Geometry of the random interlacement

Eviatar B. Procaccia; Johan Tykesson

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arXiv · 1101.1527

Geometry of the random interlacement

Eviatar B. Procaccia·Johan Tykesson

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Abstract

We consider the geometry of random interlacements on the $d$-dimensional lattice. We use ideas from stochastic dimension theory developed in \cite{benjamini2004geometry} 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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Eviatar B. Procaccia, Johan Tykesson. 2011-07-18. Geometry of the random interlacement. https://arxiv.org/abs/1101.1527

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