No Percolation at Criticality on Certain Groups of Intermediate Growth

Abstract

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.