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Journal of Statistical Physics

, Volume 158, Issue 5, pp 1083–1089 | Cite as

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

  • Jun Misumi
Article

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.

Keywords

Long-range percolation Random graph Graph diameter Fractal lattice 

Notes

Acknowledgments

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

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKochi UniversityKochiJapan

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