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

Abstract

We consider the nearest-neighbor simple random walk on ${\mathbb{Z}}^d$, $d\ge 2$, 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 ${\mathbb{Z}}^d$. 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 $d\ge 5$ 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 $d\ge 5$ 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.

22 pages. Includes a self-contained proof of isoperimetric inequality for supercritical percolation clusters. Version to appear in AIHP + additional corrections

References in corpus (1)

• Random walks on supercritical percolation clusters