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1996 Percolation Beyond $Z^d$, Many Questions And a Few Answers
Itai Benjamini, Oded Schramm
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Electron. Commun. Probab. 1: 71-82 (1996). DOI: 10.1214/ECP.v1-978


A comprehensive study of percolation in a more general context than the usual $Z^d$ setting is proposed, with particular focus on Cayley graphs, almost transitive graphs, and planar graphs. Results concerning uniqueness of infinite clusters and inequalities for the critical value $p_c$ are given, and a simple planar example exhibiting uniqueness and non-uniqueness for different $p>p_c$ is analyzed. Numerous varied conjectures and problems are proposed, with the hope of setting goals for future research in percolation theory.


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Itai Benjamini. Oded Schramm. "Percolation Beyond $Z^d$, Many Questions And a Few Answers." Electron. Commun. Probab. 1 71 - 82, 1996.


Accepted: 8 October 1996; Published: 1996
First available in Project Euclid: 25 January 2016

zbMATH: 0890.60091
MathSciNet: MR1423907
Digital Object Identifier: 10.1214/ECP.v1-978

Primary: 82B43
Secondary: 60K35

Keywords: Criticality , isoperimetericinequality , percolation , planar graph , transitive graph

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