A geometrical triumvirate of real random matrices

Abstract

We present a five-step method for the calculation of eigenvalue correlation functions for various ensembles of real random matrices, based upon the method of (skew-) orthogonal polynomials. This scheme systematises existing methods and also involves some new techniques. The ensembles considered are: the Gaussian Orthogonal Ensemble (GOE), the real Ginibre ensemble, ensembles of partially symmetric real Gaussian matrices, the real spherical ensemble (of matrices $A^{-1}B$), and the real anti-spherical ensemble (consisting of truncated real orthogonal matrices). Particular emphasis is paid to the variations in method required to treat odd-sized matrices. Various universality results are discussed, and a conjecture for an `anti-spherical law' is presented.

arXiv commentPhD thesis, submitted to the University of Melbourne, November 2011

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• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• Real eigenvalue statistics for products of asymmetric real Gaussian matrices
• How many eigenvalues of a product of truncated orthogonal matrices are real?