Non-amenable Cayley graphs of high girth have p_c < p_u and mean-field   exponents
  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. 
Asaf Nachmias, Yuval Peres
math.PR math.CO