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In this note we show that percolation on non-amenable Cayley graphs of highgirth has a phase of non-uniqueness, i.e., p_c < p_u. Furthermore, we show thatpercolation and self-avoiding walk on such graphs have mean-field criticalexponents. In particular, the self-avoiding walk has positive speed.

Non-amenable Cayley graphs of high girth have p_c < p_u and mean-field exponents