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

DOI:
https://doi.org/10.1090/S0002-9939-99-04995-3

MathSciNet review:
1622785

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**I. Benjamini and O. Schramm,*Percolation beyond , many questions and a few answers*, Electronic Communications in Probability**1**(1996), 71-82. MR**97j:60179****2.**R. Burton and M. Keane,*Density and uniqueness in percolation*, Comm. Math. Phys.**121**(1989), 501-505. MR**90g:60090****3.**G. Grimmett,*Percolation*, Springer-Verlag, New York, 1989. MR**90j:60109****4.**G. Grimmett and C. Newman,*Percolation in dimensions*, in Disorder in physical systems, (G. R. Grimmett and D. J. A. Welsh, eds.), Clarendon Press, Oxford, 1990, pp. 219-240. MR**92a:60207****5.**M. Gromov,*Volume and bounded cohomology*, Publication Mathématiques IHES**56**S-100 (1982). MR**84h:53053****6.**A. Lubotzky,*Discrete Groups, Expanding Graphs and Invariant Measures*, Birkhäuser Verlag, Basel, 1994. MR**96g:22018****7.**E. Spanier,*Algebraic Topology*, McGraw-Hill, Springer-Verlag, 1966, 1982. MR**35:1007**; MR**83i:55001****8.**J. Stallings,*Group theory and three-dimensional manifolds*, Yale University Press, New Haven 1971. MR**54:3705**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
60K35

Retrieve articles in all journals with MSC (1991): 60K35

Additional Information

**Eric Babson**

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

Email:
babson@math.ias.edu

**Itai Benjamini**

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

Email:
itai@wisdom.weizmann.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-99-04995-3

Received by editor(s):
March 13, 1997

Communicated by:
Jeffry N. Kahn

Article copyright:
© Copyright 1999
American Mathematical Society