Percolation for the vacant set of random interlacements. (English) Zbl 1168.60036

The authors continue the analysis of the model of random interlacements introduced by A. Sznitman [“Vacant set of random interlacements and percolation”, preprint, arXiv:0704.2560, to appear in Ann. Math.]. Random interlacements are a Poisson point process on the space of doubly-infinite paths on \(\mathbb Z^d\) (modulo time shifts) that tend to infinity at both positive and negative times. There is an additional parameter \(u\) which controls the typical number of paths that intersect a given compact subset. \(\mathcal{I}^u\) denotes the random interlacement at level \(u\). The main result of this paper is that for any \(d\geq 3\) and \(u>0\) sufficiently small, the complement of \(\mathcal{I}^u\) in \(\mathbb Z^d\) contains an infinite connected component (that is, the vacant set percolates). This extends the results in the previous paper of Sznitman where it was shown that the vacant set percolates for \(u\) sufficiently small when \(d\geq 7\).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
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