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.
Highlights
In the recent paper [6], the authors considered the so-called Poisson cylinder model in Euclidean space. This model can be described as a Poisson process ω on the space of bi-infinite lines in Rd
One places a cylinder c of radius one around every line L ∈ ω, and with a slight abuse of notation, we say that c ∈ ω
For Voronoi percolation, there is a form of exponentially decaying dependence, i.e. the probability that the same cell in a Voronoi tessellation contains both points x and y decays exponentially in the distance between x and y. This is not the case when dealing with the Poisson cylinder model in Euclidean space
Summary
In the recent paper [6], the authors considered the so-called Poisson cylinder model in Euclidean space. For Voronoi percolation, there is a form of exponentially decaying dependence, i.e. the probability that the same cell in a Voronoi tessellation contains both points x and y decays exponentially in the distance between x and y This is not the case when dealing with the Poisson cylinder model in Euclidean space. Since we believe that it may be of some independent interest, we present it here in the introduction, along with our main result concerning it In essence, it behaves like a branching process where every particle gives rise to an infinite number of offspring whose types can take any positive real value. Since any individual gives rise to an infinite number of offspring, the process will never die out It can still die out weakly in the sense that for any R there will eventually be no new points of type R or smaller.
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