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Transience, recurrence and critical behavior for long-range percolation. (English) Zbl 0991.82017
Summary: 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 if d<s<2d, then the walk is transient, and if s2d, 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 d1, if d<s<2d, then there is no infinite cluster at critically. This result is extended to the free random cluster model. A second corollary is that when d2 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.

MSC:
82B43Percolation (equilibrium statistical mechanics)