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

582 I. Benjamini, O. SchrammProof (of 5.1). The statement of the theorem is invariant under Möbius transformations,hence we may normalize so that ∞ /∈ carr(P). To prove that thelimit lim n z(v(n)) exists almost surely, it is enough to show that the diameterof {z(v(k)), z(v(k + 1)),...} tends to zero with probability 1 as k →∞. Letɛ>0. The lemma implies that there is a t 1 < ∞ so that the probability that|z(v(k))| > t 1 for any k is less than ɛ. There are finitely many v ∈ V with|z(v)| ≤t 1 and with d(z(v),Λ(P)) >ɛ. Since G is transient, this implies thatthere is some n 0 so that d(z(v(n 0 )),Λ(P)) ≤ ɛ with probability at least 1 − 2ɛ.Then the lemma shows that with probability at most 2ɛ + C / log t the diameterof {z(v(n 0 )), z(v(n 0 + 1)),...} is greater than tɛ. Choosing t = ɛ −1/2 , then showsthat the limit lim n z(v(n)) exists almost surely.The second part of the theorem follows immediately. ⊓⊔Remarks. 1. The methods of this section are really not particular to circles, theywould apply to a large class of other, well behaved, packings.2. It turns out that for some purposes it is better to consider a square tilingassociated with the graph, rather than a circle packing. In particular, one can getsimilar results for an especially constructed square tiling when G is not assumedto be a triangulation. We intend to study this in a forthcoming paper [3].3. The Convergence Lemma gives information about the hitting measure. Let thesituation be as in the lemma. Suppose that p is some point on ∂ carr(P), andr > 0. Let d = |p − z(v 0 )| be the distance from z(v 0 )top. The probability thatthe random walk starting at v 0 will ever reach a vertex v satisfying |z(v)−p| < ris less than C / log(d/r). This follows from the lemma by inverting the packingin the circle {|z − p| = r}.5.4. Problems Consider the random walk (v(1),v(2),...)starting at some vertexv 0 ∈ V . Then with probability 1 the limit z ∞ = lim n z(v(n)) exists and is a pointin ∂ carr(P). Let µ denote the hitting measure; that is, µ(A) is the probability thatz ∞ ∈ A. Suppose for example that carr(P) =U , the unit disk. When is µ absolutelycontinuous with respect to Lebesgue measure on ∂U ? (By further triangulatingsome of the faces of the triangulation, one can insure that the hitting measure issingular.) In general, when carr(P) /= U , is it true that µ is supported by a set ofHausdorff dimension 1, as for the harmonic measure for Brownian motion.6. <strong>Harmonic</strong> <strong>functions</strong> on almost planar manifoldsIn this section we shall prove Theorem 1.10. Namely, we shall show that atransient, bounded local geometry, almost planar manifold admits non constantDirichlet <strong>functions</strong>. This will be an easy corollary of Theorem 1.9 and the followingvery recent theorem.6.1. Theorem (Holopainen-Soardi) [13] Let X 1 be a connected, bounded localgeometry, Riemannian manifold, and suppose that X 1 is roughly isometric to a