Quenched invariance principle for simple random walk on percolation clusters

Details Quenched invariance principle for simple random walk on percolation clusters Quenched invariance principle for simple random walk on percolation clusters

2005-03-25

arxiv.org/abs/math/0503576v5

Probab. Theory Rel. Fields 137 (2007), no. 1-2, 83-120

Mathematics

Probability

38 pages (PTRF format) 4 figures. Version to appear in PTRF

Scientific paper

10.1007/s00440-006-0498-z

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in $\Z^d$ with $d\ge2$. 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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