Random interlacements at level u is a one parameter family of connected
random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the
vacant set at level u, exhibits a non-trivial percolation phase transition in
u, as shown in arXiv:0704.2560 and arXiv:0808.3344, and the infinite connected
component, when it exists, is almost surely unique, see arXiv:0805.4106.
In this paper we study local percolative properties of the vacant set of
random interlacements at level u for all dimensions d>=3 and small intensity
parameter u>0. We give a stretched exponential bound on the probability that a
large (hyper)cube contains two distinct macroscopic components of the vacant
set at level u. Our results imply that finite connected components of the
vacant set at level u are unlikely to be large. These results were proved in
arXiv:1002.4995 for d>=5. Our approach is different from that of
arXiv:1002.4995 and works for all d>=3. One of the main ingredients in the
proof is a certain conditional independence property of the random
interlacements, which is interesting in its own right.Comment: 38 pages, 4 figures; minor corrections, to appear in AIH
