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The vacant set of random interlacements at level $u>0$, introduced inarXiv:0704.2560, is a percolation model on $\mathbb{Z}^d$, $d \geq 3$ whicharises as the set of sites avoided by a Poissonian cloud of doubly infinitetrajectories, where $u$ is a parameter controlling the density of the cloud. Itwas proved in arXiv:0704.2560 and arXiv:0808.3344 that for any $d \geq 3$ thereexists a positive and finite threshold $u_*$ such that if $u<u_*$ then thevacant set percolates and if $u>u_*$ then the vacant set does not percolate. Wegive an elementary proof of these facts. Our method also gives simple upper andlower bounds on the value of $u_*$ for any $d \geq 3$.

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