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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.

On the size of a finite vacant cluster of random interlacements with small intensity