Weighted-amenability and percolation

Abstract: The automorphism group of a transitive graph defines a weight function on the vertices through the Haar modulus. Benjamini, Lyons, Peres, and Schramm introduced the notion of weighted-amenability for a transitive graph, which is equivalent to the amenability of its automorphism group. We prove that this property is equivalent to level-amenability, that is, the property that the collection of vertices of weights in a given finite set always induces an amenable graph. We then use this to prove a version of Hutchcroft's conjecture about $p_h<p_u$, relaxed `a la Pak-Smirnova-Nagnibeda, where $p_h$ is the critical probability for the regime where clusters of infinite total weight arise, and $p_u$ is the uniqueness threshold. Further characterizations are given in terms of the spectral radius and invariant spanning forests. One consequence is the continuity of the phase transition at $p_h$ for weighted-nonamenable graphs.