On fluctuations of traces of large matrices over a non-commutative algebra

TL;DR

This paper explores the asymptotic behavior of traces of large matrices with non-commutative entries, revealing connections to free, Boolean, and monotone independence structures, which differ from classical commutative cases.

Contribution

It introduces a framework for understanding asymptotics of non-commutative matrix traces using free, Boolean, and monotone cumulants, highlighting new independence relations.

Findings

Asymptotic behavior described by free and Boolean cumulants.

Relation of monotone independence derived from Boolean independence.

Differences from classical commutative matrix trace behavior.

Abstract

The paper investigates the asymptotic behavior of (non-normalized) traces of certain classes of matrices with non-commutative random variables as entries. We show that, unlike in the commutative framework, the asymptotic behavior of matrices with free circular, respectively with Bernoulli distributed Boolean independent entries is described in terms of free, respectively Boolean cumulants. We also present an exemple of relation of monotone independence arising from the study of Boolean independence.