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

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