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