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Geometry of the random interlacement. (English) Zbl 1254.60018
Summary: We consider the geometry of random interlacements on the \(d\)-dimensional lattice. We use ideas from stochastic dimension theory developed in [I. Benjamini et al., Ann. Math. (2) 160, No. 2, 465–491 (2004; Zbl 1071.60006)] to prove the following.
Given that two vertices \(x,y\) belong to the interlacement set, it is possible to find a path between \(x\) and \(y\) contained in the trace left by at most \(\lceil d/2 \rceil\) trajectories from the underlying Poisson point process. Moreover, this result is sharp in the sense that there are pairs of points in the interlacement set which cannot be connected by a path using the traces of at most \(\lceil d/2 \rceil-1\) trajectories.

MSC:
60D05 Geometric probability and stochastic geometry
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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