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

Abstract: We consider the nearest-neighbor simple random walk on $\Z<sup>d$, $d\ge2$, 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 $\omega_{xy}&gt;0$ exceeds the threshold for bond percolation on $\Z<sup>d$. 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 $P_\omega<sup>{2n}(0,0)$. We prove that $P_\omega<sup>{2n}(0,0)$ is bounded by a random constant times $n<sup>{-d/2}$ in $d=2,3$, while it is $o(n<sup>{-2})$ 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 $o(n<sup>{-2})$ 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.