Real second-order freeness and the asymptotic real second-order freeness of several real matrix ensembles

Abstract

We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, three real models of random matrices, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. We use a combinatorial approach to the matrix calculations similar to the genus expansion for complex random matrices, but in which nonorientable surfaces appear, demonstrating the commonality between the real models and the distinction from their complex analogues, motivating this distinct definition. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears.

New version with some rewriting, second-order limit distribution, and some theorems useful in future work stated explicitly

References in corpus (2)

• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Genus expansion for real Wishart matrices

Cited by in corpus (6)

• Real Second Order Freeness and Haar Orthogonal Matrices
• A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices
• Non-crossing Annular Pairings and The Infinitesimal Distribution of the GOE
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• Fluctuation Moments for Regular Functions of Wigner Matrices
• The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix