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

TL;DR

This paper introduces real second-order freeness for real matrix ensembles, demonstrating that certain real matrices are asymptotically second-order free using a combinatorial approach involving nonorientable surfaces.

Contribution

It defines real second-order freeness and proves that real Ginibre, Gaussian orthogonal, and real Wishart matrices are asymptotically second-order free, highlighting differences from complex models.

Findings

Real matrix ensembles are asymptotically second-order free.

Nonorientable surfaces appear in combinatorial calculations.

Additional terms involving matrix transpose are identified.

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.