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 \beta/|x-y|^{-s}. We show
that when s is between d and 2d, the walk is transient, and when s is
greater than or equal to 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 s is between d and 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 is at least 2
and, again, s is between d and 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.