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

The vacant set of random interlacements at level $u>0$, introduced in [Sznitman 2009], is a percolation model on Zᵈ, d ≥ 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 ≥ 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 ≥ 3.