PUB - Publikationen an der Universität Bielefeld

We consider two families of non-Hermitian Gaussian random matrices, namelythe elliptical Ginibre ensembles of asymmetric N-by-N matrices with Dyson index $\beta=1$ (real elements) and with $\beta=4$ (quaternion-real elements). Bothensembles have already been solved for finite N using the method ofskew-orthogonal polynomials, given for these particular ensembles in terms ofHermite polynomials in the complex plane. In this paper we investigate themicroscopic weakly non-Hermitian large-N limit of each ensemble in the vicinityof the largest or smallest real eigenvalue. Specifically, we derive thelimiting matrix-kernels for each case, from which all the eigenvaluecorrelation functions can be determined. We call these new kernels the"interpolating" Airy kernels, since we can recover -- as opposing limitingcases -- not only the well-known Airy kernels for the Hermitian ensembles, butalso the complementary error function and Poisson kernels for the maximallynon-Hermitian ensembles at the edge of the spectrum. Together with the knowninterpolating Airy kernel for beta=2, which we rederive here as well, thiscompletes the analysis of all three elliptical Ginibre ensembles in themicroscopic scaling limit at the spectral edge.