Non-amenable Cayley graphs of high girth have Pc < Pu and mean-field exponents

Abstract

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.