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 $p_n(v,v)\le \exp \left[-\Omega (n^{\gamma })\right]$ for some $\gamma >1/2$. The result is new in the case that the graph is of intermediate volume growth.

18 pages. V2: Minor revisions. Accepted version, to appear in IMRN