Poisson cylinders in hyperbolic space

TL;DR

This paper investigates the Poisson cylinder model in hyperbolic space, revealing a phase transition in connectivity unlike Euclidean space and showing the infinite diameter of the collection at any non-trivial intensity.

Contribution

It demonstrates a phase transition in the connectivity of Poisson cylinders in hyperbolic space and establishes the infinite diameter at any non-trivial intensity.

Findings

Phase transition in connectivity in hyperbolic space

Infinite diameter of cylinder collection at non-trivial intensities

Contrast with Euclidean case regarding connectivity

Abstract

We consider the Poisson cylinder model in $d$-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.