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The Diameter of a Long-Range Percolation Cluster on Pre-Sierpinski Gasket

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Abstract

We give several bounds for a graph diameter of a random graph created by a stochastic model called the long-range percolation, in which any pair of two distinct points is connected by a random bond independently. Such a problem is well studied on a finite subset of the \(d\)-dimensional square lattice, where \(d\) is a positive integer. In this manuscript, we consider the problem on a finite subset of a fractal lattice which is called the pre-Sierpinski gasket. We can observe that the Hausdorff dimension of the fractal lattice appears in the critical parameters of \(s\), where \(s\) is a value determining the order of probabilities that random bonds exist.

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Acknowledgments

The author thanks to the referee for reading the manuscript carefully and giving kind comments for improvements.

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  1. Department of Mathematics, Faculty of Science, Kochi University, Akebono-cho, Kochi, 780-8520, Japan

    Jun Misumi

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  1. Jun Misumi
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Correspondence to Jun Misumi.

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Misumi, J. The Diameter of a Long-Range Percolation Cluster on Pre-Sierpinski Gasket. J Stat Phys 158, 1083–1089 (2015). https://doi.org/10.1007/s10955-014-1170-0

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  • DOI: https://doi.org/10.1007/s10955-014-1170-0

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