Spectral Radii of Large Non-Hermitian Random Matrices

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 parameters 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→ α∈ (0 , ∞) and k/ n→ ∞, the liming distribution is the Gumbel distribution, a new distribution μ and the logarithmic normal distribution, respectively. The cumulative distribution function (cdf) of μ is the infinite product of some normal distribution functions. Another new distribution ν is also obtained for the spherical ensemble such that the cdf of ν is the infinite product of the cdfs of some Poisson-distributed random variables.