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Critical sponge dimensions in percolation theory

Published online by Cambridge University Press:  01 July 2016

G. R. Grimmett
G. R. Grimmett*
Affiliation:
University of Bristol
*
∗Postal address: School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U.K.
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Abstract

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.

Keywords

PERCOLATION
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 314 - 324
Copyright
Copyright © Applied Probability Trust 1981 

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