The Liouville and the intersection properties are equivalent for planar graphs

Itai Benjamini,Nicolas Curien,Agelos Georgakopoulos

doi:10.1214/ecp.v17-1913

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.

Highlights

The graph G has the intersection property if for any x, y ∈ V , the trajectories of two independent simple random walks (SRW) started respectively from x and y intersect almost surely
Let G = (V, E) be a connectedgraph
Z5 is Liouville and yet two independent simple random walk paths do not intersect with positive probability

Summary

Introduction

The graph G has the intersection property if for any x, y ∈ V , the trajectories of two independent simple random walks (SRW) started respectively from x and y intersect almost surely. We consider three independent simple random walk trajectories, and argue that if each two of them intersect only finitely many times, they divide our planar graph into three regions (Fig.1). This allows us to talk about the probability for random walk to eventually stay in one of these regions, which we use to construct a non-constant bounded harmonic function, contradicting our Liouvilleness assumption.

Results

Conclusion