Limit distributions of random matrices

TL;DR

This paper investigates the asymptotic behavior of independent random matrices and their blocks, introducing a matricial freeness framework that unifies various types of independence and extends free probability theory.

Contribution

It develops the concept of matricial freeness to analyze limit distributions of random matrices, including sums, products, and different independence structures.

Findings

Unified framework for limit distributions of sums and products of random matrices

Extension of free probability to boolean, monotone, and s-free independences

Characterization of asymptotic distributions of symmetric blocks of Hermitian matrices

Abstract

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type. This framework also leads to random matrix models for boolean, monotone and s-free independences.