Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T07:34:34.412Z Has data issue: false hasContentIssue false
  • English
  • Français

Substitution Method Critical Probability Bounds for the Square Lattice Site Percolation Model

Published online by Cambridge University Press:  12 September 2008

John C. Wierman
John C. Wierman
Affiliation:
Mathematical Sciences Department, 220 Maryland Hall, The Johns Hopkins University, Baltimore, Maryland 21218
Get access Rights & Permissions [Opens in a new window]

Abstract

The square lattice site percolation model critical probability is shown to be at most .679492, improving the best previous mathematically rigorous upper bound. This bound is derived by extending the substitution method to apply to site percolation models.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 2 , June 1995 , pp. 181 - 188
Copyright
Copyright © Cambridge University Press 1995

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.)

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]Deutscher, G., Zallen, R. and Adler, J. (eds.) (1983) Percolation Structures and Processes Annals of the Israel Physical Society 5.Google Scholar
[3]Efros, A. L. (1986) Physics and Geometry of Disorder, Mir, Moscow.Google Scholar
[4]Grimmett, G. (1985) Percolation, Springer.Google Scholar
[5]Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proceedings of the Cambridge Philosophical Society 56 13–20.CrossRefGoogle Scholar
[6]Higuchi, Y. (1982) Coexistence of the infinite (*) clusters: a remark on the square lattice site percolation. Zeitschrift fur Wdhrscheinlichkeitstheorie und Verwandte Gebiete 61 75–81.CrossRefGoogle Scholar
[7]Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals 1/2. Communications in Mathematical Physics 74 41–59.CrossRefGoogle Scholar
[8]Kesten, H. (1982) Percolation Theory for Mathematicians, Birkhäuser, Boston.CrossRefGoogle Scholar
[9]Lindvall, T. (1992) Lectures on the Coupling Method, Wiley.Google Scholar
[10]Luczak, T. and Wierman, J. C. (1988) Critical probability bounds for two-dimensional site percolation models. Journal of Physics A: Mathematical and General 21 3131–3138.CrossRefGoogle Scholar
[11]Menshikov, M. V. and Pelikh, K. D. (1989) Percolation with several defect types: An estimate of critical probability for a square lattice. Matematicheskie Zametki 46 38–47 (in Russian), 778–785 (in translation).Google Scholar
[12]Preston, C. J. (1974) A generalization of the FKG inequalities. Communications in Mathematical Physics 36 233–241.CrossRefGoogle Scholar
[13]Stauffer, D. (1985) Introduction to Percolation Theory Taylor and Francis, London.CrossRefGoogle Scholar
[14]Strassen, V. (1965) The existence of probability measures with given marginals. Annals of Mathematical Statistics 36 423–439.CrossRefGoogle Scholar
[15]Toth, B. (1985) A lower bound for the critical probability of the square lattice site percolation. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 69 19–22.CrossRefGoogle Scholar
[16]Wierman, J. C. (1990) Bond percolation critical probability bounds for the Kagomé lattice by a substitution method. In: Disorder in Physical Systems, Grimmett, G. and Welsh, D. J. A. (eds.) 349–360.Google Scholar
[17]Wierman, J. C. (1989) Bounds for critical probabilities of bond percolation models, Research Report for the Stockholm Conference on Random Graphs and Applications.Google Scholar
[18]Ziff, R. M. and Sapoval, B. (1986) The efficient determination of the percolation threshold by a frontier-generating walk in a gradient. Journal of Physics A: Mathematical and General 19 L1169–L1172.CrossRefGoogle Scholar
[19]Zuev, S. A. (1987) Bounds for the percolation threshold for a square lattice. Theory of Probability and its Applications 32 606–609 (in Russian). 551–553 (in translation).Google Scholar