  We show that simple random walks on (non-trivial) relatively hyperbolic
groups stay $O(\log(n))$-close to geodesics, where $n$ is the number of steps
of the walk. Using similar techniques we show that simple random walks in
mapping class groups stay $O(\sqrt{n\log(n)})$-close to geodesics and hierarchy
paths. Along the way, we also prove a refinement of the result that mapping
class groups have quadratic divergence.
  An application of our theorem for relatively hyperbolic groups is that random
triangles in non-trivial relatively hyperbolic groups are $O(\log(n))$-thin,
random points have $O(\log(n))$-small Gromov product and that in many cases the
average Dehn function is subasymptotic to the Dehn function.
