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

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

2008-05-27

arxiv.org/abs/0805.4106v2

Annals of Applied Probability 2009, Vol. 19, No. 1, 454-466

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP547 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP547

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

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