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

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.

44 pages

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Cited by in corpus (7)

• Nonlinear Analogue of the May-Wigner Instability Transition
• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Directional Extremal Statistics for Ginibre Eigenvalues
• Finite size corrections for real eigenvalues of the elliptic Ginibre matrices
• Wronskian structures of planar symplectic ensembles
• Real eigenvalues of elliptic random matrices
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge