 0769 Noam Berger, Marek Biskup, Christopher E. Hoffman, Gady Kozma
 Anomalous heatkernel decay for random walk among bounded random conductances
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Mar 24, 07

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Abstract. We consider the nearestneighbor 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 heatkernel 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 $d=4$.
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