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Vacant set of random interlacements and percolation. (English) Zbl 1202.60160
The paper deals with the model of random interlacements consisting of a countable collection of doubly infinite trajectories on a multidimensional integer lattice. A nonnegative parameter \(u\) measures how many trajectories enter the picture. The union of the supports of these trajectories defines the interlacement at level \(u\). It is an infinite connected translation invariant random subset of the lattice. The author introduces a critical value \(u^{\ast}\) such that the vacant set percolates for \(u\) less then \(u^{\ast}\) and does not percolate for \(u\) greater then \(u^{\ast}\). The main results of the paper show that \(u^{\ast}\) is finite for three-dimensional lattice and strictly positive when the lattice dimension is greater than 6.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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