Extension of the Ginibre Ensembles of Random Matrices

TL;DR

This paper extends the Ginibre ensembles of non-Hermitian random matrices by exploring their eigenvalue differences and comparing them with Gaussian ensembles, providing insights into dissipative quantum systems and their spectral properties.

Contribution

It introduces an extension of the Ginibre ensembles by analyzing eigenvalue differences as discrete Hessians and compares these with Gaussian Hermitian ensembles.

Findings

Eigenvalues are complex and their second differences serve as discrete Hessians.

Comparison reveals differences between non-Hermitian and Hermitian random matrix ensembles.

The extension offers new perspectives on the dynamics of dissipative quantum systems.

Abstract

The Ginibre ensemble of nonhermitean random Hamiltonian matrices $K$ is considered. Each quantum system described by $K$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. The second difference of complex eigenenergies is viewed as discrete analog of Hessian with respect to labelling index. The results are considered in view of Wigner and Dyson's electrostatic analogy. An extension of space of dynamics of random magnitudes is performed by introduction of discrete space of labeling indices. The comparison with the Gaussian ensembles of random hermitean Hamiltonian matrices $H$ is performed.