2014/07/31 by Balazs Rath
Mathematics · #math.PR #msc:60K35 #msc:82B43
paper · pdf · doi:10.1214/ecp.v20-3734
published as Electron. Commun. Probab. 20 (2015) no. 3., 1-11 · 11 pages, 1 figure; Title of paper changed
arxiv created 2015/01/22 · arxiv updated 2015/01/23
The vacant set of random interlacements at level u>0, introduced in arXiv:0704.2560, 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 arXiv:0704.2560 and arXiv:0808.3344 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.