Spectral Statistics of Non-Hermitian Random Matrix Ensembles

TL;DR

This paper explores the spectral properties of non-Hermitian random matrix ensembles, revealing new eigenvalue behaviors such as satellites and rings, extending prior Hermitian results to complex eigenvalues.

Contribution

It generalizes the $k$-checkerboard ensembles to non-Hermitian matrices, identifying novel eigenvalue distributions and deriving singular value density formulas.

Findings

Eigenvalues form multiple satellites and annular rings.

Eigenvalue distributions depend on ensemble parameters.

Singular value joint density for Complex Symmetric Gaussian Ensemble derived.

Abstract

Recently Burkhardt et. al. introduced the $k$-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but $k$ of the eigenvalues are on the order of $\sqrt{N}$ and converge to semi-circular behavior, with the remaining $k$ of size $N$ and converging to hollow Gaussian ensembles). We generalize their work to consider non-Hermitian ensembles with complex eigenvalues; instead of a blip new behavior is seen, ranging from multiple satellites to annular rings. These results are based on moment method techniques adapted to the complex plane as well as analysis of singular values, and we further isolate the singular value joint density formula for the Complex Symmetric Gaussian Ensemble.