Universal scaling limits of the symplectic elliptic Ginibre ensemble

Abstract

We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behaviour of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel-Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.

22 pages, 2 figures; v2: 24 pages, 3 figures, final version published in RMTA

References in corpus (4)

• Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Lemniscate ensembles with spectral singularity
• Real eigenvalues of elliptic random matrices

Cited by in corpus (8)

• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Finite size corrections for real eigenvalues of the elliptic Ginibre matrices
• Wronskian structures of planar symplectic ensembles
• Scaling Limits of Planar Symplectic Ensembles
• Spherical Induced Ensembles with Symplectic Symmetry
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Truncations of random symplectic unitary matrices
• A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type