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Quenched invariance principle for simple random walk on percolation clusters
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  • Published: 24 April 2006

Quenched invariance principle for simple random walk on percolation clusters

  • Noam Berger1 &
  • Marek Biskup2 

Probability Theory and Related Fields volume 137, pages 83–120 (2007)Cite this article

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Abstract

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in ℤd 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.

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Authors and Affiliations

  1. Department of Mathematics, Caltech, Pasadena, CA, 91125, USA

    Noam Berger

  2. Department of Mathematics, UCLA, Los Angeles, CA, 90095, USA

    Marek Biskup

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  1. Noam Berger
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  2. Marek Biskup
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Berger, N., Biskup, M. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Relat. Fields 137, 83–120 (2007). https://doi.org/10.1007/s00440-006-0498-z

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  • Received: 03 May 2005

  • Revised: 03 January 2006

  • Published: 24 April 2006

  • Issue Date: January 2007

  • DOI: https://doi.org/10.1007/s00440-006-0498-z

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Keywords

  • Markov Chain
  • Random Walk
  • Invariance Principle
  • Percolation Cluster
  • Simple Random Walk
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