A short proof of the phase transition for the vacant set of random interlacements

Abstract

The vacant set of random interlacements at level $u>0$, introduced in [Sznitman 2009], is a percolation model on $\mathbb{Z}^d$, $d \geq 3$ which arises as the set of sites avoided by a Poissonian cloud of doubly infinite trajectories, where $u$ is a parameter controlling the density of the cloud. It was proved in [Sznitman 2009] and [Sidoravicius, Sznitman 2010] that for any $d \geq 3$ there exists a positive and finite threshold $u_*$ such that if $u<u_*$ then the vacant set percolates and if $u>u_*$ then the vacant set does not percolate. We give an elementary proof of these facts. Our method also gives simple upper and lower bounds on the value of $u_*$ for any $d \geq 3$.