We investigate the percolative properties of the vacant set left by random
interlacements on Z^d, when d is large. A non-negative parameter u controls the
density of random interlacements on Z^d. It is known from arXiv:0704.2560, and
arXiv:0808.3344, that there is a non-degenerate critical value u_*, such that
the vacant set at level u percolates when u < u_*, and does not percolate when
u > u_*. Little is known about u_*, however for large d, random interlacements
on Z^d, ought to exhibit similarities to random interlacements on a
(2d)-regular tree, for which the corresponding critical parameter can be
explicitly computed, see arXiv:0907.0316. We prove in this article a lower
bound on u_*, which is equivalent to log(d) as d goes to infinity. This lower
bound is in agreement with the above mentioned heuristics.Comment: 31 pages, 1 figure, accepted for publication in Probability Theory
and Related Field
