Spectral Radii of Large Non-Hermitian Random Matrices

By Tiefeng Jiang, Yongcheng Qi. 2014

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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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ee42cb94ad3a30c780d19d7b27ab623e4997f25b