Johan Tykesson: The Poisson cylinder model in Euclidean space

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. First, 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) ∈ (0,∞), such that with probability 1, the vacant set has an unbounded component if u < u_*(d), and only bounded components if u > u_*(d). We then move on to study the geometry of the union of all the cylinders in the process. It turns out that this union is always a connected set. Morever, any two points x and y that are contained in the union of the cylinders, are connected via a sequence of at most d cylinders.

The talk is based on joint works with David Windisch and Erik Broman.