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

TL;DR

This paper demonstrates that non-amenable Cayley graphs with high girth exhibit a non-uniqueness phase in percolation and display mean-field critical exponents for percolation and self-avoiding walk, including positive speed.

Contribution

It establishes the existence of a non-uniqueness phase and mean-field behavior for percolation and self-avoiding walk on high-girth non-amenable Cayley graphs, advancing understanding of phase transitions.

Findings

p_c < p_u on such graphs

Self-avoiding walk has positive speed

Percolation and self-avoiding walk exhibit 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., 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.