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On the uniqueness of the infinite cluster of the vacant set of random interlacements. (English) Zbl 1158.60046
Summary: We consider the model of random interlacements on \(\mathbb Z^d\) introduced in A. Sznitman [Vacant set of random interlacements and percolation, preprint, arXiv:0704.2560, to appear in Ann. Math.]. 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 \(u<u_{*}\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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