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تسجيل مستخدم جديدConnectedness of Poisson cylinders in Euclidean space
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We consider the Poisson cylinder model in ${mathbb R}^d$, $dge 3$. 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 $d-2$.
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