Uniform infinite half-planar quadrangulations with skewness

We introduce a one-parameter family of random infinite quadrangulations of the half-plane, which we call the uniform infinite half-planar quadrangulations with skewness (UIHPQ$_p$ for short, with $p\in[0,1/2]$ measuring the skewness). They interpolate between Kesten's tree corresponding to $p=0$ and the usual UIHPQ with a general boundary corresponding to $p=1/2$. As we make precise, these models arise as local limits of uniform quadrangulations with a boundary when their volume and perimeter grow in a properly fine-tuned way, and they represent all local limits of (sub)critical Boltzmann quadrangulations whose perimeter tend to infinity. Our main result shows that the family (UIHPQ$_p$)$_p$ approximates the Brownian half-planes BHP$_\theta$, $\theta\geq 0$, recently introduced in Baur, Miermont, and Ray (2016). For $p<1/2$, we give a description of the UIHPQ$_p$ in terms of a looptree associated to a critical two-type Galton-Watson tree conditioned to survive.


Overview
The purpose of this paper is to introduce and study a one-parameter family of random infinite quadrangulations of the half-plane, which we denote by pUIHPQ p q 0ďpď1{2 and call the uniform infinite half-planar quadrangulations with skewness. Two members play a particular role: The choice p " 0 corresponds to Kesten's tree, cf. Proposition 1 below, whereas the choice p " 1{2 corresponds to the standard uniform infinite half-planar quadrangulation UIHPQ with a general boundary.
Kesten's tree [30] is a random infinite planar tree, which we may view as a degenerate quadrangulation with an infinite boundary, but no inner faces. It arises as the local limit of critical Galton-Watson trees conditioned to survive. The standard UIHPQp" UIHPQ 1{2 q forms the half-planar analog of the uniform infinite planar quadrangulation introduced by Krikun [31], after the seminal work of Angel and Schramm [7] on triangulations of the plane. Curien and Miermont [25] showed that the UIHPQ arises as a local limit of uniformly chosen quadrangulations of the two-sphere with n inner faces and a boundary of size 2σ, upon letting first n Ñ 8 and then σ Ñ 8 (see Angel [3] for the case of triangulations with a simple boundary).
We will define each UIHPQ p in Section 4 by means of an extension of the Bouttier-Di Francesco-Guitter mapping to infinite quadrangulations with a boundary. In the first part of this paper, we will discuss various local limits and scaling limits which involve the family pUIHPQ p q p . More precisely, in Theorem 1, we will see that each UIHPQ p appears as a local limit as n tends to infinity of uniform quadrangulations Q σn n with n inner faces and a boundary of size 2σ n , for an appropriate choice of σ n " σ n ppq Ñ 8. In Proposition 2, we argue that the family pUIHPQ p q p consists precisely of the infinite quadrangulations with a boundary which are obtained as local limits σ Ñ 8 of subcritical Boltzmann quadrangulations with a boundary of size 2σ. This result will prove helpful in our description of the UIHPQ p given in Theorem 4.
We will then turn to distributional scaling limits of the family pUIHPQ p q p in the so-called local Gromov-Hausdorff topology. In Theorems 2 and 3, we will clarify the connection between the (discrete) quadrangulations UIHPQ p and the family pBHP θ q θě0 of Brownian half-spaces with skewness θ introduced in [8]. More specifically, upon rescaling the graph distance by a factor a´1 n Ñ 0, we prove that each BHP θ is the distributional limit of the rescaled spaces a´1 n¨U IHPQ pn , if p n " p n pθ, a n q is adjusted in the right manner (Theorem 2). In our setting, convergence in the local Gromov-Hausdorff sense amounts to show convergence of rescaled metric balls around the roots of a fixed but arbitrarily large radius in the usual Gromov-Hausdorff topology; see Section 1.2.7.
In [8], a classification of all possible non-compact scaling limits of pointed uniform random quadrangulations with a boundary pV pQ σn n q, a´1 n d gr , ρ n q has been given, depending on the asymptotic behavior of the boundary size 2σ n and on the choice of the scaling factor a n Ñ 8 (in the local Gromov-Hausdorff topology, with the distinguished point ρ n lying on the boundary). In this paper, we address the boundary regime corresponding to the portion x ě 1 of the y " 0 axis in Figure 1 (in hashed marks), which was left untouched in [8]. As we show, it corresponds to a regime of unrescaled local limits, namely the family pUIHPQ p q p .
We finally give a branching characterization of the UIHPQ p when p ă 1{2. For that purpose, we will adapt the concept of discrete random looptrees introduced by Curien and Kortchemski [22]. We will see that the UIHPQ p admits a representation in terms of a looptree associated to a two-type version of Kesten's infinite tree. Informally, we will replace each vertex u at odd height in Kesten's tree by a cycle of length degpuq, which connects the vertices incident to u. Here, degpuq stands for the degree (i.e., the number of neighbors) of u in the tree. We then fill in the cycles of the looptree with a collection of independent quadrangulations with a simple boundary, which are drawn according to a subcritical Boltzmann law. As we show in Theorem 4, the space constructed in this way has the law of the UIHPQ p . Discrete looptrees and their scaling limits have found various applications in the study of large-scale properties of random planar maps, for instance in the description of the boundary of percolation clusters on the uniform infinite planar triangulation; see the work [23], which served as the main inspiration for our characterization of the UIHPQ p . From our description, we immediately infer that simple random walk is recurrent on the UIHPQ p for p ă 1{2.
It is well-known that the standard UIHPQ with a simple boundary satisfies the socalled spatial Markov property, which allows, in particular, the use of peeling techniques. In [5], Angel and Ray classified all triangulations (without self-loops) of the half-plane satisfying the spatial Markov property and translation invariance. They form a one-parameter asymptotic cone tangent cone fine-tuned limits UI PQ Figure 1: In [8], all possible limits for the rescaled spaces pV pQ σn n q, a´1 n d gr , ρ n q are discussed. The x-axis represents the limit values for the logarithm of the boundary length logpσ n q{ logpnq in units of logpnq, and the y-axis corresponds to the limit of the logarithm of the scaling factor logpa n q{ logpnq in units of logpnq. The focus of this paper lies on the hashed region.
family pH α q α parametrized by α P r0, 1q. The parameter α " 2{3 corresponds to the standard UIHPT with a simple boundary, the triangular equivalent of the UIHPQ with a simple boundary. When α ą 2{3 (the supercritical case), H α is of hyperbolic nature and exhibits an exponential volume growth. On the contrary, when α ă 2{3 (the subcritical case), it has a tree-like structure. We believe that the family pUIHPQ p q p is a quadrangular equivalent to the triangulations in the subcritical phase of [5]. Note that contrary to the UIHPQ p , the spaces H α for α ă 2{3 have a half-plane topology, due to the conditioning to have a simple boundary. However, there exists almost surely infinitely many cut-edges connecting the left and right boundaries; see [37,Proposition 4.11]. This should be seen as an equivalent to the branching structure formulated in Theorem 4 below. Our methods in this paper are different from [5,37] as we do not use peeling techniques.
In [21], Curien studied full-plane analogs of the family pH α q α . With similar (peeling) techniques, he constructed a (unique) one-parameter family of random infinite planar triangulations indexed by κ P p0, 2{27s, which satisfy a slightly adapted spatial Markov property. The critical case κ " 2{27 corresponds to the standard UIPT with a simple boundary of Angel and Schramm [7]. The regime κ P p0, 2{27q parallels the supercritical (or hyperbolic) phase α ą 2{3 of [5], whereas it is shown that there is no subcritical phase. Recently, a near-critical scaling limit of hyperbolic nature called the hyperbolic Brownian half-plane has been studied by Budzinski [17]. It is obtained from rescaling the triangulations of Curien [21] and letting κ Ñ 2{27 at the right speed. Theorem 1 of [17] bears some structural similarities with our Theorem 2 below, although it concerns a different regime.
Structure of the paper The rest of this paper is structured as follows. In the following section, we introduce some (standard) concepts and notation around quadrangulations, which will be used throughout this text. Moreover, we recapitulate the local topology and the local Gromov-Hausdorff topology. In Section 2, we state our main results, which concern local limits, scaling limits, and structural properties of the family pUIHPQ p q p . Section 3 reviews the definition of the family of Brownian half-planes pBHP θ q θ , and of various random trees, which are used both to describe the distributional limits of the family pUIHPQ p q p as well as their branching structure.
In Section 4, we construct the UIHPQ p . We first explain the Bouttier-Di Francesco-Guitter encoding of quadrangulations with a boundary and then define the UIHPQ p in terms of the encoding objects. We are then in position to prove our limit statements; see Section 5. In the final Section 6, we prove our main result characterizing the tree-like structure of the UIHPQ p when p ă 1{2, as well as recurrence of simple random walk.
1.2 Some standard notation and definitions 1.2
For two sequences pa n q n , pb n q n Ă N, we write a n ! b n or b n " a n if a n {b n Ñ 0 as n Ñ 8.
Given two measurable subsets U, V Ă R, we denote by CpU, V q the space of continuous functions from U to V , equipped with the usual compact-open topology, i.e., uniform convergence on compact subsets. We write }ν} TV for the total variation norm of a probability measure ν. As a general notational rule for this paper, if we drop p from the notation, we work with the case p " 1{2. For example, we write UIHPQ (and not UIHPQ 1{2 ) for the standard uniform infinite half-planar quadrangulation.

Planar maps
By a planar map we mean, as usual, an equivalence class of a proper embedding of a finite connected graph in the two-sphere, where two embeddings are declared to be equivalent if they differ only by an orientation-preserving homeomorphism of the sphere. Loops and multiple edges are allowed. Our planar maps will be rooted, meaning that we distinguish an oriented edge called the root edge. Its origin is the root vertex of the map. The faces of a planar map are formed by the components of the complement of the union of its edges.

Quadrangulations with a boundary
A quadrangulation with a boundary is a finite planar map q, whose faces are quadrangles except possibly one face called the outer face, which may an have arbitrary even degree. The edges incident to the outer face form the boundary Bq of q, and their number #Bq (counted with multiplicity) is the size or perimeter of the boundary. In general, we do not assume that the boundary edges form a simple curve. We will root the map by selecting an oriented edge of the boundary, such that the outer face lies to its right. The size of q is given by the number of its inner faces, i.e., all the faces different from the outer face.
We write Q σ n for the (finite) set of all rooted quadrangulations with n inner faces and a boundary of size 2σ, σ P N 0 . By convention, Q 0 0 " t:u consists of the unique vertex map.
More generally, Q f will denote the set of all finite rooted quadrangulations with a boundary, and Q σ f Ă Q f the set of all finite rooted quadrangulations with 2σ boundary edges, for σ P N 0 .
Similarly, we let p Q f be the set of all finite rooted quadrangulations with a simple boundary, meaning that the edges of their outer face form a cycle without self-intersection. We denote by p Q σ f Ă p Q f the subset of finite rooted quadrangulations with a simple boundary of size 2σ. Note that Q 1 0 consists of the map having one oriented edge and thus a simple boundary.

Uniform quadrangulations with a boundary
Throughout this text, we write Q σ n for a quadrangulation chosen uniformly at random in Q σ n . We denote by ρ n the root vertex of Q σ n , i.e., the origin of the root edge. By equipping the set of vertices V pQ σ n q with the graph distance d gr , we view the triplet pV pQ σ n q, d gr , ρ n q as a random rooted metric space.

Boltzmann quadrangulations with a boundary
We will also work with various Boltzmann measures. For a finite rooted quadrangulation q P Q f , we write Fpqq for the set of inner faces of q. Given non-negative weights g per inner face and ? z per boundary edge, we let When this partition function is finite, we may define the associated Boltzmann distribution The statement of Proposition 2 below deals with Boltzmann-distributed quadrangulations of a fixed boundary size 2σ, for σ P N 0 . In this case, the associated partition function and Boltzmann distribution read whenever g ě 0 is such that F σ pgq is finite. The Boltzmann distribution P σ g is related to P g,z by conditioning the latter with respect to the boundary length, i.e., P σ g pqq " P g,z pq | Q σ f q. When studying quadrangulations with a simple boundary, the partition functions are and the Boltzmann distributions take the form Remark 1. In the notation of [15], the generating function F is denoted W 0 , while p F is denotedW 0 . The index zero stands for the distance between the origin of the root edge and the marked vertex, so that these generating functions count unpointed quadrangulations.

Local topology
Our unrescaled limit results hold with respect to the local topology first studied by Benjamini and Schramm [10]: For two rooted planar maps m and m 1 , the local distance between m and m 1 is d map pm, m 1 q " p1`suptr ě 0 : Ball r pmq " Ball r pm 1 quq´1 , where Ball r pmq denotes the combinatorial ball of radius r around the root ρ of m, i.e., the submap of m consisting of all the vertices v of m with d gr pρ, vq ď r and all the edges of m between such vertices. The set Q f of all finite rooted quadrangulations with a boundary is not complete for the distance d map ; we have to add infinite quadrangulations. We shall write Q for the completion of Q f with respect to d map . The UIHPQ p will be defined as a random element in Q.

Around the Gromov-Hausdorff metric
The pointed Gromov-Hausdorff distance measures the distance between (pointed) compact metric spaces, where the latter are viewed up to isometries. More specifically, given two elements E " pE, d, ρq and E 1 " pE 1 , d 1 , ρ 1 q in the space K of isometry classes of pointed compact metric spaces, their Gromov-Hausdorff distance is defined as where the infimum is taken over all isometric embeddings ϕ : E Ñ F and ϕ 1 : E 1 Ñ F of E and E 1 into the same metric space pF, δq, and d H is the usual Hausdorff distance between compacts of F . The space pK, d GH q is complete and separable. Our results on scaling limits involve non-compact pointed metric spaces and hold in the so-called local Gromov-Hausdorff sense, which we briefly recall next. Given a pointed complete and locally compact length space E and a sequence pE n q n of such spaces, pE n q n converges in the local Gromov-Hausdorff sense to E if for every r ě 0, d GH pB r pE n q, B r pEqq Ñ 0 as n Ñ 8.
Here and in what follows, given a pointed metric space F " pF, d, ρq, B r pFq " tx P F : dpx, ρq ď ru denotes the closed ball of radius r around ρ, viewed as a subspace of F equipped with the metric structure inherited from F. For λ ą 0, λ¨F stands for the rescaled pointed metric space pF, λd, ρq, so that in particular λ¨B r pFq " B λr pλ¨Fq.
As a discrete map, the UIHPQ p is not a length space in the sense of [18]. However, by identifying each edge with a copy of the unit interval r0, 1s (and by extending the metric isometrically), one obtains a complete locally compact length space (pointed at the root vertex). By construction, balls of the same radius and around the same points in the UIHPQ p and in the approximating length space are at Gromov-Hausdorff distance at most 1 from each other. Therefore, local Gromov-Hausdorff convergence for the (rescaled) UIHPQ p , see Theorems 2 and 3 below, follows indeed from the convergence of balls as stated above.

Local limits
Our first result states that each member of the family pUIHPQ p q 0ďpď1{2 can be seen as a local limit n Ñ 8 of uniform quadrangulations of with n inner faces and a boundary of size 2σ n , provided σ n " σ n ppq is chosen in the right manner. Theorem 1. Fix 0 ď p ď 1{2, and let pσ n , n P Nq be a sequence of positive integers satisfying σ n " 1´2p p n`opnq if 0 ă p ď 1{2, and σ n " n if p " 0.
For every n P N, let Q σn n be uniformly distributed in Q σn n . Then we have the local convergence for the metric d map as n Ñ 8, In fact, we will prove a stronger result than mere local convergence: We will establish an isometry of balls of growing radii around the roots, where the maximal growth rate of the radii is given by ξ n " mintn 1{4 , a n{γ n u, for γ n " maxtσ n´1´2 p p n, 1u. We defer to Proposition 4 for the exact statement. The case p " 1{2 corresponding to the regime σ n " opnq is already covered by [8,Proposition 3.11] and is only included for completeness.
The convergence in the case p " 0 with σ n " n is somewhat simpler. However, it is a priori not obvious that the UIHPQ 0 as defined in Section 4 is actually Kesten's tree (see Section 3.2.3 for a definition of the latter). Proposition 1. The space UIHPQ 0 has the law of Kesten's tree T 8 associated to the critical geometric probability distribution pµ 1{2 pkq, k P N 0 q given by µ 1{2 pkq " 2´p k`1q .
Interestingly, the fact that the UIHPQ 0 is Kesten's tree can also be derived as a special case from Theorem 4 below; see Remark 5. We prefer, however, to give a direct proof of the proposition based on our construction of the UIHPQ 0 .
The UIHPQ p for 0 ď p ď 1{2 is also obtained as a local limit of Boltzmann quadrangulations with growing boundary size. This result will be important to describe the tree-like structure of the UIHPQ p when p ă 1{2. More specifically, the family pUIHPQ p q p is precisely given by the collection of all local limits σ Ñ 8 of Boltzmann quadrangulations with a boundary of size 2σ and weight g ď g c " 1{12 per inner face. The value g c " 1{12 is critical (see [15,Section 4.1]) and corresponds to the choice p " 1{2. Proposition 2. Fix 0 ď p ď 1{2, and set g p " pp1´pq{3. For every σ P N 0 , let Q σ ppq be a random rooted quadrangulation distributed according to the Boltzmann measure P σ gp . Then we have the local convergence for the metric d map as σ Ñ 8, Remark 2. For p " 1{2, the above proposition states convergence of critical Boltzmann quadrangulations with a boundary towards the UIHPQ, as it was already proved in [20,Theorem 7] by means of peeling techniques. In view of the above proposition, it is moreover implicit from the same theorem that an infinite random map with the law of the UIHPQ p does exist. For the case of half-planar triangulations (with a simple boundary), see [3]. When p " 0, there is no inner quadrangle almost surely and Q σ p0q is a uniform tree with σ edges (i.e., a Galton-Watson tree with geometric offspring law conditioned to have σ edges), which converges locally towards Kesten's tree; see, for example, [28,Theorem 7.1].
Remark 3. Let us write MpQq for the set of probability measures on the completion Q, and equip it with the usual weak topology. Then it is easily seen by our methods that the mapping r0, 1{2s Q p Þ Ñ LawpUIHPQ p q P MpQq is continuous.

Scaling limits
Our next results address scaling limits of the family pUIHPQ p q p . In [8], a one-parameter family of (non-compact) random rooted metric spaces called the Brownian half-planes BHP θ with skewness θ ě 0 was introduced. See Section 3.1 for a quick reminder. The Brownian half-plane BHP 0 corresponding to the choice θ " 0 forms the half-planar analog of the Brownian plane introduced in [24] and arises from zooming-out the UIHPQ around the root vertex; see [8,Theorem 3.6], and [27,Theorem 1.10]). Here, we will see more generally that the family pUIHPQ p q p approximates the space BHP θ for each θ ě 0 in the local Gromov-Hausdorff sense, provided p is appropriately fine-tuned (depending on θ). Theorem 2. Let θ ě 0. Let pa n , n P Nq be a sequence of positive reals with a n Ñ 8 as n Ñ 8. Let pp n , n P Nq Ă r0, 1{2s be a sequence satisfying p n " p n pθ, a n q " 1 2ˆ1´2 Then, in the sense of the local Gromov-Hausdorff topology as n Ñ 8, The space BHP θ satisfies the scaling property λ¨BHP θ " d BHP θ{λ 2 . It was shown in Remark 3.19 of [8] that Aldous' infinite continuum random tree ICRT, whose definition is reviewed in Section 3.2.1, is the asymptotic cone of the BHP θ around its root, implying BHP θ Ñ ICRT in law as θ Ñ 8. In particular, formally, we may think of the BHP 8 as the ICRT. In view of Theorem 2, it is therefore natural to expect that the ICRT appears also as the scaling limit of the UIHPQ pn , provided θ in the definition of p n is replaced by a sequence θ n Ñ 8, that is, if a 2 n p1´2p n q Ñ 8 as n Ñ 8. This is indeed the case.
Theorem 3. Let pa n , n P Nq be a sequence of positive reals with a n Ñ 8. Let pp n , n P Nq Ă r0, 1{2s be a sequence satisfying a 2 n p1´2p n q Ñ 8 as n Ñ 8.
Then, in the sense of the local Gromov-Hausdorff topology as n Ñ 8, As special cases of the previous two theorems, we mention Corollary 1. Let p P r0, 1{2s, and let pa n , n P Nq be a sequence of positive reals with a n Ñ 8. Then, in the sense of the local Gromov-Hausdorff topology as n Ñ 8, For the family pH α q α of half-planar triangulations studied in [5,37], convergence towards the ICRT in the subcritical regime α ă 2{3 is conjectured in [37, Section 2.1.2]. If the volume T of BD T,σ is blown up and the perimeter σ grows linearly in T such that σpT q " θT , the space BHP θ appears as the distributional local Gromov-Hausdorff limit of the disks BD T,σpT q around their roots ( [8,Corollary 3.17]). On the other hand, BHP θ is approximated by uniform quadrangulations Q σn n ([8, Theorem 3.4]), or by the UIHPQ p when p " ppa n , θq depends in the right way on θ and a n (Theorem 2). The UIHPQ p for fixed p P r0, 1{2s in turn arises as the local limit of Q σn n , provided the boundary lengths are properly chosen (Theorem 1).

Remark 4.
We stress that the spaces BHP θ can also be understood as Gromov-Hausdorff scaling limits of uniform quadrangulations Q σn n P Q σn n ; see [8, Theorems 3.3, 3.4, 3.5]. More specifically, the BHP θ for θ P p0, 8q arises when ? n ! σ n ! n and the graph metric is rescaled by a factor a´1 n satisfying 3σ n a 2 n {p4nq Ñ θ as n tends to infinity. The Brownian half-plane BHP 0 corresponding to the choice θ " 0 appears more generally when 1 ! σ n ! n and 1 ! a n ! mint ? σ n , a n{σ n u. Finally, the ICRT corresponding to θ " 8 appears when σ n " ? n and maxt1, a n{σ n u ! a n ! ? σ n . We may as well view the spaces BHP θ as local scaling limits around the roots of the so-called Brownian disks BD T,σ of volume T ą 0 and perimeter σ ą 0 introduced in [12]. More concretely, it was proved in [8,Corollaries 3.17,3.18] that when both T and σ " σpT q tend to infinity such that σpT q{T Ñ θ P r0, 8s, then the BHP θ is the local Gromov-Hausdorff limit in law of the disk BD T,σpT q around a boundary point chosen according to the boundary measure of the latter. Figure 2 depicts some convergences involving the families UIHPQ p and BHP θ .

Tree structure
We will prove that for p ă 1{2, the UIHPQ p can be represented as a collection of independent finite quadrangulations with a simple boundary glued along a tree structure. The tree structure is encoded by the looptree associated to a two-type version of Kesten's tree, and the finite quadrangulations are distributed according to the Boltzmann distribution p P σ g on quadrangulations with a simple boundary of size 2σ. Precise definitions of the encoding objects are postponed to Section 3.
For 0 ď p ď 1{2, let g p " pp1´pq{3 and z p " p1´pq{4. Let F pg, zq be the partition function of the Boltzmann measure on finite rooted quadrangulations with a boundary, with weight g per inner face and ? z per boundary edge. Let moreover p F k pgq be the partition function of the Boltzmann measure on finite rooted quadrangulations with a simple boundary of perimeter 2k, with weight g per inner face.
We introduce two probability measures µ˝and µ ‚ on N 0 by setting with µ ‚ pkq " 0 if k even. Exact expressions for F pg p , z p q and p F k`1 pg p q are given in (18) and (19) below. The fact that µ ‚ is a probability distribution is a consequence of Identity (2.8) in [15]. We will prove in Lemma 11 that the pair pµ˝, µ ‚ q is critical for 0 ď p ă 1{2, in the sense that the product of their respective means equals one, and subcritical if p " 1{2, meaning that the product of their means is strictly less than one. Moreover, both measures have small exponential moments. Our main result characterizing the structure of the UIHPQ p for 0 ď p ă 1{2 is the following.
Theorem 4. Let 0 ď p ă 1{2, and let LooppT 8 q be the infinite looptree associated to Kesten's two-type tree T 8 pµ˝, µ ‚ q. Glue into each inner face of LooppT 8 q of degree 2σ an independent Boltzmann quadrangulations with a simple boundary distributed according to p P σ gp . Then, the resulting infinite quadrangulation is distributed as the UIHPQ p .
The gluing operation fills in each (rooted) loop a finite-size quadrangulation with a simple boundary, which has the same perimeter as the loop. The two boundaries are glued together, such that the root edges of the loop and the quadrangulation get identified; see Remark 8. Figure 3 depicts the above representation of the UIHPQ p in the case 0 ă p ă 1{2, as well as the borderline cases p " 0 and p " 1{2. The branching structure of the standard UIHPQ " UIHPQ 1{2 has been investigated by Curien and Miermont [25]. They show that the UIHPQ can be seen as the uniform infinite half-planar quadrangulation with a simple boundary (represented by the big white semicircle in Figure 3), together with a collection p = 0 0 < p < 1/2 p = 1/2 Figure 3: Schematic representation of the UIHPQ p for p P r0, 1{2s. On the left: The UIHPQ 0 , that is, Kesten's tree associated to the critical geometric offspring distribution µ 1{2 . On the right: The standard uniform infinite half-planar quadrangulation UIHPQ with a general boundary. The white parts are understood to be filled in with quadrangulations, the big white semicircle representing the half-plane. In the middle: The UIHPQ p with skewness parameter p. The white parts represent the (finite-size) quadrangulations with a simple boundary which are glued into the loops of the infinite looptree LooppT 8 q associated to a two-type version T 8 pµ˝, µ ‚ q of Kesten's tree.
of finite-size quadrangulations with a general boundary, which are attached to the infinite simple boundary.
Remark 5. In the case p " 0, the above theorem can be seen as a restatement of Proposition 1. Indeed, in this case, one finds that µ˝" µ 1{2 is the critical geometric probability law, and µ ‚ is the Dirac-distribution δ 1 . By construction, all the inner faces of LooppT 8 q have then degree 2, and the gluing of a Boltzmann quadrangulation distributed according to p P 1 g 0 "0 simply amounts to close the face, by identifying its edges. One finally recovers Kesten's (one-type) tree associated to the offspring law µ 1{2 , as already found in Proposition 1.
Remark 6. In [9], it has been proved that geodesics in the standard UIHPQ intersect both the left and right part of the boundary infinitely many times (see [9,Section 2.3.3] for the exact terminology). However, up to removing finite quadrangulations that hang off from the boundary, the UIHPQ has the topology of a half-plane. Consequently, left and right parts of the boundary intersect only finitely many times. The branching structure described in Theorem 4 implies that the left and right parts of the boundary of the UIHPQ p for p ă 1{2 have infinitely many intersection points. As a consequence, any infinite self-avoiding path intersects both boundaries infinitely many times.
Our tree-like description of the UIHPQ p for 0 ď p ă 1{2 readily implies that simple random walk on the UIHPQ p is recurrent. For p " 0, this result is due to Kesten [30]. Corollary 2. Let 0 ď p ă 1{2. Almost surely, simple random walk on the UIHPQ p is recurrent.
Somewhat informally, the tree structure describing the UIHPQ p in the case p ă 1{2 shows that there is an essentially unique way for the random walk to move to infinity. Said otherwise, the walk reduces essentially to a random walk on the half-line reflected at the origin, which is, of course, recurrent. We give a precise proof in terms of electric networks in Section 6.
Remark 7. As far as the standard uniform infinite half-planar quadrangulation UIHPQ corresponding to p " 1{2 is concerned, Angel and Ray [6] prove recurrence of the triangular analog with a simple boundary, the half-plane UIPT. They construct a full-plane extension of the half-plane UIPT using a decomposition into layers and then adapt the methods of Gurel-Gurevich and Nachmias [26], and Benjamini and Schramm [10]. It is believed that the arguments of [6] can be extended to the UIHPQ, too. Ray proves in [37] of recurrence of the half-plane models H α when α ă 2{3. In [13], Björnberg and Stefánsson prove that the (local) limit of bipartite Boltzmann planar maps is recurrent, for every choice of the weight sequence.
We believe that the mean displacement of a random walker after n steps on the UIHPQ p for p ă 1{2 is of order n 1{3 , as for Kesten's tree (case p " 0). We will not pursue this further in this paper.
Let us finally mention another consequence of Theorem 4 concerning percolation thresholds. See, e.g., [4] for the terminology of Bernoulli percolation on random lattices. Therefore, percolation on the UIHPQ p changes drastically depending on whether the skewness parameter p (not to be confused the the percolation parameter) is less or equal to 1{2: In the standard UIHPQ " UIHPQ 1{2 , the critical thresholds are known to be 5{9 for site percolation, see [38], and 1{3 for edge percolation and 3{4 for face percolation, see [4]. The proof of the corollary follows immediately from Theorem 4.

Random half-planes and trees
In this section, we begin with a review of the one-parameter family of Brownian half-planes BHP θ , θ ě 0, introduced in [8] (see also [27] for the case θ " 0).
We then gather certain concepts around trees, which play an important role throughout this paper. We properly define the ICRT, two-type Galton-Watson trees and Kesten's infinite versions thereof, looptrees and the so-called tree of components.

The Brownian half-planes BHP θ
We need some preliminary notation. Given a function f " pf t , t P Rq, we set f t " inf r0,ts f for t ě 0 and f t " inf p´8,ts f for t ă 0. Moreover, if f " pf t , t ě 0q is a real-valued function indexed by the positive reals, its Pitman transform πpf q is defined by In case B " pB t , t ě 0q is a standard one-dimensional Brownian motion, its Pitman transform πpBq " pπpBq t , t ě 0q is equal in law to a three-dimensional Bessel process, which has in turn the law of the modulus of a three-dimensional Brownian motion. Now fix θ P r0, 8q. The Brownian half-plane BHP θ with skewness θ is defined in terms of its contour and label processes X θ " pX θ t , t P Rq and W θ " pW θ t , t P Rq. They are characterized as follows.
• pX θ t , t ě 0q has the law of a one-dimensional Brownian motion B " pB t , t ě 0q with drift´θ and B 0 " 0, and pX θ t , t ě 0q has the law of the Pitman transform of an independent copy of B.
• Given X θ , the (label) function W θ has same distribution as pγ´Xθ is a continuous modification of the centered Gaussian process with covariances given by pγ x , x P Rq is a two-sided Brownian motion with γ 0 " 0 and scaled by the factor ? 3, independent of Z θ .
The process Z θ is usually called the (head of the) random snake driven by X θ´X θ , see [32] for more on this. Next, we define two pseudo-metrics d X θ and d W θ on R, The pseudo-metric D θ associated to BHP θ is defined as the maximal pseudo-metric d on R satisfying d ď d W θ and td X θ " 0u Ď tD θ " 0u. According to Chapter 3 of [18], it admits the expression (s, t P R) Definition 1. The Brownian half-plane BHP θ has the law of the pointed metric space pR{tD θ " 0u, D θ , ρ θ q, with the distinguished point ρ θ is given by the equivalence class of 0.
Note that D θ stands here also for the induced metric on the quotient space. It follows from standard scaling properties of X θ and W θ that for λ ą 0, λ¨BHP θ " d BHP θ{λ 2 . In particular, BHP 0 is scale-invariant. It was shown in [8] that for every θ ě 0, BHP θ has a.s. the topology of the closed half-plane H " RˆR`.

Random trees and some of their properties 3.2.1 The infinite continuum random tree ICRT
Introduced by Aldous in [2], the ICRT is a random rooted real tree that forms the noncompact analog of the usual continuum random tree CRT. Consider the stochastic process pX t , t P Rq such that pX t , t ě 0q and pX´t, t ě 0q are two independent one-dimensional standard Brownian motions started at zero. Define on R the pseudo-metric d X ps, tq " X s`Xt´2 min rs^t,s_ts X.

Definition 2.
The ICRT is the continuum random real tree T X coded by X, i.e., the ICRT has the law of the pointed metric space pT X , d X , r0sq, where T X " R{td X " 0u, and the distinguished point is given by the equivalence class of 0.
The ICRT is scale-invariant, meaning that λ¨ICRT " d ICRT for λ ą 0, and invariant under re-rooting. We remark that the ICRT is often defined in terms of two independent three-dimensional Bessel processes pX t , t ě 0q and pX´t, t ě 0q. Since the Pitman transform π turns a Brownian motion into a three-dimension Bessel processes, it is readily seen that both definitions give rise to the same random tree.

(Sub)critical (two-type) Galton-Watson trees
We recall the formalism of (finite or infinite) plane trees, i.e., rooted ordered trees. The size |t| P N 0 Y t8u of t is given by its number of edges, and we shall write T f for the set of all finite plane trees.
We will often use the fact that if GW ν denotes the law of a Galton-Watson tree with critical or subcritical offspring distribution ν, then where for u P V ptq, k u ptq is the number of offspring of vertex u. See, for example, [33, Proposition 1.4]). In the case where ν " µ p is the geometric offspring distribution of parameter p P r0, 1{2s, (1) becomes GW µp ptq " p |t| p1´pq |t|`1 .
From the last display, the connection to random walks is apparent. Namely, let pS ppq pmq, m P N 0 q be a random walk on the integers starting from S ppq p0q " 0 with increments distributed according to pδ 1`p 1´pqδ´1. Define the first hitting time of´1, Then it is readily deduced from (2) that the size |t| of t under GW µp and pT ppq 1´1 q{2 are equal in distribution; see, e.g., [36, Section 6.3].
Given a finite or infinite plane tree, it will be convenient to say that vertices at even height of t are white, and those at odd height are black. We use the notation t˝and t ‚ for the associated subset of vertices. We next define two-type Galton-Watson trees associated to a pair pν˝, ν ‚ q of probability measures on N 0 . Definition 3. The two-type Galton-Watson tree with a pair of offspring distributions pν˝, ν ‚ q is the random plane tree such that vertices at even height have offspring distribution ν˝, vertices at odd height have offspring distribution ν ‚ , and the numbers of children of the different vertices are independent.
More formally, if GW ν˝,ν‚ denotes the law of such a tree, then In this context, the pair pν˝, ν ‚ q is said to be critical if and only if the mean vector pm˝, m ‚ q satisfies m˝m ‚ " 1.

Kesten's tree and its two-type version
We next briefly review critical Galton-Watson trees conditioned to survive; see [30] or [35], and [39] for the multi-type case.
Proposition 3 (Theorem 3.1 in [39]). Let GW be the law of a critical (either one or twotype) Galton-Watson tree. For every n P N, assume that GWpt#V ptq " nuq ą 0, and let T n be a tree with law GW conditioned to have n vertices. Then, we have the local convergence for the metric d map as n Ñ 8 to a random infinite tree T 8 , In the case GW " GW ν for ν a critical one-type offspring distribution, T 8 is often called Kesten's tree associated to ν, and simply Kesten's tree if ν " µ 1{2 . We will use the same terminology if pν˝, ν ‚ q is a critical pair of offspring distributions and GW " GW ν˝,ν‚ . In this case, we write T 8 pν˝, ν ‚ q for Kesten's tree associated to pν˝, ν ‚ q. Note that the condition GWpt#V ptq " nuq ą 0 can be relaxed, provided we can find a subsequence along which this condition is satisfied.
Galton-Watson trees conditioned to survive enjoy an explicit construction, which we briefly recall for the two-type case. Details can be found in [39]. Let pν˝, ν ‚ q be a critical pair of offspring distributions with mean pm˝, m ‚ q, and recall that the size-biased distributions ν˝andν ‚ are defined bȳ Kesten's tree T 8 associated to pν˝, ν ‚ q is an infinite locally finite (two-type) tree that has a.s. a unique infinite self-avoiding path called the spine. It is constructed as follows. The root vertex (white) is the first vertex on the spine. It has size-biased offspring distribution ν˝. Among its offspring, a child (black) is chosen uniformly at random to be the second vertex on the spine. It has size-biased offspring distributionν ‚ , and a child (white) chosen uniformly at random among its offspring becomes the third vertex on the spine. The spine is constructed by iterating this procedure. The construction of the tree is completed by specifying that vertices at even (resp. odd) height lying not on the spine have offspring distribution ν˝(resp. ν ‚ ), and that the numbers of offspring of the different vertices are independent.
The construction is similar in the mono-type case. In the particular case when ν " µ 1{2 is the geometric distribution with parameter 1{2, Kesten's tree can be represented by an infinite half-line (isomorphic to N) and a collection of independent Galton-Watson trees with law GW µ 1{2 grafted to the left and to the right of every vertex on the spine; see, for instance, [28, Example 10.1]. We will exploit this representation in our proof of Proposition 1.

Random looptrees
Our description of the UIHPQ p in Theorem 4 makes use of so-called looptrees, which were introduced in [22]. A looptree can informally be seen as a collection of loops glued along a tree structure. The following presentation is inspired by [23, Section 2.3]. We use, however, slightly different definitions which are better suited to our purpose. In particular, given a plane tree t, we will only replace vertices v P V pt ‚ q at odd height by loops of length degpuq. Consequently, several loops may be attached to one and the same vertex (at even height).
Let us now make things more precise. Let t be a finite plane tree, and recall that vertices at even height are white, and those at odd height are black (with respective subsets of vertices t˝and t ‚ ). We associate to t a rooted looptree Loopptq as follows. Around every (black) vertex in t ‚ , we connect its incident white vertices in cyclic order, so that they form a loop. Then Loopptq is the planar map obtained from erasing the black vertices and the edges of t. We root Loopptq at the edge connecting the origin of t to the last child of its first sibling in t; see Figure 4.
The reverse application associates to a looptree l a plane tree, which we call the tree of components Treeplq. In order to obtain Treeplq from l, we add a new vertex in every internal face of l and connect this vertex to all the vertices of the face. Treeplq is rooted at the corner adjacent to the target of the root edge of l. The procedures Tree and Loop extend to infinite but locally finite trees, by considering the consistent sequence of maps tB 2k ptq : k P N 0 u. We will be interested in the random infinite looptree associated to Kesten's two-type tree.
Definition 4. If pν˝, ν ‚ q is a critical pair of offspring laws and T 8 the corresponding Kesten's tree, we call the random infinite looptree LooppT 8 q Kesten's looptree associated to T 8 .
Note that a formal way to construct LooppT 8 q is to define it as the local limit of LooppT n q, where T n is a two-type Galton-Watson tree with offspring distribution pν˝, ν ‚ q conditioned to have n vertices.
Remark 8. In a looptree l, every loop is naturally rooted at the edge whose origin is the closest vertex to the origin of l, such that the outer face of l lies on the right of that edge. The gluing of a (rooted) quadrangulation with a simple boundary of perimeter 2σ into a loop of the same length is then determined by the convention that the root edge of the quadrangulation is glued on the root edge of the loop.

Construction of the UIHPQ p
A Schaeffer-type bijection due to Bouttier, Di Francesco and Guitter [14] encodes quadrangulations with a boundary in terms of labeled trees that are attached to a bridge. We shall first describe a bijective encoding of finite-size planar quadrangulations, and then extend it to infinite quadrangulations with an infinite boundary. This will allow us to construct and define the UIHPQ p for p P r0, 1{2s in terms of the encoding objects, which we define first.

The encoding objects
We briefly review well-labeled trees, forests, bridges and contour and label functions. Our notation bears similarities to [25,19,8], differs, however, at some places. Each of these references already contains the construction of the standard UIHPQ.

Forest and bridges
A well-labeled tree pt, q is a pair consisting of a finite rooted plane tree t and a labeling p puqq uPV ptq of its vertices V ptq by integers, with the constraints that the root vertex receives label zero, and | puq´ pvq| ď 1 if u and v are connected by an edge.
A well-labeled forest with σ P N trees is a pair pf, lq, where f " pt 0 , . . . , t σ´1 q is a sequence of σ rooted plane trees, and l : V pfq Ñ Z is a labeling of the vertices V pfq " Y σ´1 i"0 V pt i q such that for every 0 ď i ď σ´1, the pair pt i , laeV pt i qq is a well-labeled tree. Similarly, a well-labeled infinite forest is a pair pf, lq, where f " pt i , i P Zq is an infinite collection of rooted plane trees, together with a labeling l : Y iPZ V pt i q Ñ Z such that for each i P Z, the restriction of l to V pt i q turns t i into a well-labeled tree.
Given a bridge b, an index i for which bpi`1q " bpiq´1 is called a down-step of b. The set of all down-steps of b is denoted DSpbq. If b is a bridge of length 2σ, DSpbq has σ elements, and we write d Ó b piq for the ith largest element in DSpbq, for i " 1, . . . , σ. If b is an infinite bridge and i P N, d Ó b piq denotes the ith largest element in DSpbq X N 0 , and d Ó b p´iq denotes the ith largest element in DSpbq X Z ă0 . If there is no danger of confusion, we write The size of a forest f is the number |f| P N 0 Y t8u of tree edges. If f " pt 0 , . . . , t σ´1 q and u P V pt i q, we write H f puq for the height of u in the tree t i , i.e., the graph distance to the root of t i . Moreover, I f puq " i denotes the index of the tree the vertex u belongs to. Both H f and I f extend in the obvious way to infinite forests. If it is clear which forest we are referring to, we drop the subscript f in H and I.
We let F n σ " tpf, lq : f has σ trees and size |f| " nu be the set of all well-labeled forests of size n with σ trees and write F 8 for the set of all well-labeled infinite forests. The set of all bridges of length 2σ is denoted B σ . As far as infinite bridges are concerned, it will be sufficient to consider only those bridges b which satisfy inf iPN bpiq "´8 and inf iPN bp´iq "´8, and we denote the set of them by B 8 .

Contour and label function
We first consider the case ppf, lq, bq P F n σˆB σ for some n, σ P N. By a slight abuse of notation, we write fp0q, . . . , fp2n`σ´1q for the contour exploration of f, that is, the sequence of vertices (with multiplicity) which we obtain from walking around the trees t 0 , . . . , t σ´1 of f, one after the other in the contour order. See the left side of Figure 5. We define the contour function of pf, lq by C f pjq " Hpfpjqq´Ipfpjqq, 0 ď j ď 2n`σ´1.
Note that C f p2n`σ´1q " σ´1, since the last visited vertex by the contour exploration is the root of t σ´1 . We extend C f to r0, 2n`σs by first letting C f p2n`σq "´σ, and then by linear interpolation between integers, so that C f becomes a continuous real-valued function on r0, 2n`σs starting at zero and ending at´σ. The label function associated to ppf, lq, bq is obtained from shifting the vertex label lpfpjqq by the value of the bridge b evaluated at its pIpfpjqq`1qth down-step. Formally, We let L f p2n`σq " 0 and again linearly interpolate between integer values, so that L f becomes an element of Cpr0, 2n`σs, Rq. Contour and label functions are depicted on the right side of Figure 5.
In the case ppf, lq, bq P F 8ˆB8 , we explore the trees of f in the following way: First, pfp0q, fp1q, . . .q is the sequence of vertices of the contour paths of the trees t i , i P N 0 , in the left-to-right order, starting from the root of t 0 . Then, we let pfp´1q, fp´2q, . . .q be the sequence of vertices of the contour paths t´1, t´2, . . ., in the counterclockwise or right-to-left order, starting from the root of t´1; see the left side of Figure 6. Contour and label functions C f and L f are defined similarly to the finite case, namely Note that the asymmetry in the definition of L f stems from the numbering of the trees. By linear interpolation between integer values, we interpret C f , L f , and sometimes also l, as continuous functions (from R to R). f (1) f (2) f (3) f (4) f (5) f (6) f (7) f (8) f (9) f (10) f (11) f (12) f (13) f (14) f (15) f (16) f (17) Figure 5: Contour and label functions C f and L f of an element ppf, lq, bq P F 7 4ˆB 4 . The left side depicts the contour exploration of f. The labels on the vertices are given by L f pjq, j " 0, . . . , 18. Note that the values of b at its four down-steps are equal to the values of L f at the tree roots: In this example, we have bpd Ó p1qq " 0, bpd Ó p2qq "´1, and bpd Ó p3qq " bpd Ó p4qq " 1. The red dots on the right indicate the encoding of a new tree.

The Bouttier-Di Francesco-Guitter mapping
We denote the set of all rooted pointed quadrangulations with n inner faces and 2σ boundary edges by where v ‚ stands for the distinguished pointed vertex. In the following part, we briefly recall the definition of the bijection Φ n : F n σˆB σ Ñ Q σ,‚ n introduced in [14].

The encoding of finite quadrangulations
We represent an element ppf, lq, bq P F n σˆB σ in the plane as follows. Firstly, we view b as a cycle of length 2σ: We start from a distinguished vertex labeled bp0q " 0 and label the remaining 2σ´1 vertices in the counterclockwise order by the values bp1q, bp2q, . . . , bp2σ´1q. Then we graft the trees pt 0 , . . . , t σ´1 q of f to the σ down-steps 0 ď i 0 ă i 1 ă¨¨¨ă i σ´1 ď 2σ´1 of b, such that t j is grafted on the vertex corresponding to the value bpi j q, in the interior of the cycle. We do it in such a way that different trees do not intersect. The vertices of t j are equipped with their labels shifted by bpi j q. Figure 7 illustrates this procedure.
We now build a rooted and pointed quadrangulation pq, v ‚ q out of ppf, lq, bq. First, we put an extra vertex v ‚ in the interior of the cycle representing b. The set of vertices of q is given by the tree vertices V pfq Y tv ‚ u. As for the edges of q, we define for 0 ď i ď 2n`σ´1 the successor succpiq P r0, 2n`σ´1sYt8u of i to be the first element k in the list pi`1, . . . , 2nσ´1 , 0, . . . , i´1q (from left to right) which has label L f pkq " L f piq´1. If there is no such element, we put succpiq " 8. We extend the contour exploration fp0q, . . . , fp2n`σ´1q of f by setting fp8q " v ‚ . We follow the exploration starting from the vertex fp0q (which is the root of t 0 ) and draw for each 0 ď i ď 2n`σ´1 an arc between fpiq and fpsuccpiqq, such that arcs do not cross. Except for the leaves, a vertex of f is visited at least twice in the contour exploration, so that there are in general several arcs connecting the vertices fpiq and fpsuccpiqq. The edges of q are given by all these arcs between the vertices V pfq Y tv ‚ u. f (1) f (2) f (3) f (4) f (5) f (6) f (8) f (9) f (10) f (13) f (14)  It only remains to root the quadrangulation. To that aim, we observe from Figure 7 that the 2σ boundary edges of q are in a order-preserving correspondence with the 2σ cycle edges. We root q at the edge corresponding to the first edge of the cycle (starting from the distinguished edge, in the clockwise order), oriented in such a way that the face of degree 2σ becomes the outer face (i.e., lies to the right of the root edge). Upon erasing the tree and cycle edges of the representation of ppf, lq, bq, and the vertices of b corresponding to up-steps, we obtain a rooted pointed quadrangulation pq, v ‚ q. A description of the reverse mapping Φ´1 n : Q σ,‚ n Ñ F n σˆB σ can be found in [14] or [11].

The encoding of infinite quadrangulations
Recall that Q is the completion of the set of finite rooted quadrangulations with a boundary with respect to d map . The aim of this section is to extend Φ n to a mapping Φ : pY n,σPN F n σˆB σ q Y pF 8ˆB8 q ÝÑ Q.
We proceed as follows. If ppf, lq, bq P F n σˆB σ , we put Φppf, lq, bq " Φ n ppf, lq, bq. (We forget the distinguished vertex of Φ n ppf, lq, bq and view the quadrangulation as an element in Q σ n Ă Q.) Now assume ppf, lq, bq P F 8ˆB8 . We consider the following representation of ppf, lq, bq in the upper half-plane: First, we identify b with the bi-infinite line obtained from connecting i P Z to i`1 by an edge. Vertex i is labeled bpiq. We attach the trees tp0q, tp1q, . . . of f to the down-steps of b to the right of 0, and the trees tp´1q, tp´2q, . . . to the down-steps of b to the left of´1, everything in the upper half-plane. Again, the labels in a tree are shifted by the underlying bridge label.
Similarly to the finite case, the vertex set of q " Φppf, lq, bq is given by V pfq; here, we add no additional vertex. For specifying the edges, we let the successor succ 8 piq of i P Z be the smallest number k ą i such that L f pkq " L f piq´1. Since by assumption inf iPN bpiq "´8, succ 8 piq is a finite number. We next connect the vertices fpiq and fpsucc 8 piqq by an arc for any i P Z, such that the resulting map is planar. The arcs form the edges of the infinite  Figure 7: A representation of an element ppf, lq, bq P F 6 6ˆB 6 in the plane and the associated rooted pointed quadrangulation pq, v ‚ q " Φ n ppf, lq, bq. The distinguished vertex of the cycle is the down-most vertex labeled 0. The trees are grafted to the 6 down-steps of b (here, d Ó p1q " 0, d Ó p2q " 1, d Ó p3q " 8, d Ó p4q " 10, d Ó p5q " 11, and d Ó p6q " 12). The tree edges are indicated by the dashed lines in the interior of the cycle. Note that three trees (those above the first, fourth and sixth down-step) consist of a single vertex. The labels in a tree are shifted by the bridge value of the down-step above which the tree is attached. Note that the 12 boundary edges of the cycle are in a order-preserving correspondence with the 12 boundary edges of q. (The two edges of q which lie entirely in the outer face are counted twice.) rooted quadrangulation q we are about to construct. In order to root the map, we observe that the bi-infinite line Z is in correspondence with the boundary edges of q, and we choose the edge corresponding to t0, 1u as the root edge of q (oriented such that the outer face lies to its right). A representation of ppf, lq, bq and of the resulting quadrangulation Φppf, lq, bq is depicted in Figure 8

Definition of the UIHPQ p
We are now in position to construct the UIHPQ p by means of the above mapping Φ applied to a (random) element in F 8ˆB8 , which we introduce first.
Let t be a finite random plane tree. Conditionally on t, we assign to t a random uniform labeling of its vertices, so that the pair pt, q becomes a well-labeled tree. Namely, given t, we first equip each edge of t with an independent random variable uniformly distributed in t´1, 0, 1u. Then we define the label puq of a vertex u P V ptq to be the sum over all labels along the unique (non-backtracking) path from the tree root to u.
We consider Galton-Watson trees with a (sub-)critical geometric offspring law µ p of parameter p, p P r0, 1{2s, that is, µ p pkq " p k p1´pq, k P N 0 . If t is such a tree, we call it a p-Galton-Watson tree. Equipped with a random uniform labeling as described before, we say that the pair pt, p puqq uPV ptq q is a uniformly labeled p-Galton-Watson tree.
A uniformly labeled infinite p-forest is a random element pf ppq 8 , l ppq 8 q taking values in F 8 , such that pt i , l ppq 8 aeV pt i qq, i P Z, are independent uniformly labeled p-Galton-Watson trees. f (2) f (3) f (4) f (5) f (6) f (7) f(8) A uniform infinite bridge is a random element b 8 " pb 8 piq, i P Zq in B 8 such that b 8 has the law of a two-sided simple symmetric random walk starting from b 8 p0q " 0. We stress that our wording differs from [8], where a uniform infinite bridge refers to a two-sided random walk with a geometric offspring law of parameter 1{2. See also Lemma 2 below. Definition 5. Fix p P r0, 1{2s. Let pf ppq 8 , l ppq 8 q be a uniformly labeled infinite p-forest, and independently of pf ppq 8 , l ppq 8 q, let b 8 be a uniform infinite bridge. Then the UIHPQ p with skewness parameter p is given by the (rooted) random infinite quadrangulation Q 8 8 ppq " pV pQ 8 8 ppqq, d gr , ρq with an infinite boundary, which is obtained from applying the Bouttier-Di Francesco-Guitter mapping Φ to ppf ppq 8 , l ppq 8 q, b 8 q. In case p " 1{2, we simply write Q 8 8 , which denotes then the (standard) uniform infinite half-planar quadrangulation with a general boundary.
Remark 9. Let f ppq 8 be the encoding forest of the UIHPQ p . Instead of working with metric balls around the root vertex in the UIHPQ p , it will -due to the specific construction of the latter -often be more practical to consider metric balls around the vertex corresponding to the tree root f ppq 8 p0q in the UIHPQ p . Similarly, if Q σ n P Q σ n is a uniform quadrangulation and f n its encoding forest, it will be more natural to consider balls around f n p0q in Q σ n . Since the distance between f ppq 8 p0q or f n p0q and the root of the map is stochastically bounded (it may also be zero), this makes no difference in terms of scaling limits whatsoever; see [8,Lemma 5.6]. We shall use the notation B p0q r pQ 8 8 ppqq for the metric ball of radius r around f ppq 8 p0q in the UIHPQ p . Analogously, we define B p0q r pQ σ n q.

The UIHPQ p as a local limit of uniform quadrangulations
In this part, we prove Theorem 1 and Proposition 1. We begin with the former. The case p " 1{2 has already been treated in [8], and the case p " 0 will be considered afterwards, so we first fix 0 ă p ă 1{2 and let pσ n , n P Nq be a sequence of positive integers satisfying σ n " 1´2p p n`opnq. Recall that rooted pointed quadrangulations in Q ‚ n,σn are in one-to-one correspondence with elements in F n σnˆB σn . For proving Theorem 1, the key step is to control the law of the first k trees in a forest f n chosen uniformly at random in F n σn , for k arbitrarily large but fixed. We will see in Lemma 1 below that their law is close to the law of k independent p-Galton-Watson trees when n is sufficiently large. Together with a convergence result of bridges (Lemma 2), this allows us to couple contour and label functions of Q σn n and the UIHPQ p , such that with high probability, we have equality of balls of a constant radius around the roots in Q σn n and the UIHPQ p , respectively. This readily implies the theorem. We begin with the necessary control over the trees. Since the result on the tree convergence is of some interest on its own, we formulate an optimal version, which is stronger than we what need for mere local convergence as stated in Theorem 1. The exact formulation depends on the error term in the expression for σ n . Let us put Lemma 1. Fix 0 ă p ă 1{2, and let pσ n , n P Nq be a sequence of positive integers satisfying σ n " 1´2p p n`opnq. Define γ n in terms of σ n and p. Let pt i q 1ďiďσn be a family of σ n independent 1{2-Galton-Watson trees, and let pt ppq i q 1ďiďσn be a family of σ n independent p-Galton-Watson trees. Then, if pk n , n P Nq is a sequence of positive integers satisfying k n ď σ n and k n " o`mintn 1{2 , n{γ n u˘as n Ñ 8, we have Remark 10. We stress that if we only know γ n " opnq as assumed in the statement of Theorem 1, we can at least choose k n equals an (arbitrary) large constant k P N. This suffices in any case to show local convergence towards the UIHPQ p ; see Proposition 4 below. Lemma 1 may be seen as a complement to the results on coupling of trees in [8]; it treats a regime not considered in that work.
Proof. We write P n for the conditional law of pt i q 1ďiďkn given ř σn i"1 |t i | " n, and Q n for the (unconditioned) law of pt ppq i q 1ďiďkn . Given a family f of k n trees, we write vpfq for the sum of their sizes, i.e., the total number of edges. Note that supppP n q " supppQ n q X tf : vpfq ď nu.
We now proceed in two steps. First, we show that for each ε ą 0, there exists a constant K ą 0 such that Q n ptf : vpfq ą Kk n uq ď ε, P n ptf : vpfq ą Kk n uq ď ε.
We then show that for large enough n, we have for any f P supppP n q of total size vpfq ď Kk n , 1´ε ďˇˇˇˇP n pfq Q n pfqˇˇˇˇď 1`ε.
Clearly, (4) and (5) imply the claim of the lemma. We first prove (4). Let pS ppq pmq, m P N 0 q be a random walk on the integers starting from S ppq p0q " 0 with increments distributed according to pδ 1`p 1´pqδ´1. Set, for P Z, Note that S ppq pmq`p1´2pqm, m P N 0 , is a martingale. We now use that the total size of k n trees under Q n and pT ppq kn´k n q{2 are equal in distribution; see the discussion in Section 3.2.2. Applying Markov's inequality in the second and the optional stopping theorem in the third step, we obtain for K large enough For bounding the second probability in (4), we let pSpmq, m P N 0 q be a simple symmetric random walk started from Sp0q " 0 and write T for its first hitting time of P Z. Then P n pv ą Kk n q " P pT´k n ą Kk n | T´σ n " 2n`σ n q .
Let us abbreviate N " 2n`σ n , and K n " rKk n s. We recall Kemperman's formula; see [36, Section 6.1]. Applying first the Markov property at time K n and then Kemperman's formula to both the nominator and denominator gives, for large n, P pT´k n ą K n | T´σ n " N q " E " 1 tT´k n ąKnu P pT´σ n " N | SpK n qq P pT´σ n " N q  " E " 1 tT´k n ąKnu σ n pσ n`S pK n qq N pN´K n q P pSpN q "´σ n | SpK n qq P pSpN q "´σ n q  ď 2 P pSpK n q ą´k n | SpN q "´σ n q .
Let pSpmq, m P N 0 q be the random walk starting fromSp0q " 0 with steps Clearly, pSpmq, m P N 0 q is a martingale, and we find the relation P`SpK n q ě´k nˇS pN q "´σ n˘" P´SpK n q ě K n σ n N´k nˇS pN q " 0¯.
We now estimate P´SpK n q ě K n σ n N´k nˇS pN q " 0ď P´SpN q ě 0¯´1 P´SpK n q ě K n σ n N´k n¯ď 3 E "S pK n q 2 ı pK n σn N´k n q 2 ď 12K n pK n σn N´k n q 2 .
Here, in the next to last inequality, we have used Doob's inequality, as well as the bound PpSpN q ě 0q ě 1{3 for large n, which is a direct consequence of the martingale central limit theorem. Since σ n {N remains bounded away from zero (recall that p ă 1{2), the last expression on the right hand side can be made arbitrarily small, upon choosing K large. This proves P n pv ą Kk n q ď ε for K large enough, and hence (4) holds. We turn to (5). First, observe that for a fixed f in the support of P n , P n pfq is the probability to see k n particular trees of total size vpfq in a forest consisting of σ n trees with total size n. An application of Kemperman's formula gives P n pfq " σn´kn 2pn´vpfqq`σn´kn 2 2pn´vpfqq`σn´kn P pSp2pn´vpfqq`σ n´kn q " σ n´kn q σn 2n`σn 2 2n`σn P pSp2n`σ n q " σ n q .
On the other hand, we know from (2) that Q n pfq " p vpfq p1´pq vpfq`kn . Display (5) will therefore follow if we show that for large n and all f with vpfq ď Kk n , 1´ε ďˇˇˇˇP pSp2pn´vpfqq`σ n´kn q " σ n´kn q P pSp2n`σ n q " σ n q pp2pq´v pfq p2p1´pqq´p vpfq`knqˇď 1`ε. (6) For a given f with vpfq ď Kk n , let us abbreviate y n " 2pn´vpfqq`σ n´kn , x n " σ n´kn , and v n " vpfq. Clearly, P pSp2pn´v n q`σ n´kn q " σ n´kn q P pSp2n`σ n q " σ n q "`y n yn´xn 22 n`σn n˘2 2vn`kn .
Combining the last two displays, it remains to show that 1´ε ďˇˇˇˇy n !pn`σ n q!n! yn´xn 2 ! yn`xn 2 !p2n`σ n q! p´v n p1´pq´p vn`knqˇď 1`ε.
The constants in the following error terms are uniform in the choice of f satisfying v n " vpfq ď Kk n . By Stirling's formula and a rearrangement of the terms, we obtain y n !pn`σ n q!n! yn´xn 2 ! yn`xn 2 !p2n`σ n q! " p1`op1qqˆy Recall that k n " o`mintn 1{2 , n{γ n u˘and v n ď Kk n . We replace x n and y n by their values and obtain for the product I of the first three factors I " exp p´p2v n`kn qq exp ppv n`kn qq exp pv n q`1`Opk 2 n {nq˘" 1`op1q.
Recalling the particular form of σ n for the product II of the last two factors, we arrive at II " p vn p1´pq vn`kn p1`Opγ n {nqq 2vn`kn " p vn p1´pq vn`kn p1`op1qq .
This proves (6) and hence the lemma.
We continue with a convergence result for uniform bridges b n P B σn towards b 8 .
Lemma 2. Let pσ n , n P Nq be a sequence of positive integers satisfying σ n Ñ 8 as n Ñ 8. Let b n be uniformly distributed in B σn , and let b 8 be a uniform infinite bridge as specified in Section 4. Then, if k n is a sequence of positive integers with k n ď σ n and k n " opσ n q as n Ñ 8, lim nÑ8 }Lawppb n p2σ n´kn q, . . . , b n p2σ n´1 q, b n p0q, b n p1q, . . . , b n pk n qqq Lawppb 8 p´k n q, . . . , b 8 p´1q, b 8 p0q, b 8 p1q, . . . , b 8 pk n qqq} TV " 0.
The proof follows from a small adaption of [8, Proof of Lemma 5.5] and is left to the reader. We stress, however, that in [8], b n and b 8 were defined in a slightly different manner, by grouping the`1-steps between two subsequent down-steps together to one "big" jump. Clearly, this does change the argument only in a minor way.
We are now in position to formulate an appropriate coupling of balls.
Proposition 4. Fix 0 ă p ă 1{2, and let pσ n , n P Nq be a sequence of positive integers satisfying σ n " 1´2p p n`opnq. Define γ n in terms of σ n as under (3), and put ξ n " mintn 1{4 , a n{γ n u. Then, given any ε ą 0, there exist δ ą 0 and n 0 P N such that for every n ě n 0 , we can construct on the same probability space copies of Q σn n and the UIHPQ p such that with probability at least 1´ε, the metric balls B δξn pQ σn n q and B δξn pUIHPQ p q of radius δξ n around the roots in the corresponding spaces are isometric.
The local convergence of Q σn n towards UIHPQ ppq is a weaker statement, hence Theorem 1 in the case 0 ă p ă 1{2 will follow from the proposition.
Proof. The proof is in spirit of [8, Proof of Proposition 3.11], requires, however, some modifications. We will indicate at which place we may simply adopt the reasoning. We consider a random uniform element ppf n , l n q, b n q P F n σn , and a triplet ppf ppq 8 , l ppq 8 q, b 8 q consisting of a uniformly labeled infinite p-forest together with an (independent) uniform infinite bridge b 8 . We let pQ σn n , v ‚ q " Φ n ppf n , l n q, b n q and Q 8 8 ppq " Φppf ppq 8 , l ppq 8 q, b 8 q be the quadrangulations obtained from applying the Bouttier-Di Francesco-Guitter mapping to ppf n , l n q, b n q and ppf ppq 8 , l ppq 8 q, b 8 q, respectively. Recall that f n " pt 0 , . . . , t σn´1 q consists of σ n trees. For 0 ď k ď σ n´1 , we let tpf n , kq " t k , i.e., tpf n , kq is the tree of f n with index k, and we put tpf n , σ n q " tpf n , 0q. In a similar manner, tpf ppq 8 , kq denotes the tree of f ppq 8 indexed by k P Z. By Lemma 1, we find δ 1 ą 0 and n 1 0 P N such that for n ě n 1 0 , we can construct ppf n , l n q, b n q and ppf ppq 8 , l ppq 8 q, b 8 q on the same probability space such that with A n " tδ 1 ξ 2 n u, the event E 1 pn, δ 1 q " ! tpf n , iq " tpf ppq 8 , iq, tpf n , σ n´i q " tpf ppq 8 ,´iq for all 0 ď i ď A n ) X ! l n | tpf n ,iq " l ppq 8 | tpf ppq 8 ,iq , l n | tpf n ,σn´iq " l ppq 8 | tpf ppq 8 ,´iq for all 0 ď i ď A n ) has probability at least 1´ε{8. We now fix such a δ 1 for the rest of the proof. Recall that by our construction of the Bouttier-Di Francesco-Guitter bijection, the trees of f n are attached to the down-steps d Ó n piq " d Ó bn piq of b n , 1 ď i ď σ n , and similarly, the trees of f ppq 8 are attached to the down-steps d Ó 8 piq " d Ó b8 piq of b 8 , where now i P Z. In view of the above event, this incites us to consider additionally the event Note that on E 2 pn, δ 1 q, we automatically have d Ó n piq " d Ó 8 piq for 1 ď i ď A n`1 , and d Ó n pσ n´i`1 q " d Ó 8 p´iq for 1 ď i ď A n . Trivially, we have that d Ó 8 pA n`1 q ě A n`1 and d Ó 8 p´A n q ď´A n , but also, with probability tending to 1, d Ó 8 pA n`1 q ď 3A n and d Ó 8 p´A n q ě´3A n . Since, in any case, A n " opσ n q, we can ensure by Lemma 2 that the event E 2 pn, δ 1 q has probability at least 1´ε{8 for large n.
Now for δ ą 0, n P N, define the events By invoking Donsker's invariance principle together with Lemma 2 for the event E 3 (and again the fact that A n`1 ď d Ó 8 pA n`1 q ď 3A n and´3A n ď d Ó 8 p´A n q ď´A n with high probability), we deduce that for small δ ą 0, provided n is large enough, P`E 3 pn, δq˘ě 1´ε{8, and P`E 4 pn, δq˘ě 1´ε{8.
We will now assume that n 0 ě n 1 0 and δ ą 0 are such that for all n ě n 0 , the above bounds hold true, and work on the event E 1 pn, δ 1 q X E 2 pn, δ 1 q X E 3 pn, δq X E 4 pn, δq of probability at least 1´ε{2. We consider the forest obtained from restricting f n to the first A n`1 and the last A n trees, f 1 n " ptpf n , 0q, . . . , tpf n , A n q, tpf n , σ n´An q, . . . , tpf n , σ n´1 qq .
Similarly, we define f 1 ppq 8 . We recall the cactus bounds in the version stated in [8, (4.4) of Section 4.5]. Applied to Q σn n , it shows that for vertices v P V pf n qzV pf 1 n q, with d n denoting the graph distance, Applying now the analogous cactus bound [8, (4.6) of Section 4.5] to the infinite quadrangulation Q 8 8 ppq, we obtain the same lower bound for vertices v P V pf ppq 8 qzV pf With the same arguments as in [8, Proof of Proposition 3.11], we then deduce that vertices at a distance at most 5δξ n´1 from f n p0q in Q σn n agree with those at a distance at most 5δξ n´1 from f ppq 8 p0q in Q 8 8 ppq. Moreover, d n pu, vq " d ppq 8 pu, vq whenever u, v P B p0q 2δξn pQ σn n q.
This proves that the balls B p0q 2δξn pQ σn n q and B p0q 2δξn pQ 8 8 ppqq are isometric on an event of probability at least 1´ε{2. In order to conclude, it suffices to observe that the distances from f n p0q resp. f ppq 8 p0q to the root vertex in Q σn n resp. Q 8 8 ppq are stochastically bounded; see again Remark 9. Clearly, this implies that with probability tending to 1 as n increases, we have the inclusions B δξn pQ σn n q Ă B p0q 2δξn pQ σn n q and B δξn pQ 8 8 ppqq Ă B p0q 2δξn pQ 8 8 ppqq. As mentioned at the beginning, the case p " 1{2 has already been treated in [8, Proof of Proposition 3.11]: It is proved there that for δ small, balls of radius δ mint ? σ n , a n{σ n u in Q σn n and in the standard UIHPQ " UIHPQ 1{2 can be coupled with high probability, implying of course again local convergence of Q σn n towards the UIHPQ. Finally, it remains to consider the case p " 0 corresponding to σ n " n. This case is easy. We have the following coupling lemma.
Lemma 3. Let pσ n , n P Nq be a sequence of positive integers satisfying σ n " n. Put ξ n " σ n {n. Then, given any ε ą 0, there exist δ ą 0 and n 0 P N such that for every n ě n 0 , we can construct on the same probability space copies of Q σn n and the UIHPQ 0 such that with probability at least 1´ε, the metric balls B δξn pQ σn n q and B δξn pUIHPQ 0 q of radius δξ n around the roots in the corresponding spaces are isometric.
Proof. Let ppf n , l n q, b n q P F n σnˆB σn be uniformly distributed. By exchangeability of the trees, it follows that if k n " opσ n {nq, then the first and last k n trees of f n are all singletons with a probability tending to one. Applying Lemma 2, we can ensure that the event tb n piq " b 8 piq, b n p2σ n´i q " b 8 p´iq, 1 ď i ď k n u has a probability as large as we wish, provided n is large enough. Given ε ą 0, the same arguments as in the proof of Proposition 4 yield an equality of balls B δξn pQ σn n q and B δξn pUIHPQ 0 q for δ small and n large enough, on an event of probability at least 1´ε.
Let us now show that the space UIHPQ 0 defined in terms of the Bouttier-Di Francesco-Guitter mapping in Section 4.3 is nothing else than Kesten's tree associated to the critical geometric offspring law µ 1{2 .
Proof of Proposition 1. Let b 8 " pb 8 piq, i P Zq be a uniform infinite bridge, and let pf p0q 8 , l p0q 8 q be the infinite forest where all trees are just singletons (with label 0); see Section 4.3. The UIHPQ 0 is distributed as the infinite map Q 8 8 p0q " Φppf p0q 8 , l p0q 8 q, b 8 q. Since every vertex in f p0q 8 defines a single corner, properties of the Bouttier-Di Francesco-Guitter mapping (Section 4.2) imply that Q 8 8 p0q is a tree almost surely. Moreover, the set of vertices of Q 8 8 p0q is identified with the set of down-steps DSpb 8 q of the bridge. Following [9, Section 2.2.3], conditionally on b 8 , we introduce a function ϕ : Z Ñ DSpb 8 q that associates to i P Z the next down-step ě i with label b 8 piq (and i is mapped to itself if i P DSpb 8 q). According to our rooting convention, the root edge of Q 8 8 p0q connects ϕp0q to ϕp1q. Note that ϕ is not injective almost surely.
We recall that Kesten's tree can be represented by a half-line of vertices s 0 , s 1 , . . . , together with a collection of independent Galton-Watson trees with offspring law µ 1{2 grafted to the left and right side of each vertex s i , i P N 0 . We will now argue that the UIHPQ 0 Q 8 8 p0q has the same structure. In this regard, let us introduce the stopping times and denote by s i the vertex of Q 8 8 p0q given by ϕpS i q. Together with their connecting edges, the collection ps i , i P N 0 q forms a spine (i.e., an infinite self-avoiding path) in Q 8 8 p0q. The subtree rooted at s i on the left side of the spine is encoded by the excursion tb 8 pkq : S i ď k ď S i`1 u, in a way we describe next; see Figure 9 for an illustration. First note that by the Markov property, these subtrees for i P Z are i.i.d.. In order to determine their law, let us consider the subtree encoded by the excursion tb 8 pkq : 0 ď k ď S 1 u of b 8 . This subtree is rooted at s 0 " H, and the number of offspring of s 0 is the number of down-steps with label 1 between 0 and S 1 . Otherwise said, this is the number #t0 ă k ă S 1 : b 8 pkq " 0u of excursions of b 8 above 0 between 0 and S 1 . By the Markov property, this quantity follows the geometric distribution µ 1{2 of parameter 1{2. One can now repeat the argument for each child of s 0 , by considering the corresponding excursion above 0 encoding its progeny tree, inside the mother excursion. We obtain that the subtree stemming from s 0 on the left of the spine has indeed the law of a Galton-Watson tree with offspring distribution µ 1{2 .
The subtrees attached to the vertices s i , i P N 0 , on the right of the spine can be treated by a symmetry argument. Namely, letting we observe that the subtree rooted at s i to the right of the spine is coded by the (reversed) excursion tb 8 pkq :´S 1 i`1 ď k ď´S 1 i u. With the same argument as above, we see that it has the law of an (independent) µ 1{2 -Galton Watson tree. This concludes the proof.

The UIHPQ p as a local limit of Boltzmann quadrangulations
This section is devoted to the proof of Proposition 2. It is convenient to first prove the analogous result for pointed maps. For that purpose, we first extend the definitions of Boltzmann measures from Section 1.2.5 to pointed maps and then use a "de-pointing" argument. We use the notation Q ‚ f for the set of finite rooted pointed quadrangulations, and we write Q ‚,σ f for the set of finite pointed rooted quadrangulations with 2σ boundary edges. The corresponding partition functions read and the associated pointed Boltzmann distributions are defined by We will need the following enumeration result for pointed rooted maps. From [16, (23)] and [15, Section 3.3], we have for every 0 ď p ď 1{2 Note that the result (3.29) in [15] cannot be used directly, due to a difference in the rooting convention (there, the root vertex has to be chosen among the vertices of the boundary that are closest to the marked point).
Recall that g p " pp1´pq{3 for 0 ď p ď 1{2. The first step towards the proof of Proposition 2 is the following convergence result for pointed Boltzmann quadrangulations.
Proposition 5. Let 0 ď p ď 1{2. For every σ P N 0 , let Q ‚ σ ppq be a random rooted pointed quadrangulation distributed according to P ‚,σ gp . Then, we have the local convergence for the metric d map as σ Ñ 8 Proof. Let q P Q σ f , and ppf, lq, bq P Y ně0 F n σˆB σ such that q " Φppf, lq, bq. Moreover, let pf ppq σ , l ppq σ q be a uniformly labeled p-forest with σ trees, i.e., a collection of σ independent uniformly labeled p-Galton-Watson trees, and let b σ be uniformly distributed in B σ and independent of pf ppq σ , l ppq σ q. We have Here, for the first equality in the second line, we have used (2), the fact that the label differences are i.i.d. uniform in t´1, 0, 1u, and |B σ | "`2 σ σ˘. The last equality follows from the enumeration result (7) and the fact that the number of edges of f equals the number of faces of q. Thus, Q ‚ σ ppq is distributed as Φppf ppq σ , l ppq σ q, b σ q.
Now observe that f ppq σ is already a collection of σ independent p-Galton-Watson trees, and Lemma 2 allows us to couple the first and last opσq steps of b σ with those of a uniform infinite bridge b 8 . With exactly the same reasoning as in Proposition 4, we therefore obtain with high probability an isometry of balls B δ ? σ pQ ‚ σ ppqq and B δ ? σ pUIHPQ p q for all σ sufficiently large, provided δ is small enough. The stated local convergence follows. Proposition 2 is a consequence of the foregoing result and the following de-pointing argument inspired by [1,Proposition 14]. According to Remark 2, it suffices to consider the case p P r0, 1{2q.
In the following, by a small abuse of notation, we interpret P ‚,σ gp as a probability measure on Q f by simply forgetting the marked point.
Proof. Let #V be the mapping q Þ Ñ #V pqq, which assigns to a finite quadrangulation q its number of vertices. We have the absolute continuity relation [12, (5)] dP σ gp pqq " where K σ " pE ‚,σ gp r1{#V qsq´1. Then, Let pt ppq 0 , . . . , t ppq σ´1 q be a collection of independent p-Galton-Watson trees. The proof of Proposition 5 shows that under P ‚,σ gp , #V has the same law as Note that the summand`1 accounts for the pointed vertex, which is added to the tree vertices in the Bouttier-Di Francesco-Guitter mapping. Using the fact that #V pt ppq 0 q has the same law as pT ppq 1`1 q{2, where T ppq 1 is the first hitting time of´1 of a random walk with step distribution pδ 1`p 1´pqδ´1, an application of the optional stopping theorem gives Moreover, using p ă 1{2 and the description in terms of T ppq 1 , it is readily checked that the random variable #V pt ppq 0 q has small exponential moments. Cramer's theorem thus ensures that for every δ ą 0, there exists a constant C δ ą 0 such that P ‚,σ gp´ˇ# V´E ‚,σ gp r#V sˇˇą δσ¯ď expp´C δ σq.

5.3
The BHP θ as a local scaling limit of the UIHPQ p 's In this section, we prove Theorem 2. For the reminder, we fix a sequence pa n , n P Nq of positive reals tending to infinity and let r ą 0 be given. Similarly to [8, Proof of Theorem 3.4], the main step is to establish an absolute continuity relation of balls around the roots of radius ra n between the UIHPQ p for p P p0, 1{2s and the UIHPQ " UIHPQ 1{2 . To this aim, we compute the Radon-Nikodym derivative of the encoding contour function of the UIHPQ p with respect to that of the UIHPQ on an interval of the form r´sa 2 n , sa 2 n s for s ą 0. From Theorem 3.8 of [8] we know that a´1 n¨U IHPQ Ñ BHP 0 in distribution in the local Gromov-Hausdorff topology, jointly with a uniform convergence on compacts of (rescaled) contour and label functions. An application of Girsanov's theorem shows that the limiting Radon-Nikodym derivative turns the contour function of BHP 0 into the contour function of BHP θ , which allows us to conclude.
In order to make these steps rigorous, we begin with some notation specific to this section. Let f P CpR, Rq and x P R. We define the last (first) visit to x to the left (right) of 0, U x pf q " inftt ď 0 : f ptq " xu P r´8, 0s, T x pf q " inftt ě 0 : f ptq " xu P r0, 8s.
We agree that U x pf q "´8 if the set over which the infimum is taken is empty, and, similarly, T x pf q " 8 if the second set is empty. We will also apply U x to functions in Cpp´8, 0s, Rq, and T x to functions in Cpr0, 8q, Rq.
If f P CpR, Rq is the contour function of an infinite p-forest for some p P p0, 1{2s (or part of it defined on some interval), and if x P N, we use the notation vpf, xq " 1 2 pT´xpf q´U x pf q´2xq for the total number of edges of the 2x trees encoded by f along the interval rU x pf q, T´xpf qs. We set vpf, xq " 8 if U x pf q or T´xpf q is unbounded.
Given s ą 0, we put for n P N s n " tp3{2qsa 2 n u. Now let p P p0, 1{2s. Throughout this section and as usual, we assume that ppf ppq 8 , l ppq 8 q, b 8 q and ppf 8 , l 8 q, b 8 q encode the UIHPQ p Q 8 8 ppq and the standard UIHPQ Q 8 8 , respectively (see Definition 5). We stress that since the skewness parameter p does not affect the law of the infinite bridge b 8 , we can and will use the same bridge in the construction of both Q 8 8 ppq and Q 8 8 . We denote by pC ppq 8 , L ppq 8 q and pC 8 , L 8 q the associated contour and label functions, viewed as elements in CpR, Rq.
For understanding how the balls of radius ra n for some r ą 0 around the roots in Q 8 8 ppq and Q 8 8 are related to each other, we need to control the contour functions C ppq 8 and C 8 on rU sn , T´s n s for a suitable choice of s " sprq. In this regard, we first formulate an absolute continuity relation between the probability laws P ppq s,n and P s,n on CpR, Rq defined as follows: P ppq n,s " Law``C ppq 8 pt _ U sn pC ppq 8 q^T´s n pC ppq 8 qq, t P R˘˘, P n,s " Law ppC 8 pt _ U sn pC 8 q^T´s n pC 8 qq, t P Rqq .
Lemma 5. Let p P p0, 1q and s ą 0. The laws P ppq n,s and P n,s are absolutely continuous with respect to each other: For any f P supppP ppq n,s qp" supppP n,s qq, with s n as above, P ppq n,s pf q " p4pp1´pqq vpf,snq p2p1´pqq 2sn P n,s pf q.
Proof. By definition of C ppq 8 and C 8 , each element f P CpR, Rq in the support of P ppq n,s lies also in the support of P n,s and vice versa (note that p R t0, 1u).
More specifically, for such an f supported by these laws, P ppq n,s pf q resp. P n,s pf q is the probability of a particular realization of 2s n independent p-Galton-Watson trees resp. p1{2q-Galton-Watson trees with vpf, s n q tree edges in total. Therefore, by (2), P ppq n,s pf q " p vpf,snq p1´pq vpf,snq p1´pq 2sn , and P n,s pf q " 2´2 pvpf,snq`snq .
This proves the lemma.
We turn to the proof of Theorem 2. To that aim, we will work with rescaled and stopped versions of pC ppq 8 , L ppq 8 q and pC 8 , L 8 q, which encode the information of the first s n " tp3{2qsa 2 n u trees to the right of zero, and of the first s n trees to the left zero. Specifically, we let C 8,p n,s "`C 8,p n,s ptq, t P R˘"ˆ1 p3{2qa 2 n C ppq 8`p 9{4qa 4 n t _ U sn pC ppq 8 q^T´s n pC ppq 8 q˘, t P R˙, L 8,p n,s "`L 8,p n,s ptq, t P R˘"ˆ1 a n L ppq 8`p 9{4qa 4 n t _ U sn pC ppq 8 q^T´s n pC ppq 8 q˘, t P R˙, C 8 n,s "`C 8 n,s ptq, t P R˘"ˆ1 p3{2qa 2 n C 8`p 9{4qa 4 n t _ U sn pC 8 q^T´s n pC 8 q˘, t P R˙, L 8 n,s "`L 8 n,s ptq, t P R˘"ˆ1 a n L 8`p 9{4qa 4 n t _ U sn pC 8 q^T´s n pC 8 q˘, t P R˙.
Following our notation from Section 3.1, we denote by X θ " pX θ ptq, t P Rq and W θ " pW θ ptq, t P Rq the contour and label functions of the limit space BHP θ . We also put X θ,s "`X θ,s ptq, t P R˘"`X θ`t _ U s pX θ q^T´spX θ q˘, t P R˘, W θ,s "`W θ,s ptq, t P R˘"`W θ`t _ U s pX θ q^T´spX θ q˘, t P R˘.
Accordingly, we write X 0 , W 0 and X 0,s , W 0,s for the corresponding functions associated to BHP 0 . We will make use of the following joint convergence.
Lemma 6. Let r, s ą 0. Then, in the notation from above, we have the joint convergence in law in CpR, RqˆCpR, RqˆK, Proof. Both statements are proved in [8]; to give a quick reminder, first note by standard random walk estimates that for each δ ą 0, there exists a constant c δ ą 0 such that PpvpC 8 , s n q ą c δ a 4 n q ď δ; see [8, Proof of Lemma 6.18] for details. Together with the joint convergence in law in CpR, Rq 2ˆK obtained in [8, (6.30) of Remark 6.17], which readŝ the first claim of the statement follows, and the second is then a consequence of this.
We turn now to the Proof of Theorem 2.
Proof of Theorem 2. We fix a sequence pp n , n P Nq Ă p0, 1{2s of the form By Remark 9 and the observations in Section 1.2.7, the claim follows if we show that for all r ą 0, as n Ñ 8, We define a similar event in terms of the two-sided Brownian motion γ " pγptq, t P Rq scaled by the factor ? 3, which forms part of the construction of the space BHP θ given in Section 3.1, * .
Using the cactus bound, it was argued in [8, Proof of Theorem 3.4] that on the event E 1 pn, sq, for any p P p0, 1{2s, the ball B p0q ran pQ 8 8 ppqq viewed as a submap of Q 8 8 ppq is a measurable function of pC 8,p n,s , L 8,p n,s q. (In [8], only the case p " 1{2 was considered, but the argument remains exactly the same for all p, since the encoding bridge b 8 does not depend on the choice of p.) Similarly, on E 2 psq, the ball B r pBHP θ q for any θ ě 0 is a measurable function of pX θ,s , W θ,s q. Now let ε ą 0 be given. By the (functional) central limit theorem, we find that for s ą 0 and n 0 P N sufficiently large, it holds that for all n ě n 0 , PpE 1 pn, sqq ě 1´ε. By choosing s possibly larger, we can moreover ensure that PpE 2 psqq ě 1´ε. We fix such s ą 0 and n 0 P N such that for all n ě n 0 , both events E 1 pn, sq and E 2 psq have probability at least 1´ε.
Next, consider the laws P ppnq n,s and P n,s defined just above Lemma 5, and put for f P CpR, Rq λ n,s pf q " p4p n p1´p n qq vpf,snq p2p1´p n qq 2sn .
Then, with F : CpR, Rq 2ˆK Ñ R measurable and bounded, Lemma 5 shows Note that on the left side, we consider the closed ball of radius ra n around the vertex f 8 p0q in the UIHPQ pn Q 8 8 pp n q, whereas on the right side, we look at the corresponding ball in the standard UIHPQ Q 8 8 with contour and label functions C 8 and L 8 . Plugging in the value of p n in (9), we get λ n,s pf q "ˆ1`2 θ 3a 2 Applying both statements of Lemma 6, and using (11), it follows that for large n ě n 1 pεqˇˇE " λ n,s pC 8 qF`C 8 n,s , L 8 n,s , B p0q r`a´1 n¨Q

8˘˘‰É
" exp`2sθ´pT´s´U s qpX 0 qθ 2 {2˘F`X 0,s , W 0,s , B r pBHP 0 q˘‰ˇˇď ε. (12) The rest of the proof is now similar to [8,Proof of Theorem 3.4]. Applying Pitman's transform and Girsanov's theorem, we have for continuous and bounded G : CpR, On E 2 psq, B r pBHP 0 q is a measurable function of pX 0,s , W 0,s q, and B r pBHP θ q is given by the same measurable function of pX θ,s , W θ,s q. Consequently, Recall that the events E 1 pn, sq and E 2 psq have probability at least 1´ε. Using this fact together with (10), (12), (13) and the triangle inequality, we find a constant C " CpF, s, θq such that for sufficiently large n,ˇE This implies the theorem.

5.4
The ICRT as a local scaling limit of the UIHPQ p 's Theorem 3 states that the ICRT appears as the distributional limit of a´1 n¨U IHPQ pn when a n Ñ 8 and p n P r0, 1{2s satisfies a 2 n p1´2p n q Ñ 8 as n Ñ 8. In essence, the idea behind the proof is the following. Fix r ą 0, and sequences pa n q n and pp n q n with the above properties. It turns out that in the UIHPQ pn , vertices at a distance less than ra n from the root are to be found at a distance of order opa n q from the boundary. Therefore, upon rescaling the graph distance by a factor a´1 n , the scaling limit of the UIHPQ pn in the local Gromov-Hausdorff sense will agree with the scaling limit of its boundary. Upon a rescaling by a 2 n in time and a´1 n in space, the encoding bridge b 8 converges to a two-sided Brownian motion, which in turn encodes the ICRT.
The above observations are most naturally turned into a proof using the description of the Gromov-Hausdorff metric in terms of correspondences between metric spaces; see [18,Theorem 7.3.25]. Lemma 10 below captures the kind of correspondence we need to construct. Our strategy of showing convergence of quadrangulations with a boundary towards a tree has already been successfully implemented before; see, for instance, [11,Proof of Theorem 5].
For the reminder of this section, we write ppf pnq 8 , l pnq 8 q, b 8 q for a uniformly labeled infinite p n -forest together with an (independent) uniform infinite bridge b 8 , and we assume that the UIHPQ pn is given in terms of ppf pnq 8 , l pnq 8 q, b 8 q, via the Bouttier-Di Francesco-Guitter mapping. We interpret the associated contour function C The core of the argument lies in the following lemma, which gives the necessary control over distances to the boundary, via a control of the labels l pnq 8 . We will use it at the very end of the proof of Theorem 3, which follows afterwards.
Lemma 7. Let pa n , n P Nq be a sequence of positive reals tending to infinity, and pp n , n P Nq Ă r0, 1{2q be a sequence satisfying a 2 n p1´2p n q Ñ 8 as n Ñ 8. Then, in the notation from above, we have the distributional convergence in CpR, R 2 q as n Ñ 8,´1 a 2 n C pnq 8´a 2 n 1´2p n s¯, 1 a n l pnq 8´a 2 n 1´2p n s¯, s P R˙p dq ÝÑ pp´s, 0q, s P Rq .
Proof. We have to show joint convergence of C pnq 8 and l pnq 8 on any interval of the form r´K, Ks, for K ą 0. Due to an obvious symmetry in the definition of the contour function, we may restrict ourselves to intervals of the form r0, Ks. Fix K ą 0, and put θ n " p1´2p n q´1a 2 n . We first show that a´2 n C pnq 8 pθ n sq, s P R, converges on r0, Ks to gpsq "´s in probability. For that purpose, recall that C pnq 8 on r0, 8q has the law of an linearly interpolated random walk started from 0 with step distribution p n δ 1`p 1´p n qδ´1. Set K n " rKθ n s, and let δ ą 0. By using Doob's inequality in the second line, Thanks to our assumption on p n , the right hand side converges to zero, and the convergence of the contour function is established. Showing joint convergence together with the (rescaled) labels l pnq 8 is now rather standard: First, we may assume by Skorokhod's theorem that a´2 n C pnq 8 pθ n sq converges on r0, Ks almost surely. Now fix 0 ď s ď K. Conditionally given C pnq 8 on r0, Kθ n s, we have by construction, for pη i , i P Nq a sequence of i.i.d. uniform random variables on t´1, 0, 1u, and with C pnq 8 ptθ n suq " min r0,tθnsus C pnq 8 , Conditionally given C By our assumption, a´2 n pC pnq 8 ptθ n suq´C pnq 8 ptθ n suqq converges to zero almost surely, and we conclude´a´2 n C pnq 8 ptθ n suq, a´1 n l pnq 8 ptθ n suq¯p dq ÝÑ p´s, 0q as n Ñ 8.
Since both C pnq 8 and l pnq 8 are Lipschitz almost surely, the claim follows with tθ n su replaced by θ n s. Joint finite-dimensional convergence can now be shown inductively: As for twodimensional convergence on r0, Ks, we simply note that when 0 ď s 1 ă s 2 ď K are such that C where pη 1 i , i P Nq is an i.i.d. copy of pη i , i P Nq. Using almost sure convergence of a´2 n C pnq 8 pθ n sq on r0, Ks and an argument similar to that in the one-dimensional convergence considered above, we get two-dimensional convergence of pa´2 n C pnq 8 pθ n sq, a´1 n l pnq 8 pθ n sqq on r0, Ks, as wanted. Some more details can be found in [34,Proof of Theorem 4.3]. Higher-dimensional convergence is now shown inductively and is left to the reader. It remains to show tightness of the rescaled labels. We begin with the following lemma. Lemma 8. Let K ą 0, pa n , n P Nq and pp n , n P Nq be as above. Then, for any q ě 2, there exists a constant C q ą 0 such that for any n P N and any 0 ď s 1 , s 2 ď K, we have (with θ n " p1´2p n q´1a 2 n , as before) a´2 q n E "ˇˇC pnq 8 pθ n s 1 q´C pnq 8 pθ n s 2 qˇˇq ı ď C q |s 1´s2 | q{2 .
Since a´2 q n θ q{2 n ď a´q n p1´2p n q´q {2 Ñ 0 by assumption on p n , the claim of the lemma follows in this case. Now let |s 1´s2 | ą θ´1 n . We may assume s 2 ě s 1 . Using the triangle inequality and again the assumption on p n , we see that it suffices to establish the claim in the case where θ n s 1 and θ n s 2 are integers. In this case, by definition of C pnq 8 , where pϑ i , i P Nq are (centered) i.i.d. random variables with distribution p n δ 2p1´pnq`p 1ṕ n qδ´2 pn . Using that |a`b| q ď 2 q´1 p|a| q`| b| q q for reals a, b, we get The second term within the parenthesis is equal to a 2q n |s 2´s1 | q ď K q{2 a 2q n |s 2´s1 | q{2 . As for the sum, we apply Rosenthal's inequality and obtain for some constant C 1 q ą 0, Using once more that a´2 q n θ q{2 n Ñ 0 by assumption on p n , the lemma is proved.
Let κ ą 0. By the theorem of Kolmogorov-Čentsov (see [29,Theorem 2.8]), it follows from the above lemma that there exists M " M pκq ą 0 such that for all n P N, the event ď M + has probability at least 1´κ. We will now work conditionally given E n .
Tightness of the conditional laws of a´1 n l pnq 8 pθ n sq, 0 ď s ď K, given E n is a standard consequence of this lemma; see [29,Problem 4.11]). Since κ in the definition of E n can be chosen arbitrarily small, tightness of the unconditioned laws of the rescaled labels follows, and so does Lemma 7.
It therefore only remains to prove Lemma 9.
Proof of Lemma 9. With arguments similar to those in the proof of Lemma 8, we see that it suffices to prove the claim in the case where θ n s 1 and θ n s 2 are integers (and s 1 ď s 2 ). Let ∆C pnq 8 ps 1 , s 2 q " C pnq 8 pθ n s 1 q`C pnq 8 pθ n s 2 q´2 min By definition of pC On E n , we have the bound a´2 n |∆C n 8 ps 1 , s 2 q| ď 2 sup 0ďsătďK |C n 8 pθ n sq´C n 8 pθ n tq| a 2 n |s´t| 2{5 |s 1´s2 | 2{5 ď M |s 1´s2 | 2{5 , and the claim of the lemma follows.
Finally, for proving Theorem 3, we will make use of the following lemma.
Lemma 10 (Lemma 5.7 of [8]). Let r ě 0. Let E " pE, d, ρq and E 1 " pE 1 , d 1 , ρ 1 q be two pointed complete and locally compact length spaces. Consider a subset R Ă EˆE 1 which has the following properties: • pρ, ρ 1 q P R, • for all x P B r pEq, there exists x 1 P E 1 such that px, x 1 q P R, • for all y 1 P B r pE 1 q, there exists y P E such that py, y 1 q P R.
A proof is given in [8]. Although R is not necessarily a correspondence in the sense of [18], we might call the supremum on the right side of the inequality the distortion of R.
Proof of Theorem 3. We let pa n , n P Nq and pp n , n P Nq Ă r0, 1{2s be two sequences as in the statement, and, as mentioned at the beginning of this section, we assume that the UIHPQ pn Q 8 8 pp n q with skewness parameter p n is encoded in terms of ppf pnq 8 , l pnq 8 q, b 8 q. Local Gromov-Hausdorff convergence in law of a´1 n¨Q 8 8 pp n q towards the ICRT follows if we prove that for each r ě 0, in distribution in K, where we recall again that B p0q r pa´1 n¨Q 8 8 pp n qq denotes the ball of radius r around the vertex f pnq 8 p0q in the rescaled UIHPQ pn . We will show the claim for r " 1. The proof follows essentially the line of argumentation in [8, Proof of Theorem 3.5]; since the argument is short, we repeat the main steps for completeness. We will apply Lemma 10 in the following way. The ICRT takes the role of the space E 1 , with the equivalence class r0s of zero being the distinguished point. Then, we consider for each n P N the space a´1 n¨Q 8 8 pp n q pointed at f pnq 8 p0q, which takes the role of E in the lemma. We construct a subset R n Ă EˆE 1 with the properties of Lemma 10, such that its distortion, that is, the quantity sup t|dpx, yq´d 1 px 1 , y 1 q| : px, x 1 q, py, y 1 q P R n u, is of order op1q for n tending to infinity. By Lemma 10, this will prove (16). We remark that Q 8 8 pp n q is not a length space, hence Lemma 10 seems not applicable at first sight. However, as explained in Section 1.2.7, by identifying each edge with a copy of r0, 1s and upon extending the graph metric isometrically, we may identify Q 8 8 pp n q with the (associated) length space, which we denote by Q 8 8 pp n q " pV pQ 8 8 pp n qq, d gr , ρq.
Here and in what follows, d gr is the graph metric isometrically extended to Q 8 8 pp n q. Note that the vertex set V pf pnq 8 q may be viewed as a subset of Q 8 8 pp n q, and between points of V pf pnq 8 q, the distances d gr and d gr agree. Moreover, as a matter of fact, every point in Q 8 8 pp n q is at distance at most 1{2 away from a vertex of f pnq 8 . Recall that pb 8 ptq, t P Rq has the law of a (linearly interpolated) two-sided symmetric simple random walk with b 8 p0q " 0. Let X " pX t , t P Rq be a two-sided Brownian motion with X 0 " 0. By Donsker's invariance principle, we deduce that as n tends to infinity, Using Skorokhod's representation theorem, we can assume that the above convergence holds almost surely on a common probability space, uniformly over compacts. Now let δ ą 0, and fix α ą 0 and n 0 P N such that the event ă´a n * has probability at least 1´δ for n ě n 0 . From now on, we argue on the event Epn, αq. We moreover assume that the ICRT pT X , d X , r0sq is defined in terms of X, and we write p X : R Ñ T X for the canonical projection.
Recall that the vertices of f pnq 8 " pt i , i P Zq are identified with the vertices of Q 8 8 pp n q. The mapping Ipvq P Z gives back the index of the tree a vertex v P V pf pnq 8 q belongs to. We extend I to the elements of the length space Q 8 8 pp n q as follows. By viewing V pf pnq 8 q as a subset of Q 8 8 pp n q as explained above, we associate to every point u of Q 8 8 pp n q its closest vertex v P V pf pnq 8 q satisfying d gr pf pnq 8 p0q, vq ě d gr pf pnq 8 p0q, uq. Note again d gr pv, uq ď 1{2. Put A n " tu P Q 8 8 pp n q : Ipuq P r´αa 2 n , αa 2 n su.
A direct application of the cactus bound [8, (4.6) of Section 4.5] shows that on Epn, αq, d gr pf pnq 8 p0q, uq ą a n whenever Ipuq R A n , implying that the set A n contains the ball B p0q 1 pQ 8 8 pp n qq of radius 1 around the vertex f pnq 8 p0q. Moreover, still on Epn, αq, d X pr0s, tq ą 1 whenever |t| ą α.
We now define R n Ă Q 8 8 pp n qˆT X by R n " pu, p X ptqq : u P A n , t P r´α, αs with Ipuq " tta 2 n u ( . Letting E " pQ 8 8 pp n q, a´1 n d gr , f pnq 8 p0qq, E 1 " pT X , d X , r0sq, r " 1, we find that given the event Epn, αq, the set R n satisfies the requirements of Lemma 10. We are now in the setting of [8, Proof of Theorem 3.5]: All what is left to show is that on Epn, αq, the distortion of R n converges to 0 in probability. However, with the same arguments as in the cited proof and using the convergence (17), we obtain lim sup nÑ8 sup t|d gr px, yq´d X px 1 , y 1 q| : px, x 1 q, py, y 1 q P R n u ď lim sup nÑ8 5´sup An l pnq 8´i nf An l pnq 8ā n .
An appeal to Lemma 7 shows that the right hand side is equal to zero, and the proof of the theorem is completed.
6 Proofs of the structural properties 6.1 The branching structure behind the UIHPQ p In this section, we describe the branching structure of the UIHPQ p and prove Theorem 4. We will first study a similar mechanism behind Boltzmann quadrangulations Q and Q σ drawn according to P gp,zp and P σ gp , respectively (Proposition 6 and Corollary 4), and then pass to the limit σ Ñ 8 using Proposition 2.
To begin with, we follow an idea of [23]: We associate to a (finite) rooted map a tree that describes the branching structure of the boundary of the map. Precisely, for every finite rooted quadrangulation q with a boundary, we define the so-called scooped-out quadrangulation Scooppqq as follows. We keep only the boundary edges of q and duplicate those edges which lie entirely in the outer face (i.e., whose both sides belong to the outer face). The resulting object is a rooted looptree; see Figure 10.
To a scooped-out quadrangulation we associate its tree of components TreepScooppqqq as defined in Section 3.2.4. Following [23], we call this tree, by a slight abuse of terminology, the tree of components of q and use the notation t " Treepqq. It is seen that vertices in t ‚ have even degree in t, due to the bipartite nature of q.
By gluing the appropriate rooted quadrangulation with a simple boundary into each cycle of Scooppqq, we recover the quadrangulation q. This provides a bijection Ψ : q Þ Ñ pTreepqq, pp q u : u P V pTreepqq ‚ qq q ∂q Scoop(q) Figure 10: A rooted quadrangulation, its boundary and the associated scooped-out quadrangulation.
between, on the one hand, the set Q f of finite rooted quadrangulations with a boundary and, on the other hand, the set of plane trees t with vertices at odd height having even degree, together with a collection pp q u : u P V pt ‚ qq of rooted quadrangulations with a simple boundary and respective perimeter degpuq, for degpuq the degree of u in t. We remark that the inverse mapping Ψ´1 can be extended to an infinite but locally finite tree together with a collection of quadrangulations with a simple boundary attached to vertices at odd height, yielding in this case an infinite rooted quadrangulation q. We recall from Section 1.2.5 the definitions of the Boltzmann laws P g,z and P σ g , and their analogs with support on quadrangulations with a simple boundary, p P g,z and p P σ g . Their corresponding partition functions are F , F σ and p F , p F σ . We are now interested in the law of the tree of components under P g,z . To begin with, we adapt some enumeration results from [15] to our setting. For every 0 ď p ď 1{2, recall that g p " pp1´pq{3 and z p " p1´pq{4. Then, (3.15), (3.27) and (5.16) of [15] all together provide the identities for 0 ď p ď 1{2 and σ P N 0 . Moreover, for σ P N and 0 ă p ď 1{2, while p F 0 pg p q " 1. If p " 0 and hence g p " 0, then p F k p0q " δ 0 pkq`δ 1 pkq for all k P N 0 . (Indeed, under the maps with no inner faces, the vertex map and the map consisting of one oriented edge are the only maps with a simple boundary. ) We already introduced in Section 2.3 two probability measures µ˝and µ ‚ on N 0 given by µ˝pkq " 1 F pg p , z p qˆ1´1 F pg p , z p q˙k , k P N 0 , µ ‚ p2k`1q " 1 F pg p , z p q´1 " z p F 2 pg p , z p q ‰ k`1 p F k`1 pg p q, k P N 0 , with µ ‚ pkq " 0 if k even. The tree of components of the scooped-out quadrangulation ScooppQq when Q is drawn according to P gp,zp may now be characterized as follows.
Proposition 6. Let 0 ď p ď 1{2, and let Q be distributed according to P gp,zp . Then the tree of components TreepQq is a two-type Galton-Watson tree with offspring distribution pµ˝, µ ‚ q as given above. Moreover, conditionally on TreepQq, the quadrangulations with a simple boundary associated to Q via the bijection Ψ are independent with respective Boltzmann distribution p P degpuq gp for u P V pTreepQq ‚ q, where degpuq denotes the degree of u in TreepQq.
Proof. Note that vertices at even height of TreepQq have an odd number of offspring almost surely. Let t be a finite plane tree satisfying this property. Let also pp q u : u P V pt ‚ qq be a collection of rooted quadrangulations with a simple boundary and respective perimeters degpuq, and set q " Ψ´1pt, pp q u : u P V pt ‚ qqq. Then, writing Ψ˚P for the push-forward measure of P by Ψ, Ψ˚P gp,zp pt, pp q u : u P V pt ‚ qqq " z Applying the first equality with c " 1´1{F pg p , z p q and the second one with c " F pg p , z p q gives Ψ˚P gp,zp pt, pp q u : u P tqq " ź uPt˝1 F pg p , z p qˆ1´1 F pg p , z p q˙d egpuq´1 ź uPt‚ 1 F pg p , z p q´1`z p F 2 pg p , z p q˘d egpuq{2 p F degpuq{2 pg p q ź uPt‚ g #Fpp q u q p p F degpuq{2 pg p q , where we agree that 0{0 " 0. Therefore, Ψ˚P gp,zp pt, pp q u : u P tqq " ź uPt˝µ˝p k u q ź uPt‚ µ ‚ pk u q ź uPt‚ p P degpuq gp pp q u q, which is the expected result.
Corollary 4. Let 0 ď p ď 1{2, σ P N, and let Q be distributed according to P σ gp . Then the tree of components TreepQq is a two-type Galton-Watson tree with offspring distribution pµ˝, µ ‚ q conditioned to have 2σ`1 vertices. Moreover, conditionally on TreepQq, the quadrangulations with a simple boundary associated to Q via the bijection Ψ are independent with respective Boltzmann distribution p P degpuq gp , for u P V pTreepQq ‚ q.
Proof. Observing that #V pTreepqqq " #Bq`1 for every rooted quadrangulation q, we obtain P gp,zp pQ σ f q " Ψ˚P gp,zp ptt P T f : #V ptq " 2σ`1uq " GW µ˝,µ‚ ptt P T f : #V ptq " 2σ`1uq. Now let t be a finite plane tree with an odd number of offspring at even height, and let pp q u : u P V pt ‚ qq and q be as in the proof of Proposition 6. Then, Ψ˚P σ gp pt, pp q u : u P tqq " 1 t#Bq"2σu P gp,zp pQ σ f q ź uPt˝µ˝p k u q ź uPt‚ µ ‚ pk u q ź uPt‚ p P degpuq gp pp q u q, which concludes the proof.
Proof. Recall that pµ˝, µ ‚ q is critical if and only if the product of their respective means ma nd m ‚ equals one. Since by (20), µ˝is the geometric law with parameter 1´1{F pg p , z p q, we have m˝" F pg p , z p q´1.
Then, Identity (2.8) of [15] ensures that p F pg, zF 2 pg, zqq " F pg, zq for all non-negative weights g and z. When differentiating this relation with respect to the variable z, we obtain B z p F pg, zF 2 pg, zqq " B z F pg, zq F 2 pg, zq`2zF pg, zqB z F pg, zq .
Writing B z F pg p , z p q " ÿ σě0 σF σ pg p qz σ´1 p , and using the exact expression for F σ pg p q from (18), we see by means of Stirling's formula that B z F pg p , z p q " 8 for p P r0, 1{2q, and B z F pg p , z p q ă 8 for p " 1{2. Thus, for p P r0, 1{2q, B z p F pg p , z p F 2 pg p , z p qq " 1 2z p F pg p , z p q , whereas if p " 1{2, the derivative on the left-hand side in (22) is strictly smaller than the right-hand side for g " g p , z " z p . Finally, applying Identity (2.8) of [15] once again, we get m ‚ "G 1 µ‚ p1q " 1 F pg p , z p q´1´´´p F " g p , z p F 2 pg p , z p q ‰´1¯`2 z p F 2 pg p , z p qB z p F " g p , z p F 2 pg p , z p q ‰¯.
As a consequence, m˝m ‚ " 1 if p ă 1{2, and m˝m ‚ ă 1 if p " 1{2. The fact that µ˝has exponential moments is clear. For µ ‚ , one sees from (19) that the power series ÿ kě0 x k p F k pg p q has radius of convergence p r p " 4p1´pq 2 {p9pq, while (18) ensures that z p F 2 pg p , z p q " p1´4pq 2 9p1´pq .
Again, for p P r0, 1{2q, p r p ą z p F 2 pg p , z p q, and these quantities are equal for p " 1{2. Thus, there exists s ą 1 such that G µ‚ psq ă 8 if and only if p ă 1{2, which concludes the proof.
We are now ready to prove Theorem 4.
Proof of Theorem 4. Fix 0 ď p ă 1{2. Let us denote by Q 8 the random quadrangulation with an infinite boundary as constructed in the statement of Theorem 4, and let Q σ be distributed according to P σ gp . In view of Proposition 2, it is sufficient to prove that in the local sense, as σ Ñ 8, For every real r ě 1 and every (finite or infinite) plane tree t, we define Cut r ptq as the finite plane tree obtained from pruning all the vertices at a height larger than 2r in t. If q P Q is a quadrangulation with a boundary such that Ψpqq " pt, pp q u : u P t ‚ qq, we define Cut r pqq to be the quadrangulation obtained from gluing the maps pp q u : u P Cut r ptq ‚ q in the associated loops of LooppCut r ptqq. With this definition, we have B r pqq Ă Cut r pqq for every r ě 1, where we recall that B r pqq stands for the closed ball of radius r around the root in q.
We proved that for every r ě 1, as σ Ñ 8, Cut r pQ σ q pdq ÝÑ Cut r pQ 8 q .
Since B r pqq Ă Cut r pqq for every r ě 1 and q P Q, (23) holds and the theorem follows.

Recurrence of simple random walk
In this final part, we prove Corollary 2, stating that simple random walk on the UIHPQ p for 0 ď p ă 1{2 is almost surely recurrent. We will use a criterion from the theory of electrical networks; see, e.g., [35,Chapter 2] for an introduction into these techniques.
Proof of Corollary 2. Fix 0 ď p ă 1{2. We interpret the UIHPQ p as an electrical network, by equipping each edge with a resistance of strength one. A cutset C between the root vertex and infinity is a set of edges that separates the root from infinity, in the sense that every infinite self-avoiding path starting from the root has to pass through at least one edge of C. By the criterion of Nash-Williams, cf. [35, (2.13)], it suffices to show that there is a collection pC n , n P Nq of disjoint cutsets such that ř 8 n"1 p1{#C n q " 8 almost surely, i.e., for almost every realization of the UIHPQ p .
We recall the construction of the UIHPQ p in terms of the looptree associated to Kesten's two-type tree T 8 " T 8 pµ˝, µ ‚ q. Note that the white vertices in T 8 , i.e., the vertices at even height, represent vertices in the UIHPQ p . More precisely, by construction, they form the boundary vertices of the latter. In particular, the white vertices on the spine of T 8 are to be found in the UIHPQ p , and we enumerate them by v 1 , v 2 , v 3 , . . ., such that v 1 is the root vertex, and d gr pv j , v 1 q ě d gr pv i , v 1 q for j ě i. Now observe that for i P N, v i and v i`1 lie on the boundary of one common finite-size quadrangulation with a simple boundary, which we denote by p q v i , in accordance with notation in the proof of Theorem 4.
We define C i to be the set of all the edges of p q v i . Clearly, for each i P N, C i is a cutset between the root vertex and infinity, and for i ‰ j, C i and C j are disjoint. The sizes #C i , i P N, are i.i.d. random variables. More specifically, using the construction of the UIHPQ p in terms of Kesten's looptree, the law of #C 1 can be described as follows: First, draw a random variable Y according to the size-biased offspring distributionμ ‚ , and then, conditionally on Y , #C 1 is distributed as the number of edges of a Boltzmann quadrangulation with law p P pY`1q{2 gp , where g p " pp1´pq{3. Obviously, #C 1 is finite almost surely, implying ř 8 n"1 p1{#C n q " 8 almost surely, and recurrence of the simple symmetric random walk on the UIHPQ p follows.
Remark 11. Let us end with a remark concerning the structure of the UIHPQ p for p ă 1{2. Note that with probabilityμ ‚ p1q ą 0, a cutset C i as constructed in the above proof consists exactly of one edge. By independence and Borel-Cantelli, we thus find with probability one an infinite sequence of such cutsets C i 1 , C i 2 , . . . consisting of one edge only. In particular, this proves that the UIHPQ p for p ă 1{2 admits a decomposition into a sequence of almost surely finite i.i.d. quadrangulations Q i ppq with a non-simple boundary (whose laws can explicitly be derived from Theorem 4), such that Q i ppq and Q j ppq get connected by a single edge if and only if |i´j| " 1. This parallels the decomposition of the spaces H α for α ă 2{3 found in [37,Display (2.3)].