Random walk on the incipient infinite cluster for oriented percolation in high dimensions

Martin T. Barlow*, Antal A. Járai, Takashi Kumagai, Gordon Slade

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Citations (Scopus)

Abstract

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on ℤd × ℤ+. 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 frac {4}{3}, 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.

Original languageEnglish
Pages (from-to)385-431
Number of pages47
JournalCommunications in Mathematical Physics
Volume278
Issue number2
DOIs
Publication statusPublished - 2008 Mar
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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