Tracking rates of random walks

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 ( 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.