Transience, Recurrence and Critical Behavior for Long-Range Percolation

We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions d = 1, 2, where x and y are connected with probability ∼ β/‖x − y‖−s. We show that when d < s < 2d the walk is transient, and when s ≥ 2d, the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d, if d < s < 2d, then critical percolation has no infinite clusters. This result is extended to the free random cluster model. A second corollary is that when d ≥ 2 and d < s < 2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network.