Abstract
A comprehensive study of percolation in a more general context than the usual \(\mathbb{Z}^d \) 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.
AMS 1991 Subject classification: 60K35, 82B43
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Benjamini, I., Schramm, O. (2011). Percolation Beyond \(\mathbb{Z}^d \), Many Questions and a Few Answers. In: Benjamini, I., Häggström, O. (eds) Selected Works of Oded Schramm. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9675-6_21
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