Every planar graph with the Liouville property is amenable. Issue 3 (7th June 2020) - Carmesin, Johannes; Georgakopoulos, Agelos
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 function s proved by Benjamini and Schramm. As a corollary we obtain that every planar nonamenable graph admits nonconstant Dirichlet harmonic function s27.