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 [8], is a percolation model on Z(d), 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 [6, 8] 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.