Connectedness of Poisson cylinders in Euclidean space

2013/04/23 by Broman, Erik I., Tykesson, Johan · 2 citations
#60D05 #60K35 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1304.6357

Abstract

We consider the Poisson cylinder model in \mathbb R^d, d≥ 3. We show that given any two cylinders \mathfrak c₁ and \mathfrak c₂ in the process, there is a sequence of at most d-2 other cylinders creating a connection between \mathfrak c₁ and \mathfrak c₂. In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in a previous paper. We also show that there are cylinders in the process that are not connected by a sequence of at most d-3 other cylinders. Thus, the diameter of the cluster of cylinders equals d-2.

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• The Poisson stick model in hyperbolic space
• Multi-hop visibility through the vacant set of Poissonian obstacles

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