Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels

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.

34 pages; v2 uniqueness of odd polynomials clarified, minor corrections; v3 final version to appear in CMP

References in corpus (5)

• Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems
• Truncations of Random Orthogonal Matrices
• Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry
• Unquenched complex Dirac spectra at nonzero chemical potential: two-colour QCD lattice data versus matrix model
• Universal scaling limits of the symplectic elliptic Ginibre ensemble

Cited by in corpus (9)

• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Wronskian structures of planar symplectic ensembles
• Scaling Limits of Planar Symplectic Ensembles
• Spherical Induced Ensembles with Symplectic Symmetry
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type