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.

arXiv comment34 pages; v2 uniqueness of odd polynomials clarified, minor corrections; v3 final version to appear in CMP

References in corpus (24)

• Universal results from an alternate random matrix model for QCD with a baryon chemical potential
• Truncations of random unitary matrices
• The Ginibre ensemble of real random matrices and its scaling limits
• Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems
• Laughlin's wave functions, Coulomb gases and expansions of the discriminant
• Classical skew orthogonal polynomials and random matrices
• Products and Ratios of Characteristic Polynomials of Random Hermitian Matrices
• Characteristic Polynomials of Complex Random Matrix Models
• Eigenvalue correlations in non-Hermitean symplectic random matrices
• The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices
• Induced Ginibre ensemble of random matrices and quantum operations
• Truncations of Random Orthogonal Matrices
• Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry
• The random normal matrix model: insertion of a point charge
• Universality in the random matrix spectra in the regime of weak non-Hermiticity
• Products of Independent Quaternion Ginibre Matrices and their Correlation Functions
• Random-matrix theory of thermal conduction in superconducting quantum dots
• A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
• Non-Hermitian Random Matrix Ensembles
• Distribution of complex eigenvalues for symplectic ensembles of non-Hermitian matrices
• 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
• The Pfaff lattice and skew-orthogonal polynomials
• Coherence, squeezing and entanglement -- an example of peaceful coexistence

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