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

The Ginibre ensemble of nonhermitean random Hamiltonian matrices K is considered. Each quantum system described by K is a dissipative system and the eigenenergies Zi 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.