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On the infinite cluster of Bernoulli bond percolation in Scherk's graph

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Dayue Chen
Dayue Chen*
Affiliation:
Peking University
*
∗ Postal address: School of Mathematical Sciences, Peking University, Beijing, 100871, China. Email address: dayue@math.pku.edu.cn
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Abstract

Scherk's graph is a subgraph of the three-dimensional lattice. It was shown by Markvorsen, McGuinness and Thomassen (1992) that Scherk's graph is transient. Consider the Bernoulli bond percolation in Scherk's graph. We prove that the infinite cluster is transient for p > ½ and is recurrent for p < ½. This implies the well-known result of Grimmett, Kesten and Zhang (1993) on the transience of the infinite cluster of the Bernoulli bond percolation in the three-dimensional lattice for p > ½. On the other hand, Scherk's graph exhibits a new dichotomy in the supercritical region.

Keywords

Bernoulli bond percolationScherk's graphtransience

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 828 - 840
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

Supported in part by Grant 19631060 from the NSF of China, Grant G1999075106 from the Ministry of Science and Technology, and a Research Fund from the Ministry of Education.

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