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

Ráth, Balázs (2015) A short proof of the phase transition for the vacant set of random interlacements. Electronic Journal of Probability, 20 (3). pp. 1-11.

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Abstract

The vacant set of random interlacements at level u>0, introduced in [Sznitman 2009], 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 [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.

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