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Percolation beyond \(\mathbb{Z}^ d\), many questions and a few answers. (English) Zbl 0890.60091
Summary: 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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