The Annals of Probability

Simple random walk on long-range percolation clusters II: Scaling limits

Nicholas Crawford and Allan Sly

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Abstract

We study limit laws for simple random walks on supercritical long-range percolation clusters on $\mathbb{Z}^{d}$, $d\geq1$. For the long range percolation model, the probability that two vertices $x$, $y$ are connected behaves asymptotically as $\|x-y\|_{2}^{-s}$. When $s\in(d,d+1)$, we prove that the scaling limit of simple random walk on the infinite component converges to an $\alpha$-stable Lévy process with $\alpha=s-d$ establishing a conjecture of Berger and Biskup [Probab. Theory Related Fields 137 (2007) 83–120]. The convergence holds in both the quenched and annealed senses. In the case where $d=1$ and $s>2$ we show that the simple random walk converges to a Brownian motion. The proof combines heat kernel bounds from our companion paper [Crawford and Sly Probab. Theory Related Fields 154 (2012) 753–786], ergodic theory estimates and an involved coupling constructed through the exploration of a large number of walks on the cluster.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 445-502.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750933

Digital Object Identifier
doi:10.1214/12-AOP774

Mathematical Reviews number (MathSciNet)
MR3077517

Zentralblatt MATH identifier
1268.60056

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G52: Stable processes 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Random walk in random environment long rang percolation stable process

Citation

Crawford, Nicholas; Sly, Allan. Simple random walk on long-range percolation clusters II: Scaling limits. Ann. Probab. 41 (2013), no. 2, 445--502. doi:10.1214/12-AOP774. https://projecteuclid.org/euclid.aop/1362750933


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