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

TL;DR

This paper provides a simple, elementary proof of the phase transition in the vacant set of random interlacements, establishing the existence of a critical threshold and bounds for any dimension $d \

Contribution

It introduces an elementary proof of the phase transition and offers simple bounds on the critical parameter $u_*$ for all dimensions $d \

Findings

Existence of a finite critical threshold $u_*$ for percolation.

Elementary proof of the phase transition.

Bounds on the critical threshold $u_*$ for all $d \

Abstract

The vacant set of random interlacements at level $u>0$, introduced in arXiv:0704.2560, is a percolation model on $\mathbb{Z}^d$, $d \geq 3$ 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 arXiv:0704.2560 and arXiv:0808.3344 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 $d \geq 3$.