Harmonic Dirichlet Functions on Planar Graphs
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Abstract
Benjamini and Schramm (Invent Math 126(3):565–587, 1996) used circle packing to prove that every transient, bounded degree planar graph admits nonconstant harmonic functions of finite Dirichlet energy. We refine their result, showing in particular that for every transient, bounded degree, simple planar triangulation T and every circle packing of T in a domain D, there is a canonical, explicit bounded linear isomorphism between the space of harmonic Dirichlet functions on T and the space of harmonic Dirichlet functions on D.
Keywords
Circle packing Planar graphs Harmonic functions Dirichlet space Electrical networks1 Introduction
A circle packing is a collection P of discs in the Riemann sphere \({\mathbb {C}}\cup \{\infty \}\) such that distinct discs in P do not overlap (i.e., have disjoint interiors), but may be tangent. Given a circle packing P, its tangency graph (or nerve) is the graph whose vertices are the discs in P and where two vertices are connected by an edge if and only if their corresponding discs are tangent. The Circle Packing Theorem [24, 39] states that every finite, simple^{1} planar graph may be represented as the tangency graph of a circle packing, and that if the graph is a triangulation (i.e., every face has three sides) then the circle packing is unique up to Möbius transformations and reflections. See e.g. [32, 38] for further background on circle packing.
The Circle Packing Theorem was extended to infinite, simple planar triangulations by He and Schramm [18, 19, 20, 34]. In particular, they showed that if the triangulation is simply connected, meaning that the surface formed by gluing triangles according to the combinatorics of the triangulation is homeomorphic to the plane, then the triangulation can be circle packed in either the disc or the plane, but not both^{2}; we call the triangulation CP parabolic or CP hyperbolic accordingly. More generally, they showed that, in the CP hyperbolic case, the triangulation can be circle packed in any simplyconnected domain \(D \subsetneq {\mathbb {C}}\). These results can be viewed as discrete analogue of the Riemann mapping theorem and of the uniformization theorem for Riemann surfaces. Indeed, the theory of circle packing is closely related to the theory of conformal mapping and geometric function theory, see e.g. [6, 19, 31, 32, 38] and references therein.
He and Schramm also pioneered the use of circle packing to study probabilistic questions about planar graphs, showing in particular that a bounded degree, simply connected, planar triangulation is CP parabolic if and only if it is recurrent for simple random walk [20]. This result was recently generalised by GurelGurevich et al. [16], who proved that a (not necessarily simply connected) bounded degree planar triangulation admitting a circle packing in a domain D is recurrent for simple random walk if and only if the domain is recurrent for Brownian motion.
More recently, Angel et al. [4] showed that every bounded harmonic function and every positive harmonic function on a bounded degree, simply connected, simple planar triangulation can be represented geometrically in terms of the triangulation’s circle packing in the unit disc. A similar representation theorem for bounded (but not positive) harmonic functions using a different embedding, the square tiling, was obtained slightly earlier by Georgakopoulos [14]. Simpler proofs of both results for bounded harmonic functions have since been obtained by Peres and the author [23].
In this paper we establish a similar representation theorem for harmonic Dirichlet functions. We begin with a simple form of the result that can be stated with minimum preparation. We say that two functions \(\phi \) and \(\psi \) on the vertex set of a graph are asymptotically equal if the set \(\{v\in V: \phi (v)\psi (v)\ge \varepsilon \}\) is finite for every \(\varepsilon >0\).
Theorem 1.1
 1.
For each bounded harmonic Dirichlet function \(h \in \mathbf {BHD}(T)\), there exists a unique harmonic Dirichlet function \(H \in \mathbf {HD}(\mathbb {D})\) such that h and \(H\circ z\) are asymptotically equal.
 2.
For each bounded harmonic Dirichlet function \(H \in \mathbf {BHD}(\mathbb {D})\), there exists a unique harmonic Dirichlet function \(h \in \mathbf {HD}(T)\) such that h and \(H \circ z\) are asymptotically equal.
By a bounded linear isomorphism we mean a bounded linear map with a bounded inverse; such an isomorphism need not be an isometry. A more general form of our theorem, applying in particular to bounded degree, multiplyconnected planar triangulations circle packed in arbitrary domains, is given in Theorem 1.5. See (2.11) and (2.12) for an explicit description of the isomorphism.
Note that Theorems 1.1 and 1.5 are much stronger than those available for bounded and or positive harmonic functions. For example, the representation theorem for bounded harmonic functions [4] requires one to take integrals over the harmonic measure on the boundary, which is not particularly well understood and can be singular with respect to the corresponding measure for Brownian motion. As a consequence, there can exist bounded harmonic functions h on T such that h is not asymptotically equal to \(H \circ z\) for any bounded harmonic function H on \(\mathbb {D}\). The difference in strength between these theorems is unsurprising given that the existence of nonconstant harmonic Dirichlet functions is known to be stable under various perturbations of the underlying space [13, 21, 36], while the existence of nonconstant bounded harmonic functions is known to be unstable in general under similar perturbations [7].
1.1 Applications
Corollary 1.2
We remark that there is a generalization of the Douglas integral formula to other domains due to Doob [8], and that related results for graphs have been announced by Georgakopoulos and Kaimanovich [15]. The results of Doob could be combined with Theorem 1.5 to obtain versions of Corollary 1.2 for more general domains. We do not pursue this here.
Similarly, we can immediately deduce the following very strong boundary convergence result from Theorem 1.1 together with a theorem of Nagel et al. [30].
Corollary 1.3
See [12] and references therein for several further results concerning the boundary behaviour of harmonic Dirichlet functions on the unit disc.
Together with the Poisson boundary identification result of [4], Corollary 1.2 gives us a good understanding of the relationship between the space of bounded Harmonic Dirichlet functions \(\mathbf {BHD}(T)\) and the space of all bounded harmonic functions, denoted \(\mathbf {BH}(T)\): The latter is identified with the space of bounded Borel functions \(L^\infty (\partial \mathbb {D})\), while the former is identified with the space of bounded Douglas integrable functions on \(\partial \mathbb {D}\). In particular, this allows us to easily generate many examples of bounded harmonic functions on T that are not Dirichlet, such as harmonic extensions of indicator functions. Moreover, since the identification of \(\mathbf {BH}(T)\) and \(L^\infty (\partial \mathbb {D})\) is easily seen to be a homeomorphism when \(\mathbf {BH}(T)\) is equipped with the topology of pointwise convergence and \(L^\infty (\partial \mathbb {D})\) is given the subspace topology from \(L^1(\partial \mathbb {D})\), and since the Lipschitz functions are dense in \(L^1(\partial \mathbb {D})\), we obtain the following interesting corollary concerning harmonic functions on triangulations.
Corollary 1.4
Let T be a bounded degree, simply connected, simple, planar triangulation. Then \(\mathbf {BHD}(T)\) is dense in \(\mathbf {BH}(T)\) with respect to the topology of pointwise convergence.
A nice feature of this corollary is that it is an ‘intrinsic’ result, whose statement does not make any reference to circle packing. Corollaries 1.2–1.4 all have straightforward extensions to simply connected, weighted, polyhedral planar with bounded codegrees and bounded local geometry, both of which follow from Theorem 1.5.
Theorem 1.1 and its generalization Theorem 1.5 are also useful in the study of uniform spanning forests of planar graphs, for which closed linear subspaces of \(\mathbf {HD}(T)\) correspond, roughly speaking, to possible boundary conditions at infinity for the spanning forest measure. In particular, Theorem 1.5 will be applied in forthcoming work with Nachmias on uniform spanning forests of multiplyconnected planar maps.
1.2 The Dirichlet Space
We begin by reviewing the definitions of the Dirichlet spaces in both the discrete and continuous cases, as well as some of their basic properties. For further background, we refer the reader to [27, 37] in the discrete case, and [2] and references therein for the continuous case.
1.3 Planar Maps and Double Circle Packing
Let us briefly recall the definitions of planar maps; see e.g. [5, 25, 29] for detailed definitions. Recall that a (locally finite) mapM is a connected, locally finite graph G together with an equivalence class of proper embeddings of G into orientable surfaces, where two such embeddings are equivalent if there is an orientation preserving homeomorphism between the two surfaces sending one embedding to the other. Equivalently, maps can be defined combinatorially as graphs equipped with cyclic orderings of the oriented edges emanating from each vertex, see [25] or [5, Sect. 2.1]. We call a graph endowed with both a map structure and a network structure (i.e., specified conductances) a weighted map. A map is planar if the surface is homeomorphic to an open subset of the sphere, and is simply connected if the surface is simply connected, that is, homeomorphic to either the sphere or the plane.
Given a specified embedding of a map M, the faces of M are defined to be the connected components of the complement of the embedding. We write F for the set of faces of M, and write \(f\perp v\) if the face f is incident to the vertex v. Given an oriented edge e of M, we write \(e^\ell \) for the face to the left of e and \(e^r\) for the face to the right of E. Every map M has a dual map \(M^\dagger \) that has the faces of M as vertices, the vertices of M as faces, and for each oriented edge e of M, \(M^\dagger \) has an oriented edge \(e^\dagger \) from \(e^\ell \) to \(e^r\). The definitions of F and \(M^\dagger \) are independent of the choice of embedding of M, as different embeddings give rise to face sets that are in canonical bijection with each other and dual maps that are canonically isomorphic to each other. It is also possible to define F and \(M^\dagger \) entirely combinatorially, see [25] or [5, Sect. 2.1] for details.
 1.
M is the tangency map of \(P=\{P(v) : v \in V\}\) and \(M^\dagger \) is the tangency map of \(P^\dagger =\{P^\dagger (f) : f \in F\}\).
 2.
If v is a vertex of M and f is a face of M, then the discs P(v) and \(P^\dagger (f)\) intersect if and only if v is incident to f, and in this case their boundaries intersect orthogonally.
1.4 The Isomorphism
We are now ready to describe our isomorphism theorem in its full generality. We say that a weighted map (or more generally a network) has bounded local geometry if it has bounded degree and the conductances of its edges are bounded between two positive constants. We say that a map has bounded codegree if its dual has bounded degree.
Theorem 1.5
 1.
For every harmonic Dirichlet function \(h \in \mathbf {HD}(M)\), there exists a unique harmonic Dirichlet function \(H \in \mathbf {HD}(D)\) such that \(hH\circ z \in \mathbf {D}_0(M)\). We denote this function H by \(\mathsf {Cont}[h]\).
 2.
For every harmonic Dirichlet function \(H \in \mathbf {HD}(D)\), there exists a unique harmonic Dirichlet function \(h \in \mathbf {HD}(M)\) such that \(hH \circ z \in \mathbf {D}_0(M)\). We denote this function h by \({\mathsf {Disc}}[H]\).
Note that even in the simply connected case there are many choices of domain D and double circle packing \((P,P^\dagger )\) for any given map M, and the theorem should be understood as giving us an isomorphism for each such choice of D and \((P,P^\dagger )\).
 For each \(h \in \mathbf {HD}(M)\), \(H={\mathsf {Cont}}[h]\) is the unique harmonic Dirichlet function on D such thatalmost surely when \(\langle X_n \rangle _{n \ge 0}\) is a random walk on G. Similarly, for each \(H\in \mathbf {HD}(D)\), \(h={\mathsf {Disc}}[H]\) is the unique harmonic Dirichlet function on M such that (1.9) holds almost surely. Given Theorem 1.5, both statements are implied by (1.4).$$\begin{aligned} \lim _{n\rightarrow \infty } \big h(X_n)  H \circ z(X_n)\big =0 \end{aligned}$$(1.9)
 For each \(h\in \mathbf {HD}(M)\), \(H={\mathsf {Cont}}[h]\) is the unique harmonic Dirichlet function on D such that h and \(H\circ z\) are quasiasymptotically equal, meaning thatfor every \(\varepsilon >0\). See Sect. 2.1 for the definition of capacity. Similarly, for each \(H\in \mathbf {HD}(D)\), \(h={\mathsf {Disc}}[H]\) is the unique harmonic Dirichlet function on M such that h is quasiasymptotically equal to \(H \circ z\). Given Theorem 1.5, both statements are implied by Proposition 2.1.$$\begin{aligned} \mathrm {Cap}\big (\big \{v\in V : h(v)H\circ z(v) \ge \varepsilon \big \}\big )<\infty \end{aligned}$$(1.10)

IfDis uniformly transient, then for each bounded\(h\in \mathbf {BHD}(M)\), \(H={\mathsf {Cont}}[h]\) is the unique harmonic Dirichlet function on D such that h and \(H\circ z\) are asymptotically equal. Similarly, for each bounded\(H\in \mathbf {BHD}(D)\), \(h={\mathsf {Disc}}[H]\) is the unique harmonic Dirichlet function on M such that h is asymptotically equal to \(H \circ z\). As we will see, given Theorem 1.5, both statements are implied by Proposition 2.11, and yield Theorem 1.1 as a special case.
1.5 Related Work and an Alternative Proof
A related result concerning linear isomorphisms between harmonic Dirichlet spaces induced by rough isometries between bounded degree graphs was shown by Soardi [36], who proved that if \(G_1\) and \(G_2\) are bounded degree, rough isometric graphs, then \(G_1\) admits nonconstant harmonic Dirichlet functions if and only if \(G_2\) does. See e.g. [27, 37] for definitions of and background on rough isometries. Soardi’s result was subsequently generalized by Holopainen and Soardi [21] to rough isometries between bounded degree graphs and a certain class of Riemannian manifolds. This result was then strengthened by Lee [26], who showed that the dimension of the space of harmonic Dirichlet functions is preserved under rough isometry.
By a small improvement on the methods in the works mentioned (or, alternatively, using the methods of this paper), it is not difficult to show the stronger result that for each rough isometry \(\rho :G_1 \rightarrow G_2\), we have that \(h \mapsto (h \circ \rho )_{\mathbf {HD}}\) is a bounded linear isomorphism \(\mathbf {HD}(G_2)\rightarrow \mathbf {HD}(G_1)\). Similar statements hold for rough isometries between graphs and manifolds and between two manifolds (under appropriate assumptions on the geometry in both cases). Indeed, in the discrete case the fact that \(h \mapsto (h \circ \rho )_{\mathbf {HD}}\) is a bounded linear isomorphism can easily be read off from the proof of Soardi’s result presented in [27].
Using these ideas, one could obtain an alternative, less direct proof of Theorem 1.5, sketched as follows: First, let S be the ‘piecewise flat’ surface obtained by gluing regular polygons according to the combinatorics of the map M, which is Riemannian apart from having conical singularities at its vertices. The assumption that M has bounded degrees and codegrees readily implies that the function i sending each vertex of M to the corresponding point of S is a rough isometry. One can then show that \(H \mapsto (h \circ i)_{\mathbf {HD}}\) is a bounded linear isomorphism \(\mathbf {HD}(S)\rightarrow \mathbf {HD}(M)\), similar to the above discussion. Next, the Ring Lemma easily allows us to construct, facebyface, a quasiconformal map \(q:S\rightarrow D\) such that \(q\circ i = z\). One can then arrive at Theorem 1.5 by composing the isomorphism \(\mathbf {HD}(S)\rightarrow \mathbf {HD}(M)\), \(H \mapsto (H \circ i)_{\mathbf {HD}}\) and the isomorphism \(\mathbf {HD}(D)\rightarrow \mathbf {HD}(S)\), \(H \mapsto (H \circ q)_{\mathbf {HD}}\).
2 Proof
2.1 Capacity Characterisation of \(\mathbf {D}_0\)
The following characterisation of \(\mathbf {D}_0\) is presumably wellknown to experts.
Proposition 2.1
 1.Let G be a network and let \(\phi \in \mathbf {D}(G)\). Then \(\phi \in \mathbf {D}_0(G)\) if and only if it is quasiasymptotically equal to the zero function, that is, if and only iffor every \(\varepsilon >0\).$$\begin{aligned} \mathrm {Cap}\big (\{v\in V : \phi (v) \ge \varepsilon \}\big )<\infty \end{aligned}$$
 2.Let D be a domain and let \(\Phi \in \mathbf {D}(D)\). Then \(\Phi \in \mathbf {D}_0(D)\) if and only if it is quasiasymptotically equal to the zero function, that is, if and only iffor every \(\varepsilon >0\).$$\begin{aligned} \mathrm {Cap}\big (\{z\in D : \Phi (z) \ge \varepsilon \text { a.e. on an open neighbourhood of }z \}\big )<\infty . \end{aligned}$$
Proof
2.2 Proof of the Main Theorems
We begin by recalling the Ring Lemma of Rodin and Sullivan [31], which was originally proven for circle packings of triangulations and was generalized to double circle packings of polyhedral maps in [22]. See [1, 17] for quantitative versions in the case of triangulations. Given a double circle packing \((P,P^\dagger )\) in a domain \(D \subseteq {\mathbb {C}}\) of a map M we write r(v) for the radius of P(v) and r(f) for the radius of \(P^\dagger (f)\) for each \(v\in V\) and \(f\in F\).
Theorem 2.2
A consequence of the Ring Lemma is that the embedding of M given by drawing straight lines between the centres of circles in its double circle packing is good^{6} in the sense of [4], meaning that adjacent edges have comparable lengths and that the faces in the embedding have internal angles uniformly bounded away from zero and \(\pi \). We will require the following useful geometric property of good embeddings of planar graphs, stated here for double circle packings. For each \(v\in V\) and \(\delta >0\), we write \(P_{\delta }(v)\) for the disc that has the same centre as P(v) but has radius \(\delta r(v)\). Given a set of vertices \(A \subseteq V\), we write \(P_{\delta }(A)\) for the union \(P_{\delta }(A)=\bigcup _{v\in A} P_{\delta }(v)\).
Lemma 2.3
(The Sausage Lemma [4]) There exists a positive constant \(\delta _1=\delta _1(\mathbf {M})\) such that for each two oriented edges \(e_1,e_2\in E^\rightarrow \) of M that do not share an endpoint, the convex hull of \(P_{\delta _1}(e^_1)\cup P_{\delta _1}(e_1^+)\) and the convex hull of \(P_{\delta _1}(e_2^)\cup P_{\delta _1}(e^+_2)\) are disjoint.
We now define the two operators that will be the key players in the proof of Theorem 1.5.
Definition 2.4
If \(H\in \mathbf {HD}(D)\), then it follows from harmonicity that \({\mathsf {R}}[H](v) = H \circ z(v)\) for every \(v\in V\).
Definition 2.5
We fix a root vertex o of M with which to define the inner product on \(\mathbf {D}(M)\) in (1.2), and take the interior of \(P_{\delta _0}(o)\) to be the precompact open set O used to define the inner product on \(\mathbf {D}(D)\) in (1.6).
Lemma 2.6
The main estimates needed for this lemma are implicit in [16], and our proof is closely modeled on the arguments in that paper.
Proof of Lemma 2.6
A second immediate corollary is the following.
Corollary 2.7
If \(\phi \in \mathbf {D}_0(M)\) then \({\mathsf {A}}[\phi ] \in \mathbf {D}_0(D)\). Similarly, if \(\Phi \in \mathbf {D}_0(D)\) then \({\mathsf {R}}[\Phi ] \in \mathbf {D}_0(M)\).
Proof
We prove the first sentence, the second being similar. It is immediate from the definitions that if \(\phi \in \mathbf {D}_0(M)\) is finitely supported, then \({\mathsf {A}}[\phi ]\) is compactly supported. We conclude by applying the boundedness of \({\mathsf {A}}\). \(\square \)
The following lemma, which is proved below and is also an easy corollary of Lemma 2.6, is also implicit in [16]; indeed, it can be thought of as a quantitative form of the main result of that paper.
Lemma 2.8
We will require the following simple estimates.
Lemma 2.9
 1.Let \(\phi :V\rightarrow \mathbb {R}\) be a function. Thenfor every \(v\in V\) and \(0<\delta <1\).$$\begin{aligned}&\sup _{z\in P_\delta (v)}\big  {\mathsf {A}}[\phi ](z)  \phi (v)\big \\&\qquad \le \delta \sup \big \{ \phi (u)\phi (v) : u\text { and }v\text { share a face of }M\big \} \preceq \delta \sqrt{\mathcal {E}(\phi )} \end{aligned}$$
 2.Let \(H:D\rightarrow \mathbb {R}\) be a harmonic function. Then for every \(r>0\), \(\alpha >1\), and \(z_0 \in D\) such that \(B(z_0,\alpha r) \subseteq D\) we have that$$\begin{aligned} \sup _{z\in B(z_0,r)} H(z)H(z_0)^2 \le \frac{1}{\pi } \log \bigg [ \frac{\alpha ^2}{\alpha ^21} \bigg ] \int _{B(z_0,\alpha r)} \Vert \nabla H(z) \Vert ^2 \mathrm{d}z. \end{aligned}$$
Proof
Proof of Lemma 2.8
There is one more lemma to prove before we prove Theorem 1.5.
Lemma 2.10
 1.
If \(\phi \in \mathbf {D}(M)\), then \(\phi {\mathsf {R}}[{\mathsf {A}}[\phi ]]\in \mathbf {D}_0(M)\).
 2.
If \(\phi \in \mathbf {D}(M)\), then \({\mathsf {A}}[\phi ] \in \mathbf {D}_0(D)\) if and only if \(\phi \in \mathbf {D}_0(M)\).
 3.
If \(\Phi \in \mathbf {D}(D)\), then \({\mathsf {R}}[\Phi ] \in \mathbf {D}_0(M)\) if and only if \(\Phi \in \mathbf {D}_0(D)\).
Proof of Lemma 2.10
We are now ready to prove Theorem 1.5.
Proof of Theorem 1.5
 1.
For each \(H \in \mathbf {HD}(D)\), \(h={\mathsf {Disc}}[H]=({\mathsf {R}}[H])_{\mathbf {HD}}\) is the unique element of \(\mathbf {HD}(M)\) such that \({\mathsf {R}}[H]  h \in \mathbf {D}_0(M)\).
 2.
For each \(h \in \mathbf {HD}(M)\), \(H={\mathsf {Cont}}[h]\) is the unique element of \(\mathbf {HD}(D)\) such that \(h  {\mathsf {R}}[H] \in \mathbf {D}_0(M)\).
 3.
\(h={\mathsf {Disc}}[{\mathsf {Cont}}[h]]\) and \(H={\mathsf {Cont}}[{\mathsf {Disc}}[H]]\) for every \(h\in \mathbf {HD}(M)\) and \(H\in \mathbf {HD}(D)\) respectively.
 1.
This follows immediately from the uniqueness of the Royden decomposition (i.e., the fact that \(\mathbf {D}(D)=\mathbf {D}_0(D)\oplus \mathbf {HD}(D)\)).
 2.
We first wish to prove that \(h{\mathsf {R}}{\mathsf {Cont}}[h] = h  {\mathsf {R}} {\mathsf {P}}_{\mathbf {HD}(D)} {\mathsf {A}} h \in \mathbf {D}_0(M)\) for every \(h\in \mathbf {D}(M)\). To see this, note that \(h{\mathsf {R}}{\mathsf {P}}_{\mathbf {HD}(D)} {\mathsf {A}} h = [h {\mathsf {R}}{\mathsf {A}}h] + {\mathsf {R}} {\mathsf {P}}_{\mathbf {D}_0(D)} {\mathsf {A}} h\). Since \(h{\mathsf {R}}{\mathsf {A}}h \in \mathbf {D}_0(M)\) by item 1 of Lemma 2.10 and \({\mathsf {R}} {\mathsf {P}}_{\mathbf {D}_0(D)} {\mathsf {A}} h \in \mathbf {D}_0(M)\) by Corollary 2.7, we deduce that \(h{\mathsf {R}}{\mathsf {Cont}}[h] \in \mathbf {D}_0(M)\) as claimed.
We now prove uniqueness. Suppose that \(H\in \mathbf {HD}(D)\) is such that \(h{\mathsf {R}}[H]\) is in \(\mathbf {D}_0(M)\). Then we must have that \({\mathsf {R}}[{\mathsf {Cont}}[h]H] = (h{\mathsf {R}}[H])  (h{\mathsf {R}}[{\mathsf {Cont}}[h]])\) is in \(\mathbf {D}_0(M)\) also, and it follows from Lemma 2.10 (more specifically the ‘only if’ implication of item 3 of that lemma) that \({\mathsf {Cont}}[h]H \in \mathbf {D}_0(D)\). But since \({\mathsf {Cont}}[h]H \in \mathbf {HD}(D)\) we deduce that \(H={\mathsf {Cont}}[h]\) as claimed.
 3.
We first prove that \(h={\mathsf {Disc}}[{\mathsf {Cont}}[h]]\) for every \(h\in \mathbf {HD}(M)\). We have that \(h{\mathsf {Disc}}[{\mathsf {Cont}}[h]] =h {\mathsf {R}} {\mathsf {Cont}}[h] + {\mathsf {P}}_{\mathbf {D}_0(M)} {\mathsf {R}} {\mathsf {Cont}}[h]\), and since, by item 2, \(h {\mathsf {R}} {\mathsf {Cont}}[h]\) and \({\mathsf {P}}_{\mathbf {D}_0} {\mathsf {R}} {\mathsf {Cont}}[h]\) are both in \(\mathbf {D}_0(M)\), it follows that \(h{\mathsf {Disc}}[{\mathsf {Cont}}[h]] \in \mathbf {D}_0(M)\) and hence that \(h{\mathsf {Disc}}[{\mathsf {Cont}}[h]]=0\) as claimed.
It remains to prove that \(H={\mathsf {Cont}}[{\mathsf {Disc}}[H]]\) for every \(H\in \mathbf {HD}(D)\). By item 2 we have that \({\mathsf {Disc}}[H]  {\mathsf {R}} {\mathsf {Cont}}[{\mathsf {Disc}}[H]] \in \mathbf {D}_0(M)\), and hence thatalso. It follows by Lemma 2.10 that \(H  {\mathsf {Cont}}[{\mathsf {Disc}}[H]]\in \mathbf {D}_0(D)\) and hence that \(H  {\mathsf {Cont}}[{\mathsf {Disc}}[H]]=0\) as claimed.$$\begin{aligned} {\mathsf {R}}\bigl [H  {\mathsf {Cont}}[{\mathsf {Disc}}[H]] \bigr ]= {\mathsf {P}}_{\mathbf {D}_0(M)}{\mathsf {R}} [H] + {\mathsf {Disc}}[H]  {\mathsf {R}} {\mathsf {Cont}}[{\mathsf {Disc}}[H]] \in \mathbf {D}_0(M) \end{aligned}$$
2.3 Asymptotic Equality in the Uniformly Transient Case
We now prove the following proposition, which, together with Proposition 2.1, allows us to deduce Theorems 1.1 from 1.5.
Proposition 2.11
Let M be a transient weighted polyhedral planar map with bounded codegrees and bounded local geometry, let \((P,P^\dagger )\) be a double circle packing of M in a domain \(D \subset {\mathbb {C}}\), and let \(z:V\rightarrow D\) be the function sending each circle to the centre of its corresponding disc. Let h and H be bounded harmonic functions on M and D respectively. If D is uniformly transient, then h and \(H\circ z\) are asymptotically equal if and only if they are quasiasymptotically equal.
Theorem 2.12
Proof
The following lemma is presumably wellknown to experts, but we were not able to find a reference.
Lemma 2.13
Let G be a transient network and suppose that A is a set of vertices for which there exists \(\varepsilon >0\) and infinitely many disjoint sets \(A_1,A_2, \ldots \subseteq A\) such that \(\mathrm {Cap}(A_i)\ge \varepsilon \) for every \(i\ge 1\). Then \(\mathrm {Cap}(A)=\infty \).
Proof
First note that if A has finite capacity then we must have that simple random walk on G visits A at most finitely often almost surely. Indeed, if \(\mathrm {Cap}(A)<\infty \) then there exists \(\psi \in \mathbf {D}_0(G)\) with \(\psi _A\ge 1\), and it follows from (1.5) that if X is a random walk then \(\psi (X_n)\rightarrow 0\) a.s. and hence that X visits A at most finitely often a.s. Thus, it suffices to consider the case that the simple random walk visits A at most finitely often almost surely.
Proof of Proposition 2.11
Footnotes
 1.
A graph is said to be simple if it does not contain any loops or multiple edges.
 2.
Here the word in is being used in a technical sense to mean that the carrier of the circle packing is equal to either the disc or the plane, see Sect. 1.3.
 3.
Recall that a function or vector field \(\Phi :D\rightarrow \mathbb {R}^d\), \(d\ge 1\), is said to be locally integrable if \(\int _A \Vert \Phi (z)\Vert \mathrm{d}z<\infty \) for every precompact open subset A of D, and locally\(L^2\) if \(\int _A \Vert \Phi (z)\Vert ^2\mathrm{d}z<\infty \) for every precompact open subset A of D. A locally integrable vector field \(W:D\rightarrow \mathbb {R}^2\) is said to be a weak gradient of the locally integrable function \(\Phi :D\rightarrow \mathbb {R}\) if the identity \(\int _D \Psi W \mathrm{d}z = \int _D \Phi \nabla \Psi \mathrm{d}z\) holds for every smooth, compactly supported function \(\Psi \) on D. We say that a locally integrable function \(\Phi :D\rightarrow \mathbb {R}\) is weakly differentiable if it admits a weak gradient. The weak gradient of a locally integrable, weakly differentiable \(\Phi :D\rightarrow \mathbb {R}^2\) is unique up to almosteverywhere equivalence, and is denoted by \(\nabla \Phi \). The weak gradient coincides with the usual gradient of \(\Phi \) at z if \(\Phi \) is differentiable on an open neighbourhood of z.
 4.
Strictly speaking, since \(\Phi \) is only defined up to almost everywhere equivalence, we choose a quasicontinuous version of \(\Phi \) before applying it to the Brownian motion \(B_t\). This ensures that \(\Phi (B_t)\) is welldefined and continuous in t almost surely. See [2] for details.
 5.
He worked in a more general setting, see [22, Sect. 2.5] for a discussion of how his results imply those claimed here.
 6.
We remark that all our results hold more generally for good straightline embeddings of M, not just those produced using double circle packing. However, we are not aware of any general method of producing good embeddings that does not rely on double circle packing.
Notes
Acknowledgements
The author was supported by a Microsoft Research PhD Fellowship. We thank the anonymous referees for their comments and corrections.
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