Wronskian structures of planar symplectic ensembles. (1st February 2023)

• Record Type: Journal Article
• Authors: Byun, Sung-Soo; Ebke, Markus; Seo, Seong-Mi
• Is Part Of: Nonlinearity. Volume 36:Number 2(2023)
• Page Start: 809
• Page End: 844
• Publication Date: 2023-02-01
• Subjects: planar symplectic ensemble -- Pfaffian point process -- skew-orthogonal polynomials -- Wronskian structures -- almost-Hermitian scaling limits; Primary 60B20; Secondary 33C45; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aca3f4
• Languages: English
• ISSNs: 0951-7715

Abstract

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.