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Bond percolation on honeycomb and triangular lattices

Published online by Cambridge University Press:  01 July 2016

John C. Wierman
John C. Wierman*
Affiliation:
University of Minnesota
*
∗Postal address: School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, MN 55455, U.S.A.
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Abstract

The two common critical probabilities for a lattice graph L are the cluster size critical probability pH(L) and the mean cluster size critical probability pT(L). The values for the honeycomb lattice H and the triangular lattice T are proved to be pH(H) = pT(H) = 1–2 sin (π/18) and PH(T) = pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

Keywords

BOND PERCOLATIONCRITICAL PROBABILITYSTAR-TRIANGLE RELATIONSPONGE-CROSSING PROBABILITYPERCOLATION THEORY
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 298 - 313
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

Research supported by NSF MCS 78-01168.

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