Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Cut sets and normed cohomology
with applications to percolation

Authors: Eric Babson and Itai Benjamini
Journal: Proc. Amer. Math. Soc. 127 (1999), 589-597
MSC (1991): Primary 60K35
MathSciNet review: 1622785
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Abstract: We discuss an inequality for graphs, which relates the distances between components of any minimal cut set to the lengths of generators for the homology of the graph. Our motivation arises from percolation theory. In particular this result is applied to Cayley graphs of finite presentations of groups with one end, where it gives an exponential bound on the number of minimal cut sets, and thereby shows that the critical probability for percolation on these graphs is neither zero nor one. We further show for this same class of graphs that the critical probability for the coalescence of all infinite components into a single one is neither zero nor one.

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Additional Information

Eric Babson
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350

Itai Benjamini
Affiliation: Department of Mathematics, The Weizmann Institute, Rehovot 76100, Israel

Received by editor(s): March 13, 1997
Communicated by: Jeffry N. Kahn
Article copyright: © Copyright 1999 American Mathematical Society