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

  • T. E. Harris (a1)

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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.

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(1)Ahlfors, L. V.Complex analysis (New York, 1953).
(2)Broadbent, S. R. and Hammersley, J. M.Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc. 53 (1957), 629–41.
(3)Hammersley, J. M.Percolation processes. II. The connective constant. Proc. Camb. Phil. Soc. 53 (1957), 642–5.
(4)Hammersley, J. M.Percolation processes: lower bounds for the critical probability. Ann. Math. Stat. 28 (1957), 790–5.
(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.
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