Percolation of fat Poisson cylinders in hyperbolic space

TL;DR

This paper investigates the percolation properties of fat Poisson cylinders in hyperbolic space, revealing a phase transition and connecting it to fractal models on spheres and Euclidean spaces.

Contribution

It establishes the existence of a percolation phase transition for fat Poisson cylinders in hyperbolic space and links this to semi-scale invariant fractal models.

Findings

Percolation phase transition proven for fat Poisson cylinders in hyperbolic space

Connection established between cylinder process and fractal models on spheres and Euclidean space

Fractal ball model in Euclidean space exhibits a non-empty sheet phase

Abstract

In this paper we study Poisson processes of so-called "fat" cylinders in hyperbolic space. As our main result we show that this model undergoes a percolation phase transition. We prove this by establishing a novel link between the fat Poisson cylinder process and semi-scale invariant random fractal models on the unit sphere and in $\mathbb{R}^d.$ As a secondary result, we show that the semi-scale invariant fractal ball model in $\mathbb{R}^d$ has a non-empty sheet phase.