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

Akemann G, Phillips MJ (2014)
Journal of Statistical Physics 155(3): 421-465.

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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.
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Akemann G, Phillips MJ. The Interpolating Airy Kernels for the $\beta=1$ and $\beta=4$ Elliptic Ginibre Ensembles. Journal of Statistical Physics. 2014;155(3):421-465.
Akemann, G., & Phillips, M. J. (2014). The Interpolating Airy Kernels for the $\beta=1$ and $\beta=4$ Elliptic Ginibre Ensembles. Journal of Statistical Physics, 155(3), 421-465.
Akemann, G., and Phillips, M. J. (2014). The Interpolating Airy Kernels for the $\beta=1$ and $\beta=4$ Elliptic Ginibre Ensembles. Journal of Statistical Physics 155, 421-465.
Akemann, G., & Phillips, M.J., 2014. The Interpolating Airy Kernels for the $\beta=1$ and $\beta=4$ Elliptic Ginibre Ensembles. Journal of Statistical Physics, 155(3), p 421-465.
G. Akemann and M.J. Phillips, “The Interpolating Airy Kernels for the $\beta=1$ and $\beta=4$ Elliptic Ginibre Ensembles”, Journal of Statistical Physics, vol. 155, 2014, pp. 421-465.
Akemann, G., Phillips, M.J.: The Interpolating Airy Kernels for the $\beta=1$ and $\beta=4$ Elliptic Ginibre Ensembles. Journal of Statistical Physics. 155, 421-465 (2014).
Akemann, Gernot, and Phillips, M. J. “The Interpolating Airy Kernels for the $\beta=1$ and $\beta=4$ Elliptic Ginibre Ensembles”. Journal of Statistical Physics 155.3 (2014): 421-465.
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