Electronic Communications in Probability

Percolation Beyond $Z^d$, Many Questions And a Few Answers

Itai Benjamini and Oded Schramm

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Abstract

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.

Article information

Source
Electron. Commun. Probab. Volume 1 (1996), paper no. 8, 71-82.

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

Permanent link to this document
http://projecteuclid.org/euclid.ecp/1453756499

Digital Object Identifier
doi:10.1214/ECP.v1-978

Zentralblatt MATH identifier
0890.60091

Subjects
Primary: 82B43: Percolation [See also 60K35]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Percolation criticality planar graph transitive graph isoperimetericinequality

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Benjamini, Itai; Schramm, Oded. Percolation Beyond $Z^d$, Many Questions And a Few Answers. Electron. Commun. Probab. 1 (1996), paper no. 8, 71--82. doi:10.1214/ECP.v1-978. http://projecteuclid.org/euclid.ecp/1453756499.


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