Real Second Order Freeness and Haar Orthogonal Matrices

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].

50 pages, revision has refreshed references and corrected typos

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• Jucys-Murphy elements and Weingarten matrices
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Cited by in corpus (7)

• Non-crossing Annular Pairings and The Infinitesimal Distribution of the GOE
• Matrix denoising: Bayes-optimal estimators via low-degree polynomials
• Freeness and The Partial Transposes of Wishart Random Matrices
• Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$
• Commutators of random matrices from the unitary and orthogonal groups
• Fluctuation moments induced by conjugation with asymptotically liberating random matrix ensembles
• The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix