On the uniqueness of the infinite cluster of the vacant set of random interlacements

We consider the model of random interlacements on ${\mathbb{Z}}^d$ introduced in Sznitman [Vacant set of random interlacements and percolation (2007) preprint]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in $u$ of the probability that the origin belongs to the infinite component of the vacant set at level $u$ in the supercritical phase $u<u_{\ast }$.