Every planar graph with the Liouville property is amenable

TL;DR

This paper introduces a stronger form of transience for planar maps, relaxing bounded degree conditions, and shows that non-amenable planar graphs admit Dirichlet harmonic functions, expanding understanding of harmonic analysis on graphs.

Contribution

It proposes a new transience concept for planar maps and proves that non-amenable planar graphs admit Dirichlet harmonic functions, broadening previous results.

Findings

Introduces a strengthened transience notion for planar maps.

Shows non-amenable planar graphs admit Dirichlet harmonic functions.

Relaxes the bounded degree condition in harmonic function existence proofs.

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.