Connectedness of Poisson cylinders in Euclidean space

TL;DR

This paper studies the connectivity properties of Poisson cylinders in Euclidean space, proving that any two cylinders can be connected through a chain of at most d-2 cylinders, establishing the union's connectedness and cluster diameter.

Contribution

It establishes the exact diameter of the cylinder cluster and confirms the union's connectedness, answering a previously open question.

Findings

The union of cylinders is connected in ${ m R}^d$ for d ≥ 3.

Any two cylinders can be connected via a chain of at most d-2 cylinders.

The cluster diameter of cylinders is exactly d-2.

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 $d-2$.