Quenched invariance principle for simple random walk on percolation clusters

By Noam Berger, Marek Tomasz Biskup. 2005

238 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/59ae80dfd7aa3f76270f5885edfe7354a02569cd

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In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Invariance principle for the random conductance model in a degenerate ergodic environment (2013, background): “Nevertheless, the proof relies on arguments— inspired by [10] for the percolation case—which only work in the very special two-dimensional case (see below) and do not seem to be extendable to higher dimensions.”
Quenched invariance principle for random walks in balanced random environment (2010, background): “2 in [4]) show that the quenched invariance principle holds whenever an invariant measure Q ∼ P of {ω̄(n)} exists.”
The Alexander-Orbach conjecture holds in high dimensions (2008, background): “In this paper we establish their conjecture in high dimensions.”