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

Nicholas Crawford, Allan Sly

Research output: Contribution to journalArticle

9 Scopus citations

Abstract

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||-s 2 . 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.

Original languageEnglish (US)
Pages (from-to)445-502
Number of pages58
JournalAnnals of Probability
Volume41
Issue number2
DOIs
StatePublished - Mar 1 2013
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Long rang percolation
  • Random walk in random environment
  • Stable process.

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