Wronskian structures of planar symplectic ensembles

TL;DR

This paper uncovers Wronskian structures in the correlation kernels of symplectic Ginibre ensembles, revealing universal scaling limits and connections between different symmetry classes of non-Hermitian random matrices.

Contribution

It introduces a unified Wronskian framework for the correlation kernels of symplectic Ginibre ensembles and their universality classes, advancing understanding of non-Hermitian random matrix eigenvalue distributions.

Findings

Derived scaling limits for symplectic Ginibre variants.

Established Wronskian structures in universality classes.

Linked symplectic and complex symmetry kernels.

Abstract

We consider the eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which are known to form a Pfaffian point process in the plane. It was recently discovered that the limiting correlation kernel of the symplectic Ginibre ensemble in the vicinity of the real line can be expressed in a unified form of a Wronskian. We derive scaling limits for variations of the symplectic Ginibre ensemble and obtain such Wronskian structures for the associated universality classes. These include almost-Hermitian bulk/edge scaling limits of the elliptic symplectic Ginibre ensemble and edge scaling limits of the symplectic Ginibre ensemble with boundary confinement. Our proofs follow from the generalised Christoffel-Darboux formula for the former and from the Laplace method for the latter. Based on such a unified integrable structure of Wronskian form, we also provide an intimate relation between the function in the argument of the Wronskian in the symplectic symmetry class and the kernel in the complex symmetry class which form determinantal point processes in the plane.