  Consider the time T_oz when the random walk on a weighted graph started at
the vertex o first hits the vertex set z. We present lower bounds for T_oz in
terms of the volume of z and the graph distance between o and z. The bounds are
for expected value and large deviations, and are asymptotically sharp. We
deduce rate of escape results for random walks on infinite graphs of
exponential or polynomial growth, and resolve a conjecture of Benjamini and
Peres.
