Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-09T06:51:04.375Z Has data issue: false hasContentIssue false
  • English
  • Français

A lower bound for the critical probability in a certain percolation process

Published online by Cambridge University Press:  24 October 2008

T. E. Harris
T. E. Harris
Affiliation:
The Rand Corporation, Santa Monica, California
Get access Rights & Permissions [Opens in a new window]

Extract

Consider a lattice L in the Cartesian plane consisting of all points (x, y) such that either x or y is an integer. Points with integer coordinates (positive, negative, or zero) are called vertices and the sides of the unit squares (including endpoints) are called links. Each link of L is assigned the designation active with probability p or passive with probability 1 − p, independently of all other links. To avoid trivial cases, we shall always assume 0 < p < 1. The lattice L, with the designations active or passive attached to the links, is called a random maze. A set of links is called connected if the points comprising the links (including endpoints) form a connected point set in the plane.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 56 , Issue 1 , January 1960 , pp. 13 - 20
Copyright
Copyright © Cambridge Philosophical Society 1960

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

REFERENCES

(1)Ahlfors, L. V.Complex analysis (New York, 1953).Google Scholar
(2)Broadbent, S. R. and Hammersley, J. M.Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc. 53 (1957), 629–41.Google Scholar
(3)Hammersley, J. M.Percolation processes. II. The connective constant. Proc. Camb. Phil. Soc. 53 (1957), 642–5.CrossRefGoogle Scholar
(4)Hammersley, J. M.Percolation processes: lower bounds for the critical probability. Ann. Math. Stat. 28 (1957), 790–5.CrossRefGoogle Scholar
(5)Hammersley, J. M. Bornes supérieures de la probabilité critique dans un processus de fitration. Le calcul des probabilitiés et ses applications. Centre national de la recherche scientifique, Paris (1959), 17–37.Google Scholar
(6)Halmos, P. R.Measure theory (New York, 1950).Google Scholar
(7)Hopf, E.Ergodentheorie (Berlin, 1937 and New York, 1948).Google Scholar
(8)Moore, R. L.Foundations of point set theory (New York, 1932).CrossRefGoogle Scholar
(9)Whitney, H.Planar graphs. Fund. Math. 21 (1933), 73–84.CrossRefGoogle Scholar