In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., $p_c< p_u$. 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.
"Non-amenable Cayley graphs of high girth have $p_c < p_u$ and mean-field exponents." Electron. Commun. Probab. 17 1 - 8, 2012. https://doi.org/10.1214/ECP.v17-2139