Electronic Journal of Probability

On the internal distance in the interlacement set

Jiří Černý and Serguei Popov

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We prove a shape theorem for the internal (graph) distance on the interlacement set $\mathcal{I}^u$ of the random interlacement model on $\mathbb Z^d$, $d\ge 3$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 29, 25 pp.

Accepted: 12 April 2012
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35] 60G50: Sums of independent random variables; random walks

Random interlacement Internal distance Shape theorem Simple random walk Capacity

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Černý, Jiří; Popov, Serguei. On the internal distance in the interlacement set. Electron. J. Probab. 17 (2012), paper no. 29, 25 pp. doi:10.1214/EJP.v17-1936. https://projecteuclid.org/euclid.ejp/1465062351

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