Simple Random Walk on Long Range Percolation Clusters II: Scaling Limits

2009/11/30 by Crawford, Nicholas, Sly, Allan · 1 citation
#60F17 #82B41 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.0911.5668

Abstract

We study limit laws for simple random walks on supercritical long range percolation clusters on \Z^d, d ≥ 1. For the long range percolation model, the probability that two vertices x, y are connected behaves asymptotically as ‖x-y‖₂^-s. When s∈(d, d+1), we prove that the scaling limit of simple random walk on the infinite component converges to an α-stable Lévy process with α= s-d establishing a conjecture of Berger and Biskup. 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, ergodic theory estimates and an involved coupling constructed through the exploration of a large number of walks on the cluster.

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