Fluctuations of the process of moduli for the Ginibre and hyperbolic ensembles

TL;DR

This paper studies the statistical fluctuations of the moduli in Ginibre and hyperbolic ensembles, revealing Gaussian and Poisson behaviors at different scales, including white noise and correlated fluctuations.

Contribution

It provides a detailed analysis of the fluctuation phenomena of moduli in these ensembles, identifying various types of Gaussian fluctuations and their covariance structures.

Findings

Gaussian and Poisson fluctuations observed

White noise and non-trivial covariance fluctuations identified

Fluctuation behaviors depend on the scale and location

Abstract

We investigate the point process of moduli of the Ginibre and hyperbolic ensembles. We show that far from the origin and at an appropriate scale, these processes exhibit Gaussian and Poisson fluctuations. Among the possible Gaussian fluctuations, we can find white noise but also fluctuations with non-trivial covariance at a particular scale.