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Disproof of the conjectured subexponentiality of certain functions in percolation theory

Published online by Cambridge University Press:  01 July 2016

J. Van den berg
J. Van den berg*
Affiliation:
Delft University of Technology
*
∗Postal address: Department of Mathematics, Delft University of Technology, Julianalaan 132, 2628 BL Delft, The Netherlands.
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Abstract

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Consider bond-percolation on a graph G with sites S(G). We disprove the conjecture of Hammersley (1957) that the function n → sups ϵ S(G)E [the number of sites s′ at distance n from s which can be reached from s by an open path which, except for s′, only passes through sites at distance smaller than n from s] is always subexponential.

Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 16 , Issue 3 , September 1984 , pp. 690 - 691
Copyright
Copyright © Applied Probability Trust 1984 

Footnotes

Research supported by the Netherlands Foundation for Mathematics SMC with financial aid from the Netherlands Organization for the Advancement of Pure Research (ZWO).

References

Broadbent, S. R. and Hammersley, J. M. (1957) Percolation processes. Proc. Camb. Phil. Soc. 53, 629–641; 642-645.CrossRefGoogle Scholar
Hammersley, J. M. (1957) Percolation processes: lower bounds for the critical probability. Ann. Math. Statist. 28, 790–795.CrossRefGoogle Scholar
Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhäuser, Boston.CrossRefGoogle Scholar