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On the infinite cluster of Bernoulli bond percolation in Scherk's graph
Published online by Cambridge University Press: 14 July 2016
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.
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- Copyright © by the Applied Probability Trust 2001
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.