Quenched invariance principle for simple random walk on percolation clusters

2005/03/31 by Noam Berger, Marek Biskup · 7 citations
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60F17 #msc:60K37 #msc:82C41

paper · pdf · doi:10.1007/s00440-006-0498-z

published as Probab. Theory Rel. Fields 137 (2007), no. 1-2, 83-120 · 38 pages (PTRF format) 4 figures. Version to appear in PTRF

arxiv created 2006/02/20 · crossref issued 2006/04/24 · crossref published 2006/04/24 · crossref published-online 2006/04/24 · openalex publication_date 2006/04/24 · crossref created 2006/04/24 · crossref published-print 2007/01/01 · arxiv updated 2009/12/01 · crossref deposited 2022/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · crossref indexed 2026/08/04

Abstract

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in \Z^d with d≥2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

Citations

• Random walks on supercritical percolation clusters
• On the scaling of the chemical distance in long-range percolation models

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