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We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$,driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$. Theconductance law is i.i.d. subject to the condition that the probability of$\omega_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$. Forenvironments in which the origin is connected to infinity by bonds withpositive 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 randomconstant 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-kerneldecay approaching $1/n^2$ we prove that the $o(n^{-2})$ bound in $d\ge5$ is thebest possible. We also construct natural $n$-dependent environments thatexhibit the extra $\log n$ factor in $d=4$. See also math.PR/0701248.Comment: 22 pages. Includes a self-contained proof of isoperimetric inequality  for supercritical percolation clusters. Version to appear in AIHP +  additional correction

Anomalous heat-kernel decay for random walk among bounded random conductances