Open Access
2006 Uniqueness and non-uniqueness in percolation theory
Olle Häggström, Johan Jonasson
Probab. Surveys 3: 289-344 (2006). DOI: 10.1214/154957806000000096


This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ℤd and, more generally, on transitive graphs. For iid percolation on ℤd, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models – most prominently the Fortuin–Kasteleyn random-cluster model – and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.


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Olle Häggström. Johan Jonasson. "Uniqueness and non-uniqueness in percolation theory." Probab. Surveys 3 289 - 344, 2006.


Published: 2006
First available in Project Euclid: 30 December 2006

zbMATH: 1189.60175
MathSciNet: MR2280297
Digital Object Identifier: 10.1214/154957806000000096

Primary: 60K35 , 82B43

Keywords: amenability , percolation , transitive graphs , uniqueness of the infinite cluster

Rights: Copyright © 2006 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.3 • 2006
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