article, nous nous intéressons au $q$ -tuple de matrices $N \times N$ qui ont une distribution limite forte (i.e., pour tout polynôme non commutatif en les matrices et leurs adjoints, sa trace normalisée et sa norme convergent). Nous partons d'une telle suite de matrices aléatoires et montrons que cette propriété persiste si on rajoute au $q$ -tuple des matrices indépendantes unitaires distribuées suivant la mesure de Haar. Par ailleurs, la limite des normes et des traces en des polynômes non commutatifs en la suite élargie peut être calculée avec la construction du produit libre réduit. Ceci étend les résultats d'un des auteurs (C.M.) et de Haagerup et Thorbjørnsen. Nous montrons aussi qu'un $p$ -tuple de matrices indépendantes orthogonales et symplectiques a une distribution limite forte, étendant par là-même un résultat de Schultz. Nous passons aussi en revue quelques applications de notre résultat aux matrices aléatoires et à la théorie des espaces d'opérateur.

In this paper, we are interested in sequences of $q$ -tuple of $N \times N$ random matrices having a strong limiting distribution (i.e., given any non-commutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the $q$ -tuple is enlarged with independent unitary Haar distributed random matrices. Besides, the limit of norms and traces in non-commutative polynomials in the enlarged family can be computed with reduced free product construction. This extends results of one author (C. M.) and of Haagerup and Thorbjørnsen. We also show that a $p$ -tuple of independent orthogonal and symplectic Haar matrices have a strong limiting distribution, extending a recent result of Schultz. We mention a couple of applications in random matrix and operator space theory.

Publié le : 2014-09-16

MR Zbl

DOI : 10.24033/asens.2211

Classification : 46L54, 60B20, 15B52.
Keywords: Matrices aléatoires, probabilités libres, convergence forte.
Mot clés : Random matrices, free probability, strong convergence.

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     author = {Collins, Beno{\^\i}t and Male, Camille},
     title = {The strong asymptotic freeness of {Haar}  and deterministic matrices},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {147--163},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
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Collins, Benoît; Male, Camille. The strong asymptotic freeness of Haar  and deterministic matrices. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 1, pp. 147-163. doi : 10.24033/asens.2211. http://www.numdam.org/articles/10.24033/asens.2211/

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