Simple random walk on long range percolation clusters I: Heat kernel bounds

Nicholas Crawford, Allan Sly

Research output: Contribution to journalArticle

13 Scopus citations

Abstract

In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any d ≥ 1 and for any exponent s giving the rate of decay of the percolation process, we show that the return probability decays like t-d/s-d up to logarithmic corrections, where t denotes the time the walk is run. Our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. The bounds and accompanying understanding of the geometry of the cluster play a crucial role in the companion paper (Crawford and Sly in Simple randomwalk on long range percolation clusters II: scaling limit, 2010) where we establish the scaling limit of the random walk to be α-stable Lévy motion.

Original languageEnglish (US)
Pages (from-to)753-786
Number of pages34
JournalProbability Theory and Related Fields
Volume154
Issue number3-4
DOIs
StatePublished - Dec 1 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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