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Connectedness of Poisson cylinders in Euclidean space. (English. French summary) Zbl 1333.60197
Summary: We consider the Poisson cylinder model in \(\mathbb{R}^{d}\), \(d\geq3\). We show that given any two cylinders \({\mathfrak{c}}_{1}\) and \({\mathfrak{c}}_{2}\) in the process, there is a sequence of at most \(d-2\) other cylinders creating a connection between \({\mathfrak{c}}_{1}\) and \({\mathfrak{c}}_{2}\). In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in [J. Tykesson and D. Windisch,Probab. Theory Relat. Fields 154, No. 1–2, 165–191 (2012; Zbl 1263.82027)]. 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 \(d-2\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
82B43 Percolation
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