The Interpolating Airy Kernels for the beta=1 and beta=4 Elliptic Ginibre Ensembles

TL;DR

This paper derives new interpolating Airy kernels for the elliptic Ginibre ensembles with beta=1 and beta=4, unifying edge eigenvalue statistics across Hermitian and non-Hermitian limits in large-N matrices.

Contribution

It introduces the limiting matrix-kernels, called interpolating Airy kernels, for beta=1 and beta=4 ensembles, completing the microscopic edge analysis for all three elliptical Ginibre ensembles.

Findings

Derived the interpolating Airy kernels for beta=1 and beta=4 ensembles.

Unified the edge eigenvalue statistics between Hermitian and non-Hermitian cases.

Rederived the beta=2 interpolating Airy kernel, completing the set.

Abstract

We consider two families of non-Hermitian Gaussian random matrices, namely the elliptical Ginibre ensembles of asymmetric N-by-N matrices with Dyson index beta=1 (real elements) and with beta=4 (quaternion-real elements). Both ensembles have already been solved for finite N using the method of skew-orthogonal polynomials, given for these particular ensembles in terms of Hermite polynomials in the complex plane. In this paper we investigate the microscopic weakly non-Hermitian large-N limit of each ensemble in the vicinity of the largest or smallest real eigenvalue. Specifically, we derive the limiting matrix-kernels for each case, from which all the eigenvalue correlation functions can be determined. We call these new kernels the "interpolating" Airy kernels, since we can recover -- as opposing limiting cases -- not only the well-known Airy kernels for the Hermitian ensembles, but also the complementary error function and Poisson kernels for the maximally non-Hermitian ensembles at the edge of the spectrum. Together with the known interpolating Airy kernel for beta=2, which we rederive here as well, this completes the analysis of all three elliptical Ginibre ensembles in the microscopic scaling limit at the spectral edge.