Simple Random Walk on Long Range Percolation Clusters II: Scaling Limits

Details Simple Random Walk on Long Range Percolation Clusters II: Scaling Limits Simple Random Walk on Long Range Percolation Clusters II: Scaling Limits

2009-11-30

arxiv.org/abs/0911.5668v2

Mathematics

Probability

47 pages. Minor Revision

Scientific paper

We study limit laws for simple random walks on supercritical long range percolation clusters on $\Z^d, d \geq 1$. For the long range percolation model, the probability that two vertices $x, y$ are connected behaves asymptotically as $\|x-y\|_2^{-s}$. When $s\in(d, d+1)$, we prove that the scaling limit of simple random walk on the infinite component converges to an $\alpha$-stable L\'evy process with $\alpha = s-d$ establishing a conjecture of Berger and Biskup. The convergence holds in both the quenched and annealed senses. In the case where $d=1$ and $s>2$ we show that the simple random walk converges to a Brownian motion. The proof combines heat kernel bounds from our companion paper, ergodic theory estimates and an involved coupling constructed through the exploration of a large number of walks on the cluster.

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