  The divergence of a group is a quasi-isometry invariant defined in terms of
pairs of points and lengths of paths avoiding a suitable ball around the
identity. In this paper we study "random divergence'', meaning the divergence
at two points chosen according to independent random walks or Markov chains;
the Markov chains version can be turned into a quasi-isometry invariant. We
show that in many cases, such as for relatively hyperbolic groups, mapping
class groups, and right-angled Artin groups, the divergence at two randomly
chosen points is with high probability equivalent to the divergence of the
group. That is, generic points realise the largest possible divergence.
