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

Abstract

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.