  We study limit laws for simple random walks on supercritical long-range
percolation clusters on the integer lattice. For the long range percolation
model, the probability that two vertices are connected behaves asymptotically
as a negative power of distance between them. We prove that the scaling limit
of simple random walk on the infinite component converges to an isotropic
alpha-stable Levy process. This complements the work of Crawford and Sly, who
proved the corresponding result for alpha between 0 and 1. The convergence
holds in both the quenched and annealed senses.
