On symmetric random walks with random conductances on ${\mathbb{Z}}^d$

We study models of continuous time, symmetric, ${\mathbb{Z}}^d$-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.