No percolation at criticality on certain groups of intermediate growth

TL;DR

This paper proves that on certain groups with intermediate growth, critical percolation almost surely does not produce infinite clusters, extending understanding of percolation behavior on complex graph structures.

Contribution

It establishes the absence of infinite clusters at criticality for unimodular quasi-transitive graphs with specific return probability bounds, especially in the case of intermediate volume growth.

Findings

Critical percolation has no infinite clusters on specified graphs.

The result applies to graphs with intermediate volume growth.

It extends percolation theory to new classes of graphs.

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.