We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmüller distance for random walks on mapping class groups and on Cayley graphs of a large class of finitely generated groups.
"Sublinear deviation between geodesics and sample paths." Duke Math. J. 164 (3) 511 - 539, 15 February 2015. https://doi.org/10.1215/00127094-2871197