Critical sponge dimensions in percolation theory
- 1 March 1981
- journal article
- research article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 13 (02) , 314-324
- https://doi.org/10.1017/s000186780003603x
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
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