Wronskian structures of planar symplectic ensembles

Byun SS, Ebke M, Seo SM (2023)

Publication Type: Journal article

Publication year: 2023


Book Volume: 36

Pages Range: 809-844

Journal Issue: 2

DOI: 10.1088/1361-6544/aca3f4


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.

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Byun, S.S., Ebke, M., & Seo, S.M. (2023). Wronskian structures of planar symplectic ensembles. Nonlinearity, 36(2), 809-844. https://dx.doi.org/10.1088/1361-6544/aca3f4


Byun, Sung Soo, Markus Ebke, and Seong Mi Seo. "Wronskian structures of planar symplectic ensembles." Nonlinearity 36.2 (2023): 809-844.

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