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

• Autores: Nicholas Crawford, Allan Sly
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 2, 2013, págs. 445-502
• Idioma: inglés
• DOI: 10.1214/12-AOP774
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• Resumen

  • We study limit laws for simple random walks on supercritical long-range percolation clusters on Zd, d≥1. For the long range percolation model, the probability that two vertices x, y are connected behaves asymptotically as ∥x−y∥−s2. 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 [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.