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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
##### Keywords:
Poisson cylinder model; continuum percolation
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