Connectedness of Poisson cylinders in Euclidean space

الأصل · EN

We consider the Poisson cylinder model in Rᵈ, d≥ 3. We show that given any two cylinders c₁ and c₂ in the process, there is a sequence of at most d-2 other cylinders creating a connection between c₁ and 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.