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