This talk will discuss random walks on percolation clusters.
The first case is supercritical ($p>p_c$) bond percolation in
$Z^d$. Here one can obtain Aronsen type bounds on the transition
probabilities, using analytic methods based on ideas of Nash.
For the critical case ($p=p_c$) one needs to study the incipient
infinite cluster (IIC). The easiest situation is the IIC on trees -
where the methods described above lead to an alternative approach to
results of Kesten (1986). (This case is joint work with T. Kumagai).