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

Abstract

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 $d\ge 3$.

11 pages, 1 figure; Title of paper changed

References in corpus (1)

• On the critical parameter of interlacement percolation in high dimension

Cited by in corpus (2)

• Results on the contact process with dynamic edges or under renewals
• Percolation Perspective on Sites Not Visited by a Random Walk in Two Dimensions