  We study models of continuous-time, symmetric, $\Z^{d}$-valued random walks
in random environments, driven by a field of i.i.d. random nearest-neighbor
conductances $\omega_{xy}\in[0,1]$ with a power law with an exponent $\gamma$
near 0. We are interested in estimating the quenched decay of the return
probability $P_\omega^{t}(0,0)$, as $t$ tends to $+\infty$. We show that for
$\gamma> \frac{d}{2}$, the standard bound turns out to be of the correct
logarithmic order. As an expected concequence, the same result holds for the
discrete-time case.
