Percolation in the vacant set of Poisson cylinders

TL;DR

This paper studies the percolation properties of the vacant space in a high-dimensional Poisson cylinder model, identifying a phase transition at a critical intensity in dimensions four and higher.

Contribution

It establishes the existence of a critical intensity for percolation in dimensions four and above, and analyzes non-percolation in three dimensions.

Findings

Percolation occurs in the vacant set for u<u_*(d) in d>=4.

Vacant set does not percolate for large u in d=3.

Vacant set intersected with a 2D subspace does not percolate for small u in d=3.

Abstract

We consider a Poisson point process on the space of lines in R^d, where a multiplicative factor u>0 of the intensity measure determines the density of lines. Each line in the process is taken as the axis of a bi-infinite cylinder of radius 1. We investigate percolative properties of the vacant set, defined as the subset of R^d that is not covered by any such cylinder. We show that in dimensions d >= 4, there is a critical value u_*(d) \in (0,\infty), such that with probability 1, the vacant set has an unbounded component if u<u_*(d), and only bounded components if u>u_*(d). For d=3, we prove that the vacant set does not percolate for large u and that the vacant set intersected with a two-dimensional subspace of R^d does not even percolate for small u>0.