Journal of Statistical Physics

, Volume 165, Issue 1, pp 153–163 | Cite as

The Mixing Time of a Random Walk on a Long-Range Percolation Cluster in Pre-Sierpinski Gasket



We consider a random graph created by the long-range percolation on the nth stage finite subset of a fractal lattice called the pre-Sierpinski gasket. The long-range percolation is a stochastic model in which any pair of two points is connected by a random bond independently. On the random graph obtained as above, we consider a discrete-time random walk. We show that the mixing time of the random walk is of order \(2^{(s-d)n}\) if \(d<s<2d\) in a sense. Here, s is a parameter which determines the order of probabilities that random bonds exist, and \(d=\log 3/\log 2\) is the Hausdorff dimension of the pre-Sierpinski gasket.


Long-range percolation Mixing time Random walk Fractal lattice 



The author thanks to the referee for giving comments to the first version of the manuscript. The author was supported by JSPS KAKENHI Grant Number 16K17615.


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKochi UniversityAkebono-choJapan

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