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

N. Berger; M. Biskup; C. E. Hoffman; G. Kozma

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 2, page 374-392
  • ISSN: 0246-0203

Abstract

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We consider the nearest-neighbor simple random walk on ℤd, d≥2, driven by a field of bounded random conductances ωxy∈[0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy>0 exceeds the threshold for bond percolation on ℤd. 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 2 n ( 0 , 0 ) . We prove that 𝖯 ω 2 n ( 0 , 0 ) is bounded by a random constant timesn−d/2 in d=2, 3, while it is o(n−2) in d≥5 and O(n−2log n) in d=4. By producing examples with anomalous heat-kernel decay approaching 1/n2, we prove that the o(n−2) bound in d≥5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d=4.

How to cite

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Berger, N., et al. "Anomalous heat-kernel decay for random walk among bounded random conductances." Annales de l'I.H.P. Probabilités et statistiques 44.2 (2008): 374-392. <http://eudml.org/doc/77975>.

@article{Berger2008,
abstract = {We consider the nearest-neighbor simple random walk on ℤd, d≥2, driven by a field of bounded random conductances ωxy∈[0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy&gt;0 exceeds the threshold for bond percolation on ℤd. 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 $\mathsf \{P\}_\{\omega \}^\{2n\}(0,0)$. We prove that $\mathsf \{P\}_\{\omega \}^\{2n\}(0,0)$ is bounded by a random constant timesn−d/2 in d=2, 3, while it is o(n−2) in d≥5 and O(n−2log n) in d=4. By producing examples with anomalous heat-kernel decay approaching 1/n2, we prove that the o(n−2) bound in d≥5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d=4.},
author = {Berger, N., Biskup, M., Hoffman, C. E., Kozma, G.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {heat kernel; random conductance model; random walk; percolation; isoperimetry; simple random walk; random environments},
language = {eng},
number = {2},
pages = {374-392},
publisher = {Gauthier-Villars},
title = {Anomalous heat-kernel decay for random walk among bounded random conductances},
url = {http://eudml.org/doc/77975},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Berger, N.
AU - Biskup, M.
AU - Hoffman, C. E.
AU - Kozma, G.
TI - Anomalous heat-kernel decay for random walk among bounded random conductances
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 2
SP - 374
EP - 392
AB - We consider the nearest-neighbor simple random walk on ℤd, d≥2, driven by a field of bounded random conductances ωxy∈[0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy&gt;0 exceeds the threshold for bond percolation on ℤd. 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 $\mathsf {P}_{\omega }^{2n}(0,0)$. We prove that $\mathsf {P}_{\omega }^{2n}(0,0)$ is bounded by a random constant timesn−d/2 in d=2, 3, while it is o(n−2) in d≥5 and O(n−2log n) in d=4. By producing examples with anomalous heat-kernel decay approaching 1/n2, we prove that the o(n−2) bound in d≥5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d=4.
LA - eng
KW - heat kernel; random conductance model; random walk; percolation; isoperimetry; simple random walk; random environments
UR - http://eudml.org/doc/77975
ER -

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