  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.
