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 [15], [14], that there is a nondegenerate 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, random interlacements on Z d, for large d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, where the corresponding critical parameter can be explicitly computed, see [19]. We show in this article that lim infd u∗/log d ≥ 1. This lower bound is in agreement with the above mentioned heuristics.

We study the percolative properties of random interlacements on G × Z, where G is a weighted graph satisfying certain sub-Gaussian estimates attached to the parameters α> 1 and 2 ≤ β ≤ α + 1. We develop decoupling inequalities, which are a key tool in showing that the critical level u ∗ for the percolation of the vacant set of random interlacements is always finite in our set-up, and that it is positive when α ≥ 1 + β 2. We also obtain several stretched exponential controls both in the percolative and non-percolative phases of the model. Even in the case where G = Zd, d ≥ 2, several of these results are new.