Abstract
Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.
Highlights
Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials
We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane
The study of orthogonal and skew-orthogonal polynomials in the complex plane is closely related to the question of integrability of determinantal and Pfaffian point processes in the plane
Summary
The study of orthogonal and skew-orthogonal polynomials in the complex plane is closely related to the question of integrability of determinantal and Pfaffian point processes in the plane. It is one of the goals of this article to develop the theory of SOP in the complex plane This will allow us to construct further examples of Pfaffian point processes with symplectic symmetry that are integrable, where this extended universality in the complex plane can be studied. Current explicit examples for SOP include Hermite [21] and Laguerre polynomials [5] for the elliptic symplectic Ginibre ensemble and its chiral counterpart, respectively. These formulas imply the large-N limit at the origin of the spectrum for all correlation functions at strong nonHermiticity in the corresponding ensembles. Appendix C contains a collection of integrals needed throughout the article
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