We prove diffusive lower bounds on the rate of escape of the random walk on infinite transitive graphs. Similar estimates hold for finite graphs, up to the relaxation time of the walk. Our approach uses nonconstant equivariant harmonic mappings taking values in a Hilbert space. For the special case of discrete, amenable groups, we present a more explicit proof of the Mok–Korevaar–Schoen theorem on the existence of such harmonic maps by constructing them from the heat flow on a Følner set.
"Harmonic maps on amenable groups and a diffusive lower bound for random walks." Ann. Probab. 41 (5) 3392 - 3419, September 2013. https://doi.org/10.1214/12-AOP779