Percolation Beyond Z^d, Many Questions And a Few Answers
Itai Benjamini, Weizmann Institute of Science
Oded Schramm, Microsoft Research
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.
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