No percolation at criticality on certain groups of intermediate growth

Abstract

We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form $p_n(v,v) \leq \exp\left[-\Omega(n^\gamma)\right]$ for some $\gamma>1/2$. The result is new in the case that the graph is of intermediate volume growth.