Cylinders' percolation in three dimensions

TL;DR

This paper investigates the percolation properties of the complement of a Poissonian ensemble of infinite cylinders in three-dimensional space, identifying a phase transition and percolation behavior in slabs versus planes.

Contribution

It establishes a non-trivial phase transition for percolation in the cylinder complement and shows percolation in slabs despite non-percolation in planes.

Findings

Existence of a critical intensity u* for percolation

Percolation occurs in slabs but not in planes

Phase transition is non-degenerate

Abstract

We study the complementary set of a Poissonian ensemble of infinite cylinders in R^3, for which an intensity parameter u > 0 controls the amount of cylinders to be removed from the ambient space. We establish a non-trivial phase transition, for the existence of an unbounded connected component of this set, as u crosses a critical non-degenerate intensity u*. We moreover show that this complementary set percolates in a sufficiently thick slab, in spite of the fact that it does not percolate in any given plane of R^3, regardless of the choice of u.