Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-29T12:09:10.849Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Benjamini, I., Pemantle, R., and Peres, Y. (1998). Unpredictable paths and percolation. Ann. Prob. 26, 1198–1211.Google Scholar
Burton, R. M., and Keane, M. (1989). Density and uniqueness of percolation. Commun. Math. Phys. 121, 501–505.Google Scholar
Doyle, P., and Snell, J. L. (1984). Random Walks and Electric Networks (Carus Math. Monogr. 22). MAA, Washington, DC.Google Scholar
Grimmett, G. R. (1989). Percolation. Springer, New York.Google Scholar
Grimmett, G. R., Kesten, H., and Zhang, Y. (1993). Random walk on the infinite cluster of percolation model. Prob. Theory Relat. Fields 96, 33–44.CrossRefGoogle Scholar
Häggström, O., and Mossel, E. (1998). Nearest-neighbor walks with low predictability profile and percolation in 2+ η dimensions. Ann. Prob. 26, 1212–1231.Google Scholar
Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.CrossRefGoogle Scholar
Lyons, T. (1983). A simple criterion for transience of a reversible Markov chain. Ann. Prob. 11, 393–402.Google Scholar
Markvorsen, S., McGuinness, S., and Thomassen, C. (1992). Transient random walks on graphs and metric spaces with applications to hyperbolic surfaces. Proc. London Math. Soc. 64, 1–20.Google Scholar