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March 2013 Simple random walk on long-range percolation clusters II: Scaling limits
Nicholas Crawford, Allan Sly
Ann. Probab. 41(2): 445-502 (March 2013). DOI: 10.1214/12-AOP774

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.

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Nicholas Crawford. Allan Sly. "Simple random walk on long-range percolation clusters II: Scaling limits." Ann. Probab. 41 (2) 445 - 502, March 2013. https://doi.org/10.1214/12-AOP774

Information

Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1268.60056
MathSciNet: MR3077517
Digital Object Identifier: 10.1214/12-AOP774

Subjects:
Primary: 60G50
Secondary: 60G52 , 82B41

Keywords: long rang percolation , Random walk in random environment , Stable process

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • March 2013
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