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

578 I. Benjamini, O. SchrammThe construction of the sequence r 1 , r 2 ,... insures that the supports of dφ n anddφ n ′ are disjoint when n /= n ′ . It is easy to see that the definition of φ n showsthat there is a finite constant C such that(4.1)|dφ n (e)| 2 ≤ C area((P u ∪ P v ) ∩ B(3r n )),where {u,v} = ∂e. Let Ω be an upper bound on the degrees of the vertices inG. Since the interiors of the sets in P are disjoint, (4.1) implies‖dφ n ‖ 2 = ∑ e∈Eφ n (e) 2 ≤ 9πC Ωr 2 n .Now set∞∑ φ nφ = .nr nn=1As we have noted, the supports of the different dφ n are disjoint, and therefore,‖dφ‖ 2 =∞∑n=1‖dφ n ‖ 2n 2 rn2 ,and the above estimate for ‖dφ n ‖ 2 shows that |dφ| ∈L 2 (E). Therefore, there issome metric m p ∈ L 2 (E) such that m p (e) ≥|dφ n (e)|for every e ∈ E. (Technically,we cannot take m p = |dφ|, since |dφ| is not positive, and hence not ametric.)Now let γ =(γ(1),γ(2),...) be any path in Γ p , and let E(γ) denote its edges.We have lim n z(γ(n)) = z(p). Therefore,This gives∑ ψ rj (z)lim φ(γ(n)) = lim= ∑n z→z(p) jrj jjlength mp(γ) = ∑e∈E(γ)So Γ p is null, and m is resolving.m p (e)≥ ∑⊓⊔e∈E(γ)1j= ∞.|dφ(e)|=∞.Proof of 1.9 and 1.1.(1). These follow immediately from 4.1 and 3.1.⊓⊔5. The Dirichlet problem for circle packing graphsLet G =(V,E) be the 1-skeleton of a triangulation of an open disk, and supposethat G has bounded valence. Suppose that P =(P v :v∈V) is any disk packingin the Riemann sphere Ĉ whose contacts graph is G. A point z ∈ Ĉ is a limitpoint of P if every open neighborhood of z intersects infinitely many disksof P, and the set of all limit points of P is denoted Λ(P). The carrier of P,carr(P), is the connected component of Ĉ − Λ(P) that contains P. It is easy tosee that carr(P) is homeomorphic to a disk. From [12] we know that ∂ carr(P) is