The Liouville and the intersection properties are equivalent for planar graphs

TL;DR

This paper proves that for planar graphs, the absence of non-constant bounded harmonic functions implies that two independent random walks will almost surely intersect, establishing an equivalence between harmonic function properties and intersection behavior.

Contribution

It establishes a new equivalence between harmonic function properties and intersection behavior of random walks in planar graphs.

Findings

Graphs with no non-constant bounded harmonic functions have almost sure intersection of independent random walks.

The result links harmonic analysis and probabilistic intersection properties in planar graphs.

Provides a characterization of planar graphs based on harmonic and random walk properties.

Abstract

It is shown that if a planar graph admits no non-constant bounded harmonic functions then the trajectories of two independent simple random walks intersect almost surely.