Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on . In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infi...

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Bibliographic Details
Published in:Communications in mathematical physics 2008-03, Vol.278 (2), p.385-431
Main Authors: Barlow, Martin T., Járai, Antal A., Kumagai, Takashi, Slade, Gordon
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Combinatorics
Combinatorics. Ordered structures
Complex Systems
Exact sciences and technology
Graph theory
Mathematical and Computational Physics
Mathematical methods in physics
Mathematical Physics
Mathematics
Other topics in mathematical methods in physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Sciences and techniques of general use
Theoretical
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Summary:We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on . In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is , and thereby prove a version of the Alexander–Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d > 6, by extending results about critical oriented percolation obtained previously via the lace expansion.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-007-0410-4