Weighted-amenability and percolation

In 1999, Benjamini, Lyons, Peres, and Schramm introduced a notion of weighted-amenability for transitive graphs that is equivalent to the amenability of its automorphism group. For unimodular graphs this notion coincides with classical graph-amenability and has been intensely studied. In the present work, we show that many classical unimodular results can be extended to the nonunimodular setting, which is further motivated by recent progress in the mcp (measure class preserving or quasi-pmp) setting of measured group theory. To this end, we prove new characterizations of weighted-amenability, in particular that it is equivalent to all finite unions of levels inducing amenable graphs. Hutchcroft conjectured that the latter property implies that $p_h<p_u$, where $p_h$ is the critical probability for the regime where clusters of Bernoulli percolation are infinite total weight and $p_u$ is the uniqueness threshold. We prove a relaxed version of his conjecture \`a la Pak--Smirnova-Nagnibeda. Further characterizations are given in terms of the spectral radius and invariant spanning forests. One of the consequences is the continuity of the phase transition at $p_h$ for weighted-nonamenable graphs.