Spectral Radii of Large Non-Hermitian Random Matrices

Abstract

By using the independence structure of points following a determinantal point process, we study the radii of the spherical ensemble, the truncation of the circular unitary ensemble and the product ensemble with parameter n and k. The limiting distributions of the three radii are obtained. They are not the Tracy-Widom distribution. In particular, for the product ensemble, we show that the limiting distribution has a transition phenomenon: when k/n -> 0, k/n -> a in (0,infty) and k/n -> infty, the liming distribution is the Gumbel distribution, a new distribution $\mu$ and the logarithmic normal distribution, respectively. The cumulative distribution function (cdf) of mu is the infinite product of some normal distribution functions. Another new distribution nu is also obtained for the spherical ensemble such that the cdf of nu is the infinite product of the cdfs of some Poisson-distributed random variables.