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On the size of a finite vacant cluster of random interlacements with small intensity. (English) Zbl 1231.60117
The paper deals with some properties of percolation for the vacant set of random interlacements on $$\mathbb Z^d$$ for $$d$$ greater than or equal to 5 and small intensity $$u$$. The main result of the paper is a theorem, which proves a stretched exponential bound on the probability that the interlacement-set separates two macroscopic connected sets in a large cube. By applying this theorem, the author estimates the distribution of the diameter and the volume of the vacant component at level $$u$$ containing the origin, given that it is finite. As another application, the author shows that with high probability, the unique infinite connected component of the vacant set is “ubiquitous” in large neighbourhoods of the origin.

##### 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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