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

TL;DR

This paper proves the uniqueness of the infinite cluster in the vacant set of random interlacements on integer lattices and establishes the continuity of the probability of the origin being in this cluster in the supercritical phase.

Contribution

It demonstrates the uniqueness of the infinite component in the vacant set of random interlacements and shows the probability's continuity with respect to the parameter u.

Findings

Uniqueness of the infinite cluster in the vacant set.

Continuity of the probability that the origin is in the infinite cluster.

Results hold in the supercritical phase u<u_*.

Abstract

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 $u<u_*$.