Quantum symmetrical statistical system: Ginibre-Girko ensemble

TL;DR

This paper analyzes the distribution of eigenenergy differences in the Ginibre ensemble of complex random matrices, providing analytical formulas and exploring their implications in quantum dissipative systems.

Contribution

It introduces analytical formulas for the distribution of second differences of eigenenergies in the Ginibre ensemble, extending understanding of eigenvalue behavior in non-Hermitian quantum systems.

Findings

Derived formulas for second difference distributions of eigenenergies.

Provided explicit distributions for real and imaginary parts at N=3.

Discussed homogenization law of eigenenergies across ensembles.

Abstract

The Ginibre ensemble of complex random Hamiltonian matrices $H$ is considered. Each quantum system described by $H$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. For generic $N$-dimensional Ginibre ensemble analytical formula for distribution of second difference $\Delta^{1} Z_{i}$ of complex eigenenergies is presented. The distributions of real and imaginary parts of $\Delta^{1} Z_{i}$ and also of its modulus and phase are provided for $N$=3. The results are considered in view of Wigner and Dyson's electrostatic analogy. General law of homogenization of eigenergies for different random matrix ensembles is formulated.