Quenched invariance principle for simple random walk on percolation clusters

DOI10.1007/S00440-006-0498-ZzbMATH Open1107.60066arXivmath/0503576OpenAlexW1970611197MaRDI QIDQ866943

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Publication date: 14 February 2007

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Abstract: We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in $$ with $dge2$. 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.

Full work available at URL: https://arxiv.org/abs/math/0503576

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