Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T03:13:30.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixed percolation on the square lattice

Published online by Cambridge University Press:  14 July 2016

John C. Wierman
John C. Wierman*
Affiliation:
The Johns Hopkins University
*
∗ Postal address: Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In a planar percolation model, faces of the underlying graph, as well as the sites and bonds, may be viewed as random elements. With this viewpoint, Whitney duality allows construction of a planar dual percolation model for each planar percolation model, which applies to mixed models with sites, bonds, and faces open or closed at random. Using self-duality for percolation models on the square lattice, information is obtained about the percolative region in the mixed model.

Keywords

CRITICAL PROBABILITYPERCOLATION THEORYDUALITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 2 , June 1984 , pp. 247 - 259
Copyright
Copyright © Applied Probability Trust 1984 

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

Research supported by the National Science Foundation under Grant No. MCS 78–01168 and MCS 81–18229.

References

Broadbent, S. R. and Hammersley, J. M. (1957) Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc. 53, 629–641.Google Scholar
Fisher, M. E. (1961) Critical probabilities for cluster size and percolation problems. J. Math. Phys. 2, 620–627.Google Scholar
Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103.Google Scholar
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
Hammersley, J. M. (1980) A generalization of McDiarmid's theorem for mixed Bernoulli percolation. Math. Proc. Camb. Phil. Soc. 88, 167–170.Google Scholar
Hammersley, J. M. and Welsh, D. J. A. (1980) Percolation theory and its ramifications. Contemporary Phys. 21, 593–605.Google Scholar
Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 13–20.Google Scholar
Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals . Comm. Math. Phys. 74, 41–59.Google Scholar
Kesten, H. (1981) Analyticity properties and power law estimates of functions in percolation theory. J. Statist. Phys. 25, 717–756.CrossRefGoogle Scholar
Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhäuser, Boston.Google Scholar
Russo, L. (1978) A note on percolation theory. Z. Wahrscheinlichkeitsth. 43, 39–48.Google Scholar
Seymour, P. D. and Welsh, D. J. A. (1978) Percolation probabilities on the square lattice. Ann. Discrete Math. 3, 227–245.CrossRefGoogle Scholar
Shante, V. K. S. and Kirkpatrick, S. (1971) An introduction to percolation theory. Adv. Phys. 20, 325–357.Google Scholar
Smythe, R. T. and Wierman, J. C. (1978) First-passage Percolation on the Square Lattice. Springer Lecture Notes in Mathematics 671, Springer-Verlag, Berlin.Google Scholar
Sykes, M. F. and Essam, J. W. (1964) Exact critical percolation probabilities for site and bond problems in two dimensions. J. Math. Phys. 5, 1117–1127.Google Scholar
Whitney, H. (1932) Nonseparable and planar graphs. Trans. Amer. Math. Soc. 34, 339–362.Google Scholar
Whitney, H. (1933) Planar graphs. Fundamenta Math. 21, 73–84.Google Scholar
Wierman, J. C. (1978) On critical probabilities in percolation theory. J. Math. Phys. 19, 1979–1982.Google Scholar
Wierman, J. C. (1981) Bond percolation on honeycomb and triangular lattices. Adv. Appl. Prob. 13, 298–313.Google Scholar
Wierman, J. C. (1982) Percolation theory. Ann. Prob. 10, 509–524.Google Scholar