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On the internal distance in the interlacement set. (English) Zbl 1245.60090
Summary: We prove a shape theorem for the internal (graph) distance on the interlacement set $$\mathcal{I}^u$$ of the random interlacement model on $$\mathbb Z^d, d\geq 3$$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation 60G50 Sums of independent random variables; random walks
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