Every planar graph with the Liouville property is amenable

2015/02/09 by Carmesin, Johannes, Georgakopoulos, Agelos
#05C63 #05C81 #31C20 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1502.02542

Abstract

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic functions proved by Benjamini and Schramm. As a corollary we obtain that every planar non-amenable graph admits Dirichlet harmonic functions.

Related

• Harmonic Dirichlet Functions on Planar Graphs
• Nonamenable Liouville Graphs
• The Liouville and the intersection properties are equivalent for planar graphs
• Hyperbolicity vs. Amenability for planar graphs
• A Liouville Theorem for Domains in Graphs