Hostname: page-component-797576ffbb-tx785 Total loading time: 0 Render date: 2023-12-09T22:33:44.802Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false

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

Published online by Cambridge University Press:  24 October 2008

T. E. Harris
The Rand Corporation, Santa Monica, California


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.

Research Article
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.)



(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.Google Scholar
(4)Hammersley, J. M.Percolation processes: lower bounds for the critical probability. Ann. Math. Stat. 28 (1957), 790–5.Google 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), 1737.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).Google Scholar
(9)Whitney, H.Planar graphs. Fund. Math. 21 (1933), 7384.Google Scholar