Preface

The model of random interlacements was introduced in 2007 by A.-S. Sznitman in the seminal paper [41], motivated by questions about the disconnection of discrete cylinders and tori by the trace of simple random walk. In fact, random interlacements is a random subset of Zd , d ≥ 3, which on a mesoscopic scale does appear as the limiting distribution of the trace of simple random walk on a large torus when it runs up to times proportional to the volume. It serves as a model for corrosion and in addition gives rise to interesting and challenging percolation problems. Random interlacements can be constructed via a Poisson point process of labeled doubly infinite random walk trajectories in Zd . In fact, there is a one-parameter family of random interlacements. For u > 0, the random interlacements at level u, denoted by I u, is the random subset of Zd obtained as the union of the ranges of all the trajectories with labels at most u. Thus, the bigger the u, the more trajectories enter into the picture, the bigger the I u. The law of I u has nice properties such as invariance and ergodicity with respect to lattice shifts. It also exhibits long-range correlations, which leads to interesting challenges in its investigation. By construction, the graph induced by I u consists of only infinite connected components. In fact, it is almost surely connected for any level u. In contrast, the complement of I u, the so-called vacant set V u = Zd\I u, exhibits a percolation phase transition. Namely, there exists u∗ ∈ (0,∞) such that