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On some percolation results of J. M. Hammersley

Published online by Cambridge University Press:  14 July 2016

J. G. Oxley  and
D. J. A. Welsh
J. G. Oxley
Affiliation:
University of Oxford
D. J. A. Welsh*
Affiliation:
University of Oxford
*
∗∗ Postal address: Merton College, Oxford 0X1 4JD, U.K.
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Abstract

We examine how much classical percolation theory on lattices can be extended to arbitrary graphs or even clutters of subsets of a finite set. In the process we get new short proofs of some theorems of J. M. Hammersley. The FKG inequality is used to get an upper bound for the percolation probability and we also derive a lower bound. In each case we characterise when these bounds are attained.

Keywords

ATOM AND BOND PERCOLATIONCLUTTERSINTERACTION PROBABILITYFKG INEQUALITYBETHE TREE
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 526 - 540
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

∗

Present address: Department of Mathematics, The Australian National University, P.O. Box 4, Canberra A.C.T. 2600, Australia.

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