Harmonic Dirichlet Functions on Planar Graphs

Benjamini and Schramm (1996) used circle packing to prove that every transient, bounded degree planar graph admits non-constant 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 $D$.


Introduction
A circle packing is a collection P of discs in the Riemann sphere C ∪ {∞} 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 given by 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 [19,29] 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. [28,24] for further background on circle packing.
The Circle Packing Theorem was extended to infinite, simple planar triangulations by He and Schramm [14,15,25,13]. 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. This can be viewed as a discrete analogue of the uniformization theorem for Riemann surfaces. He and Schramm also pioneered the use of circle packing to study probabilistic questions about planar graphs, showing that a bounded degree, simply connected, planar triangulation is CP parabolic if and only if it is recurrent for simple random walk [15]. This result was recently generalised by Gurel-Gurevich, Nachmias, and Suoto [11], 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.
A more detailed study of the relationship between circle packing and random walks was initiated by Benjamini and Schramm [5], who proved in particular that if T is a bounded degree triangulation circle packed in the unit disc D, then the random walk on T converges almost surely to a point in the boundary ∂D, and the law of this limit point is non-atomic. They used this to deduce the existence of various kinds of harmonic functions on transient, bounded degree planar graphs. Recall that a function h on the vertex set of a locally finite graph G = (V , E) is said to be harmonic if for every ∈ V . Three particularly important and probabilistically meaningful classes of harmonic functions are the bounded harmonic functions, the positive harmonic functions, and the harmonic Dirichlet functions. It is an easy consequence of the Benjamini-Schramm convergence theorem that every bounded degree, transient planar graph admits non-constant harmonic functions in each of these three classes. Here, a harmonic Dirichlet function on a graph with oriented edge set E → is a harmonic function h such that We denote the space of harmonic Dirichlet functions on a graph G by HD(G) and the space of bounded harmonic Dirichlet functions on G by BHD(G). For each vertex of G, h = h( ) 2 + E(h) is a norm on HD(G), and BHD(G) is dense in HD(G) with respect to this norm. Harmonic Dirichlet functions and function spaces on domains are defined similarly; see Section 1.1 for details. More recently, Angel, Barlow, Gurel-Gurevich, and Nachmias [3] 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 its circle packing in the unit disc. A similar representation theorem for bounded (but not positive) harmonic functions using a different embedding, the square tiling, were obtained slightly earlier by Georgakopoulos [9]. Simpler proofs of both results for bounded harmonic functions have since been obtained by Peres and the author [18].
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 ϕ and ψ on the vertex set of a graph are asymptotically equal if the set { ∈ V : |ϕ( ) − ψ ( )| ≥ ε} is finite for every ε > 0. Theorem 1.1. Let T be a bounded degree, simply connected, simple, planar triangulation, let P be a circle packing of T in the unit disc D, and let z : V → D be the function sending vertices to the centres of their corresponding discs. Moreover, the function assigning each h ∈ BHD(T ) to the unique H ∈ HD(D) such that H • z is asymptotically equal to h can be uniquely extended to a bounded linear isomorphism from HD(T ) to HD(D).
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, multiply-connected planar triangulations circle packed in arbitrary domains, is given in Theorem 1.3. See (14) and (15) for an explicit description of the isomorphism.
Note that Theorems 1.1 and 1.3 are much stronger than those available for bounded and or positive harmonic functions. For example, the representation theorem for bounded harmonic functions [3] 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 • z for any bounded harmonic function H on D. The difference in strength between these theorems is unsurprising given that the existence of non-constant harmonic Dirichlet functions is known to be stable under various perturbations of the underlying space [26,16,8], while the existence of non-constant bounded harmonic functions is known to be unstable in general under similar perturbations [5].
We remark, however, that by combining Theorem 1.1 with the classical theory of harmonic Dirichlet functions on the disc, we immediately obtain a representation theorem for the harmonic Dirichlet functions on T in terms of boundary functions, similar to that obtained for bounded harmonic functions in [3]. We say that a Borel function ϕ : It is a classical theorem of Douglas [7] that a harmonic function H : D → R is Dirichlet if and only if it is the extension of a Douglas integrable function ϕ : ∂D → R, and in this case D(ϕ) = E(h). This equality is known as the Douglas integral formula. Thus, we obtain the following corollary to Theorem 1.1. We remark that there is a generalization of the Douglas integral formula to other domains due to Doob [6], and that related results for graphs have been announced by Georgakopoulos and Kaimanovich [10]. The results of Doob could be combined with Theorem 1.3 to obtain versions of Corollary 1.2 for more general domains. We do not pursue this here.

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 [21,27] in the discrete case, and [2] and references therein for the continuous case.
Recall that a network is a locally finite graph G = (V , E) (which in this paper will always be locally finite and connected) together with an assignment c : E → (0, ∞) of positive conductances to the edges of G. The random walk on a network is the Markov process that, at each step, chooses an edge to traverse from among those edges emanating from its current position, weighted by their conductances. Let G = (V , E) be a network, and let E → be the set of oriented edges of G. The Dirichlet energy of a function ϕ : V → R is defined to be We say that ϕ is a Dirichlet function (or equivalently that ϕ has finite energy) if E (ϕ) < ∞. The space of Dirichlet functions on G and the space of harmonic Dirichlet functions on G are denoted by D(G) and HD(G) respectively. These spaces are both Hilbert spaces with respect to the inner product where o is a fixed root vertex. (It is easily seen that different choices of o yield equivalent norms.) We denote the space of bounded Dirichlet functions by BD(G) and the space of bounded harmonic Dirichlet functions by BHD(G). These spaces are dense in D(G) and HD(G) respectively. Let D 0 (G) be the closure in D(G) of the space of finitely supported functions. If G is transient, then every Dirichlet function ϕ ∈ D(G) has a unique decomposition where ϕ D 0 ∈ D 0 (G) and ϕ HD ∈ HD(G), known as the Royden decomposition of ϕ. In other words, D(G) = D 0 (G) ⊕ HD(G). Note that it is not necessarily an orthogonal decomposition. Let X n n ≥0 be a random walk on G. It is a theorem of Ancona, Lyons, and Peres [2], which complements earlier results of Yamasaki [30], that the limit lim n→∞ ϕ(X n ) exists almost surely for each ϕ ∈ D(G), that almost surely, and moreover that ϕ HD can be expressed as ϕ HD ( ) = E lim n→∞ ϕ(X n ) , where E denotes the expectation with respect to the random walk X n n ≥0 started at . A similar theory holds in the continuum. If D ⊆ C is a domain, the Dirichlet energy of a locally L 2 , weakly differentiable 3 function Φ : D → R on D is defined to be As in the discrete case, we say that Φ is a Dirichlet function (or equivalently that Φ has finite energy) if it is locally L 2 , weakly differentiable, and satisfies E (Φ) < ∞. We let D(D), and HD(D) be the spaces of Dirichlet functions (modulo almost everywhere equivalence) and harmonic Dirichlet functions respectively. The spaces D(D) and HD(D) are Hilbert spaces with respect to the inner product   motion stopped at the first time it hits ∂D, denoted T ∂D . Anconca, Lyons, and Peres [2] also proved that if Φ ∈ D(D), then the limit lim t →T ∂D Φ(B t ) exists almost surely 4 , that

Planar maps and double circle packing
Let us briefly recall the definitions of planar maps; see e.g. [22,4] for detailed definitions. Recall that a (locally finite) map M 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. 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.
The faces of the map are the connected component of the complement of the embedding. We write F for the set of faces of M, and write f ⊥ if the face f is incident to the vertex . Given an oriented edge e of M, we write e 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 † that has the faces of M as vertices, the vertices of M as faces, and for each oriented edge e of M, M † has an edge e † from e to e r .
The carrier of a circle packing P, carr(P), is defined to be union of the discs in P together with the components of C ∪ {∞} \ P whose boundaries are contained in a union of finitely many discs in P. Note that every circle packing P in the Riemann sphere whose tangency graph is locally finite also defines a locally finite tangency map as well as a tangency graph, where we embed the tangency graph into the carrier of P by drawing straight lines between the centres of the circles.
Let M be a locally finite map with locally finite dual M † . A double circle packing of M is a pair of circle packings (P, P † ) in the Riemann sphere such that the following conditions hold (see

If is a vertex of M and f is a face of M, then the discs P( ) and P † (f ) intersect if and only if
is incident to f , and in this case their boundaries intersect orthogonally.
Observe that if (P, P † ) is a double circle packing of a locally finite map with locally finite dual then carr(P) = carr(P † ) = P ∪ P † . It follows from Thurston's interpretation of Andreev's theorem that a finite planar map has a double circle packing in the Riemann sphere if and only if it is polyhedral, that is, simple and 3-connected. The corresponding infinite theory 5 was developed by He [13], who proved that every simple connected, locally finite map M with locally finite dual admits a double circle packing in either the plane or the disc, that this packing is unique up to Möbius transformations 6 . An easy compactness argument implies that every locally finite polyhedral planar map with locally finite dual admits a double circle packing in some domain, possibly a very wild one.

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 degrees, 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 degrees.  Note that the weighted map M is not required to be uniformly transient.

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 [26], who proved that if G 1 and G 2 are bounded degree, rough isometric graphs, then G 1 admits non-constant harmonic Dirichlet functions if and only if G 2 does. See e.g. [27,21] for definitions of and background on rough isometries. Soardi's result was subsequently generalized by Holopainen and Soardi [16] to rough isometries between bounded degree graphs and a certain class of Riemannian manifolds. This result was then strengthened by Lee [20], 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 ρ : G 1 → G 2 , we have that h → (h • ρ) HD is a bounded linear isomorphism HD(G 2 ) → 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 → (h • ρ) HD is a bounded linear isomorphism can easily be read off from the proof of Soardi's result presented in [21].
Another setting in which one very easily obtains an isomorphism between harmonic Dirichlet spaces is given by quasi-conformal mapping between domains (or other Riemannian manifolds).
Recall that a homeomorphism q : D → D is said to be quasi-conformal if it is orientation preserving, weakly differentiable, and there exists a constant C such that Dq(z) 2 ≤ C |det Dq(z) | for a.e. z ∈ D. It is trivial to verify by change of variables that E(ϕ • q) ≤ CE(ϕ) for every ϕ ∈ D(D) and E(ψ • q −1 ) ≤ CE(ψ ) for every ψ ∈ D(D ), so that composition with q defines a bounded linear isomorphism from D(D ) to D(D). Moreover, it is immediate that ψ • q ∈ D 0 (D) if and only if ψ ∈ D 0 (D ), and it follows that H → (H • q) HD is a bounded linear isomorphism from HD(D ) to HD(D).
Using these ideas, one could obtain an alternative, less direct proof of Theorem 1.3, 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 → (h • i) HD is a bounded linear isomorphism HD(S) → HD(M), similar to the above discussion. Next, the Ring Lemma easily allows us to construct, face-by-face, a quasiconformal map q : S → D such that q • i = z. One can then arrive at

Capacity characterisation of D 0
Recall that the capacity of a finite set of vertices A in a network G is defined to be where P (τ + A = ∞) is the probability that a random walk on G started at A never returns to  The following characterisation of D 0 is presumably well-known to experts.

Let G be a network and let ϕ ∈ D(G). Then ϕ ∈ D 0 (G) if and only if it is quasi-asymptotically equal to the zero function, that is, if and only if
Cap { ∈ V : |ϕ( )| ≥ ε} < ∞ for every ε > 0.

Let D be a domain and let Φ ∈ D(D). Then Φ ∈ D 0 (D) if and only if it is quasi-asymptotically equal to the zero function, that is, if and only if
Cap {z ∈ D : |Φ(z)| ≥ ε a.e. on an open neighbourhood of z} < ∞.

Proof of the main theorems
We begin by recalling the Ring Lemma of Rodin and Sullivan [23], which was originally proven for circle packings of triangulations and was generalized to double circle packings of polyhedral maps in [17]. See [12,1] for quantitative versions in the case of triangulations. For the rest of this section M will be a transient weighted polyhedral map with bounded codegrees and bounded local geometry, (P, P † ) will be a double circle packing of M in a domain D ⊆ C ∪ {∞}, and z will be the associated embedding of M. By applying a Möbius transformation if necessary, we can and will assume that D ⊆ C. We write M = M(M) for the data We say that two quantities are comparable if they differ up to positive multiplicative constants depending only on M, and write , , and for equalities and inequalities that hold up to positive multiplicative constants depending only on the data M. We also use standard big-O notation, where again the implicit positive multiplicative constants depend only on M.
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 7 in the sense of [3], meaning that adjacent edges have comparable lengths and that the faces in the embedding have internal angles uniformly bounded away from zero and π . We will require the following useful geometric property of good embeddings of planar graphs, stated here for double circle packings. For each ∈ V and δ > 0, we write P δ ( ) for the disc that has the same centre as P( ) but has radius δr ( ). Given a set of vertices A ⊆ V , we write P δ (A) for the union P δ (A) = ∈A P δ ( ). [3]). There exists a positive constant δ 1 = δ 1 (M) such that for each two oriented edges e 1 , e 2 ∈ E → of M that do not share an endpoint, the convex hull of P δ 1 (e − 1 )∪P δ 1 (e + 1 ) and the convex hull of P δ 1 (e − 2 ) ∪ P δ 1 (e + 2 ) are disjoint.

Lemma 2.3 (The Sausage Lemma
We now define the two operators that will be the key players in the proof of Theorem 1.3. Definition (The operator R). Fix δ 0 = δ 0 (M) ≤ 1/2 sufficiently small that δ 0 is less than or equal to the sausage lemma constant δ 1 and that 1 4 |z(u) − z( )| ≥ δ 0 r ( ) for every adjacent pair u, ∈ V . For each locally integrable Φ : D → R, we define R[Φ] : V → R by setting R[ϕ]( ) to be the average value of Φ on the disc P δ 0 ( ) for each ∈ V ,  The main estimates needed for this lemma are implicit in [11], and our proof is closely modeled on the arguments in that paper. Integrating over z ∈ T e and summing over e ∈ E → , we obtain that and hence by Cauchy-Schwarz we have that Since each oriented edge is counted at most a constant number of times in this sum we obtain that A simple Cauchy-Schwarz argument similar to the above then shows that and combining (8) and (9)  Φ, Φ and in particular that E(R[Φ]) E(Φ) for every Φ ∈ D(D). Let us first suppose that Φ is continuously differentiable. It is well known, and can be seen by a simple mollification argument, that such Φ are dense in D(D) (as indeed are the smooth Dirichlet functions). For each ∈ V , let X be a random point chosen uniformly from the disc P δ 0 ( ), independently from each other, so that R[Φ]( ) = EΦ(X ). For each u, ∈ V , let Γ u, be the random line segment connecting X u to X . By Jensen's inequality and the assumption that Φ is continuously differentiable we have that For each adjacent u, ∈ V , conditional on Γ u, , let Z u, be a random point chosen uniformly on the line segment Γ u, . The Cauchy-Schwarz inequality implies that O r(v) Figure 2: Illustration of the proof of the boundedness of R. Suppose that z (green square) is closer to z( ) (navy disc) than to z(u) (brown disc). Then conditional on the location of X u (red square), in order for Z u, to be located in B(z, δr ( )) (purple disc), X must be located in the intersection (blue segment) of P δ 0 ( ) with the cone whose vertex is at X u and that is tangent to B(z, δr ( )).
The dashed line is the perpendicular bisector of the line from z(u) to z( ). This intersection is contained within a triangle (grey) whose sides have lengths of order O(r ( )), O(r ( )) and O(δr ( )), and consequently has area O(δr ( ) 2 ).
Next, the Ring Lemma implies that |Γ u, | r ( ), and we deduce that Let A u, be the the support of Z , i.e., the convex hull of P δ 0 (u)∪P δ 0 ( ). We claim that the Radon-Nikodym derivative of the law of Z with respect to the Lebesgue measure on A u, is O(r ( ) −2 ), that is, that P Z u, ∈ B(z, δr ( )) δ 2 for every z ∈ A u, and δ > 0. Suppose without loss of generality that |z − z( )| ≤ |z − z(u)|, and condition on the value of X u , so that |X u − z| ≥ |z(u) − z( )|/4 r ( ) by definition of δ 0 . In order for Z u, to be in the ball B(z, δr ( )), we must have that X is in the cone K that has its vertex at X u and that is tangent to B(z, δr ( )), see Figure 2. Since |X u − z| ≥ |z(u) − z( )|/4, it follows by elementary trigonometry that the internal angle at the vertex of K is O(δ ), and consequently that the intersection of K with P δ 0 ( ) (or indeed with all of A u, ), being contained inside a triangle with height O(r ( )) and width O(δr ( )), has area at most O(δr ( ) 2 ). Thus, the probability that X lies in this region is at most O(δ ). Conditioned on the event that X lies in K, the intersection of Γ u, with B(z, δ ) has length at most 2δr ( ), and so the conditional probability that Z lies in this segment is O(δ ). The estimate (10) follows.
Integrating over the Radon-Nikoydm estimate (10) we obtain that and hence that for every adjacent u, ∈ V . Since (11) holds uniformly for all continuously differentiable Φ ∈ D(D) and the expressions on both sides of the inequality are continuous functions of Φ ∈ D(D), we deduce by density that the inequality holds for all Φ ∈ D(D). Since δ 0 was taken to be less than the Sausage Lemma constant, we have that each point z is in at most max ∈V deg( ) different regions of the form A u, , and so as required. The other term in R[Φ], R[Φ] can be bounded using Jensen's inequality, which yields that Combining (12) and (13) are also well defined and bounded.
A second immediate corollary is the following.
Proof. We prove the first sentence, the second being similar. It is immediate from the definitions that if ϕ ∈ D 0 (M) is finitely supported, then A[ϕ] is compactly supported. We conclude by applying boundedness of A.
The following lemma, which is an easy corollary of Lemma 2.4, is also implicit in [11]; indeed, it can be thought of as a quantitative form of the main result of that paper. Lemma 2.6. For every 0 < δ ≤ 1/2, we have that for every set of vertices A in M.
We will require the following simple estimates.
Proof. Item 1 is immediate from the definition of A[ϕ]. Item 2 follows immediately from the elliptic Harnack inequality (see (16)).
Proof of Lemma 2.6. We start with the upper bound. Let ϕ ∈ D 0 (M) be such that ϕ | A ≥ 1, and let ψ = (ϕ ∧ 1) ∨ 0. It is easily verified that E(ψ ) ≤ E(ϕ) and ψ | A ≥ 1, and it follows from Proposition 2.1 that ψ ∈ D 0 (M) (this is also easy to verify directly and the upper bound of Lemma 2.4 follows by taking the infimum over ϕ. We now turn to the lower bound. Let Φ ∈ D 0 (D) be such that Φ ≥ 1 on an open neighbourhood of P δ (A), and let Ψ = (Φ ∧ 1) ∨ 0. As before, we have that E(Ψ) ≤ E(Φ) and that Ψ ≥ 1 on an open neighbourhood of A. For every ∈ A we have that Thus, by Lemma 2.5, the function δ 2 0 R[Ψ]/δ 2 ∈ D 0 (M) is at least 1 on A, and so, by Dirichlet's principle and Lemma 2.4, The claimed lower bound follows by taking the infimum over Φ.
There is one more lemma to prove before we prove Theorem 1.3.    This proposition is a simple consequence of the elliptic Harnack inequality, which we now discuss. For each z ∈ C and r > 0, let B(z, r ) denote the Euclidean ball of radius r around z. Recall the classical elliptic Harnack inequality for the plane, which implies that for every z 0 ∈ C, every harmonic function h : B(z 0 , r ) → R, and every z ∈ B(z 0 , C), we have that Angel, Barlow, Gurel-Gurevich, and Nachmias [3] established a version of the elliptic Harnack inequality that holds for double circle packings with respect to the Euclidean metric. The version of the theorem that we state here follows from that stated in [3] by a simple rearrangement and iteration argument, below. for every harmonic function h on V , every ∈ V , every r ≤ d(z( ), ∂D), and every u ∈ V with z( ) ∈ B(z( ), αr ).
Proof. Let X be the union of the straight lines between the centres of circles in P. The Ring Lemma implies that the path metric on X is comparable to the subspace metric on X [3, Proposition 2.5].
Given a function f on the vertex set of M, we extend f to X by linear interpolation along each edge. The version of the elliptic Harnack inequality stated in [3,Theorem 5.4] implies that for each A > 1, there exists a constant C = C(A, M) > 1 such that for every x ∈ X with d(x, ∂D) ≥ Ar , and every harmonic function h on M such that the extension of h to X is positive on B(x, Ar ), we have that sup This implies that, for an arbitrary harmonic function h on M, we have that for every harmonic function h on M, every r > 0, every n ≥ 1, and every x ∈ X such that d(x, ∂D) ≥ r . This is easily seen to imply the claimed inequality Proof of Proposition 2.10. Asymptotic equality clearly implies quasi-asymptotic equality. Suppose that h and H • z are not asymptotically equal, so that there exists ε > 0 such that the set A ε = { ∈ V : |h( ) − H • z( )/h( )| ≥ ε} is infinite. Since h and H are bounded, it follows from the elliptic Harnack inequality that there exists δ > 0 such that if ∈ A ε then u ∈ V : z(u) ∈ B z( ), δd z( ), ∂D ⊆ A ε /2 .
Since D is uniformly transient, Lemma 2.6 implies that the sets on the left-hand side have capacity bounded below by some positive constant η. Since any set of vertices in a network containing infinitely many disjoint subsets with capacity at least η > 0 must have infinite capacity, we deduce that h and H • z are not quasi-asymptotically equal.