Geometry of the random interlacement

Eviatar Ben Procaccia (Weizmann Institute of science)
Johan Tykesson (Weizmann Institute of science)

Abstract


We consider the geometry of random interlacements on the $d$-dimensional lattice. We use ideas from stochastic dimension theory developed in [1] 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.

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Pages: 528-544

Publication Date: September 26, 2011

DOI: 10.1214/ECP.v16-1660

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