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Asymptotic freeness of unitary matrices in tensor product spaces for invariant states

Benoît Collins

Department of Mathematics, Kyoto University, Kyoto, Japan

E-mail Address: collins@math.kyoto-u.ac.jp

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Pierre Yves Gaudreau Lamarre

Department of Statistics, University of Chicago, Chicago, USA

E-mail Address: pyjgl@uchicago.edu

Corresponding author.

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Camille Male

Institut de Mathematiques de Bordeaux, Université de Bordeaux & CNRS, Talence, France

E-mail Address: camille.male@math.u-bordeaux.fr

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https://doi.org/10.1142/S2010326322500526Cited by:0 (Source: Crossref)

Abstract

In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In [6], the first and second authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. As a byproduct, we explore two additional generalizations: (i) we state results of freeness in a context of general sequences of representations of the unitary group — the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.

Keywords:

AMSC: 60B20, 46L54, 60B15

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