A limit theorem at the edge of a non-Hermitian random matrix ensemble

The study of the edge behaviour in the classical ensembles of Gaussian Hermitian matrices has led to the celebrated distributions of Tracy-Widom. Here we take up a similar line of inquiry in the non-Hermitian setting. We focus on the family of N x N random matrices with all entries independent and distributed as complex Gaussian of mean zero and variance 1/N. This is a fundamental non-Hermitian ensemble for which the eigenvalue density is known. Using this density, our main result is a limit law for the (scaled) spectral radius as N up arrow infinity. As a corollary, we get the analogous statement for the case where the complex Gaussians are replaced by quaternion Gaussians.