Limit Theorems For Non-Hermitian Ensembles

TL;DR

This paper studies the asymptotic behavior of extreme eigenvalues in non-Hermitian random matrix ensembles, revealing their limiting distributions and independence properties in large dimensions.

Contribution

It provides new limit theorems for the distribution and tail behavior of extreme eigenvalues in complex Ginibre and induced Ginibre ensembles, including independence results.

Findings

Limiting distribution of spectral radius and minimum modulus is Gumbel.

Extreme eigenvalues are asymptotically independent.

Tail distributions of minimum modulus are Rayleigh and Weibull.

Abstract

The distribution of the modulus of the extreme eigenvalues is investigated for the complex Ginibre and complex induced Ginibre ensembles in the limit of large dimensions of random matrices. The limiting distribution of the scaled spectral radius and the scaled minimum modulus for the complex induced Ginibre ensemble, with a proportional rectangularity index, is the Gumbel distribution. The independence of these extrema is established, at appropriate scaling, for large matrices from the complex Ginibre ensemble as well as from the complex induced Ginibre ensemble for fixed and proportional rectangularity indexes. In the limit of a large size of the complex Ginibre matrices, the left and right tail distributions of the minimum modulus are the Rayleigh and Weibull distributions, respectively. The limiting left tail distribution of the minimum modulus is the same for these non-Hermitian ensembles when the rectangularity index of the complex induced Ginibre ensemble is equal to zero. This phenomenon is also verified for the right tail distribution of this minimum.