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A short proof of the phase transition for the vacant set of random interlacements. (English) Zbl 1307.60146

Summary: The vacant set of random interlacements at level \(u>0\), introduced in [A.-S. Sznitman, Ann. Math. (2) 171, No. 3, 2039–2087 (2010; Zbl 1202.60160)], 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 [loc. cit.] and [A.-S. Sznitman and V. Sidoravicius, Commun. Pure Appl. Math. 62, No. 6, 831–858 (2009; Zbl 1168.60036)] 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 \(d \geq 3\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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