Freeness and The Transposes of Unitarily Invariant Random Matrices

TL;DR

This paper demonstrates that real second order freeness emerges in the context of Haar unitary and unitarily invariant matrices when considering transposes, revealing that such matrices are asymptotically free from their transposes.

Contribution

It introduces the novel result that unitarily invariant random matrices are asymptotically free from their transposes, expanding the understanding of freeness in random matrix theory.

Findings

Real second order freeness appears with transposes.

Unitarily invariant matrices are asymptotically free from their transposes.

New connections between freeness and matrix transposes are established.

Abstract

We show that real second order freeness appears in the study of Haar unitary and unitarily invariant random matrices when transposes are also considered. In particular we obtain the unexpected result that a unitarily invariant random matrix will be asymptotically free from its transpose.