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We consider the eigenvalues of symplectic elliptic Ginibre matrices which areknown to form a Pfaffian point process whose correlation kernel can beexpressed in terms of the skew-orthogonal Hermite polynomials. We derive thescaling limits and the convergence rates of the correlation functions at thereal bulk/edge of the spectrum, which in particular establishes the localuniversality at strong non-Hermiticity. Furthermore, we obtain the subleadingcorrections of the edge correlation kernels, which depend on thenon-Hermiticity parameter contrary to the universal leading term. Our proofsare based on the asymptotic behaviour of the complex elliptic Ginibre ensembledue to Lee and Riser as well as on a version of the Christoffel-Darbouxidentity, a differential equation satisfied by the skew-orthogonal polynomialkernel.

Universal scaling limits of the symplectic elliptic Ginibre ensemble