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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
##### Keywords:
random walks; percolation; random interlacements
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##### References:
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