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 similar to beta/parallel tox - yparallel to(-s). We show that if d < s < 2d, then the walk is transient, and if s greater than or equal to 2d, then 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 greater than or equal to 1, if d < s < 2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d greater than or equal to 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.