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

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dc.creatorBerger, Noam
dc.creatorBiskup, Marek
dc.creatorHoffman, Christopher E.
dc.creatorKozma, Gady
dc.date2006-11-22
dc.date2007-06-26
dc.date.accessioned2026-07-07T13:08:13Z
dc.date.available2026-07-07T13:08:13Z
dc.descriptionWe consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $ω_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $ω_{xy}>0$ 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 $P_ω^{2n}(0,0)$. We prove that $P_ω^{2n}(0,0)$ 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.
dc.description22 pages. Includes a self-contained proof of isoperimetric inequality for supercritical percolation clusters. Version to appear in AIHP + additional corrections
dc.identifierhttps://arxiv.org/abs/math/0611666
dc.identifierhttp://arxiv.org/abs/math/0611666
dc.identifierAnn. Inst. H. Poincare Probab. Statist. 274 (2008), no. 2, 374-392
dc.identifierdoi:10.1214/07-AIHP126
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/228348
dc.subjectProbability
dc.subjectDiscrete Mathematics
dc.subjectMathematical Physics
dc.subject60G50; 58J35; 80A20
dc.titleAnomalous heat-kernel decay for random walk among bounded random conductances
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