Real Second Order Freeness and Haar Orthogonal Matrices

TL;DR

This paper proves that Haar orthogonal matrices and certain independent random matrices become asymptotically real second order free, extending known examples and providing a theoretical foundation for their joint behavior.

Contribution

It establishes the asymptotic real second order freeness of Haar orthogonal matrices with independent ensembles under conjugation invariance, generalizing previous specific cases.

Findings

Haar orthogonal matrices are asymptotically real second order free with certain ensembles.

The result applies to ensembles with a real second order limit distribution.

It captures and generalizes known examples of real second order freeness.

Abstract

We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier [R1, R2].