## Electronic Communications in Probability

### Geometry of the random interlacement

#### 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.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465262003

Digital Object Identifier
doi:10.1214/ECP.v16-1660

Mathematical Reviews number (MathSciNet)
MR2836759

Zentralblatt MATH identifier
1254.60018

Subjects
Primary: Probability

Rights

#### Citation

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

#### References

• Benjamini, Itai; Kesten, Harry; Peres, Yuval; Schramm, Oded. Geometry of the uniform spanning forest: transitions in dimensions $4,8,12,dots$. Ann. of Math. (2) 160 (2004), no. 2, 465–491.
• Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8
• Lawler, Gregory F.; Limic, Vlada. Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, 123. Cambridge University Press, Cambridge, 2010. xii+364 pp. ISBN: 978-0-521-51918-2
• 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
• B. R'ath and A. Sapozhnikov.. Connectivity properties of random interlacement and intersection of random walks. Arxiv preprint arXiv:1012.4711}, 2010.
• Sznitman, Alain-Sol. Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 (2010), no. 3, 2039–2087.