Asymptotically liberating sequences of random unitary matrices

TL;DR

This paper introduces new systems of random unitary matrices that induce asymptotic freeness through conjugation, extending classical results by leveraging general conditions and Hadamard matrices.

Contribution

It establishes a general framework for asymptotic liberation of unitary matrices and applies it to novel constructions using Hadamard matrices, broadening the scope of free probability.

Findings

New systems of random unitary matrices induce asymptotic freeness.

General conditions for asymptotic liberation are proven.

Applications recover and extend previous results in the field.

Abstract

A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting "asymptotic liberation" under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the second-named author and of Tulino-Caire-Shamai-Verd\'{u}.