Cylinders percolation in three dimensions

In this talk we will discuss the existence of unbounded clusters in the Euclidean space ${\bf R}^3$, after we remove from it a Poissonian cloud of infinite cylinders of radius one. We will mention a recent result, showing that this so-called vacant set undergoes a phase transition as the density of the cylinders crosses a critical threshold. This happens despite of the fact that for this model there is never percolation in two dimensional planes, no matter how small the density of cylinders is chosen.

This talk is based on a joint work with M. Hilario and V. Sidoravicius.