A combinatorial result on asymptotic independence relations for random matrices with non-commutative entries

TL;DR

This paper establishes conditions under which non-commutative random matrices, like semicircular and Bernoulli matrices, exhibit asymptotic independence or freeness, extending classical results to broader non-commutative settings.

Contribution

It provides a general permutation-based condition for asymptotic independence in non-commutative random matrices, including semicircular and Bernoulli types.

Findings

Semicircular matrices are asymptotically free from their transposes.

The paper characterizes asymptotic second order relations for semicircular matrices.

The same conditions imply Boolean independence for Bernoulli matrices.

Abstract

The paper gives a general condition on permutations, condition under which a semicircular matrix is free independent, or asymptotically free independent from the semicircular matrix obtained by permuting its entries. In particular, it is shown that semicircular matrices are asymptotically free from their transposes, a result similar to the case of Gaussian random matrices. There is also an analysis of asymptotic second order relations between semicircular matrices and their transposes, with results not very similar to the commutative (i.e. Gaussian random matrices) framework. The paper also presents an application of the main results to the study of Gaussian random matrices and furthermore it is shown that the same condition as in the case of semicircular matrices gives Boolean independence, or asymptotic Boolean independence when applied to Bernoulli matrices.