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Advances in Applied Probability Advances in Applied Probability

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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
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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 PERCOLATION CRITICAL PROBABILITY STAR-TRIANGLE RELATION SPONGE-CROSSING PROBABILITY PERCOLATION 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.

References

[1] Broadbent, S. R. and Hammersley, J. M. (1957) Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc. 53, 629–641.CrossRefGoogle Scholar
[2] Dean, P. and Bird, N. F. (1967) Monte Carlo estimates of critical probabilities. Proc. Camb. Phil. Soc. 63, 477–479.CrossRefGoogle Scholar
[3] Essam, J. W. (1972) Percolation and cluster size. In Phase Transitions and Critical Phenomena, Volume 2, ed. Domb, C. and Sykes, M. S., Academic Press, New York.Google Scholar
[4] Fortuin, C. M, Kasteleyn, P. W., and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103.CrossRefGoogle Scholar
[5] Frisch, H. L. and Hammersley, J. M. (1963) Percolation processes and related topics. J. SIAM 11, 894–918.Google Scholar
[6] Frisch, H. L., Hammersley, J. M., and Welsh, D. J. A. (1962) Monte Carlo estimates of percolation probabilities for various lattices. Phys. Rev. 126, 949–951.Google Scholar
[7] Grimmett, G. R. (1976) On the number of clusters in the percolation model. J. London Math. Soc. (2) 13, 346–350.CrossRefGoogle Scholar
[8] Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 13–20.CrossRefGoogle Scholar
[9] Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals ½. Commun. Math. Phys. 74, 41–59.CrossRefGoogle Scholar
[10] Russo, L. (1978) A note on percolation. Z. Wahrscheinlichkeitsth. 43, 39–48.CrossRefGoogle Scholar
[11] Seymour, P. D. and Welsh, D. J. A. (1978) Percolation probabilities on the square lattice Ann. Discrete Math. 3, 227–245.Google Scholar
[12] Shante, V. K. S. and Kirkpatrick, S. (1971) An introduction to percolation theory. Adv. Phys. 20, 325–357.CrossRefGoogle Scholar
[13] Smythe, R. T. and Wierman, J. C. (1978) First-Passage Percolation on the Square Lattice Springer Lecture Notes in Mathematics 671, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[14] Sykes, M. F. and Essam, J. W. (1964) Exact critical percolation probabilities for the site and bond problems in two dimensions, J. Math. Phys. 5, 1117–1132.CrossRefGoogle Scholar
[15] Wierman, J. C. (1978) On critical probabilities in percolation theory. J. Math. Phys. 19, 1979–1982.CrossRefGoogle Scholar

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