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.

PhD thesis, submitted to the University of Melbourne, November 2011

References in corpus (6)

• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• Random matrices beyond the Cartan classification
• Universality for mathematical and physical systems
• Truncations of Random Orthogonal Matrices
• Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry
• Pfaffian Expressions for Random Matrix Correlation Functions

Cited by in corpus (2)

• Real eigenvalue statistics for products of asymmetric real Gaussian matrices
• How many eigenvalues of a product of truncated orthogonal matrices are real?