Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T00:18:03.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Percolation on subsets of the square lattice

Published online by Cambridge University Press:  14 July 2016

Colin McDiarmid
Colin McDiarmid*
Affiliation:
London School of Economics and Political Science
*
∗Postal address: London School of Economics and Political Science, Houghton St., London WC 2A 2AE, U.K. Research partially supported by NRC grant A9211.
Get access Rights & Permissions [Opens in a new window]

Abstract

We adapt arguments from a paper of Seymour and Welsh concerning percolation probabilities on the infinite square lattice L to show that for certain regions R in L, if there is a positive probability of having an infinite open path in L starting at the origin then there is also a positive probability of having such a path within R.

Keywords

PERCOLATIONCRITICAL PROBABILITYSQUARE LATTICE
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 278 - 283
Copyright
Copyright © Applied Probability Trust 1980 

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] Hammersley, J. M. (1957) Bornes supérieures de la probabilité dans un processus de filtration. Proc. 87th Internat. Colloq. CNRS, Paris , 17–37.Google Scholar
[2] 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
[3] Seymour, P. D. and Welsh, D. J. A. (1978) Percolation probabilities on the square lattice. Ann. Discrete Math. 3, 227–245.Google Scholar
[4] Smythe, R. T. and Wierman, J. C. (1978) First Passage Percolation on the Square Lattice. Lecture Notes in Mathematics 671, Springer-Verlag, Berlin.Google Scholar
[5] 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
[6] Wierman, J. C. (1978) On critical probabilities in percolation theory. J. Math. Phys. 19, 1979–1982.Google Scholar