Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry

Abstract

We compute all massive partition functions or characteristic polynomials and their complex eigenvalue correlation functions for non-Hermitean extensions of the symplectic and chiral symplectic ensemble of random matrices. Our results are valid for general weight functions without degeneracies of the mass parameters. The expressions we derive are given in terms of the Pfaffian of skew orthogonal polynomials in the complex plane and their kernel. They are much simpler than the corresponding expressions for symplectic matrix models with real eigenvalues, and we explicitly show how to recover these in the Hermitean limit. This explains the appearance of three different kernels as quaternion matrix elements there in terms of derivatives of a single kernel here. We compare to the Hermitean limit of complex matrix models with unitary symmetry leading to some determinantal identities.

24 pages, v2 published version: discussion + refs added, typos corected

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Cited by in corpus (19)

• Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem
• A new Chiral Two-Matrix Theory for Dirac Spectra with Imaginary Chemical Potential
• Characteristic polynomials in real Ginibre ensembles
• Random matrix theory of unquenched two-colour QCD with nonzero chemical potential
• A new approach to derive Pfaffian structures for random matrix ensembles
• Matrix averages relating to the Ginibre ensembles
• The Interpolating Airy Kernels for the beta=1 and beta=4 Elliptic Ginibre Ensembles
• The chiral Gaussian two-matrix ensemble of real asymmetric matrices
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
• A geometrical triumvirate of real random matrices
• Scaling Limits of Planar Symplectic Ensembles
• Universality Conjecture for all Airy, Sine and Bessel Kernels in the Complex Plane
• Spherical Induced Ensembles with Symplectic Symmetry
• Asymmetric random matrices: What do we need them for?
• Pfaffian point process for the Gaussian real generalised eigenvalue problem
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• CDT as a scaling limit of matrix models
• Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics