  We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$,
driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$. The
conductance law is i.i.d. subject to the condition that the probability of
$\omega_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$. For
environments in which the origin is connected to infinity by bonds with
positive conductances, we study the decay of the $2n$-step return probability
$P_\omega^{2n}(0,0)$. We prove that $P_\omega^{2n}(0,0)$ is bounded by a random
constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and
$O(n^{-2}\log n)$ in $d=4$. By producing examples with anomalous heat-kernel
decay approaching $1/n^2$ we prove that the $o(n^{-2})$ bound in $d\ge5$ is the
best possible. We also construct natural $n$-dependent environments that
exhibit the extra $\log n$ factor in $d=4$. See also math.PR/0701248.
