The strong asymptotic freeness of Haar and deterministic matrices

TL;DR

This paper proves that sequences of random matrices with strong limiting distributions retain this property when combined with independent Haar-distributed matrices, and provides methods to compute limits using free product constructions.

Contribution

It extends the strong asymptotic freeness results to include Haar matrices and orthogonal/symplectic matrices, with explicit limit computation methods.

Findings

Strong asymptotic freeness is preserved when enlarging matrix tuples with Haar matrices.

Limits of norms and traces can be computed via reduced free product construction.

Orthogonal and symplectic Haar matrices also exhibit strong limiting distributions.

Abstract

In this paper, we are interested in sequences of q-tuple of N-by-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 Thorbjornsen. 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.