Anomalous heat-kernel decay for random walk among bounded random conductances
<|reference_start|>Anomalous heat-kernel decay for random walk among bounded random conductances: 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.<|reference_end|>
@article{berger2006anomalous,
    title={Anomalous heat-kernel decay for random walk among bounded random
  conductances},
    author={Noam Berger, Marek Biskup, Christopher E. Hoffman, Gady Kozma},
    journal={Ann. Inst. H. Poincare Probab. Statist. 274 (2008), no. 2, 374-392},
    year={2006},
    doi={10.1214/07-AIHP126},
    archivePrefix={arXiv},
    eprint={math/0611666},
    primaryClass={math.PR cs.DM math-ph math.MP}
}