Limit Theorems For Non-Hermitian Ensembles

Abstract

The complex Ginibre ensemble and its generalisation, the complex induced Ginibre ensemble, have been well-studied in the field of Random Matrix Theory. In the present work, the asymptotic distribution and the independence of the extreme eigenvalue moduli are studied in the limit of large dimensions of random matrices for these non-Hermitian ensembles. They are derived with the use of Andreief's integration formula and the known methodological approach defined for the study of the limiting distribution of the scaled spectral radius at the edge of the complex Ginibre ensemble. The limiting distribution of the scaled spectral radius and the scaled minimum modulus for the complex induced Ginibre ensemble, with a proportional rectangularity index different from zero, is the Gumbel distribution. In the limit of a large size of complex Ginibre matrices, the left and right tails of the distribution of the minimum modulus are the Rayleigh and Weibull distributions, respectively. The limiting left tail of the distribution of the minimum modulus is the same for these random matrix ensembles, with a rectangularity index of the complex induced Ginibre ensemble equal to zero. This phenomenon is also verified for the right tail of the distribution of this random minimum. The independence of the extreme moduli is formally established, at appropriate scaling, for large matrices from the complex Ginibre ensemble and the complex induced Ginibre ensemble with a proportional rectangularity index. This study extends knowledge in the field of Random Matrix Theory for random variables, like the minimum modulus of matrices from non-Hermitian ensembles, whose limiting stochastic dynamics have never been explored.