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The Diameter of a Long-Range Percolation Cluster on Generalized Pre-Sierpinski Carpet and Regular Tree

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Abstract

We consider some kinds of graphs obtained by generalizing the pre-Sierpinski carpet, which is one of well known fractal lattices. Here, we deal with not only some fractal lattices, but also many irregular graphs which do not belong to fractal lattices. Under several assumptions, we present estimates on a diameter of the long-range percolation cluster on such graphs. In Misumi (J Stat Phys 158:1083–1089, 2015), estimates on the graph diameter of the long-range percolation cluster are established on the pre-Sierpinski gasket, which is another fundamental fractal lattice, and the result of this paper is its extension in a certain sense. We will also discuss the behavior of the diameter of the long-range percolation graph on a regular tree, which seems to be quite different from the result on the generalized pre-Sierpinski carpet.

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Acknowledgements

The author thanks to the referees for reading the manuscript carefully and giving valuable comments. The author was supported by JSPS KAKENHI Grant Number 16K17615.

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  1. Department of Mathematics, Faculty of Science and Technology, Kochi University, Akebono-cho, 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 Generalized Pre-Sierpinski Carpet and Regular Tree. J Stat Phys 174, 276–286 (2019). https://doi.org/10.1007/s10955-018-2181-z

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  • DOI: https://doi.org/10.1007/s10955-018-2181-z

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