Summary: 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.