In this paper we establish some properties of percolation for the vacant set
of random interlacements, for d at least 5 and small intensity u. The model of
random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560.
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 arXiv:0808.3344 and arXiv:0805.4106. 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.Comment: Accepted for publication in Probability Theory and Related Field
