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 ${\mathbb{Z}}^d,d\ge 1$. For the long range percolation model, the probability that two vertices $x,y$ are connected behaves asymptotically as $\Vert x-y{\Vert }_2^{-s}$. 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. 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.

47 pages. Minor Revision

References in corpus (3)

• Random walks on supercritical percolation clusters
• On the scaling of the chemical distance in long-range percolation models
• Heat Kernel Upper Bounds on Long Range Percolation Clusters