Harmonic functions on planar and almost planar graphs and ...

572 I. Benjamini, O. Schramm3. Stability and instability of harmonic <strong>functions</strong> on graphsA graph G is said to have the weak-Liouville property if every bounded harmonicfunction on G is constant.M. Kanai [15] and Markvorsen et al [21] have shown that recurrence on abounded valence graph is invariant under rough isometries. The weak-Liouvilleproperty is not: T. Lyons [19] constructed two mutually bilipschitz metrics m, m ′on a graph G, such that (G, m) is weak-Liouville, while (G, m ′ ) is not. (Replacingthe edges by tubes produces a Riemannian example). We will describe below arelatively easy recipe for making such examples.While the weak-Liouville property is not stable under rough isometries,Soardi [27] has shown that the existence of non-constant, harmonic <strong>functions</strong>with finite Dirichlet energy is invariant. Below, we introduce the notion of aresolvable graph, and will see that a transient resolvable graph has non-constant,bounded harmonic <strong>functions</strong> with finite Dirichlet energy. Moreover, the propertyof being resolvable is very stable: if f : G ◦ → G is a quasimonomorphism andG is resolvable, then so is G ◦ .Definitions. Let G =(V,E)be some graph, and let Γ be a collection of (infinite)paths in G. Then Γ is null if there is an L 2 (E) metric on G such that length m (γ) =∞for every γ ∈ Γ . It is easy to see that Γ is null iff its extremal lengthEL(Γ ) = supminfγ∈Γlength m (γ) 2‖m‖ 2 ,is infinite. (Extremal length was imported to the discrete setting by Duffin [9].See [28] for more about extremal length on graphs.) When Γ is a collection ofpaths and a property holds for every γ ∈ Γ , except for a null family, we shall saythat the property holds for almost every γ ∈ Γ .Let m be a metric on G, and recall that d m is the associated distance function.Let C m (G) denote the completion of (V , d m ), and let the m-boundary of G be∂ m G = C m (G)−V . We use d m to also denote the metric of the completion C m (G).The metric m will be called resolving if it is in L 2 (E) and for every x ∈ ∂ m Gthe collection of half infinite paths in G that converge to x in C m (G) is null. G isresolvable if it has a resolving metric.Note that if m is a resolving metric and m ′ is another L 2 metric satisfyingm ′ ≥ m, then m ′ is also resolving.Theorem 3.2 below shows that any recurrent graph is resolvable, in fact,there is a metric m with ∂ m G = ∅. On the other hand, the next theorem showsthat a transient graph with no non-constant, harmonic <strong>functions</strong> in D(G) isnotresolvable, for example, Z 3 or any lattice in hyperbolic n-space n > 2isnotresolvable. We shall see that any bounded valence planar graph is resolvable.3.1. Theorem Let G =(V,E)be an infinite, connected, locally finite, resolvablegraph.