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

TL;DR

This paper establishes the scaling limits of simple random walks on long range percolation clusters, showing convergence to stable Lévy processes or Brownian motion depending on parameters, thus confirming a conjecture and extending understanding of such models.

Contribution

It proves the convergence of random walks on long range percolation clusters to stable Lévy processes for certain parameters, confirming a conjecture and providing new insights into their scaling behavior.

Findings

Convergence to an $ ext{alpha}$-stable Lévy process for $s ext{ in } (d, d+1)$.

Convergence to Brownian motion when $d=1$ and $s>2$.

Established both quenched and annealed convergence results.

Abstract

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.