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.
Formulation of the main resultsThe main results of this paper, Theorems 1.1-1.6, establish limit theorems for additive statistics of the Ginibre ensemble and the hyperbolic ensembles, introduced by Krishnapur, including the determinantal point process with the Bergman kernel, which, by the Peres-Virág theorem, is the zero set of the Gaussian analytic function on the unit disc.In this section, we begin by recalling the notion of determinantal point process, which are point processes where the correlation functions take the form of a determinant. Afterwards, the specific examples we are interested in, namely the Ginibre point process and the hyperbolic ensembles, are discussed. Finally, we state the main results of this note, Theorems 1.1-1.6.1.1. Determinantal point process. Let X be a locally compact Polish space and B 0 (X) the collection of all pre-compact Borel subsets of X. We shall denote by Conf(X), the space of all locally finite configurations over X, that is,We shall consider this set endowed with the vague topology, i.e., the weakest topology on Conf(X) such that for any compactly supported continuous function f on X, the map Conf(X) ∋ ξ → X f dξ is continuous. It can be seen that the configuration space Conf(X) equipped with the vague topology turns out to be a Polish space. Additionally, it can be seen that the Borel σ-algebra F on Conf(X) is generated by the cylinder sets C ∆ n = ξ ∈ Conf(X) | ξ(∆) = n , where n ∈ N = {0, 1, 2, • • • } and ∆ ∈ B 0 (X). Finally, we will say that a measurable map X : (Ω, F (Ω), P) → (Conf(X), F )
