Percolation beyond Z D , Many Questions and a Few Answers

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 innnite clusters and inequalities for the critical value p c are given, and a simple planar example exhibiting uniqueness and non-uniqueness for diierent p > p c is analyzed. Numerous varied conjectures and problems are proposed, with the hope of setting goals for future research in percolation theory.