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 arXiv:0704.2560, is a percolation model on ${\mathbb{Z}}^d$, $d\ge 3$ 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 arXiv:0704.2560 and arXiv:0808.3344 that for any $d\ge 3$ there exists a positive and finite threshold $u_{\ast }$ such that if $u<u_{\ast }$ then the vacant set percolates and if $u>u_{\ast }$ 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 $u_{\ast }$ for any $d\ge 3$.