Universal scaling limits of the symplectic elliptic Ginibre ensemble

TL;DR

This paper analyzes the eigenvalue distributions of symplectic elliptic Ginibre matrices, establishing local universality and detailed asymptotics of correlation functions at spectrum edges, with implications for non-Hermitian random matrix theory.

Contribution

It derives the scaling limits and convergence rates of correlation functions for symplectic elliptic Ginibre matrices, including subleading corrections depending on non-Hermiticity.

Findings

Established local universality at strong non-Hermiticity.

Derived subleading correction terms for edge correlation kernels.

Provided asymptotic analysis of the eigenvalue distribution.

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.