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August 2021 Graph distances of continuum long-range percolation
Ercan Sönmez
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Braz. J. Probab. Stat. 35(3): 609-624 (August 2021). DOI: 10.1214/21-BJPS500

Abstract

We consider a version of continuum long-range percolation on finite boxes of Rd in which the vertex set is given by the points of a Poisson point process and each pair of two vertices at distance r is connected with probability proportional to rs for a certain constant s. We explore the graph-theoretical distance in this model. The aim of this paper is to show that this random graph model undergoes phase transitions at values s=d and s=2d in analogy to classical long-range percolation on Zd, by using techniques which are based on an analysis of the underlying Poisson point process.

Acknowledgments

The author would like to thank the anonymous referees and the Editor for their constructive comments that improved the quality of this paper.

Citation

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Ercan Sönmez. "Graph distances of continuum long-range percolation." Braz. J. Probab. Stat. 35 (3) 609 - 624, August 2021. https://doi.org/10.1214/21-BJPS500

Information

Received: 1 June 2020; Accepted: 1 March 2021; Published: August 2021
First available in Project Euclid: 22 July 2021

Digital Object Identifier: 10.1214/21-BJPS500

Rights: Copyright © 2021 Brazilian Statistical Association

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Vol.35 • No. 3 • August 2021
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