Simple random walk on long-range percolation clusters II: Scaling limits

Abstract

We study limit laws for simple random walks on supercritical long-range percolation clusters on $\mathbb{Z}^{d}$, $d\geq1$. 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 Lévy process with $\alpha=s-d$ establishing a conjecture of Berger and Biskup [Probab. Theory Related Fields 137 (2007) 83–120]. 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 [Crawford and Sly Probab. Theory Related Fields 154 (2012) 753–786], ergodic theory estimates and an involved coupling constructed through the exploration of a large number of walks on the cluster.