  We study the behavior of the random walk in a continuum independent
long-range percolation model, in which two given vertices $x$ and $y$ are
connected with probability that asymptotically behaves like $|x-y|^{-\alpha}$
with $\alpha>d$, where $d$ denotes the dimension of the underlying Euclidean
space. More precisely, focus is on the random connection model in which the
vertex set is given by the realization of a homogeneous Poisson point process.
We show that this random graph exhibits the same properties as classical
discrete long-range percolation models studied in [3] with regard to recurrence
and transience of the random walk. The recurrence results are valid for every
intensity of the Poisson point process while the transience results hold for
large enough intensity. Moreover, we address a question which is related to a
conjecture in [16] for this graph.

Probability