Discrete Hessians in study of Quantum Statistical Systems: Complex Ginibre Ensemble

TL;DR

This paper investigates the properties of complex eigenenergies in the Ginibre ensemble, using discrete Hessians to analyze dissipative quantum systems and extending the space of dynamics through discrete labeling indices.

Contribution

It introduces the concept of discrete Hessians for complex eigenenergies and extends the dynamical space with discrete labeling indices in the context of non-Hermitian random matrices.

Findings

Eigenenergies are modeled as complex random variables.

Discrete Hessians provide insights into eigenenergy structure.

Extension of dynamical space enhances analysis of dissipative 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.