Limit distributions of Gaussian block ensembles

TL;DR

This paper explores the asymptotic behavior of Gaussian block ensembles using the newly introduced concept of matricial freeness, extending free probability theory to matrix blocks.

Contribution

It introduces and applies the concept of matricial freeness to analyze the limit distributions of Gaussian block ensembles.

Findings

Describes asymptotic distributions of Gaussian block matrices

Extends free probability to matricial structures

Provides new tools for analyzing matrix ensembles

Abstract

It has been shown by Voiculescu that important classes of square independent random matrices are asymptotically free, where freeness is a noncommutative analog of classical independence. Recently, we introduced the concept of matricial freeness, which is similar to freeness in free probability, but it also has some matricial features. Using this new concept of noncommutative independence, we described the asymptotics of blocks and symmetric blocks of certain classes of independent random matrices. In this paper, we present the main results obtained in this framework, concentrating on the ensembles of blocks of Gaussian random matrices.