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In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., pc< pᵤ. Furthermore, we show that percolation and self-avoiding walk on such graphs have mean-field critical exponents. In particular, the self-avoiding walk has positive speed.
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Nachmias et al. (2012) studied this question.
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