  We study models of continuous time, symmetric, $\Z^d$-valued random walks in
random environments. One of our aims is to derive estimates on the decay of
transition probabilities in a case where a uniform ellipticity assumption is
absent. We consider the case of independent conductances with a polynomial tail
near 0, and obtain precise asymptotics for the annealed return probability and
convergence times for the random walk confined to a finite box.
