Electronic Communications in Probability

Geometry of the random interlacement

Eviatar Procaccia and Johan Tykesson

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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.

Article information

Electron. Commun. Probab., Volume 16 (2011), paper no. 47, 528-544.

Accepted: 26 September 2011
First available in Project Euclid: 7 June 2016

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Primary: Probability

Random Interlacements Stochastic dimension

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Procaccia, Eviatar; Tykesson, Johan. Geometry of the random interlacement. Electron. Commun. Probab. 16 (2011), paper no. 47, 528--544. doi:10.1214/ECP.v16-1660. https://projecteuclid.org/euclid.ecp/1465262003

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  • Resnick, Sidney I. Extreme values, regular variation and point processes. Reprint of the 1987 original. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2008. xiv+320 pp. ISBN: 978-0-387-75952-4
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