Weighted-amenability and percolation

2025/02/04 by Terlov, Grigory, Timár, Ádám · 1 citation
#60B99 Secondary: 37A20 #60C05 #FOS: Mathematics #Primary: 60K35 #Probability (math.PR)

paper · doi:10.48550/arxiv.2502.02560

Abstract

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

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