Random walk on the high-dimensional IIC

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Abstract

We study the asymptotic behavior of the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by Kumagai and Misumi (J Theor Probab 21:910–935, 2008). We do this by getting bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander–Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation.
Original languageEnglish
Pages (from-to)57-115
Number of pages59
JournalCommunications in Mathematical Physics
Volume329
Issue number1
DOIs
Publication statusPublished - 2014

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