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

Abstract

We consider the model of random interlacements on 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_*.