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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
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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

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References

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