Boundaries of Planar Graphs: A Unified Approach

We give a new proof that the Poisson boundary of a planar graph coincides with the boundary of its square tiling and with the boundary of its circle packing, originally proven by Georgakopoulos and Angel, Barlow, Gurel-Gurevich and Nachmias respectively. Our proof is robust, and also allows us to identify the Poisson boundaries of graphs that are rough-isometric to planar graphs. We also prove that the boundary of the square tiling of a bounded degree plane triangulation coincides with its Martin boundary. This is done by comparing the square tiling of the triangulation with its circle packing.

Boundaries of planar graphs straight lines between the centres of tangent discs in the packing. The Koebe-Andreev-Thurston Circle Packing Theorem [15,22] states conversely that every finite, simple (i.e., containing no self-loops or multiple edges), planar graph may be represented as the tangency graph of a circle packing. If the graph is a triangulation (i.e., every face has three sides), its circle packing is unique up to Möbius transformations and reflections. We refer the reader to [21] and [18] for background on circle packing.
The carrier of a circle packing is defined to be the union of all the discs in the packing together with the bounded regions that are disjoint from the discs in the packing and are enclosed by some set of discs in the packing corresponding to a face of the tangency graph. Given some planar domain D, we say that a circle packing is in D if its carrier is D.
The circle packing theorem was extended to infinite planar graphs by He and Schramm [12,13], who proved that every proper plane triangulation admits a locally finite circle packing in the plane or the disc, but not both. We call a triangulation of the plane CP parabolic if it can be circle packed in the plane and CP hyperbolic otherwise. He and Schramm also proved that a bounded degree simple triangulation of the plane is CP parabolic if and only if it is recurrent for the simple random walk.
Benjamini and Schramm [4] proved that, when a bounded degree, CP hyperbolic triangulation is circle packed in the disc, the simple random walk converges to a point in the boundary of the disc and the law of the limit point is non-atomic and has full support. Angel, Barlow, Gurel-Gurevich and Nachmias [2] later proved that, under the same assumptions, the boundary of the disc is a realisation of the Poisson boundary of the triangulation. These results were extended to unimodular random rooted triangulations of unbounded degree by Angel, Hutchcroft, Nachmias and Ray [3]. Our proof of Theorem 1.1 is adapted from the proof of [3], and also yields a new proof of the Poisson boundary result of [2], which follows as a special case of both Theorems 1.2 and 1.3 below.

Robustness under rough isometries
The proof of Theorem 1.1 is quite robust, and also allows us to characterise the Poisson boundaries of certain non-planar networks. Let G = (V, E) and G = (V , E ) be two graphs, and let d and d denote their respective graph metrics. Recall that a function φ : V → V is a rough isometry if there exist positive constants α and β such that the following conditions are satisfied (in which case we say φ is an (α, β)-rough isometry): for every v ∈ V . That is, the geometric boundary of the disc coincides with the Poisson boundary of G.
The part of Theorem 1.2 concerning the existence of an embedding is implicit in [4].
A further generalisation of Theorem 1.1 concerns embeddings of possibly irreversible planar Markov chains: The only changes required to the proof of Theorem 1.1 in order to prove the following are notational. We say that a Markov chain (V, P ) is locally finite if the set {u ∈ V : P (v, u) > 0} is finite for every vertex v ∈ V . Theorem 1.3. Let (V, P ) be a locally finite Markov chain such that the graph G = V, {(u, v) ∈ V 2 : P (u, v) > 0 or P (v, u) > 0} is planar. Suppose further that there exists a vertex ρ ∈ V such that for every v ∈ V there exists n such that P n (ρ, v) > 0, and let X n n≥0 be a trajectory of the Markov chain. Let z be a (not necessarily proper) embedding of G into the unit disc D such that z(X n ) n≥0 converges to a point in ∂D almost surely and the law of the limit point is non-atomic.
Then for every bounded harmonic function h on (V, P ), there exists a bounded Borel The assumption that (V, P ) is locally finite is not really necessary, but removing it would require a non-cosmetic modification of the proof.

The Martin boundary
In [2] it was also proven that the Martin boundary of a bounded degree CP hyperbolic triangulation can be identified with the geometric boundary of its circle packing. Recall that a function g : for every vertex u ∈ V . Let ρ be a fixed vertex of G and consider the space S + of positive superharmonic functions g on G such that g(ρ) = 1, which is a convex, compact subset of the space of functions V → R equipped with the product topology (i.e. the topology of pointwise convergence). For each v ∈ V , let P v be the law of the random walk on G started at v. We can embed V into S + by sending each vertex u of G to its Martin The Martin compactification M(G) of the network G is defined as the closure of {M u : u ∈ V } in S + , and the Martin boundary ∂M(G) of the network G is defined to be the complement of the image of V , See [8,17,23] for background on the Martin boundary. Our next result is that, for a triangulation of the plane with bounded local geometry, the geometric boundary of the square tiling coincides with the Martin boundary. in T converges to a point in the Martin boundary of T if and only if y(v n ) → 1 and θ(v n ) converges to a point in R/ηZ.

The map
which is well-defined by (1), is a homeomorphism from R/ηZ to the Martin boundary ∂M(T ) of T .
That is, the geometric boundary of the rectangle tiling of G coincides with the Martin boundary of G.
This will be deduced from the analogous statement for circle packings [2, Theorem 1.2] together with the following theorem, which states that for a bounded degree triangulation T , the square tiling and circle packing of T define equivalent compactifications of T . Theorem 1.5 (Comparison of square tiling and circle packing). Let T be a transient, simple, proper plane triangulation with bounded local geometry. Let S ρ be a rectangle tiling of T in the cylinder R/ηZ × [0, 1] and let C be a circle packing of T in D with associated embedding z. Then 1. The sequence of points z(v n ) n≥0 associated to a sequence of vertices v n n≥0 in T converges to a point in ∂D if and only if y(v n ) → 1 and θ(v n ) converges to a point in R/ηZ.

The map
which is well-defined by (1), is a homeomorphism from ∂D to R/ηZ.  for each Borel set A ⊆ R/ηZ, and let λ = ω ρ denote the Lebesgue measure on R/ηZ. Then for every v of T , the density of the harmonic measure from v with respect to Lebesgue measure is given by which is continuous with respect to θ.
for every v ∈ V .

Notation
We use e to denote both oriented and unoriented edges of a graph or network. An oriented edge e is oriented from its tail e − to its head e + . Given a network G and a vertex v of G, we write P v for the law of the random walk on G started at v and E v for the associated expectation operator.

Embeddings of planar graphs
Let G = (V, E) be a connected graph. For each edge e, choose an orientation of e arbitrarily and let I(e) be an isometric copy of the interval [0, 1]. The metric space G = G(G) is defined to be the quotient of the union e I(e) ∪ V , where we identify the endpoints of I(e) with the vertices e − and e + respectively, and is equipped with the path metric. An embedding of G into a surface S is a continuous, injective map z : G → S. The embedding is proper if every compact subset of S intersects at most finitely many edges and vertices of z(G). A connected graph is planar if and only if it admits an embedding into R 2 . A plane graph is a connected planar graph G together with an embedding of G = G(G) into a surface S homeomorphic to R 2 ; It is a proper plane graph if G is locally finite and the embedding z is proper. A (proper) plane network is a (proper) locally finite plane graph together with an assignment of conductances c : E → (0, ∞). A proper, simple plane triangulation is a proper, simple plane network such that for each face f of the network (i.e., connected component of S \ z(G)), there exist three edges e 1 , e 2 , e 3 such that the boundary of f is equal to the union I(e 1 ) ∪ I(e 2 ) ∪ I(e 3 ).
If C is a circle packing of a proper simple plane triangulation T with carrier D, then we can define a proper embedding z of T into D by setting z(v) to be the centre of the disc the center of the disc of C corresponding to v for every vertex v of T , and embedding each edge of G as a straight line between the centers of the circles corresponding to its endpoints. The fact that this embedding is proper is an easy consequence of the definition of the carrier (and the implicit assumption that T is locally finite).
A set of vertices W ⊆ V is said to be absorbing if with positive probability the random walk X n n≥0 on G is contained in W for all n greater than some random N . A plane graph G is said to be uniquely absorbing if for every finite subgraph G 0 of G, there is exactly one connected component D of R 2 \ {z(I(e)) : e ∈ G 0 } such that the set of vertices {v ∈ V : z(v) ∈ D} is absorbing. (If G has locally finite dual, meaning that every face has finitely many edges incident to it, then it is uniquely absorbing if any only if G \ G 0 has exactly one absorbing connected component for every finite subgraph G 0 .)

Square tiling
Let G be a transient, uniquely absorbing plane network and let ρ be a vertex of G. For each v ∈ V let y(v) denote the probability that the random walk on G started at v never visits ρ, and let η := u∼ρ c(ρ, u)y(u).
Let R/ηZ be the circle of length η. Then there exists a set If e is such that y(e + ) < y(e − ), we define I(e) = I(−e) and S(e) = S(−e). In particular, the aspect ratio of S(e) is equal to the conductance c(e) for every edge e ∈ E.
2. The interiors of the rectangles S(e) are disjoint, and the union e S(e) is equal to R/ηZ × [0, 1).
3. For every vertex v ∈ V , the set I(v) = e − =v I(e) is an interval and is equal to e + =v I(e). 4. For almost every θ ∈ R/ηZ and for every t ∈ [0, 1), the line segment {θ} × [0, t] intersects only finitely many rectangles of S ρ .
Note that the rectangle corresponding to an edge through which no current flows is degenerate, consisting of a single point. The existence of the above tiling was proven by Benjamini and Schramm [5]. Their proof was stated for the case c ≡ 1 but extends immediately to our setting, see [9]. Let us also note the following property of the rectangle tiling, which follows from the construction given in [5]. For each v ∈ V , we let θ(v) be a point chosen arbitrarily from I(v).
Let G be a uniquely absorbing proper plane network. Benjamini and Schramm [5] proved that if G has bounded local geometry and X n n≥0 is a random walk on G started at ρ, then θ(X n ) converges to a point in R/ηZ and the law of the limit point is equal to the Lebesgue measure (their proof is given for bounded degree plane graphs but extends immediately to this setting). An observation of Georgakopoulos [9, Lemma 6.2] implies that, more generally, whenever G is such that θ(X n ) converges to a point in R/ηZ and length(I(X n )) converges to zero almost surely, the law of the limit point is equal to the Lebesgue measure.

The Poisson boundary
Let (V, P ) be a Markov chain. Harmonic functions on (V, P ) encode asymptotic behaviours of a trajectory X n n≥0 as follows. We call a sequence x i and let B denote the Borel σ-algebra for the product topology on Ω. Let I denote the invariant σ-algebra Assume that there exists a vertex ρ ∈ V from which all other vertices are reachable: which is always satisfied when P is the transition operator of the random walk on a connected network G. Then there exists an invertible linear transformation H between L ∞ (Ω, I, P ρ ) and the space of bounded harmonic functions on G defined as follows; see [6] and [16,Proposition 14.12]: where the above limit exists for P ρ -a.e. sequence x i i≥0 by the martingale convergence theorem. Proposition 3.1 (Path-hitting criterion for the Poisson boundary). Let (V, P ) be a locally finite Markov chain and let X n n≥0 be a trajectory of the Markov chain. Suppose that ψ : V → M is a function from V to a metric space M such that ψ(X n ) converges to a point in M almost surely. For each k ≥ 0, let Z k j j≥0 be a trajectory of the Markov chain started at X k that is conditionally independent of X n n≥0 given X k , and let P denote the joint distribution of X n n≥0 and each of the Z k m m≥0 . Suppose that almost surely, Then for every bounded harmonic function h on (V, P ), there exists a bounded Borel Note that the function sending X to lim n→∞ (X n ) is an almost sure limit of measurable functions and is therefore measurable. Note also that it suffices to define f on the support of the law of lim n→∞ ψ(X n ), which is contained in the set of accumulation points Proof of Proposition 3.1. Since indicator functions of Borel sets have dense linear span in L ∞ (Ω, I, P ρ ), and using the correspondence (3.1), it suffices to prove that for every invariant event A ∈ I there exists a Borel set B ⊆ M such that where denotes the symmetric difference.
To this end, let A ∈ I be an invariant event and let h be the harmonic function Lévy's 0-1 law implies that and so it suffices to exhibit a Borel set B ⊆ M such that We may assume that P ρ (A ) > 0, otherwise the claim is trivial.
For each n, let m(n) be the largest m such that n ≥ N (m), so that m(n) → ∞ as n → ∞.
for all but finitely many n almost surely.
Define a set B ⊆ M by To see that B is Borel, observe that, by an obvious compactness argument using the fact that (V, P ) is locally finite, B may be written as where B k,j,I is defined to be the closed set It follows from the discussion in the paragraph before last that lim n→∞ ψ(X n ) ∈ B almost surely on the event that h(X n ) converges to 1: Simply take v i i≥0 = X i i≥0 as the required path. In particular, the event {lim n→∞ ψ(X n ) ∈ B} has positive probability. We now prove conversely that lim inf n→∞ h(X n ) > 0 almost surely on the event that lim n→∞ ψ(X n ) ∈ B. Condition on this event, so that there exists a path v i i≥0 in G starting at ρ such that lim i→∞ ψ(v i ) = lim n→∞ ψ(X n ) and inf i≥0 h(v i ) > 0. Fix one such path. Applying the optional stopping theorem to h(Z k m ) m≥0 , we have and so, by our assumption (3.2), we have that is positive almost surely.
We remark that controlling the rate of convergence of the path in the definition of B can be avoided by invoking the theory of universally measurable sets.

Proof of Theorem 1.1
Proof of Theorem 1.1. Let X n n≥0 and Y n n≥0 be independent random walks on G started at ρ, and for each k ≥ 0 let Z k j j≥0 be a random walk on G started at X k that is conditionally independent of X n n≥0 and Y n≥0 given X k . Let P denote the joint distribution of X n n≥0 , Y n n≥0 and all of the random walks Z k m m≥0 . Given two points θ 1 , θ 2 ∈ R/ηZ, we denote by (θ 1 , θ 2 ) ⊂ R/ηZ the open arc between θ 1 and θ 2 in the counter-clockwise direction. For each such interval (θ 1 , θ 2 ) ∈ R/ηZ, let be the probability that a random walk started at v converges to a point in the interval (θ 1 , θ 2 ).
Since the law of lim n→∞ θ(X n ) is equal to the Lebesgue measure and hence nonatomic, the two random variables θ + := lim n→∞ θ(X n ) and θ − := lim n→∞ θ(Y n ) are Figure 2: Illustration of the proof. Conditioned on the random walk X n n≥0 , there exists a random ε > 0 such that almost surely, for infinitely many k, a new random walk Z k m m≥0 (red) started from X k has probability at least ε of hitting the path v i i≥0 (blue).
almost surely distinct. We can therefore write R/ηZ \ {θ + , θ − } as the union of the two disjoint non-empty intervals R/ηZ be the probability that a random walk started at X k , that is conditionally independent of X n n≥0 and Y n n≥0 given X k , converges to a point in the interval (θ − , θ + ). We claim that the random variable Q k is uniformly distributed on [0, 1] conditional on X n k n=0 and Y n n≥0 . Indeed, since the law of θ + given X k is non-atomic, for each s ∈ [0, 1] there exists 1 θ s = θ s (X k , θ − ) ∈ R/ηZ such that The claim follows by observing that As a consequence, Fatou's lemma implies that for every ε > 0,  Let v i i≥0 be a path in G started at ρ such that lim i→∞ θ(v i ) = θ + and the height y(v i ) tends to 1. Observe (Figure 1) that for every k ≥ 0, the union of the traces {v i : i ≥ 0} ∪ {Y n : n ≥ 0} either contains X k or disconnects X k from at least one of the two intervals (θ − , θ + ) or (θ + , θ − ). That is, for at least one of the intervals (θ − , θ + ) or (θ + , θ − ), any path in G started at X k whose image under θ converges to this interval must intersect {v i : i ≥ 0} ∪ {Y n : n ≥ 0}. It follows that P Z k m m≥0 hits {v i : i ≥ 0} ∪ {Y n : n ≥ 0} X n n≥0 , Y n n≥0 ≥ min{Q k , 1 − Q k }.

Proof of Theorem 1.2
In this subsection we prove Theorem 1.2. We begin with the following simple lemma, which is implicit in [19]. Recall that the Dirichlet energy of a function f : Lemma 3.2. Let φ : V → V be a rough isometry from a network G with bounded local geometry to a network G with bounded local geometry. Then there exists a constant C such that for all functions f : V → R.
Proof. Let α and β be constants such that φ is an (α, β)-rough isometry. For each pair of adjacent vertices u and v in G, we have that d (φ(u), φ(v)) ≤ α + β, and so there exists a path in G from φ(u) to φ(v) of length at most α + β. Fix one such path Φ(u, v) for each u and v. Since there at most α + β edges of G in each path Φ(u, v), and the conductances of G are bounded, there exists a constant C 1 such that Let e be an edge of G . If e 1 and e 2 are two edges of G such that Φ(e − 1 , e + 1 ) and Φ(e − 1 , e + 1 ) both contain e , then d (φ(e − 1 ), φ(e − 2 )) ≤ 2(α + β) and so Thus, the set of oriented edges e of G such that Φ(e − , e + ) contains e is contained a ball of radius α(2α + 3β) in G. Since G has bounded degrees, the number of edges contained in such a ball is bounded by some constant C 2 . Combining this with the assumption that the conductances of G are bounded below by some constant C 3 , we have that Now suppose that G = (V, E) is a transient network with bounded local geometry such that there exists a proper plane graph G = (V , G ) with bounded degrees and a rough isometry φ : V → V . Then G is also transient [16, §2.6]. We may assume that G is simple, since φ remains a rough isometry if we modify G by deleting all self-loops and identifying all multiple edges between each pair of vertices. In this case, there exists a simple, bounded degree, proper plane triangulation T containing G as a subgraph [4,Lemma 4.3]. The triangulation T is transient by Rayleigh's monotonicty principle, and since it has bounded degrees it is CP hyperbolic by the He-Schramm Theorem [13]. Let C be a circle packing of T in the disc D and let z be the embedding of G defined by this circle packing, so that for each v ∈ V , z(v ) is the center of the circle of C corresponding to v , and each edge of G is embedded as a straight line between the centers of the circles corresponding to its endpoints.
The proofs of Lemmas 3.3 and 3.4 below are adapted from [2]. Lemma 3.3. Let X n n≥0 denote the random walk on G. Then z • φ(X n ) converges to a point in ∂D almost surely.
Proof. Ancona, Lyons and Peres [1] proved that for every function f of finite Dirichlet energy on a transient network G, the sequence f (X n ) converges almost surely as n → ∞. Thus, it suffices to prove that each coordinate of z •φ has finite Dirichlet energy. Applying the inequality (3.6), it suffices to prove that each coordinate of z has finite energy. For each vertex v of G , let r(v ) denote the radius of the circle corresponding to v in C. Then, letting z = (z 1 , z 2 ), 2 ≤ π is the total area of all the circles in the packing.
Lemma 3.4. The law of lim n→∞ z • φ(X n ) does not have any atoms.
The proof will require the following well-known inequality [16, Exercise 2.36]: If G is a finite network, v is a vertex of G, and A and B are two sets of vertices of G, then .
Proof of Lemma 3.4. Let B k (ρ) be the set of vertices of G at graph distance at most k from ρ, and let G k be the subnetwork of G induced by B k (ρ) (i.e. the induced subgraph together with the conductances inherited from G). Recall that the free effective conductance between a set A and a set B in an infinite graph G is given by The same variational formula also defines the effective conductance between two sets A and B in a finite network G, denoted C eff (A ↔ B; G). A related quantity is the effective conductance from a vertex v to infinity in G for each two sets of vertices A and B in G. Let ρ ∈ V and ξ ∈ ∂D be fixed, and let In [2,Corollary 5.2], it is proven that Applying the inequality (3.9) and Rayleigh's monotonicity principle, we have that Applying the inequality (3.8) yields that Applying the exhaustion characterisation of the free effective conductance [16, §9.1], which states that (3.11)

Boundaries of planar graphs
Combining (3.10) and (3.11) we deduce that Remark 3.5. The statement that P(hit A ε (ξ)) converges to zero as ε tends to zero for every ξ ∈ ∂D can also be used to deduce convergence of z • φ(X n ) without appealing to the results of [1]: Suppose for contradiction that with positive probability z • φ(X n ) does not converge, so that with positive probability there exist two (random) boundary points ξ 1 , ξ 2 ∈ ∂D in the closure of {z • φ(X n ) : n ≥ 0}. It is not hard to see that in this case the closure of {z • φ(X n ) : n ≥ 0} must contain at least one of the boundary intervals [ξ 1 , ξ 2 ] or [ξ 2 , ξ 1 ]. However, if the closure of {z • φ(X n ) : n ≥ 0} contains an interval of positive length with positive probability, then there exists a point ξ ∈ ∂D that is contained in the closure of {z • φ(X n ) : n ≥ 0} with positive probability. This contradicts the convergence of P(hit A ε (ξ)) to zero.
Proof of Theorem 1.2. The part of the theorem concerning the existence of an embedding z of G in D such that z • φ(X n ) converges to a point in ∂D almost surely and that the law of the limit is non-atomic follows immediately from Lemmas 3.3 and 3.4. Fix one such embedding z. Let X n n≥0 and Y n n≥0 be independent random walks on G started at ρ, and for each k ≥ 0 let Z k j j≥0 be a random walk on G started at X k that is conditionally independent of X n n≥0 and Y n≥0 given X k . Let P denote the joint distribution of X n n≥0 , Y n n≥0 and all of the random walks Z k m m≥0 .
Since the law of lim n→∞ z•φ(X n ) is non-atomic, the random variables ξ + = lim n→∞ z• φ(X n ) and ξ − = lim n→∞ z • φ(Y n ) are almost surely distinct. Let Using the non-atomicity of the law of lim n→∞ z • φ(X n ), the same argument as in the proof of Theorem 1.1 also shows that lim sup k→∞ min{Q k , 1 − Q k } > 0 almost surely. (3.12) We now come to a part of the proof that requires more substantial modification. Let v i i≥0 be a path in G starting at ρ such that z • φ(v i ) → ξ + . Let Φ be as in the proof of Lemma 3.2. Given a path u i i≥0 in G, let Φ( u i i≥0 ) denote the path in G formed by concatenating the paths Φ(u i , u i+1 ) for i ≥ 0. For each k ≥ 0, let τ k be the first time t such that the path Φ(Z k t−1 , Z k t ) intersects the union of the traces of the paths Φ( v i i≥0 ) and Φ( Y n n≥0 ). Observe that for every k ≥ 0, the union of the traces of the paths Φ( v i i≥0 ) and Φ( Y n n≥0 ) either contains φ(X k ) or disconnects φ(X k ) from at least one of the two intervals (ξ − , ξ + ) or (ξ + , ξ − ), meaning that any path in G starting at φ(X k ) whose image under z converges to this interval must intersect one of the paths Φ( v i i≥0 ) or Φ( Y n n≥0 ). Thus, we have that P τ k < ∞ X n n≥0 , Y n n≥0 ≥ min{Q k , 1 − Q k } and hence that lim sup k→∞ P τ k < ∞ X n n≥0 , Y n n≥0 > 0 (3.13) almost surely by (3.12).

Identification of the Martin boundary
In this section, we prove Theorem 1.5 and deduce Theorem 1.4. We begin by proving that the rectangle tiling of a bounded degree triangulation with edge conductances bounded above and below does not have any accumulation points of rectangles other than at the boundary circle R/ηZ × {1}. Proposition 4.1. Let T be a transient, simple, proper plane triangulation with bounded local geometry. Then for every vertex v of T and every ε > 0, there exist at most finitely many vertices u of T such that the probability that a random walk started at u visits v is greater than ε.
Proof. Let C be a circle packing of T in the unit disc D, with associated proper embedding z of T into D. Benjamini and Schramm [4,Lemma 5.3] proved that for every ε > 0 and κ > 0, there exists δ > 0 such that for any v ∈Ṽ with |z(v)| ≥ 1 − δ, the probability that a random walk from v ever visits a vertex u such that |z(u)| ≤ 1 − κ is at most ε. (Their proof is given for c ≡ 1 but extends immediately to this setting.) By setting κ = 1 − |z(v)|, it follows that for every ε > 0, there exists δ > 0 such that for every vertex u with |z(u)| ≥ 1 − δ, the probability that a random walk started at u hits v is at most ε. The claim follows since |z(u)| ≥ 1 − δ for all but finitely many vertices u of T .
Since the height y(v) of each vertex v in the rectangle tiling is equal to the probability that the random walk started at v never visits ρ, we obtain the following immediate corollary.

Corollary 4.2.
Let T be a transient proper plane triangulation with bounded local geometry, and let S ρ be a rectangle tiling of T . Then for every t ∈ [0, 1), the cylinder R/ηZ × [0, t] intersects only finitely many rectangles S(e) ∈ S ρ .
Proof. Define a sequence of vertices v n n≥0 as follows. If the interval I(v n ) is nondegenerate (i.e. has positive length), let v n = v n . Otherwise, let η j j≥0 be a path from v n to ρ in T , let j(n) be the smallest j > 0 such that the interval I(η j(n) ) is nondegenerate, and set v n = η j(n) . Observe that y(v n ) = y(v n ), that the interval I(v n ) contains the singleton I(v n ), and that the pathη n = η n j j(n) j=0 from v n to v n in T satisfies (I(η j ), y(η j )) = (I(v n ), y(v n )) for all j < j(n).
Let θ (v n ) ∈ I(v n ) be chosen so that the line segment n between the points (θ (v n ), y(v n )) and (θ (v n+1 ), y(v n+1 )) in the cylinder R/ηZ × [0, 1] does not intersect any degenerate rectangles of S ρ or a corner of any rectangle of S ρ (this is a.s. the case if θ (v n ) is chosen uniformly from I(v n ) for each n). By Corollary 4.2, the line segment n intersects some finite sequence of non-degenerate rectangles of S ρ , corresponding to a finite sequence of edges e n 1 , . . . , e n l(n) of G. Let e n i be oriented so that y(e n+ i ) > y(e n− i ) for every i and n.  Let w be a vertex visited by the pathη n • γ n • (−η n+1 ). By construction, w has an edge emanating from it such that the associated rectangle intersects the line segment n , and consequently a neighbouring vertex w such that y(w ) ≥ min{y(v n ), y(v n+1 )}.
We also have the following similar lemma for circle packings.

Lemma 4.4.
Let T be a CP hyperbolic plane triangulation, and let C be a circle packing of T in D with associated embedding z. Then for every sequence v n n≥0 such that z(v n ) converges as n → ∞, there exists a path γ n n≥0 in T containing {v n : n ≥ 0} such that z(γ n ) converges as n → ∞.
Proof. The proof is similar to that of Lemma 4.3, and we provide only a sketch. Draw a straight line segment in D between the centres of the circles corresponding to each consecutive pair of vertices v n and v n+1 . The set of circles intersected by the line segment contains a path γ n in T from v n to v n+1 . (If this line segment is not tangent to any of the circles of C, then the set of circles intersected by the segment is exactly a path in T .) The path γ is defined by concatenating the paths γ n . For every ε > 0, there are at most finitely many v for which the radius of the circle corresponding to v is greater than ε, since the sum of the squared radii of all the circles in the packing is at most 1. Thus, for large n, all the circles corresponding to vertices used by the path γ n are small. The circles that are intersected by the line segment between z(v n ) and z(v n+1 ) therefore necessarily have centers close to ξ 0 for large n. We deduce that z(γ i ) → ξ 0 as i → ∞.
Proof of Theorem 1.5. Let X n n≥0 be a random walk on T . Our assumptions guarantee that θ(X n ) and z(X n ) both converge almost surely as n → ∞ and that the laws of these limits are both non-atomic and have support R/ηZ and ∂D respectively. Suppose that γ is a path in T that visits each vertex at most finitely often. We claim that θ(γ i ) converges if and only if z(γ i ) converges. To prove this, it suffices to show that both forms of convergence are equivalent to the property that the random walk X n almost surely does not hit γ i infinitely often.
We now prove that θ(γ i ) converges if and only if X n almost surely does not hit γ i infinitely often. The proof of the corresponding statement for z(γ i ) is similar. If θ(γ i ) converges, then X n almost surely does not hit γ i infinitely often, since otherwise lim i→∞ θ(γ i ) would be an atom in the law of lim n→∞ θ(X n ). Conversely, if θ(γ i ) does not converge, then there exist at least two distinct points θ 1 , θ 2 ∈ R/ηZ such that (θ 1 , 1) and (θ 2 , 1) are in the closure of {(θ(γ i ), y(γ i )) : n ≥ 0}. Let Y n n≥0 be a random walks started at ρ independent of X n n≥0 . Since the law of lim n→∞ θ(X n ) has full support, we have with positive probability that both lim n→∞ θ(X n ) ∈ (θ 1 , θ 2 ) and lim n→∞ θ(Y n ) ∈ (θ 2 , θ 1 ). On this event, the union of the traces {X n : n ≥ 0} ∪ {Y n : n ≥ 0} disconnects (θ 1 , 1) from (θ 2 , 1), and consequently the path γ i n≥0 must hit {X n : n ≥ 0} ∪ {Y n : n ≥ 0} infinitely often. By symmetry, there is a positive probability that X n n≥0 hits the trace {γ i : n ≥ 0} infinitely often as claimed.
It follows from the above claim together with Lemmas 4.3 and 4.4 that for any sequence of vertices v i i≥0 in T that includes each vertex of T at most finitely often, the sequence θ(v i ) converges if and only if z(v i ) converges, and hence that the map θ : ξ −→ θ(ξ) := lim n→∞ θ(v n ) where v n n≥0 is a sequence of vertices such that z(v n ) → ξ is well defined and bijective. To see that this map is a homeomorphism, suppose that ξ n is a sequence of points in ∂D converging to a point ξ ∈ ∂D. For every n, there exists a vertex v n ∈ V such that |ξ n − z(v n )| ≤ 1/n and |θ(ξ n ) − θ(v n )| ≤ 1/n. Thus, z(v n ) → ξ and