PERCOLATION CRITICAL PROBABILITIES OF MATCHING LATTICE-PAIRS

. A necessary and sufficient condition is established for the strict inequality p c ( G ∗ ) < p c ( G ) between the critical probabilities of site percolation on a one-ended, quasi-transitive, plane graph G and on its matching graph G ∗ . When G is transitive, strict inequality holds if and only if G is not a triangulation. The basic approach is the standard method of enhancements, but its implemen-tion has complexity arising from the non-Euclidean (hyperbolic) space, the study of site (rather than bond) percolation, and the generality of the assumption of quasi-transitivity. Thisresultiscomplementary to the work of the authors (“Hyperbolic site per-colation”, arXiv:2203.00981 ) on the equality p u ( G )+ p c ( G ∗ ) = 1, where p u is the critical probability for the existence of a unique infinite open cluster. It implies for transitive, one-ended G that p u ( G ) + p c ( G ) ≥ 1, with equality if and only if G is a triangulation.


Strict inequalities for percolation probabilities
It is fundamental to the percolation model on a graph G that there exists a 'critical probability' p c (G) marking the onset of infinite open clusters.Two questions arise immediately.
(a) What can be said about the value of p c (G)? (b) For what values of the percolation density p is there a unique infinite cluster?These questions have attracted a great deal of attention since percolation was introduced by Broadbent and Hammersley [7] in 1957.They turn out to be more tractable when G is planar.
Amongst exact calculations of p c (G), those for bond percolation on the square, triangular, and hexagonal lattices have been especially influential (see [16,23], and also the book [11]).Earlier discussion (falling short of rigorous proof) of these values was provided by Sykes and Essam [22] in 1964.The last paper includes also an account of site percolation on the triangular lattice, and a discussion of site percolation on a so-called 'matching pair' of planar lattices.This term is explained in the companion paper [13]; the current work is concerned with the matching pair (G, G * ), where the so-called matching graph G * is defined as follows.
Let G = (V, E) be a planar graph, embedded in the plane R 2 in such way that two edges may intersect only at their endpoints.A face of G is a connected component of R 2 \ E. The boundary of a bounded face F is comprised of edges of G.The matching graph of G, denoted G * , is obtained from G by adding all diagonals to all faces.See Figure 1.1.Evidently, G * = G when G is a triangulation.A graph with connectivity 1 or 2 may have a multiplicity of non-homeomorphic planar embeddings, and therefore there is potential ambiguity over the definition of its matching and dual graphs (see Theorem 2.1(c)).
Remark 1.1.A face F of the above graph G may be unbounded, in which case its boundary comprises infinitely many edges and vertices.Such F generates an infinite complete subgraph of G * , on which a percolation process is trivial.We shall usually assume that all faces are bounded.Since our graphs are assumed quasi-transitive, this is equivalent to assuming that G is one-ended.(See [14], [2, Prop.2.1].)For quasi-transitive graphs with two or infinitely many ends, see Remark 1.5.
Sykes and Essam presented motivation for the exact relationship (1.1) p site c (G) + p site c (G * ) = 1, and this has been verified in a number of cases when G is amenable (see [6,16]).Note that, since G is a subgraph of G * , it is trivial that (1.2) p site c (G * ) ≤ p site c (G).It is less trivial to prove strict inequality in (1.2) for non-triangulations, and indeed this sometimes fails to hold.
Suppose that G is planar, quasi-transitive, one-ended, and possibly non-amenable.If we are to embed G in a plane in an appropriate fashion, the plane in question may need to be hyperbolic rather than Euclidean.Site percolation in the hyperbolic plane is the subject of the recent paper [13], where it is proved, amongst other things, that (1.3) p site u (G) + p site c (G * ) = 1, where p site u is the critical probability for the existence of a unique infinite open cluster.When G is amenable, we have p site c (G) = p site u (G), in agreement with (1.1) (see [18,Chap. 7] for a discussion of critical points of quasi-transitive, amenable graphs).By (1.2), we have p site u (G) + p site c (G) ≥ 1, and it becomes desirable to know when strict inequality holds.(When G is non-amenable, it is proved in [5] that p site c (G) < p site u (G).)Let T (respectively, Q) be the set of all infinite, connected, locally finite, plane, 2connected, simple graphs that are in addition transitive (respectively, quasi-transitive).(It is explained in [13,Rem. 3.4] that the assumption of 2-connectedness is innocent in the context of site percolation.)A path (. . ., x −1 , x 0 , x 1 , . . . ) of G * is called nonself-touching if, for all i, j, two vertices x i and x j are adjacent if and only if |i−j| = 1.
Here is the main theorem of the current work, followed by a corollary.Theorem 1.2.Let G ∈ Q be one-ended.Then p site c (G * ) < p site c (G) if and only if G * contains some doubly-infinite, non-self-touching path that includes some diagonal of G.
Theorem 1.2 is proved in Section 5 using methods derived in Section 4. □ We turn to examples of Theorem 1.2 in action.Firstly, the condition of the theorem is satisfied by all transitive, one-ended non-triangulations G ∈ T , as in the next theorem.
Theorem 1.4.Let G ∈ T be one-ended but not a triangulation.Then G satisfes the condition of Theorem 1.2, and therefore p c (G * ) < p c (G).This is essentially the assertion of the forthcoming Theorem 3.1, which is proved in Section 6.2 by the so-called metric method.The inequality of Theorem 1.4 then holds by Theorem 1.2.
The situation for quasi-transitive graphs G is more complicated, and we have no useful necessary and sufficient condition for the inequality p c (G * ) < p c (G).Instead, we include in Section 3 a sufficient (but not necessary) condition.(Note added before publication: the quasi-transitive case is treated in [12].)Remark 1.5.The above results are subject to the assumption that G is one-ended.By [14] and [2, Prop.2.1], the number η of ends of G ∈ Q lies in the set {1, 2, ∞}.As in Remark 1.1, we have that p c (G * ) = 0 if η ̸ = 1.On the other hand, it is standard that p c (G) ≥ 1/(∆ − 1) where ∆ is the maximum vertex-degree of G.The inequality There follow some remarks about the proof of Theorem 1.2.The general approach of the proof is to use the method of enhancements, as introduced and developed in [1] (though there is earlier work of relevance, including [19]).While this approach is fairly standard, and the above result natural, the proof turns out to have substantial complexity arising from the generality of the assumptions on G, and the fact that we are studying site (rather than bond) percolation (see [3]); the proof is, in contrast, fairly immediate for the amenable, planar lattices mentioned above.
We remark that the version of (1.3) for bond percolation, namely was proved by Benjamini and Schramm [5,Thm 3.8] for one-ended, non-amenable, plane, transitive graphs.Here, G + denotes the dual graph of G. (The amenable case is standard.)The basic difference between the bond and site problems is the following.In the study of bond percolation, one is interested in open self-avoiding paths, whereas for site percolation we study open, non-self-touching paths -given an infinite path (. . ., x −1 , x 0 , x 1 , . . . ) such that, for some i + 1 < j, x i and x j are adjacent, the states of vertices x i+1 , . . ., x j−1 are independent of the event that the path contains an infinite, open sub-path.That is, one can cut out the loop.The central idea of the proof of Theorem 1.2 is as follows.Suppose G satisfies the given assumptions, and write π for the given doubly-infinite path containing the diagonal d.In order to apply the enhancement method, one needs to show that, if z is a pivotal vertex for the existence of a long (but finite) open path of G * between given regions A, B of space, after making local changes to the configuration one may find a pivotal diagonal near z.This is achieved by a surgery of paths.First, one cuts a finite subpath π ′ from π containing the diagonal d.Then one inserts a translate of π ′ into an open path ν from A to B in which z is pivotal.Such insertion requires 'adjustments' near the interfaces of these two paths, and it must be achieved without sacrifice of the non-self-touching property.It is an impediment to this surgery that G * is non-planar (unless G is a triangulation), and thus one works instead with a graph, denoted G, that is obtained from G by placing a new vertex within each non-triangular face of G and joining this new vertex to each vertex of the face.
Turning to the contents of the current article, after the introductory Section 2, we explain the relevance of Theorem 1.2 to transitive and quasi-transitive graphs in Section 3. The proofs begin with some preliminary observations in Section 4, and the main theorem is proved in Section 5.The claim of Section 3 for quasi-transitive graphs is proved in Section 6.

Notation and basic properties
2.1.Graph embeddings.We shall assume familiarity with basic graph theory and its notation, and refer the reader to [13] for relevant definitions.Let Q be given as prior to Theorem 1.2, and let T be the subset of Q comprising the transitive graphs.
An embedding of a graph G = (V, E) (with underling 1-complex denoted |G|) in a surface S is a continuous map ϕ : |G| → S such that the induced map |G| → ϕ(|G|) is a homeomorphism.An embedding ϕ is called cellular if S \ϕ(G) is a disjoint union of spaces homeomorphic to an open disc.(See [20] and [21,Sect. 3.2].) We are concerned here with embeddings of planar graphs in either the Euclidean or hyperbolic planes, and we shall use H to denote either of these as appropriate for the setting.A useful summary of hyperbolic geometry may be found in [8] (see also [15]).An embedding of a graph G in H is called proper if every compact subset of H contains only finitely many vertices of G and intersects only finitely many edges.Henceforth, all embeddings will be assumed to be proper.
An Archimedean tiling (or uniform tiling) of a two-dimensional Riemannian manifold is a tiling by regular polygons such that its isometry group (of the tiling) acts transitively on its vertex-set.The edges of the tiling are geodesics.A discussion of amenability may be found in [18,Sect. 6  We give a formal definition of the matching graph of a planar graph G = (V, E).Firstly, one embeds G in the plane in such a way that two edges intersect only at their endpoints; such an embedded graph is called a plane graph.A face of a plane graph G is a connected component of H \ E. In this work we shall treat only one-ended graphs, for which all faces G are bounded with (topological) boundaries ∂F comprised of finitely many edges; the size of F is the number of edges in its boundary.A cycle C of a simple graph G = (V, E) is a sequence v 0 , v 1 , . . ., v n+1 = v 0 of vertices v i such that n ≥ 3, e i := ⟨v i , v i+1 ⟩ satisfies e i ∈ E for i = 0, 1, . . ., n, and v 0 , v 1 , . . ., v n are distinct.Let G be a plane graph, duly embedded in the Euclidean or hyperbolic plane.In this case we write C • for the bounded component of H \ C, and Let V (∂F ) be the set of vertices lying along the boundary of the face F .For each face F and each non-adjacent pair x, y ∈ V (∂F ), we add an edge inside F between x and y.We write G * = (V, E * ) for the ensuing matching graph of G.Note that G * depends on the particular embedding of G.If G is 3-connected then, by Theorem 2.1(b), it has a unique embedding up to homeomorphism.If G is 2-connected but not 3-connected, we need to be definite about the choice of embedding, and we require it henceforth to be 'canonical' in the sense of Theorem 2.1(c).

Further notation.
A plane graph G is called a triangulation it every face is bounded by a 3-cycle.The automorphism group of the graph G = (V, E) is denoted Aut(G).The orbit of v ∈ V is written Aut(G)v, and we let and d G denotes graph-distance in G.We write u ∼ v if u, v ∈ V are adjacent, which is to say that d G (u, v) = 1.For any G, we fix some vertex denoted v 0 .
We shall work with one-ended graphs G ∈ Q.Since G is assumed one-ended and 2-connected, all its faces are bounded, with boundaries which are cycles of G (see Remark 2.2(d)).
Non-self-touching paths and cycles arise naturally when studying site percolation (such paths were called stiff in [1], and self-repelling in [11, p. 66]).
We shall consider non-self-touching paths in two graphs derived from a given G ∈ Q, namely its matching graph G * , and the graph G obtained by adding a site within each face F of size 4 or more, and connecting every vertex of F to this new site.The graph G * may possess parallel edges.The property of being non-selftouching is indifferent to the existence of parallel edges, since it is given in terms of the vertex-set of π and the adjacency relation of H.
Here is the fundamental property of graphs that implies strict inequality of critical points.This turns out to be equivalent to a more technical 'local' property, as described in Section 4.2; see Theorem 4.8.As a shorthand, henceforth we abbreviate 'doubly-infinite non-self-touching path' to '2∞-nst path'.Let p ∈ [0, 1].We endow Ω with the product measure P p with density p.For v ∈ V , let θ v (p) be the probability that v lies in an infinite open cluster.It is standard that there exists p c (G) ∈ (0, 1] such that and p c (G) is called the critical probability of G.
For background and notation concerning percolation theory, the reader is referred to the book [11], the article [13], and to Section 5.

Two criteria for property Π
In this section we present the 'metric criterion' for a one-ended graph G ∈ Q to have the property Π of Definition 2.4.This criterion is valid for one-ended, nontriangulations G ∈ T , and thus we arrive in particular at the following.
Theorem 3.1.Let G ∈ T be one-ended but not a triangulation.Then G has property Π.
The criterion holds for a certain class of quasi-transitive graphs, and the outcome is a sufficient but not necessary condition for a quasi-transitive graph G ∈ Q to have property Π, namely Theorems 3.4.
The embedding results of Section 2 may be used in proofs of the existence of 2∞nst paths in one-ended graphs G ∈ Q satisfying the following forthcoming metric criterion.First, recall the relevant embedding property.By Theorem 2.1(a, c), every quasi-transitive, one-ended G ∈ Q has a canonical embedding in H.
Throughout this section we shall work with the Poincaré disk model of hyperbolic geometry (also denoted H), and we denote by ρ the corresponding hyperbolic metric.For definiteness, we consider only graphs G embedded in the hyperbolic plane; the Euclidean case is similar, subject to the simplification that the geometry of the space is Euclidean rather than hyperbolic.
Let G ∈ Q be one-ended and not a triangulation.By 2-connectedness and Remark 2.2(d), the faces of G are bounded by cycles.As before, we restrict ourselves to the case when G is non-amenable, and we embed G canonically in the Poincaré disk H.The edges of G are hyperbolic geodesics, but its diagonals are not generally so.The hyperbolic length of an edge e ∈ E * \ E does not generally equal the hyperbolic distance between its endvertices, denoted ρ(e).
For e ∈ E * , let Γ e denote the doubly-infinite hyperbolic geodesic of H passing though the endvertices of e, and denote by π ρ(e) ≥ ρ(π e (x), π e (y)), for all f = ⟨x, y⟩ ∈ E.
The graph G is said to satisfy the metric criterion if G has a canonical embedding in H for which some diagonal d ∈ E * \ E is maximal.There always exists some maximal edge of E * , but it is not generally unique, and it may not be a diagonal.The following lemma is proved in the same manner as the forthcoming Lemma 6.1.
Here is the main theorem for quasi-transitive graphs using the metric method.Remark 3.5.The condition of Theorem 3.4 is sufficient but not necessary, as indicated by the following example.Let G be the canonical tiling of R 2 illustrated in Figure 3.1.By inspection, no diagonal is maximal, whereas G has property Π.The sufficient condition in question can be weakened as explained in Remark 6.4, and the above example satisfes the weaker condition.

Some observations
4.1.Oxbow-removal.We begin by describing a technique of loop-removal (henceforth referred to as 'oxbow-removal').Let H be a simple graph embedded in the Euclidean/hyperbolic plane H (possibly with crossings).(a) Let C be a plane cycle of H that surrounds a point x / ∈ H.There exists a nonempty subset C ′ of the vertex-set of C that forms a plane, non-self-touching cycle of H and surrounds x.(b) Let π be a finite (respectively, infinite) path with endpoint v.There exists a non-empty subset π ′ of the vertex-set of π that forms a finite (respectively, infinite) non-self-touching path of H starting at v. If π is finite, then π ′ can be chosen with the same endpoints as π.
Proof.(a) Let C = (v 0 , v 1 , . . ., v n , v n+1 = v 0 ) be a plane cycle of H that surrounds x / ∈ H; we shall apply an iterative process of 'loop-removal' to C, and may assume n ≥ 4. We start at v 0 and move around C in increasing order of vertex-index.Let J be the least j ≤ n such that there exists i ∈ {1, 2, . . ., j − 2} with v i ∼ v J , and let I be the earliest such i.Consider the two cycles (These cycles are called oxbows since they arise through cutting across a bottleneck of the original cycle C.) Since C surrounds x, so does at least one of C ′ and C ′′ , and we suppose for concreteness that C ′′ surrounds x.We replace C by C ′′ .This process is iterated until no such oxbows remain.
(b) This part is proved by a similar argument.When the endpoints v 0 , v n of π are not neighbours, we use oxbow-removal as above; otherwise, we set π ′ = (v 0 , v n ).□ Remark 4.2.Lemma 4.1 will be used in the following context.Firstly, one may apply oxbow-removal to certain paths of a planar graph in order to obtain a non-selftouching subpath (see the forthcoming Lemma 4.3).Similarly, oxbow-removal may sometimes be used to generate a non-self-touching subpath of a concatenation of two non-self-touching paths.
Path-surgery will be used in the forthcoming proofs: that is, the replacement of certain paths by others.Consider a one-ended G ∈ Q, embedded canonically in the hyperbolic plane H, which for concreteness we consider here in the Poincaré disk model (see [8]), also denoted H.By Theorem 2.1(c), every automorphism of G extends to an isometry of H. Let F be the set of faces of G.For F ∈ F and x, y ∈ V (∂F ), let L x,y be the set of rectifiable curves with endpoints x, y whose interiors are subsets of F • \ E, and write l x,y for the infimum of the hyperbolic lengths of all l ∈ L x,y .Let for any arc γ that crosses L δ laterally and intersects no vertex of G, the number of intersections between γ and π, if finite, is odd.
A more refined result may be found in Section 6.
Proof.(a) Since all faces of G are bounded, there exist vertices of G in both components of H \ L δ .Now, L δ fails to be crossed in the long direction if and only if it contains some arc γ that traverses it laterally and that intersects no edge of G. To see the 'only if' statement, let V − and V + be the subsets of V ∩ L δ that are joined in G ∩ L δ to the two boundary points of L, respectively; if V − ∩ V + = ∅, then there exists such γ separating V + and V − in L δ .For this γ, there exists a face F and A square of the square lattice, its matching graph, and with its facial site added.points x, y ∈ V (∂F ), such that γ ⊆ λ for some λ ∈ L x,y .Let ϵ ∈ (0, 2δ − Φ), and find λ ′ ∈ L x,y with length not exceeding l x,y + ϵ.We may replace γ by some subarc γ ′ of λ ′ ∩ L δ .The length of γ ′ is no greater than Φ + ϵ < 2δ, a contradiction since L δ has width 2δ.Therefore, L δ contains some path π of G that crosses L δ in the long direction.
We apply oxbow-removal in G to π as described in the proof of Lemma 4.1.For any arc γ that crosses L δ laterally and intersects no vertex of G, the number of intersections between γ and π, if finite, decreases by a non-negative, even number whenever an oxbow is removed.It follows that the non-self-touching path π ′ (obtained after oxbow-removal) crosses L δ in the long direction.Proof.Let F be a face.The path π cannot contain three or more vertices of F , since that contradicts the non-self-touching property.Similarly, if π contains two such vertices, it must contain also the corresponding edge.If π is non-plane, it contains two or more diagonals of some face, which, by the above, cannot occur.□ As a device in the proof of Theorem 1.2, we shall work with the graph G obtained from G = (V, E) by adding a vertex at the centre of each face F , and adding an edge from every vertex in the boundary of F to this central vertex.These new vertices are called facial sites, or simply sites in order to distinguish them from the vertices of G.The facial site in the face F is denoted ϕ(F ).See [17, Sec.2.3], and also Figure 4.2.If ⟨v, w⟩ is a diagonal of G * , it lies in some face F , and we write ϕ(v, w) = ϕ(F ) for the corresponding facial site.
The main reason for working with G is that it serves to interpolate between G and G * in the sense of (5.2) below: we shall assign a parameter s ∈ [0, 1] to the facial sites in such a way that s = 0 corresponds to G and s = 1 to G * .It will also be useful that G is planar whereas G * is not.
Next, we specify some desirable properties of the graphs G * and G. Recall the property Π of Definition 2.4.Proof.Let G have property Π and let π be a 2∞-nst path of G * .For any two consecutive vertices u, v of Π such that δ(u, v) is a diagonal, we add between u and v the facial site ϕ(u, v).The result is a doubly-infinite path π ′ of G.By Lemma 4.4, ν ′ is non-self-touching in G, whence G has property Π. □ The properties of Definition 4.5 are 'global' in that they concern the existence of infinite paths.It is sometimes preferable to work in the proofs with finite paths, and to that end we introduce corresponding 'local' properties.
Let ζ(G) be as in Lemma 4.3(b).We shall make reference to the non-self-touching cycles σ r (v), σ * r (v) given in that lemma.We write σ r (v) for the non-self-touching cycle of G obtained from σ * r (v) by replacing any diagonal by a path of length 2 passing via the appropriate facial site of G.We abbreviate the closure of the region surrounded by σ * r (respectively, σ r ) to σ * r (respectively, σ r ).Let A(G) be the real number given as The graph G is said to have property Π A if there exists a vertex v ∈ V and a non-self-touching path π = (x 0 , x 1 , . . ., x n ) of G * such that (i) every vertex of π lies in σ * A (v), and x 0 , x n ∈ σ * A (v), (ii) there exists i such that The graph G is said to have property Π A if there exist vertices v, w ∈ V and a non-self-touching path π = (x 0 , x 1 , . . ., x n ) of G such that (i) every vertex of π lies in σ A (v), and 3. An illustration of the property Π A : a non-self-touching path of G * containing a diagonal near its middle.
(ii) there exists i such that That is to say, G has property Π A (respectively, Π A ) if G * (respectively, G) contains a finite, non-self-touching path of sufficient length that contains some diagonal (respectively, facial site).This definition is illustrated in Figure 4.3.Note that Π A+1 (respectively, Π A+1 ) implies Π A (respectively, Π A ) for sufficiently large A. The proof of this useful theorem utilises some methods of path-surgery that will be important later, and it is given next.

Proof of Theorem 4.8. (a) Let A > A(G)
. First, we prove that Π ⇔ Π A .Evidently, Π ⇒ Π A .Assume, conversely, that Π A holds for some A > A(G).Let the non-self-touching path π = (x 0 , x 1 , . . ., x n ) of G * , the vertex v = x i , and the diagonal d = ⟨v, x i+1 ⟩ be as in Definition 4.7(a); think of π as a directed path from x 0 to x n , and note by Lemma 4.4 that π is a plane graph.We abbreviate Let π 1 be the subpath of π from v to x 0 , and π 2 that from x i+1 to x n .Let a i be the earliest vertex/site of π i lying in ∂ − σ A .See the central circle of In the easiest case when D ≥ 2, one finds (green) nontouching subarcs σ i A of σ A to which v may be connected by non-selftouching paths.These subarcs may be connected to the boundary of H using subpaths of a doubly-infinite path constructed using Lemma 4.3(a).claim the following.

(4.3)
There exist two non-touching subpaths σ 1 , σ 2 of σ * A , each of length at least 1  2 |σ * A | − 4, such that: (i) for i = 1, 2, the subpath of π i leading to a i may be extended beyond a i along σ i to form a non-self-touching path ending at any prescribed y i ∈ σ i , and (ii) the composite path thus created (after oxbow-removal if necessary) is non-self-touching.
The proof of (4.3) follows.Let (4.4) As illustrated in the centre of Figure 4.4, we may find a non-touching pair of non-self-touching subpaths of σ * A such that the conclusion of (4.3) holds.Some oxbow-removal may be needed at the junctions of paths (see Remark 4.2).Suppose D = 1.We may picture σ * A as a (topological) circle with centre v, and for concreteness we assume that a 2 lies clockwise of a 1 around σ * A (a similar argument holds if not).See Figure 4.5.
A. Suppose the path π 1 , when continued beyond a 1 , passes at the next step to some b 1 ∈ A 1 , and add b 1 to obtain a path denoted π ′ 1 .Since D = 1, the next step of π 2 beyond a 2 is not into A 2 .On following π 2 further, it moves inside (σ * A ) • until it arrives at some point a we then include the subpath of π 2 between a 2 and b ′ 2 to obtain a path denoted π ′ 2 .We declare σ 1 to be the subpath of σ * A starting at b 1 and extending a total distance Let θ ∈ (0, 2π) be the angle subtended by the vector − − → a 2 a ′ 2 at the centre v.If θ < 3  4 π, say, each π ′ i may be extended along σ i to end at any prescribed y i ∈ σ i .Therefore, claim (4.3) holds in this case.
The situation can be more delicate if θ ≥ 3 4 π, since then a ′ 2 may be near to σ 1 .By the planarity of π, the region R between π ′ 2 and σ * A contains no point of π ′ 1 (R is the shaded region in Figure 4.5).We position a hyperbolic tube of width greater than Φ in such a way that it is crossed laterally by both π ′ 2 and the path σ 2 (as illustrated in Figure 4.5).By Lemma 4.3(a), this tube is crossed in the long direction by some path τ of G.The union of π ′ 2 and τ contains a non-self-touching path π ′′ 2 of G * from x i+1 to σ 2 (whose unique vertex in σ 2 is its second endpoint).Claim (4.3) follows in this situation.B. Suppose the hypothesis of part A does not hold, but instead π 2 passes from a 2 directly into σ * A .In this case we follow A above with π 1 and π 2 interchanged.C. Suppose neither π i passes from a i in one step into σ * A .We add b 2 to the subpath from x i+1 to a 2 , and continue as in part A above.
Suppose D = 0. Statement (4.3) holds by a similar argument to that above.
Having located the σ i of (4.3), we position a hyperbolic tube as in Figure 4.4, to deduce (after oxbow-removal, see Remark 4.2) the existence of a 2∞-nst path of G * that contains the diagonal d.Therefore, G has property Π, as required.
Hyperbolic tubes are superimposed on the graph at two steps of the argument above, and it is for this reason that we need A to be sufficiently large, say A > A ′ (G).
(b) It remains to show that Π ⇒ Π A for large A. By Lemma 4.6, Π ⇒ Π, and it is immediate that Π ⇒ Π A for large A.

Proof of Theorem 1.2
Consider site percolation on G with product measure P p , and fix some vertex v 0 of G.We write v ↔ w if there exists a path of G from v to w using only open sites (such a path is called open), and v ↔ ∞ if there exists an infinite, open path starting at v. The percolation probability is the function θ given by Remark 5.1.It is an old problem dating back to [4] to decide which graphs G satisfy p c (G) < 1, and there has been a series of related results since.It was proved in [9, Thm 1.3] that p c (G) < 1 for all quasi-transitive graphs G with super-linear growth (see also [10]).This class includes all G ∈ Q with either one or infinitely many ends (see [2,Sect. 1.4] and Theorem 2.1).Proposition 5.3.There exists A ′ (G) < ∞ such that the following holds.Suppose G ∈ Q is one-ended and has property Π A where A > A ′ (G).Let s ∈ (0, 1).There exists ϵ = ϵ(s) > 0 such that θ(p, s) > 0 for p c (G) − ϵ < p < p c (G).
We do not investigate the details of how A ′ (G) depends on G.An explicit lower bound on A ′ (G) may be obtained in terms of local properties of the embedding of G, but it is doubtful whether this will be useful in practice.
The rest of this proof is devoted to an outline of that of Proposition 5.3.Full details are not included, since they are very close to established arguments of [1], [11,Sect. 3.3], and elsewhere.
Let n be large, and later we shall let n → ∞.Consider site percolation on G with measure P p,s .We call a vertex (respectively, facial site) z pivotal if it is pivotal for the existence of an open path of G from v 0 to ∂Λ n (which is to say that such a path exists if z is open, and not otherwise).Let Pi n be the set of pivotal vertices, and Di n the set of pivotal facial sites.Proposition 5.3 follows in the 'usual way' (see [11,Sect. 3.3]) from the following statement.
By making changes to the configuration ω within the box Λ 4M (z) for some fixed M , (5.7) we construct a configuration in which Λ M (z) contains a pivotal facial site.
This implies (5.3) with f depending on the choice of z.Since Λ 4M (z) is finite and there are only finitely many types of vertex (by quasi-transitivity), f may be chosen to be independent of z.The above is achieved in five stages.
Assume for now that ω ∈ Ω and the pivotal vertex z satisfies For clarity of exposition, our illustrations are drawn as if G is embedded properly in the Euclidean rather than the hyperbolic plane.The principal effect of this is that hyperbolic tubes are represented as Euclidean rectangles.
Let G have property Π A .Let π = (x j ), v = x i , be as in Definition 4.7(b), and write ϕ The outline of the proof is as follows.
I. If there exist one or more open facial sites in Λ M (z), we declare them oneby-one to be closed.If at some point in this process, some facial site is found to be pivotal, then we have achieved (5.7), by changing ω within a bounded region.We may therefore assume that this never occurs, or equivalently that (5.9) ω has no open facial site in Λ M (z).
II. Find a non-self-touching open path ν in ω from v 0 to ∂Λ n .This path passes necessarily through the pivotal vertex z.III.By making changes within Λ 2M (z), we construct non-touching subpaths of ν from v 0 (respectively, ∂Λ n ) to ∂Λ M (z), that can be extended inside Λ M (z) in a manner to be specified at Stage V. This, and especially the following, stage resembles closely part of the proof in Section 4.3.IV.We splice a copy (denoted π ′ = απ) of π inside Λ A (v ′ ), and we make local changes to obtain paths π 1 , π 2 from the two endpoints of αϕ, respectively, to ∂Λ A (v ′ ) that can be extended outside Λ A (v ′ ) in a manner to be specified at Stage V. V. Between the contours ∂Λ A (v ′ ) and ∂Λ M (z), we arrange the configuration in such a way that the retained parts of ν hook up with the endpoints of the π i .
In the resulting configuration, the facial site ϕ ′ := αϕ is pivotal.
Some work is needed to ensure that ϕ ′ can be made pivotal in the final configuration.Lemma 4.3(b) will be used to traverse the annulus between the two contours at Stage V.In making connections at junctions of paths, we shall make use of the planarity of G. Rather than working with the boundaries of Λ M (z) and Λ A (v ′ ), we shall work instead with the non-self-touching cycles σ M := σ M (z) and Proof of Lemma 5.4.Stage I is first followed as stated above.
Stage II.By (5.6), we may find an open, non-self-touching path ν of G from v 0 to ∂Λ n , and we consider ν as thus directed.By (5.9), ν includes no facial site of Λ M (z).The path ν passes necessarily through z, and we let u (respectively, w) be the preceding (respectively, succeeding) vertex to z.
1.An illustration of the construction at Stages II/III.The non-self-touching path ν contains subpaths from v 0 to σ M , and from the latter set to ∂Λ n .The subpaths σ i M of σ M are indicated in green.
For y ∈ V , and the given configuration ω (satisfying (5.9)), let and write C y also for the corresponding induced subgraph of G.By (5.6), A. C u and C w are disjoint (and also non-touching), B. the subpath of ν, denoted ν(u−), from v 0 to u contains no facial site of Λ M (z), C. the subpath of ν, denoted ν(w+), from w to ∂Λ n contains no facial site of Λ M (z), D. the pair ν(z−), ν(z+) is non-touching.
Stage III.This is closely related to the proof of Theorem 4.8 given in Section 4.3.Note that the intersection of ν(u−) ∪ ν(w+) and Λ 2M (z) comprises a family of paths rather than two single paths.See Figure 5.1.
We follow ν(u−) towards u, and ν(w+) backwards towards w, until we reach the first vertices/sites, denoted a 1 , a 2 , respectively, lying in ∂ + σ M .Let ν 1 be the subpath of ν(u−) between v 0 and a 1 , and ν 2 that of ν(w+) between ∂Λ n and a 2 .We now change the states of certain vertices/sites x ∈ Λ 2M (z) by declaring (5.10) every 2. An illustration of the case D = 1 in the Stage III construction.There are two subcases, depending on whether θ > 0 (solid line) or θ < 0 (dashed line).The green lines indicate the subpaths σ i M in the subcase θ > 0. The rectangle is added in illustration of the hyperbolic tube used in the case θ ≥ 3 4 π.
We investigate next the subsets of σ M to which the a i may be connected within σ M .We shall show that: (5.11) there exist two non-touching subpaths σ 1 M , σ 2 M of σ M , each of length at least 1  2 | σ M | − 4, such that, for i = 1, 2: (i) a i has a neighbour b i ∈ σ i M , (ii) for y i ∈ σ i M , the path ν i may be extended from b i to y i along σ i M , thereby creating (after oxbow-removal if necessary) a non-self-touching path from the other endpoint of ν i , (iii) the composite path ν ′ i thus created is non-self-touching, and (iv) the pair ν ′ 1 , ν ′ 2 is non-touching.An explanation follows.Let (5.12) (5.11) follows as illustrated in Figure 5.1.Suppose D = 1.We may picture σ M as a circle with centre z, and for concreteness we assume that a 2 lies clockwise of a 1 around σ M (a similar argument holds if not) See Figure 5.2.
A. Suppose the path ν 1 , when continued along ν(z−) beyond a 1 , passes at the next step to some b 1 ∈ A 1 , and add b 1 to ν 1 (to obtain a path denoted ν ′ 1 ).Since D = 1, the next step of ν(w+) beyond a 2 is not to A 2 .On following ν(w+) further, it moves inside H\ σ M until it arrives at some point a 2 ) ≥ 2; we then add to ν 2 the subpath of ν(w+) between a 2 and b ′ 2 (to obtain an extended path ν ′ 2 ).Let θ(a ′ 2 ) be the angle subtended by the vector − − → a 2 a ′ 2 at the centre z, counted positive if ν(w+) passes clockwise around z of σ M , and negative if anticlockwise.
(i) There are two cases, depending on whether θ := θ(a ′ 2 ) is positive or negative.Assume first that θ > 0. If θ < 3  4 π, say, we declare σ 1 M to be the subpath of σ M starting at b 1 and extending a total distance 1  2 | σ M |−4 around σ M anticlockwise.We declare σ 2 M similarly to start at distance 2 clockwise of b 1 along σ M and to have the same length as σ 1 M .Each ν ′ i may be extended along σ i M to end at any prescribed y i ∈ σ i M .Therefore, claim (5.11) holds in this case.The situation can be more delicate if θ ≥ 3 4 π, since then a ′ 2 may be near to σ 1 M .By the planarity of ν, the region R between ν ′ 2 and σ M contains no point of ν ′ 1 (R is the shaded region in Figure 5.2).We position a hyperbolic tube of width greater than Φ in such a way that it is crossed laterally by both ν ′ 2 and the path σ 2 M given above.By Lemma 4.3(a), this tube is crossed in the long direction by some path τ of G.As illustrated in Figure 5.2, the union of ν ′ 2 and τ contains (after oxbow-removal) a non-self-touching path ν ′′ 2 from ∂Λ n to σ 2 M (whose unique vertex in σ 2 M is its second endpoint).We now declare each vertex/site of Λ 2M (z) \ ( σ M ) • to be open if and only if it lies in ν ′ 1 ∪ ν ′′ 2 .Claim (5.11) follows in this situation, with the σ i M given as above.(ii) Assume θ < 0, in which case there arises a complication in the above construction, as illustrated in Figure 5.3.In this case, there is a subpath L of ν ′ 2 from a 2 to a ′ 2 , that passes anticlockwise around v 0 , and ν ′ of G from ∂Λ n to a ′ 2 that does not contain a 2 (see Figure 5.3).We declare every x ∈ ν ′′ 2 open and every x ∈ ν ′ 2 \ ν ′′ 2 closed.The subpaths σ i M of σ M may now be defined as above.B. Suppose the hypothesis of part A does not hold, but instead ν 2 passes from a 2 into σ M .In this case we follow A with ν(u−) and ν(w+) interchanged.This case is slightly shorter than A since the above complication cannot occur.C. Suppose neither ν i passes from a i directly into σ M .We add b 2 to ν 2 and continue as in A above.
Suppose D = 0. Statement (5.11) holds by a similar argument to that of case (ii), Stage IV.We next pursue a similar strategy within Λ A (v ′ ).The argument is essentially that in proof of Theorem 4.8 given in Section 4.3, and the details of this are omitted here.See Stage V. Having located the subpaths σ i M of σ M , and the subpaths σ i A of σ A , we prove next that there exists j ∈ {1, 2}, and non-self-touching paths µ 1 , µ 2 , such that: (i) µ 1 , µ 2 is a non-touching pair, (ii) µ 1 has endpoints in σ 1 M and σ j A , and µ 2 has endpoints in σ 2 M and σ j ′ A , where j ′ ∈ {1, 2}, j ′ ̸ = j, and (iii) apart from their endpoints, µ 1 and µ 2 lie in ( σ M ) • \ σ A .This statement follows as in Figure 5.4 by positioning two hyperbolic tubes of width exceeding Φ, and appealing to Lemma 4.3(a).It may be necessary to remove some oxbows at the junctions of paths.
Hyperbolic tubes are superimposed on σ A above, and it is for this reason that A is assumed to be sufficiently large.
Having satisfied (5.7) subject to (5.8), we next explain how to remove the assumption (5.8).Let the pivotal vertex v satisfy v ∈ Λ 2M ; a similar argument applies if v ∈ Λ n \ Λ n−2M .Let π be an infinite, non-self-touching open path of G starting at v 0 , and declare closed every vertex of Λ 4M not lying in π. (Such a π exists by connectivity and oxbow-removal.)In the resulting configuration, every vertex/site in the subpath of π from ∂Λ 2M to ∂Λ 4M is pivotal.We pick one such vertex and apply the above arguments to obtain a pivotal facial site lying in Λ 4M .□ 6. Strict inequality using the metric method Lemma 6.1.For x, y ∈ H, we have ρ(π(x), π(y)) ≤ ρ(x, y).
Proof.We assume for simplicity that x and y are distinct and lie in the same connected component of H \ Γ; a similar proof holds if not.The points x, π(x), π(y), y form a quadrilateral with two consecutive right angles (see Figure 6.1).Let z be the orthogonal projection of x onto the geodesic containing y and π(y).The triple x, z, y forms a right-angled triangle, and the quadruple x, z, π(y), π(x) forms a Lambert quadrilateral.By the geometry of such shapes (see, for example, [15, Sect.III.5]), we have that ρ(x, y) ≥ ρ(x, z) ≥ ρ(π(x), π(y)).□ Let G = (V, E) ∈ T be one-ended but not a triangulation.We shall consider only the case when G is non-amenable, so that it is embedded as an Archimedean tiling in the Poincaré disk; the Euclidean case is similar.For an edge e of G * = (V, E * ), let ρ(e) denote the hyperbolic distance between its endvertices; since every e of G * (in its embedding) is a geodesic, ρ(e) equals the hyperbolic length of e.Since the embedding is Archimedean, every edge of G has the same hyperbolic length, and we may therefore assume for simplicity that (6.1) ρ(e) = 1, e ∈ E.
Each e ∈ E * is a sub-arc of a unique doubly-infinite geodesic, denoted Γ e , of H. Let r be the maximal number of edges in a face of G, and let F be a face of size r.Since F is a regular r-gon, by (6.1), F has some diagonal d satisfying  Proof.The first case arises when e, viewed as a geodesic, is perpendicular to Γ + d , and the second when it is not.See Figure 6.2.□ In proceeding along Γ + d , we make an ordered list (w i ) of vertices as follows.(a) Set w 0 = b.(b) Every time Γ d passes into the interior of a face F ′ , it exits either at a vertex v ′ or across the interior of some edge e ′ .In the first case we add v ′ to the list, and in the second, we add to the list an endvertex of e ′ with maximal p-value.(c) If Γ + d passes along an edge e ∈ E, we add both its endvertices to the list in the order in which they are encountered.
The following lemma is proved after the end of the current proof.Lemma 6.3.The infinite ordered list w = (w 0 , w 1 , . . . ) is a path of G * with the property that p(w i ) is strictly increasing in i.
The composite path ν obtained by following ν − towards a, then d, then ν + , fails to be non-self-touching in G * if and only if there exists s < 0 and t ≥ 0 with (s, t) ̸ = (−1, 0) such that e ′′ := ⟨ν s , ν t ⟩ ∈ E * .If the last were to occur, by (6.4)-(6. in contradiction of (6.3).Thus ν is the required non-self-touching path.The above may be regarded as a more refined version of part of Proposition 4.3.
Proof of Lemma 6.3.That w is a path of G * follows from its construction, and we turn to the second claim.Let m ≥ 0, and consider w 0 , w 1 , . . ., w m as having been identified.We claim that (6.6) p(w m ) < p(w m+1 ).
(a) Suppose w m ∈ Γ + d .(i) If Γ + d includes next an entire edge of the form ⟨w m , g⟩ ∈ E, then w m+1 = g and (6.6) holds.(ii) Suppose Γ + d enters next the interior of some face F ′ .If it exits F ′ at a vertex, then this vertex is w m+1 and (6.6) holds.Suppose it exits by crossing the interior of an edge e ′ .If w m is an endvertex of e ′ , then w m+1 is its other endvertex and (6.6) holds; if not, then w m+1 is an endvertex of e ′ with maximal p-value (recall Lemma 6.2).(b) Suppose w m is the endvertex of an edge e that is crossed (but not traversed in its entirety) by Γ + d , and let F ′ be the face thus entered.The next vertex w m+1 is given as in (a)(ii) above, and (6.6) holds.
The proof is complete.□ Finally in this section, we prove Lemma 3. where π denotes projection onto Γ.The last inequality holds by Lemma 6.1.Therefore, e is maximal.□ 6.3.The case of quasi-transitive graphs.Certain complexities arise in applying the techniques of Section 6.2 to quasi-transitive graphs.In contrast to transitive graphs, the faces are not generally regular polygons, and the longest edge need not be a diagonal.Let G ∈ Q be one-ended and not a triangulation.As before, we restrict ourselves to the case when G is non-amenable, and we embed G canonically in the Poincaré disk H.The edges of G are hyperbolic geodesics, but its diagonals need not be so.The hyperbolic length of an edge e ∈ E * \ E does not generally equal the hyperbolic distance ρ(e) between its endvertices.
The proof is an adaptation of that of Section 6.2, and full details are omitted.In identifying a path corresponding to the path w of Lemma 6.3, we use the fact that edges of E are geodesics, and concentrate on the final departures of Γ + d from the faces whose interiors it enters.Remark 6.4.The condition of Theorem 3.4 may be weakened as follows.In the above proof of Theorem 3.1 is constructed a 2∞-nst path of G * (see the discussion following Lemma 6.3).It suffices that, in the sense of that discussion, there exist a diagonal d and s < 0, t ≥ 1 such that (i) the path (ν s , ν s+1 , . . ., ν t ) is non-selftouching in G * , and (ii) for all e ∈ E we have p(ν t ) − p(ν s ) > p(π(e)).Cf.Theorem 4.8.
Note added before publication: the quasi-transitive case is treated in [12].

Corollary 1 . 3 .
Let G ∈ Q be one-ended.Then p site u (G) + p site c (G) ≥ 1, with strict inequality if and only if the condition of Theorem 1.2 holds.Proof of Corollary 1.3.The given (weak) inequality is proved at [13, Thm 1.1(b)], and the strict inequality holds by (1.3) and Theorem 1.2.

( a )
All one-ended, transitive, planar graphs are 3-connected, and all embeddings of a one-ended, quasi-transitive, planar graph have only finite faces.(b) By Theorem 2.1(b), any one-ended G ∈ Q that is in addition transitive has a unique cellular embedding in H up to homeomorphism.Hence, the matching and dual graphs of G are independent of the embedding.(c) The conclusion of part (b) holds for any one-ended, 3-connected G ∈ Q.(d) For a one-ended, 2-connected G ∈ Q, we fix a canonical embedding (in the sense of Theorem 2.1(c)).With this given, the dual graph G + and the matching graph G * are quasi-transitive, and furthermore the boundary of every face is a cycle of G.
An edge e ∈ E * \ E is called a diagonal of G or of G * , and it is denoted δ(a, b) where a, b are its endvertices.If δ(a, b) is a diagonal, a and b are called * -neighbours.
locally finite graph with bounded vertex-degrees.A site percolation configuration on G is an assignment ω ∈ Ω := {0, 1} V to each vertex of either state 0 or state 1.A vertex is called open if it has state 1, and closed otherwise.An open cluster in ω is a maximal connected set of open vertices.

Figure 3 . 1 .
Figure 3.1.The graph G is the tiling of the plane with copies of this square.Taking into account the symmetries of the square, this tiling is canonical after a suitable rescaling of the interior square.The diagonals are indicated by dashed lines.

Theorem 3 . 4 .
Let G ∈ Q be one-ended but not a triangulation.Assume that G satisfies the metric criterion of Definition 3.2.Then G has the property Π of Definition 2.4.See Sections 6.2 and 6.3 for the proofs of Theorem 3.1, Lemma 3.3, and Theorem 3.4 by the metric method.

Lemma 4 . 1 .
Let H be a graph embedded in H.

Lemma 4 . 3 .
Let G = (V, E) ∈ Q be one-ended and embedded canonically in the Poincaré disk H, and let L δ be a hyperbolic tube.(a) If 2δ > Φ, then L δ contains a 2∞-nst path of G, and a 2∞-nst path of G * , that cross L δ in the long direction.(b) There exists ζ = ζ(G) (depending on G and its embedding) such that, for r > ζ and v ∈ V , the annulus Λ r

Figure 4 .
Figure 4.2.A square of the square lattice, its matching graph, and with its facial site added.

4 . 2 .Lemma 4 . 4 .
The same conclusion applies to G * on letting π be a path ofG * .(b) Let ζ be such that ρ(u, v) ≥ 2Φ whenever d G (u, v) ≥ ζ.The proof of part (b) follows that of part (a).□Graph properties.The proofs of this article make heavy use of path-surgery which, in turn, relies in part on the property of planarity.Let G ∈ Q, and let π be a (finite or infinite) non-self-touching path of G * .(a)For every face F of G, π contains either zero or one or two vertices of F .If π contains two such vertices u, v, then it contains also the corresponding edge ⟨u, v⟩, which may be either an edge of G or a diagonal.(b) The path π is plane when viewed as a graph.

Definition 4 . 5 .
The graph G ∈ Q is said to have property Π if G has a 2∞-nst path including some facial site.Lemma 4.6.Let G ∈ Q be one-ended.Then Π ⇒ Π.

Figure 4 2 Figure 4 . 4 .
Figure 4.4.In the easiest case when D ≥ 2, one finds (green) nontouching subarcs σ i A of σ A to which v may be connected by non-selftouching paths.These subarcs may be connected to the boundary of H using subpaths of a doubly-infinite path constructed using Lemma 4.3(a).

1 2 2 Figure 4 . 5 .
Figure 4.5.An illustration of the case D = 1.The green lines indicate the subpaths σ i A .The rectangle is added in illustration of the case θ ≥ 3 4 π.
The constant A ′ (G) in part (b) depends on the embedded graph G, viewed as a subset of H, rather on the graph G alone.In advance of giving the proof of Theorem 5.2, we explain how it implies Theorem 1.2.Proof ofTheorem 1.2 (assuming Theorem 5.2).If G does not have property Π, by Theorem 4.8 for large A it does not have property Π A , whence by Theorem 5.2(a), p c (G * ) = p c (G).Conversely, if G has property Π, by Theorem 4.8 again it has property Π A for large A, whence by Theorem 5.2(b), p c ( G) < p c (G).The final claim follows by the elementary inequality p c (G * ) ≤ p c ( G); see (5.2). □ Proof of Theorem 5.2(a).Let A 0 ∈ Z. Assume G has property Π A for no A ≥ A 0 , and let p > p c (G * ).Let π be an infinite open path of G * with some endpoint x.By Lemma 4.1(b), there exists a subset π ′ of π that forms a non-self-touching path of G * with endpoint x.Let A > A 0 .Since Π A does not hold, every edge of π ′ at distance 2A or more from x is an edge of G, so that there exists an infinite open path in G. Therefore, p ≥ p c (G), whence p c (G * ) = p c (G). □ The rest of this section is devoted to the proof of Theorem 5.2(b).Let Ω = Ω V × Ω Φ where Φ is the set of facial sites and Ω Φ = {0, 1} Φ .For ω = ω × ω ′ ∈ Ω and ϕ ∈ Φ, we call ϕ open if ω ′ ϕ = 1, and closed otherwise.Let P p,s = P p × P s be the corresponding product measure on Ω V × Ω Φ , and θ(p, s) = lim n→∞ θ n (p, s) where θ n (p, s) = P p,s (v 0 ↔ ∂Λ n in G), so that (5.2) θ(p, 0) = θ(p; G), θ(p, p) = θ(p; G), θ(p, 1) = θ(p; G * ), where θ(p; H) denotes the percolation probability of the graph H.Note that θ(p, s) is non-decreasing in p and s.The following proposition implies Theorem 5.2(b).

1
contains no vertex/site outside the closed cycle comprising L followed by the subpath of σ M from b ′ 2 to b 2 .In order to overcome this problem, we alter the path ν ′ 2 as follows.Let α denote the annulus Λ M (a 2 )\Λ M −ζ (a 2 ), with ζ as in Lemma 4.3(b).(We may assume M ≥ 2ζ.)By that lemma, α contains a non-self-touching cycle β of G that surrounds a 2 .The union of ν ′ 2 and β contains (after oxbow-removal) a non-self-touching path ν ′′ 2

6. 1 . 6 . 2 .
Embeddings in the Poincaré disk.Throughout this section we shall work with the Poincaré disk model of hyperbolic geometry (also denoted H), and we denote by ρ the corresponding hyperbolic metric.Proof of Theorem 3.1.Let Γ be a doubly-infinite geodesic in the Poincaré disk.Pick a fixed but arbitrary total ordering < of Γ.Then Γ may be parametrized by any function p : Γ → R satisfying p(v) = p(u) + ρ(u, v) for u, v ∈ Γ, u < v, and we fix such p.Any x / ∈ Γ has an orthogonal projection π(x) onto Γ (for x ∈ Γ, we set π(x) = x).

(6. 2 )
ρ(d) ≥ ρ(e) ≥ 1, e ∈ E * , and we choose d accordingly.By Lemma 6.1 applied to the geodesic Γ d , (6.3) ρ(π(e)) ≤ ρ(e) ≤ ρ(d), e ∈ E * , where π denote orthogonal projection onto Γ d , and ρ(γ) is the hyperbolic distance between the endpoints of an arc γ.Let < and p be the ordering and parametrization of Γ d given at the start of this subsection.We extend the domain of p by setting p(x) = p(π(x)), x ∈ H.We construct next a doubly-infinite path of G * containing d and lying 'close' to Γ d .Write d = ⟨a, b⟩ where a < b.Let Γ + d (respectively, Γ − d ) be the sub-geodesic obtained by proceeding along Γ d from b in the positive direction (respectively, from a in the negative direction).As we proceed along Γ + d , we encounter edges and faces of G.If e ∈ E is such that e ∩ Γ + d ̸ = ∅, then the intersection is either a point or the entire edge e (this holds since both e and Γ d are geodesics).

Figure 6 . 2 .
Figure 6.2.The two cases that arise when Γ + d meets an edge which is either perpendicular or not.