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Scaling limits for random walks on long range percolation clusters
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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.
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Critical long-range percolation II: Low effective dimension
In the long-range low-dimensional regime of percolation, the cluster volume tail and k-point functions are determined up to constants, yielding the hyperscaling identities delta=(d+alpha)/(d-alpha) and d_f=(d+alpha)/2.
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