The Hyperbolic-type Point Process

TL;DR

This paper introduces a two-parameter determinantal point process on the Poincaré disc, analyzing its variance asymptotics and particle distribution, with implications for hyperbolic geometry and random matrix theory.

Contribution

It presents a novel hyperbolic point process, computes variance asymptotics using geometric methods, and extends known Euclidean results to the hyperbolic setting.

Findings

Variance of particle count grows as the disc radius approaches the boundary.

Distribution of particles inside the disc is characterized for the weighted Bergman kernel case.

Provides a shorter proof of a known Euclidean result using geometric arguments.

Abstract

In this paper, we introduce a two-parameters determinantal point process in the Poincar\'e disc and compute the asymptotics of the variance of its number of particles inside a disc centered at the origin and of radius $r$ as $r$ tends to $1$. Our computations rely on simple geometrical arguments whose analogues in the Euclidean setting provide a shorter proof of Shirai's result for the Ginibre-type point process. In the special instance corresponding to the weighted Bergman kernel, we mimic the computations of Peres and Virag in order to describe the distribution of the number of particles inside the disc.