Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations

Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble

The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian ...

Characteristic polynomials in real Ginibre ensembles

Abstract. We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a skew-symmetric kernel from which all complex eigenvalue correlations can be derived. Our results are obtained in a very simple fashion without going to an eigenvalue representation, and are co...

Generalizations of orthogonal polynomials

We give a survey of recent generalizations for orthogonal polynomials that were recently obtained. It concerns not only multidimensional (matrix and vector orthogonal polynomials) and multivariate versions, or multipole (orthogonal rational functions) variants of the classical polynomials but also extensions of the orthogonality conditions (multiple orthogonality). Most of these generalizations...

Bounds for Real Roots and Applications to Orthogonal Polynomials

Extremal laws for the real Ginibre ensemble

The real Ginibre ensemble refers to the family of n × n matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as n→∞. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs...