We will consider a class of groups that includes non-elementary (relatively) hyperbolic groups, mapping class groups, many cubulated groups and C'(1/6) small cancellation groups. Their common feature is to admit an acylindrical action on some Gromov-hyperbolic space and a collection of quasi-geodesics compatible with such action.
As it turns out, random walks (generated by measures with exponential tail) on such groups tend to stay close to geodesics in the Cayley graph in the following sense: The probability that a given point on a random path is further away than L from a geodesic connecting the endpoints of the path decays exponentially fast in L.
This kind of estimate has applications to the rate of escape of random walks (Lipschitz continuity in the measure) and its variance (linear upper bound in the length).
Joint work with Pierre Mathieu.