Every planar graph with the Liouville property is amenable
Every planar graph with the Liouville property is amenable
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 …