Finite-difference distributions for the Ginibre ensemble

TL;DR

This paper analyzes the distribution of second differences in the Ginibre ensemble of complex random matrices, providing exact formulas and comparisons with other ensembles to understand spectral fluctuations.

Contribution

It derives exact analytical formulas for the distribution of second differences in the Ginibre ensemble, including real, imaginary parts, radius, and argument, for arbitrary N.

Findings

Exact distribution formulas for N=3 Ginibre ensemble

Comparison with Gaussian and Poisson ensembles

Insights into spectral fluctuation behaviors

Abstract

The Ginibre ensemble of complex random matrices is studied. The complex valued random variable of second difference of complex energy levels is defined. For the N=3 dimensional ensemble are calculated distributions of second difference, of real and imaginary parts of second difference, as well as of its radius and of its argument (angle). For the generic N-dimensional Ginibre ensemble an exact analytical formula for second difference's distribution is derived. The comparison with real valued random variable of second difference of adjacent real valued energy levels for Gaussian orthogonal, unitary, and symplectic, ensemble of random matrices as well as for Poisson ensemble is provided.