  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 F{\o}lner set.
