Connectedness of Poisson cylinders in Euclidean space

Abstract

We consider the Poisson cylinder model in ${\mathbb{R}}^d$, $d\ge 3$. 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 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 $d-2$.

30 pages

References in corpus (1)

• Cylinders' percolation in three dimensions