Abstract
In this paper, we establish some properties of percolation for the vacant set of random interlacements, for d ≥ 5 and small intensity u. The model of random interlacements was first introduced by Sznitman in (Ann Math, arXiv:0704.2560, 2010). It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see Sidoravicius and Sznitman (Commun Pure Appl Math 62(6):831–858, 2009) and Teixeira (Ann Appl Probab 19(1):454–466, 2009). We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-product of our main theorem, which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube. As another application, we show that with high probability, the unique infinite connected component of the vacant set is “ubiquitous” in large neighborhoods of the origin.
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Teixeira, A. On the size of a finite vacant cluster of random interlacements with small intensity. Probab. Theory Relat. Fields 150, 529–574 (2011). https://doi.org/10.1007/s00440-010-0283-x
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DOI: https://doi.org/10.1007/s00440-010-0283-x