Loïc Richier
email: loic.richier@polytechnique.edu.

LIMITS OF THE BOUNDARY OF RANDOM PLANAR MAPS

We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter α ∈ (1, 2). First, in the dense phase corresponding to α ∈ (1, 3/2), we prove that the scaling limit of the boundary is the random stable looptree with parameter 1/(α -1/2). Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to α ∈ (3/2, 2), it has a component homeomorphic to the half-plane. As an application, we identify the limits of loops conditioned to be large in the rigid O(n) loop model on quadrangulations, proving thereby a conjecture of Curien & Kortchemski.

Introduction

The purpose of this work is to investigate local limits, in the sense of Angel & Schramm, and scaling limits, in the Gromov-Hausdorff sense, of the boundary of bipartite Boltzmann planar maps conditioned to have a large perimeter.

Model and motivation. Given a sequence q = (q 1 , q 2 , . . .) of nonnegative real numbers and a planar map m which is bipartite (i.e., with faces of even degree), the associated Boltzmann weight is w q (m) := f ∈Faces(m)

q deg(f )/2 .
The sequence q is admissible if these weights form a finite measure on the set of rooted bipartite maps (with a distinguished oriented edge) that we call the Boltzmann measure with weight q. We also say that q is critical if moreover the expected squared number of vertices of the map is infinite under this measure. The scaling limits of Boltzmann maps conditioned to have a large number of faces (or vertices) have attracted a lot of attention. The first model to be considered was the uniform measure on 2p-angulations, in which all faces have the same degree 2p. In this case, 1 Le Gall [START_REF] Gall | The topological structure of scaling limits of large planar maps[END_REF] proved the subsequential convergence towards a random metric space called the Brownian map, first introduced by Marckert & Mokkadem in [START_REF] Marckert | Limit of normalized quadrangulations: The Brownian map[END_REF] and whose distribution has been characterized later by Le Gall [START_REF] Gall | Uniqueness and universality of the Brownian map[END_REF] and Miermont [53]. This result has been extended by Le Gall [START_REF] Gall | Uniqueness and universality of the Brownian map[END_REF] to critical sequences q such that the degree of a typical face has exponential moments (while the first results on this model were obtained by Marckert & Miermont [START_REF] Marckert | Invariance principles for random bipartite planar maps[END_REF]). The result also holds for critical sequences q such that the degree of a typical face has a finite variance, as shown in the recent work [START_REF] Marzouk | Scaling limits of random bipartite planar maps with a prescribed degree sequence[END_REF] (such a sequence is called generic critical, see Section 2.1 for precise definitions). Convergence towards the Brownian map has also been established in the non-bipartite case in [START_REF] Miermont | An invariance principle for random planar maps[END_REF][START_REF] Miermont | Radius and profile of random planar maps with faces of arbitrary degrees[END_REF]. All these results demonstrate the universality of the Brownian map, whose geometry is now well understood [START_REF] Gall | Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere[END_REF][START_REF] Gall | Geodesics in large planar maps and in the Brownian map[END_REF].

For a different behaviour to arise, Le Gall & Miermont suggested in [START_REF] Gall | Scaling limits of random planar maps with large faces[END_REF] to assume, besides criticality, that the degree of a typical face is in the domain of attraction of a stable law with parameter α ∈ (1, 2). The weight sequence q is then called non-generic critical with parameter α. Under slightly stronger assumptions, they proved the (subsequential) convergence towards a one-parameter family of random metric spaces called the stable maps with parameter α. These are supposed to be very different from the Brownian map because of large faces that remain present in the scaling limit. Their duals have been recently studied in [START_REF] Budd | Geometry of infinite planar maps with high degrees[END_REF][START_REF] Bertoin | Martingales in self-similar growthfragmentations and their connections with random planar maps[END_REF], but their geometry remains widely unknown. The stable maps are believed to undergo a phase transition at α = 3/2. In the regime α ∈ (1, 3/2), called the dense phase, the large faces of the map are supposed to be self-intersecting in the limit, while in the regime α ∈ (3/2, 2), called the dilute phase, they are supposed to be self-avoiding. The aim of this work is twofold: first, we identify the branching structure of the large faces in the dense phase via scaling limits. Then, we establish the phase transition through local limits of large faces.

Main results. This paper adresses maps with a boundary, meaning that the face on the right of the root edge (the root face) is interpreted as the boundary ∂m of the map m. Precisely, we consider a Boltzmann map with weight q conditioned to have perimeter 2k, say M k , whose law is denoted by P (k) q . One can then interpret ∂M k as a typical face of degree 2k of a Boltzmann map. Our main result studies the scaling limit of ∂M k , equipped with its graph distance.

Theorem 1.1. Let q be a non-generic critical sequence with parameter α ∈ (1, 3/2). For every k ≥ 0, let M k be a map with law P (k) q . Then, there exists a slowly varying function Λ such that in the Gromov-Hausdorff sense,

Λ(k) (2k) α-1/2 • ∂M k (d) -→ k→∞ L β ,
where

β := 1 α -1 2 ∈ (1, 2)
and L β is the random stable looptree with parameter β.

The stable looptrees (L β : β ∈ (1, 2)) were introduced by Curien & Kortchemski in [START_REF] Curien | Random stable looptrees[END_REF], and can be seen as the stable trees of Duquesne & Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF][START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] in which branching points are turned into topological circles. Stable looptrees also appear as the scaling limits of discrete looptrees [START_REF] Curien | Random stable looptrees[END_REF], which are informally collections of cycles glued along a tree structure. They have Hausdorff dimension β a.s. [START_REF] Curien | Random stable looptrees[END_REF]Theorem 1.1].

The result of Theorem 1.1 covers the dense case. We believe that in the dilute and generic critical phases, the scaling limit of ∂M k is a circle. Furthermore, in the subcritical phase, the Continuum Random Tree [START_REF] Aldous | The Continuum Random Tree[END_REF][START_REF] Aldous | The Continuum Random Tree III[END_REF] is expected to arise as a scaling limit. We will discuss this in Section 4.

The local limits of Boltzmann maps with a boundary have been studied by Curien in [24, Theorem 7]. He proved that for any admissible weight sequence q, we have in the local sense

M k (d) -→ k→∞ M ∞ . (1)
The map M ∞ = M ∞ (q) is known as the Infinite Boltzmann Half-Planar Map with weight q (q-IBHPM for short). The infinite boundary ∂M ∞ of M ∞ is a.s. non-simple and has selfintersections, the cut vertices (or pinch points). Then, M ∞ can be decomposed into irreducible components, that are maps with a simple boundary attached by cut vertices of ∂M ∞ .

When M ∞ has a unique infinite irreducible component, it is called the core. For technical reasons, we rather deal with the scooped-out map Scoop(M ∞ ), obtained by duplicating the edges of ∂M ∞ whose both sides belong to the root face. Naturally, Scoop(M ∞ ) is a local limit version of looptrees that we briefly describe (see Section 5.2 for details). Given a pair of offspring distributions (ρ • , ρ • ), an alternated twotype Galton-Watson tree is a random tree in which vertices at even (resp. odd) height have offspring distribution ρ • (resp. ρ • ) all independently of each other. As in the monotype case, we can make sense of such trees conditioned to survive, and denote the limiting infinite tree by T

•,• ∞ = T •,• ∞ (ρ • , ρ • ). When (ρ • , ρ • ) is critical (meaning that the product of their means equals one), Stephenson established in [57] that T •,•
∞ is a two-type version of Kesten's tree, that is a.s. locally finite with a unique spine (see [START_REF] Kesten | Subdiffusive behavior of random walk on a random cluster[END_REF][START_REF] Lyons | Probability on trees and networks[END_REF][START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF] for details). Under suitable assumptions, we will prove in Proposition 5.3 that when (ρ

• , ρ • ) is subcritical, T •,•
∞ has a.s. a unique vertex of infinite degree (at odd height). This phenomenon, known as condensation, was first observed by Jonsson & Stefánsson in [START_REF] Jonsson | Condensation in Nongeneric Trees[END_REF] (see also [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF][START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the condensation case[END_REF][START_REF] Kortchemski | Limit theorems for conditioned non-generic Galton-Watson trees[END_REF]). We then define an infinite map

L ∞ = L ∞ (ρ • , ρ • ) by taking each vertex at odd height in T •,•
∞ and connecting its neighbours by edges in cyclic order. Therefore, L ∞ has only finite faces in the critical regime, and a unique infinite face in the subcritical regime. Note that ρ • dictates the size of the finite faces of L ∞ . We can now state our local limit result. Theorem 1.2. Let q be either subcritical, generic critical or non-generic critical with parameter α ∈ (1, 2). For every k ≥ 0, let M k be a map with distribution P (k) q and let M ∞ be the q-IBHPM. Then, there exists probability measures ν • (geometric) and ν • such that in the local sense

Scoop(M k ) (d) -→ k→∞ Scoop(M ∞ ) (d) = L ∞ (ν • , ν • ).
A phase transition is observed:

• If q is subcritical or non-generic critical with parameter α ∈ (1, 3/2), (ν • , ν • ) is critical
and M ∞ has only finite irreducible components.

• If q is non-generic critical with parameter α ∈ (3/2, 2) or generic critical, (ν • , ν • ) is subcritical and M ∞ has a well-defined core with an infinite simple boundary.

Moreover, ν • has finite variance if and only if q is subcritical. Otherwise, ν • is in the domain of attraction of a stable distribution, with parameter

1/(α -1/2) (if α ∈ (1, 3/2)), α -1/2 (if α ∈ (3/2, 2)) or 3/2 (if q is generic critical).
In the dense phase, M ∞ is tree-like, while in the dilute phase, it has an irreducible component homeomorphic to the half-plane on which finite maps are grafted (see Figure 1). In the subcritical and dense phases, the q-IBHPM can even be recovered from L ∞ and a collection of independent maps with a simple boundary, as shown in Proposition 5.7. Such collections of random combinatorial structures attached to a tree also appear in [START_REF] Stufler | Limits of random tree-like discrete structures[END_REF]. In the dilute and generic critical regimes, we expect the core of M ∞ to be the local limit of Boltzmann maps constrained to have a simple boundary when the perimeter goes to infinity (see Section 5.3 for more on this). The critical parameter α = 3/2 plays a special role that we discuss in Section 6.

α ∈ (1, 3/2) α ∈ (3/2, 2) Figure 1: Schematic representation of ∂M ∞ for q non-generic critical with α ∈ (1, 2).
Applications to the rigid O(n) loop model on quadrangulations. The study of Boltzmann distributions such that q is non-generic critical with parameter α ∈ (1, 2) is also motivated by the connection with statistical physics models on random maps. Here, we are interested in the rigid O(n) loop model on quadrangulations, studied by Borot, Bouttier & Guitter in [START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF], see also [START_REF] Chen | The perimeter cascade in critical Boltzmann quadrangulations decorated by an O(n) loop model[END_REF][START_REF] Budd | The peeling process on random planar maps coupled to an O(n) loop model[END_REF].

A loop-decorated quadrangulation with a boundary (q, ) is a planar map q whose faces all are quadrangles (except the root face), together with a collection of disjoint closed simple paths = ( 1 , 2 , . . .) drawn on the dual of q, called loops (which do not visit the root face). The loop configuration is rigid if all loops cross quadrangles through their opposite sides. Given n ∈ (0, 2) and g, h ≥ 0, we define a measure on loop-decorated quadrangulations by

W (n;g,h) ((q, )) := g #Faces(q)-| | h | | n # ,
where | | is the total length of the loops and # the number of loops. We say that the triplet (n; g, h) is admissible if for every k ≥ 0 this induces a probability measure P (k) (n;g,h) on loop-decorated quadrangulations with perimeter 2k (see Figure 2 for an illustration). The case k = 1 corresponds to the rigid O(n) model on quadrangulations of the sphere, by gluing the two edges of the boundary together.

In [START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF], Borot, Bouttier & Guitter introduced the gasket of a loop-decorated quadrangulation, obtained by pruning the interior of the outermost loops (with respect to the root). They proved that under P (k) (n;g,h) , the gasket is a Boltzmann map with law P (k) q , where q = q(n; g, h) is the solution of [START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF]Equation 2.3]. This leads to a classification of the parameters (n; g, h) g h n in regimes depending of the type of the sequence q. It has been argued in [START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF] (and fully justified in [START_REF] Budd | The peeling process on random planar maps coupled to an O(n) loop model[END_REF]Appendix]) that the model admits a complete phase diagram shown in [14, Figure 12]. For every n ∈ (0, 2), there exists a critical line h = h c (n; g) that separates subcritical and ill-defined parameters. The regime changes along the critical line. There is a special point (g * (n), h * (n)) such that the parameters are non-generic critical with parameter α = 3/2arccos(n/2)/π (dense) for g < g * , and generic critical for g > g * . The special point (g * , h * ) itself is non-generic critical with parameter α = 3/2 + arccos(n/2)/π (dilute).

In this work, we are motivated by the study of the geometry of large loops in the rigid O(n) model on quadrangulations. More generally, the interfaces in statistical physics models on maps are of great interest. In [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF], Curien & Kortchemski studied percolation on uniform triangulations of the sphere. They proved that the boundary of a critical percolation cluster conditioned to be large admits as a scaling limit the random stable looptree with parameter 3/2. They also conjectured that the whole family (L β : β ∈ (1, 2)) appears as scaling limit of large loops in the O(n) model on triangulations. The following application of Theorem 1.1 proves this conjecture for the rigid O(n) model on quadrangulations.

Corollary 1.3. Let n ∈ (0, 2), g ∈ [0, g * (n)) and h := h c (n; g). For every k ≥ 0, let (Q k , L k ) be a loop-decorated quadrangulation with law P (k) (n;g,h) . Then, there exists a constant C = C(n, g, h) such that in the Gromov-Hausdorff sense,

C (2k) 1/β • ∂Q k (d) -→ k→∞ L β ,
where

β := 1 1 - 1 π arccos n 2 ∈ (1, 2).
Note that the value of β fits the prediction of [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF]. We also obtain the local limits of large loops in the O(n) model from Theorem 1.2. These results are obtained by applying Theorems 1.1 and 1.2 to the gasket of the loop-decorated quadrangulation (Q k , L k ) (see also Remark 4.2). At first glance, they hold only for the boundary of loop-decorated quadrangulations. However, by the gasket decomposition, they apply to any loop conditioned to be large in the rigid O(n) loop model. To make it more concrete, on can choose any deterministic procedure to pick a loop in the rigid O(n) loop model on quadrangulations of the sphere (e.g. the loop that is the closest to the root edge) and condition this loop to have perimeter 2k. Then, the inner contour of this loop is the boundary of a loop-decorated quadrangulation with law P (k) (n;g,h) (see [START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF][START_REF] Chen | The perimeter cascade in critical Boltzmann quadrangulations decorated by an O(n) loop model[END_REF] for more details).

Overview and comments. The paper is organized as follows. We first introduce a more general framework than [START_REF] Gall | Scaling limits of random planar maps with large faces[END_REF][START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF] for Boltzmann maps, by allowing slowly varying corrections. In Section 2.2, we extend known enumerative results to this family. The proofs of the main results rely on a decomposition of maps with a general boundary into a tree of maps with a simple boundary, inspired by [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF] and described in Section 3. Then, we need enumerative results for bipartite maps with a simple boundary, which were unknown so far and are of independent interest. This is done in Section 2.3, by means of a second relation between maps with a general (resp. simple) boundary, and by using Tauberian theorems. This is a key feature of this work.

This method is quite robust, and only needs estimates on the partition function of the model as an input. For this reason, we believe that our proofs can be adapted to more general statistical physics models on random maps for which Borot, Bouttier & Guitter proved results similar to those of [START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF]. For instance, general O(n) loop models on triangulations with bending energy [START_REF] Borot | More on the O(n) model on random maps via nested loops: loops with bending energy[END_REF] or domain symmetry breaking [START_REF] Borot | Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model[END_REF]. This last case covers in particular the Potts model and Fortuin-Kasteleyn percolation on general maps, that have been studied in [START_REF] Berestycki | Critical Exponents on Fortuin-Kasteleyn Weighted Planar Maps[END_REF][START_REF] Sheffield | Quantum gravity and inventory accumulation[END_REF][START_REF] Gwynne | Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times[END_REF][START_REF] Gwynne | Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case[END_REF][START_REF] Gwynne | Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map II: local estimates and empty reduced word exponent[END_REF][START_REF] Chen | Basic properties of the infinite critical-FK random map[END_REF]. An interesting example is the critical Bernoulli percolation model on random triangulations, treated in [START_REF] Borot | More on the O(n) model on random maps via nested loops: loops with bending energy[END_REF]Section 4.2,p.23]. This corresponds to a O(n) loop model on triangulations for n = 1 and a suitable choice of the parameters. The asymptotics are similar to the quadrangular case, and we get the exponent β = 1/(1arccos(1/2)/π) = 3/2, which is consistent with the result of [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF].

Notation. Throughout this work, we use the notation N := {1, 2, . . .} and Z + := N ∪ {0}.

Boltzmann distributions 2.1 Boltzmann distributions on bipartite maps

Maps. A planar map is a proper embedding of a finite connected graph in the twodimensional sphere S 2 , considered up to orientation-preserving homeomorphisms. The faces are the connected components of the complement of the embedding, and the degree deg(f ) of a face f is the number of its incident oriented edges. The sets of vertices, edges and faces of a (planar) map m are denoted by V(m), E(m) and F(m). For technical reasons, the maps we consider are always rooted, which means that an oriented edge e * = (e -, e + ), called the root edge, is distinguished. The face f * incident on the right of the root edge is called the root face. A map with a boundary m is a map in which we consider the root face as an external face, whose incident edges and vertices form the boundary ∂m of the map. The degree #∂m of the external face is the perimeter of the map, and non-root faces are called internal.

In this paper, we only consider bipartite maps, in which all face degrees are even. We denote by M the corresponding set, and by M k be the set of (bipartite) maps with perimeter 2k, for k ≥ 0. The map † consisting of a single vertex is the only element of M 0 . We will also consider pointed maps, which have a marked vertex v * . A pointed bipartite map m such that d m (e + , v * ) = d m (e -, v * ) + 1 is said to be positive, and the corresponding set is denoted by M • + (by convention, † ∈ M • + ). Finally, M stands for the identity mapping on M. Boltzmann distributions. Given a weight sequence q = (q k : k ∈ N) of nonnegative real numbers, the Boltzmann weight of a bipartite map m is defined by

w q (m) := f ∈F(m) q deg(f )/2 .
(2)

By convention, we set w q ( †) = 1. This defines a σ-finite measure on M • + with total mass

Z q := w q M • + ∈ [1, ∞]. (3)
A weight sequence q is admissible if Z q < ∞ (or equivalently if w q (M) < ∞, see [8, Proposition 4.1]). Then, the Boltzmann measure P • q is defined by

P • q (m) := w q (m) Z q , m ∈ M • + .
Following [START_REF] Marckert | Invariance principles for random bipartite planar maps[END_REF], we introduce the function

f q (x) := ∞ k=1 2k -1 k -1 q k x k-1 , x ≥ 0, (4)
By [49, Proposition 1], a weight sequence q is admissible iff the equation

f q (x) = 1 - 1 x , x > 0 (5)
has a solution. In that case, the smallest such solution is Z q and Z 2 q f q (Z q ) ≤ 1. A classification of weight sequences was introduced in [START_REF] Marckert | Invariance principles for random bipartite planar maps[END_REF][START_REF] Gall | Scaling limits of random planar maps with large faces[END_REF], which is closely related to the Bouttier-Di Francesco-Guitter bijection [START_REF] Bouttier | Planar maps as labeled mobiles[END_REF]. This bijection associates to every map m ∈ M • + a plane tree Φ BDG (m) (together with labels on vertices at even height). The study is simplified by using additionally a bijection Φ JS due to Janson and Stefánsson [START_REF] Janson | Scaling limits of random planar maps with a unique large face[END_REF]Section 3]. This will be of independent interest, so we give a detailed presentation in Section 3.1. We are interested in the application that associates to m ∈ M • + the tree Φ(m) := Φ JS (Φ BDG (m)). By [START_REF] Marckert | Invariance principles for random bipartite planar maps[END_REF]Proposition 7] and [37, Appendix A] (see also Proposition 3.1), we get the following. Lemma 2.1. Let q be an admissible weight sequence. Under P • q , the tree Φ(M ) is a Galton-Watson tree with offspring distribution µ defined by

µ(0) = 1 -f q (Z q ) and µ(k) = Z k-1 q 2k -1 k -1 q k , k ∈ N.
Recall that the offspring distribution µ is critical (resp. subcritical) iff it has mean m µ = 1 (resp. m µ < 1). Lemma 2.1 transfers to the generating function G µ of µ, which reads

G µ (s) := ∞ k=0 s k µ(k) = 1 -f q (Z q ) + sf q (sZ q ), s ∈ [0, 1]. ( 6
)
The aforementioned classification of weight sequences can be rephrased as follows.

Definition 2.2. An admissible sequence q is critical if µ is critical, and subcritical otherwise.

A critical sequence q is generic critical if µ has finite variance. Finally, a critical sequence q is non-generic critical with parameter α ∈ (1, 2) if there exists a slowly varying function

such that µ([k, ∞)) = (k) • k -α .
Recall that a positive function is slowly varying (at infinity) if it satisfies (λx)/ (x) → 1 as x → ∞, for every λ > 0. We emphasize that Definition 2.2 is more general than that of [START_REF] Gall | Scaling limits of random planar maps with large faces[END_REF], which implies that the slowly varying function is asymptotically constant (and is also the framework in [START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF][START_REF] Budd | Geometry of infinite planar maps with high degrees[END_REF][START_REF] Bertoin | Martingales in self-similar growthfragmentations and their connections with random planar maps[END_REF][START_REF] Curien | Peeling random planar maps[END_REF]).

Remark 2.3. The classification can be translated in terms of P • q by properties of Φ BDG . Namely,

E • q (#V(M )) = ∞ iff q is critical and µ • (k) := µ(k + 1)/f q (Z q
) is interpreted as the law of (half) the degree of a typical face of the map under P • q , see [START_REF] Bouttier | Planar maps as labeled mobiles[END_REF] for more on this.

We conclude by translating Definition 2.2 in terms of the Laplace transform L µ of µ. First, if q is subcritical, µ has finite mean m µ < 1 and

L µ (t) := G µ (e -t ) = 1 -m µ t + o(t) as t → 0 + . ( 7
)
When q is generic critical, µ has mean m µ = 1 and finite variance σ 2 µ which yields

L µ (t) = 1 -t + σ 2 µ + 1 2 t 2 + o(t 2 ) as t → 0 + . (8)
For q non-generic critical with α ∈ (1, 2), Karamata's Abelian theorem [START_REF] Bingham | Regular Variation[END_REF]Theorem 8.1.6] gives

L µ (t) = 1 -t + |Γ(1 -α)|t α (1/t) + o(t α (1/t)) as t → 0 + . ( 9
)

Boltzmann distributions on maps with a boundary

We now deal with maps that have a boundary. The root face f * is then considered as external to the map, and receives no weight. This amounts to change the Boltzmann weights for

w q (m) := f ∈F(m)\{f * } q deg(f )/2 . ( 10
)
Let us introduce the partition functions for bipartite maps with a fixed perimeter

F k := m∈M k w q (m), k ∈ Z + , (11)
where we hide the dependence in q in the notation. These quantities are finite if q is admissible. The associated Boltzmann measure on maps with fixed perimeter is defined by

P (k) q (m) := 1 {m∈M k } w q (m) F k , m ∈ M, k ∈ Z + . ( 12
)
The goal of this section is to derive asymptotics of F k . We also define the generating function

F (x) := ∞ k=0 F k x k , x ≥ 0, (13)
whose radius of convergence is denoted by r q . We borrow ideas of [14, Section 3.1] and [24, Section 5.1], but we need to extend these results due to our more general definition of non-genericity. We let the (admissible) weight sequence q vary by defining q(u) := (u k-1 q k : k ∈ N). Using the generating function for pointed maps [18, Proposition 2, Section A.1] and Euler's formula, we obtain (see [START_REF] Curien | Peeling random planar maps[END_REF]Equation (5.2)])

F k = 2k k 1 0 (uZ q(u) ) k du, k ∈ Z + . (14)
In the setting of [START_REF] Gall | Scaling limits of random planar maps with large faces[END_REF], the asymptotics of F k would follow from Laplace's method, see [START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF][START_REF] Curien | Peeling random planar maps[END_REF].

Here, we use a technique based on Karamata's Abelian theorem. Let Y q be the inverse function of u → uZ q(u) on [0, 1]. Since Z q(u) is the smallest solution of ( 5) with q = q(u) and f q(u) (x) = f q (ux), we have by ( 6)

Y q (x) = x -xf q (x) = 1 + x -Z q G µ (x/Z q ), x ∈ [0, Z q ].
This proves that Y q is of class C ∞ on (0, Z q ). Coming back to the integral in ( 14),

1 0 (uZ q(u) ) k du = Zq 0 x k Y q (x)dx = Z k+1 q ∞ 0 e -t(k+1) Y q (Z q e -t )dt.
We now introduce the increasing function

U (t) := t 0 Z q e -u Y q (Z q e -u )du = 1 -Y q (Z q e -t ) = -Z q e -t + Z q L µ (t), t ≥ 0.
On the one hand, the integral is expressed in terms of the Laplace transform of U :

1 0 (uZ q(u) ) k du = Z k q ∞ 0 e -kt U (dt),
and on the other hand from ( 7), ( 8) and ( 9), as t → 0 + ,

U (t) =            Z q (1 -m µ )t + o(t) (q subcritical) Z q σ 2 µ t 2 /2 + o(t 2 ) (q generic critical) Z q |Γ(1 -α)|t α (1/t) + o(t α (1/t)) (q non-generic critical α)
.

We can thus apply Karamata's Abelian theorem [10, Theorem 1.7.1'], giving

F k ∼ k→∞                        Z q (1 -m µ )(4Z q ) k √ πk 3/2 (q subcritical) Z q σ 2 µ (4Z q ) k √ πk 5/2 (q generic critical) Z q α √ π(4Z q ) k (k) sin(π(α -1))k α+1/2 (q non-generic critical α) , (15)
where we used Stirling's formula and the identity Γ(1α)Γ(1 + α) = απ/ sin(πα) for α ∈ (1, 2). The quantity a := α + 1/2 is of particular importance, so we use the notation of [START_REF] Curien | Peeling random planar maps[END_REF].

Notation. An admissible weight sequence q is said of type a = 3/2 if it is subcritical, of type a = 5/2 if it is generic critical and of type a ∈ (3/2, 5/2) if it is non-generic critical with parameter α = a -1/2. This allows us to write [START_REF] Bouttier | Planar maps as labeled mobiles[END_REF] in a unified way. Let

c 3/2 := Z q (1 -m µ ) √ π , c 5/2 := Z q σ 2 µ √ π and c a := Z q (a -1/2) √ π sin(π(a -3/2))
, a ∈ (3/2, 5/2). ( 16)

We also set the convention that = 1 if a ∈ {3/2, 5/2}. Then,

F k ∼ k→∞ c a (4Z q ) k (k) k a , a ∈ [3/2, 5/2]. ( 17
)
From now on, the case a = 2 is excluded and will be treated apart in Section 6. Let us derive from (17) a singular expansion for F , whose radius of convergence is r q = 1/(4Z q ). For k ≥ 0, let

ζ(k) := F k r k q F (r q ) ∼ k→∞ c a (k) k a F (r q ) .
The function k → k a ζ(k) is slowly varying, so by Karamata's theorem [10, Proposition 1.5.10]

j≥k ζ(j) ∼ k→∞ kζ(k) a -1 ∼ k→∞ c a (k) (a -1)F (r q )k a-1 .
We then apply Karamata's Abelian theorem [START_REF] Bingham | Regular Variation[END_REF]Theorem 8.1.6] to get the asymptotics of the Laplace transform

L ζ of ζ. For a ∈ [3/2, 2), we find L ζ (t) = 1 - Γ(2 -a)c a (a -1)F (r q ) t a-1 (1/t) + o(t a-1 (1/t)) as t → 0 + , while for a ∈ (2, 5/2], L ζ (t) = 1 -m ζ t + |Γ(2 -a)|c a (a -1)F (r q ) t a-1 (1/t) + o(t a-1 (1/t)) as t → 0 + .
The function 1 (y) := (-1/ log(1 -1/y)) is slowly varying at infinity by stability properties of slowly varying functions [START_REF] Bingham | Regular Variation[END_REF]Proposition 1.3.6]. We obtain that for a ∈ [3/2, 2),

G ζ (s) = 1 - Γ(2 -a)c a (a -1)F (r q ) (1 -s) a-1 1 1 1 -s (1 + o(1)) as s → 1 -,
and for a ∈ (2, 5/2],

G ζ (s) = 1 -m ζ (1 -s) + |Γ(2 -a)|c a (a -1)F (r q ) (1 -s) a-1 1 1 1 -s (1 + o(1)) as s → 1 -.
The singular expansion of F follows from F (xr q ) = F (r q )G ζ (x). Note that we have m ζ = r q F (r q )/F (r q ), and let κ a := c a |Γ(2a)|/(a -1). Recall also that 1 = 1 for a ∈ {3/2, 5/2}.

Proposition 2.4. Let q be a weight sequence of type a. For a ∈ [3/2, 2),

F (x) = F (r q ) -κ a 1 - x r q a-1 1 1 1 -x rq (1 + o(1)) as x → r - q ,
and for a ∈ (2, 5/2],

F (x) = F (r q ) -r q F (r q ) 1 - x r q + κ a 1 - x r q a-1 1 1 1 -x rq (1 + o(1)) as x → r - q .
2.3 Boltzmann distributions on maps with a simple boundary

The aim of this section is to obtain enumerative results for maps with a simple boundary. A (bipartite) map with a simple boundary is a map whose boundary is a cycle with no selfintersection. Their set is denoted by M. Consistently, for k ≥ 0, M k is the set of maps with a simple boundary of perimeter 2k. A generic element of M is denoted by m, and † ∈ M 0 by convention. The associated partition function is

F k := m∈ M k w q ( m), k ∈ Z + . (18)
For admissible q, the Boltzmann measure for maps with a simple boundary is defined by

P (k) q (m) := 1 {m∈ M k } w q (m) F k , m ∈ M, k ∈ Z + , (19)
and the associated generating function by

F (x) := ∞ k=0 F k x k x ≥ 0. ( 20
)
The radius of convergence of F is denoted by r q . We will prove the following analogue of Proposition 2.4 for maps with a simple boundary, which is the technical core of this paper.

The constants ( c a : a ∈ {3/2} ∪ (2, 5/2]) and the slowly varying functions 1 (also depending on a) will be defined at the end of the section, see ( 26) and ( 27).

Proposition 2.5. Let q be a weight sequence of type a. For a = 3/2, as y → r q F 2 (r q ) -,

F (y) = F (r q ) 1 - 1 2 1 - y r q F 2 (r q ) + c 3/2 1 - y r q F 2 (r q ) 2 (1 + o(1)) .
If a ∈ (3/2, 5/2]\{2}, F has radius of convergence r q = r q F 2 (r q ). Moreover, for a ∈ (3/2, 2),

F (y) = F (r q ) 1 - 1 2 1 - y r q + 1 - y r q 1 a-1 1 1 1 -y rq (1 + o(1)) as y → r q -,
and for a ∈ (2, 5/2],

F (y) = F (r q ) 1 - c a 2 1 - y r q + 1 - y r q a-1 1 1 1 -y rq (1 + o(1)) as y → r q -.
Our approach relies on a simple relation between the generating functions F and F , which was first observed in [START_REF] Brézin | Planar diagrams[END_REF] (see also [START_REF] Bouttier | Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop[END_REF] for quadrangulations). This relation is based on the decomposition of a map with a boundary m into a map with a simple boundary m containing the root edge, and a collection of maps with a general boundary attached to vertices of ∂ m (see [START_REF] Bouttier | Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop[END_REF]Figure 11 and Equation (5.1)] for details). We then obtain the following identity.

Lemma 2.6 ( [START_REF] Brézin | Planar diagrams[END_REF][START_REF] Bouttier | Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop[END_REF]). For every weight sequence q and every x ≥ 0, we have

F (x) = F xF 2 (x) .
In particular, the radius of convergence of F satisfies r q ≥ r q F 2 (r q ). We now use this relation to prove Proposition 2.5. For x ≥ 0, let P (x) := xF 2 (x), so that P is continuous increasing on [0, r q ] with inverse P -1 . By Proposition 2.4, for a ∈ [3/2, 2),

P (x) = P (r q ) -κ a 1 - x r q a-1 1 1 1 -x rq (1 + o(1)) as x → r - q , (21)
and for a ∈ (2, 5/2],

P (x) = P (r q ) -C q 1 - x r q + κ a 1 - x r q a-1 1 1 1 -x rq (1 + o(1)) as x → r - q , ( 22
)
where C q := r q F (r q )(F (r q ) + 2r q F (r q )) and κ a := 2r q F (r q )κ a . We now invert this expansion to get that of P -1 , and treat a ∈ [3/2, 2) and a ∈ (2, 5/2] separately. Recall that a positive function f is regularly varying (at infinity) with index γ ∈ R if it satisfies f (λx)/f (x) → λ γ as x → ∞, for every λ > 0. The next lemma is a variant of [10, Theorem 1.5.12].

Lemma 2.7. Let f be a continuous decreasing regularly varying function with index -γ < 0. Then, f is invertible and the function y → f -1 (1/y) is regularly varying with index 1/γ.

Let a ∈ [3/2, 2). From ( 21), we know that R(x) := P (r q ) -P (r q (1 -1/x)) ∼ κ a x 1-a 1 (x) as x → ∞, thus R is regularly varying with index 1a < 0. Moreover, R is continuous decreasing on [1, ∞) with inverse R -1 defined by R -1 (y) = 1 1 -1 r q P -1 (P (r q )y) , y ∈ (0, P (r q )].

By Lemma 2.7, y → R -1 (1/y) is regularly varying with index 1/(a -1), so that [10, Theorem 1.4.1] ensures the existence of a positive slowly varying function ¯ 1 such that R -1 (1/y) = y 1/(a-1) ¯ 1 (y), for y ∈ [1/P (r q ), ∞). As a consequence, P -1 (y) = r qr q (P (r q )y)

1 a-1 ¯ 1 1 P (r q ) -y , y ∈ [0, P (r q )). ( 23
)
When a = 3/2, 1 = 1 so that computation can be made more explicit. Indeed, we find

R(x) ∼ κ 3/2 / √ x as x → ∞. Then, the function Q(x) := R((κ 3/2 /x) 2 ) satisfies Q -1 (y) ∼ y as y → 0 + and R -1 (y) = κ 3/2 Q -1 (y) 2 ∼ κ 3/2 y 2 as y → 0 + .
As a conclusion, P -1 (y) = r q -r q (κ 3/2 ) 2 (P (r q )y) 2 (1 + o(1)) as y → P (r q ) -.

(

) 24
We are now interested in the case where a ∈ (2, 5/2]. From [START_REF] Chen | Basic properties of the infinite critical-FK random map[END_REF], we have

R(x) := [P (r q ) -P (r q (1 -x))]/C q = x - κ a C q x a-1 1 (1/x) (1 + o(1)) as x → 0 + .
The function R is continuous increasing on [0, 1], with inverse R -1 defined by

R -1 (y) = 1 - 1 r q P -1 (P (r q ) -C q y), y ∈ [0, P (r q )/C q ].
It also satisfies R -1 (y) ∼ y as y → 0 + . In particular, y → R -1 (1/y) is regularly varying with index -1 and by [10, Proposition 1.5.7], ¯ 1 (y) := 1 (1/R -1 (1/y)) is slowly varying. We get

R -1 (y) -y ∼ κ a C q R -1 (y) a-1 1 1/R -1 (y) ∼ κ a C q y a-1 ¯ 1 (1/y) as y → 0 + ,
and as a conclusion

P -1 (y) = r q - r q C q (P (r q ) -y) - κ a C a q (P (r q ) -y) a-1 ¯ 1 C q P (r q ) -y (1 + o(1)) as y → P (r q ) -. (25)
We can now introduce the constants involved in the statement of Proposition 2.5,

c 3/2 := P (r q ) 2 2(κ 3/2 ) 2 - 1 8 and c a = 1 - P (r q ) C q ∈ (0, 1) for a ∈ (2, 5/2]. (26)
and the functions 1 (that are slowly varying by [10, Proposition 1.3.6]) defined by 1 (y) :=

P (r q ) 1 a-1 2 ¯ 1 y P (rq)
, a ∈ (3/2, 2) and 1 (y) := κ a P (r q ) a-1 2C a q ¯ 1 C q y P (r q ) , a ∈ (2, 5/2].

(

) 27
Proof of Proposition 2.5. By Lemma 2.6, we have that F (r q F 2 (r q )) = F (r q ), as well as

F (y) = y P -1 (y)
, 0 < y ≤ P (r q ).

We obtain asymptotic expansions for F around P (r q ) using ( 23), [START_REF] Curien | Peeling random planar maps[END_REF], and [START_REF] Curien | The CRT is the scaling limit of random dissections[END_REF]. These expansions are singular for a = 3/2, and thus F is not of class C ∞ at P (r q ). Together with Lemma 2.6, this proves that the radius of convergence of F is r q = P (r q ) in these cases.

Remark 2.8. From there, one expects the theory of singularity analysis [START_REF] Flajolet | Analytic Combinatorics[END_REF]Chapter 6] to give an equivalent of the partition function F k . However, it is not clear that the so-called delta-analyticity assumption is satisfied by F . We will use instead Karamata's Tauberian theorem, which provides a weaker result (see Proposition 3.6). Note also that in the subcritical case, we do not know if r q = r q F 2 (r q ) in general because the expansion of F is not singular. In the special case of quadrangulations, computations can be carried out explicitly using [START_REF] Bouttier | Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop[END_REF]. We find that r q > r q F 2 (r q ) if q is subcritical. Moreover,

F k ∼ k→∞ 2 √ 3 r q -k 27 √ πk 5/2 (q critical) and F k ∼ k→∞ c q r q -k k 3/2 (q subcritical).
3 Structure of the boundary of Boltzmann maps The total number of vertices of a tree t is denoted by |t|. The set of finite (plane) trees is denoted by T f , and T stands for the identity mapping on T f . Given a probability measure ρ on Z + with mean m ρ ≤ 1, the law GW ρ of a Galton-Watson tree with offspring distribution ρ is characterized by

GW ρ (t) = u∈t ρ(k u ), ∀ t ∈ T f . (28)
A pair (ρ

• , ρ • ) of probability measures on Z + is called critical (resp. subcritical) if m ρ• m ρ• = 1 (resp. m ρ• m ρ• < 1)
. Then, the law GW ρ•,ρ• of an (alternated) two-type Galton-Watson trees with offspring distribution (ρ • , ρ • ) is characterized by

GW ρ•,ρ• (t) = u∈t• ρ • (k u ) u∈t• ρ • (k u ), ∀ t ∈ T f . ( 29
)
The Janson-Stefánsson bijection. We now describe the Janson-Stefánsson bijection Φ JS introduced in [37, Section 3]. First, Φ JS ({∅}) = {∅}. For t = {∅}, Φ JS (t) has the same vertices as t but different edges defined as follows. For every u ∈ t • , set the convention that u0 = u (if u = ∅) and u(k u + 1) = u. Then, for every j ∈ {0, 1, . . . , k u }, add the edge (uj, u(j + 1)) to Φ JS (t). The root vertex of Φ JS (t) is 1 and its first children is chosen according to the lexicographical order of t. For further notice, we give a brief description of the inverse application Φ -1 JS . For t = {∅}, Φ -1 JS (t) has the same vertices as t, and edges defined as follows. For every leaf u ∈ t, let (u 1 , u 2 , . . .) be the sequence of vertices after u in the contour order of t, and (u) the largest index such that u 1 , . . . , u (u) all are ancestors of u in t. Then, add an edge between u and u k in Φ -1 JS (t) for every k ∈ {1, . . . , (u)}. The last leaf u of t in contour order is the root vertex, and u (u) its first child.

The application Φ JS is a bijection from T f onto itself, such that every u ∈ t • is mapped to a leaf of Φ JS (t), and every u ∈ t • with k children is mapped to a vertex of Φ JS (t) with k + 1 children. See Figure 3 for an illustration. This bijection simplifies the study of two-type Galton-Watson trees because of the following result of [START_REF] Janson | Scaling limits of random planar maps with a unique large face[END_REF] (see also [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF]Proposition 3.6]). Proposition 3.1. [37, Appendix A] Let ρ • and ρ • be probability measures on Z + such that m ρ• m ρ• ≤ 1 and ρ • has geometric distribution with parameter 1-p ∈ (0, 1):

ρ • (k) = (1-p)p k for k ≥ 0. Then, the image of GW ρ•,ρ• under Φ JS is GW ρ , where ρ(0) = 1 -p and ρ(k) = p • ρ • (k -1), k ∈ N. In particular, m ρ -p = (1 -p)m ρ• m ρ• , so that (ρ • , ρ • ) is critical iff ρ itself is critical. Φ JS Φ -1

JS

Figure 3: The Janson-Stefánsson bijection and its inverse application.

Random looptrees and scooped-out maps

We now introduce random looptrees and their tree of components to represent the boundary of a map as a tree, following the presentation of [26, Section 2.3] (see also [START_REF] Curien | Random stable looptrees[END_REF]).

Random looptrees. A looptree is a map whose edges are incident to two distinct faces, one being the root face (such a map is called edge-outerplanar). Informally, a (finite) looptree is a collection of polygons glued along a tree structure. Their set is denoted by L f . We associate to every tree t ∈ T f a looptree Loop(t) as follows. For every u ∈ t • , connect all the incident (white) vertices of u in cyclic order. Then, Loop(t) is the map obtained by discarding the black vertices and edges of t. The root edge of Loop(t) connects the origin of t to the last child of its first offspring. The inverse application associates to every looptree l ∈ L f a tree Tree(l), called the tree of components, as follows. We add an extra vertex in every internal face of l, which we connect by an edge to all the vertices of this face. The tree Tree(l) is obtained by discarding the edges of l. The root edge of Tree(l) connects the origin of l to the vertex inside the internal face incident to the root. See Figure 4 for an example.

t l Loop Tree

Figure 4: A looptree l and the associated tree of components t. Remark 3.2. Every internal face of l ∈ L f is rooted at the oriented edge whose origin is the closest to that of l, and such that the root face lies on its right. The gluing of a map with a simple boundary of perimeter k into a face of degree k is then determined by the convention that the root edges match. This definition of looptree slightly differs from that of [START_REF] Curien | Random stable looptrees[END_REF][START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF], that we now recall. Given a tree t ∈ T f , the looptree Loop(t) (or Loop (t) in [START_REF] Curien | Random stable looptrees[END_REF]) is built from t as follows. For every u, v ∈ t, there is an edge between u and v iff one of these conditions is fulfilled: u and v are consecutive siblings in t, or v is either the first or the last child of u in t. We will also need Loop(t), which is obtained from Loop(t) by contracting the edges linking a vertex of t and its last child in t. These objects are rooted at the oriented edge between the origin of t and its last child in t (resp. penultimate for Loop). See [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF]Figures 9 and 10] for an example. We use the bold print Loop to distinguish this construction from Loop. Note that contrary to Loop, Loop does not allow several loops to be glued at the same vertex.

The scooped-out map. The scooped-out map of a map m was defined in [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF] as the looptree Scoop(m) obtained from ∂m by duplicating the edges whose both sides belong to the root face. We call tree of components of m the tree Tree(m) := Tree(Scoop(m)).

A map m is recovered from Scoop(m) by gluing into its internal faces the proper maps with a simple boundary. These maps are the connected components obtained when splitting m at the pinch-points of ∂m, called irreducible components in [START_REF] Bouttier | Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop[END_REF] and [START_REF] Curien | Uniform infinite planar quadrangulations with a boundary[END_REF]Section 2.2]. They have the same rooting convention as in Remark 3.2. This construction provides a bijection

Φ TC : m → (Tree(m), ( m u : u ∈ Tree(m) • ))
that associates to a map m ∈ M the tree t = Tree(m), whose vertices at odd height have even degree, and a collection ( m u : u ∈ t • ) of maps with a simple boundary of respective perimeter deg(u). See Figure 5 for an example. The following relations will be useful:

|t| = #∂m + 1 and u∈t• deg(u) = #∂m (t = Tree(m)). ( 30
)
m Scoop(m)

Distribution of the tree of components

We now introduce the probability measure P q,rq with "free perimeter" defined by

P q,rq (m) := r #∂m/2 q w q (m) F (r q ) , m ∈ M. (31)
It is related to

P (k)
q by conditioning with respect to the perimeter of the map: for every k ≥ 0 and m ∈ M, we have P q,rq (m | M k ) = P (k) q (m). The main result of this section identifies the distribution of the tree of components (see also [START_REF] Baur | Uniform infinite half-planar quadrangulations with skewness[END_REF]Proposition 6] for quadrangulations). Proposition 3.3. Let q be a weight sequence of type a ∈ [3/2, 5/2]. Under P q,rq , Tree(M ) is a two-type Galton-Watson tree with offspring distribution (ν

• , ν • ) defined by ν • (k) = 1 F (r q ) 1 - 1 F (r q ) k and ν • (2k + 1) = 1 F (r q ) -1 r q F 2 (r q ) k+1 F k+1 , k ∈ Z + .
(With ν • (2Z + ) = 0.) Moreover, conditionally on Tree(M ), the maps with a simple boundary ( M u : u ∈ Tree(M ) • ) associated to M by Φ TC are independent with respective law P

(deg(u)/2) q .
Proof. Let us check that ν • and ν • are probability measures. This is clear for ν • , and since 1 < F (r q ) = F (r q F 2 (r q )) we get

k∈Z + ν • (k) = 1 F (r q ) -1 F (r q F 2 (r q )) -1 = 1.
Recall that Φ TC associates to m ∈ M its tree of components t = Tree(m) and maps ( m u : u ∈ t • ) with a simple boundary of perimeter deg(u). Using ( 31) and ( 30), we have

P q,rq (m) = r #∂m/2 q w q (m) F (r q ) = 1 F (r q ) u∈t• r deg(u)/2 q w q ( m u ).
Then, for every c > 0

1 = u∈t• c ku 1 c |t•| and 1 c = u∈t• c ku 1 c |t•| .
Applying the first identity with c = 1 -1/F (r q ) and the second one with c = F (r q ) yields

P q,rq (m) = u∈t• 1 F (r q ) 1 - 1 F (r q ) ku u∈t• 1 F (r q ) -1 r q F 2 (r q ) (ku+1)/2 w q ( m u ) = u∈t• ν • (k u ) u∈t• ν • (k u )w q ( m u ) 1
F (ku+1)/2 .
By convention, both sides equal zero if there exists u ∈ t • such that F (ku+1)/2 = 0. Finally,

P q,rq Tree(M ) = t, M u = m u : u ∈ t • = P q,rq (M = m) = GW ν•,ν• (t) u∈t• P (deg(u)/2) q ( m u ),
which is the expected result.

By Proposition 3.1, we obtain the following.

Corollary 3.4. Let q be a weight sequence of type a ∈ [3/2, 5/2]. Under P q,rq , Φ JS (Tree(M )) is a Galton-Watson tree with offspring distribution ν defined by

ν(2k) = 1 F (r q ) r q F 2 (r q ) k F k , k ∈ Z + (and ν(k) = 0 for k odd).
As a consequence, the generating function of ν reads

G ν (s) = 1 F (r q ) ∞ k=0 s 2k r q F 2 (r q ) k F k = 1 F (r q ) F r q F 2 (r q )s 2 , s ∈ [0, 1]. (32)
From Lemma 2.6, we easily deduce the following formula for the mean of ν

m ν = G ν (1) = 1 F (r q ) 2r q F 2 (r q ) F r q F 2 (r q ) = 1 1 + F (rq) 2rqF (rq)
.

Similarly, the generating function of ν • satisfies G ν• (0) = 0 and

G ν• (s) = 1 F (r q ) -1 • 1 s F r q F 2 (r q )s 2 -1 , s ∈ (0, 1]. ( 34
)
The next result is a consequence of ( 17), ( 33) and Proposition 3.1.

Lemma 3.5. The offspring distribution ν and the pair of offspring distributions (ν

• , ν • ) are critical if a ∈ [3/2, 2) and subcritical if a ∈ (2, 5/2].
We now describe ν and ν • using Proposition 2.5, ( 32) and [START_REF] Gwynne | Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case[END_REF]. For a = 3/2, as

t → 0 + L ν (t) = 1 -t + 1 + 4 c 3/2 t 2 + o(t 2 ), (35)
L ν• (t) = 1 - 1 F (r q ) -1 t + 1 2 + 4 c 3/2 F (r q ) F (r q ) -1 t 2 + o(t 2 ). (36)
For a ∈ (3/2, 2), as

t → 0 + L ν (t) = 1 -t + 2 1 a-1 t 1 a-1 (1/t) + o t 1 a-1 (1/t) , (37)
L ν• (t) = 1 - 1 F (r q ) -1 t + F (r q ) F (r q ) -1 2 1 a-1 t 1 a-1 (1/t) + o t 1 a-1 (1/t) . (38)
Finally, for a ∈ (2, 5/2], as

t → 0 + , L ν (t) = 1 -c a t + 2 a-1 t a-1 (1/t) + o t a-1 (1/t) , (39)
L ν• (t) = 1 -1 -c a F (r q ) F (r q ) -1 t + F (r q ) F (r q ) -1 2 a-1 t a-1 (1/t) + o t a-1 (1/t) . (40)
The function (x) := 1 (1/(1exp(-2/x))) is slowly varying from [10, Proposition 1.5.7].

For a = 3/2, ( 35) and [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF] entail that ν and ν • have finite variance equal to

σ 2 ν = 2P (r q ) κ 3/2 2 = F (r q ) Z q (1 -m µ ) 2 and σ 2 ν• = F (r q ) F (r q ) -1 F (r q ) Z q (1 -m µ ) 2 -1 . (41)
For a ∈ (3/2, 2), Karamata's Tauberian theorem [START_REF] Bingham | Regular Variation[END_REF]Theorem 8.1.6], [START_REF] Janson | Scaling limits of random planar maps with a unique large face[END_REF] and [START_REF] Jonsson | Condensation in Nongeneric Trees[END_REF] give

ν([k, ∞)) ∼ k→∞ 2 1 a-1 Γ a-2 a-1 • (k) k 1 a-1 and ν • ([k, ∞)) ∼ k→∞ F (r q ) F (r q ) -1 • 2 1 a-1 Γ a-2 a-1 • (k) k 1 a-1 . (42)
Finally, when a ∈ (2, 5/2], the same version of Karamata's Tauberian theorem gives

ν([k, ∞)) ∼ k→∞ 2 a-1 |Γ (2 -a) | • (k) k a-1 and ν • ([k, ∞)) ∼ k→∞ F (r q ) F (r q ) -1 • 2 a-1 |Γ (2 -a) | • (k) k a-1 . ( 43
)
Proposition 3.6. For a = 3/2, ν and ν • have finite variance (and exponential moments iff r q > r q F 2 (r q )). For a ∈ (3/2, 2), ν and ν • are in the domain of attraction of a stable law with parameter 1/(a -1) ∈ (1, 2) and for a ∈ (2, 5/2], ν and ν • are in the domain of attraction of a stable law with parameter a -1 ∈ (1, 3/2].

For every n ≥ 1, let

GW (n) ρ (resp. GW (n) ρ•,ρ• )

Scaling limits of the boundary of Boltzmann maps

This section deals with the scaling limits of the boundary of Boltzmann maps in the Gromov-Hausdorff sense. We refer to [START_REF] Burago | A Course in Metric Geometry[END_REF] for a complete definition of this topology. We start with a preliminary result directly adapted from [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF]Lemma 4.3]. Scaling limits: the dense regime. We first focus on the dense phase a ∈ (3/2, 2) and prove Theorem 1.1. The proof parallels that of [START_REF] Curien | Percolation on random triangulations and stable looptrees[END_REF]Theorem 1.2].

Proof of Theorem 1.1. For every k ≥ 0, let M k be a random map with law P (k) q and set T k := Φ JS (Tree(M k )). By definition of Loop, we have

d GH Loop(T k ), Loop(T k ) ≤ 2H(T k ), (44)
where H(T k ) is the overall height of T k . Indeed, the longest path of vertices of T k that are identified in Loop(T k ) has length at most H(T k ). By scaling limits results for conditioned Galton-Watson trees ([29, Theorem 3.1], [40, Theorem 3]) we have that

H(T k ) k a-1 -→ k→∞ 0 in probability. ( 45
)
The results of [START_REF] Duquesne | A limit theorem for the contour process of conditioned Galton-Watson trees[END_REF][START_REF] Kortchemski | A Simple Proof of Duquesne's Theorem on Contour Processes of Conditioned Galton-Watson Trees[END_REF] together with [START_REF] Gall | Uniqueness and universality of the Brownian map[END_REF] ensure that the invariance principle of [27, Theorem 4.1] applies: there exists a slowly varying function Λ such that in the Gromov-Hausdorff sense

Λ(k) (2k) a-1 • Loop(T k ) (d) -→ k→∞ L 1 a-1 .
Applying [START_REF] Gall | Geodesics in large planar maps and in the Brownian map[END_REF], [START_REF] Gall | Uniqueness and universality of the Brownian map[END_REF] and Lemma 4.1, we deduce that in the Gromov-Hausdorff sense

Λ(k) (2k) a-1 • Scoop(M k ) (d) -→ k→∞ L 1 a-1 .
This concludes the proof since ∂m and Scoop(m) always define the same metric space. Remark 4.2. In the setting of [START_REF] Gall | Scaling limits of random planar maps with large faces[END_REF][START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF] (in particular for applications to the O(n) model), the definition of non-generic critical sequences imply that Λ can be replaced by a constant.

Scaling limits: the subcritical regime. In the subcritical case, we expect that there exists K q > 0 such that in the Gromov-Hausdorff sense

K q √ 2k • ∂M k (d) -→ k→∞ T e ,
where T e is the Continuum Random Tree [START_REF] Aldous | The Continuum Random Tree[END_REF][START_REF] Aldous | The Continuum Random Tree III[END_REF]. When ν has exponential moments (i.e., if r q > r q F 2 (r q )) this follows from [START_REF] Curien | The CRT is the scaling limit of random dissections[END_REF]Theorem 14]. As mentioned in Remark 2.8, we do not know if this is satisfied for all subcritical sequences. However, we believe that [25, Theorem 14] holds under a finite variance assumption, by proving tightness of the sequence of laws and identifying the finite-dimensional marginals.

Scaling limits: the generic and dilute regimes. In the generic and dilute regimes, we believe that there exists K q > 0 such that in the Gromov-Hausdorff sense

K q 2k • ∂M k (d) -→ k→∞ S 1 ,
where S 1 stands for the unit circle. A proof could be adapted from [26, Theorem 1.2], which is itself based on the results of [START_REF] Jonsson | Condensation in Nongeneric Trees[END_REF][START_REF] Kortchemski | Limit theorems for conditioned non-generic Galton-Watson trees[END_REF] about condensation in non-generic trees. However, these results apply only if we have an equivalent of the partition function F k , which our techniques do not provide (see Remark 2.8).

5 Local limits of the boundary of Boltzmann maps

Local limits of Galton-Watson trees

The local topology. The local topology on the set M is induced by the local distance

d loc (m, m ) := 1/(1 + sup {R ≥ 0 : B R (m) = B R (m )}) , m, m ∈ M. (46)
Here, B R (m) is the ball of radius R in m for the graph distance, made of all the vertices of m at distance less than R from the origin, and all the edges whose endpoints are in this set. We let M be the completed space of M, so that elements of M ∞ := M \M are infinite (bipartite) maps. All the elements of M ∞ we consider can be seen as proper embeddings of a graph in the plane (up to orientation preserving homeomorphisms, see [START_REF] Curien | Peeling random planar maps[END_REF]Proposition 2]). Then, the boundary ∂m of m ∈ M ∞ is the embedding of edges and vertices of its root face. When the boundary is infinite, it is called simple if isomorphic to Z.

In order to take account of convergence towards plane trees with vertices of infinite degree, a weaker form of local convergence has been introduced in [START_REF] Jonsson | Condensation in Nongeneric Trees[END_REF] (see also [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF]Section 6]). The idea is to replace the ball B R (t) in ( 46) by the sub-tree B ← R (t), called the left ball of radius R of t. Formally, the root vertex belongs to B ← R (t), and a vertex u = uk ∈ t belongs to B ← R (t) iff u ∈ t, k ≤ R and |u| ≤ R.

For our purposes, a slightly stronger form of convergence is needed. For every t ∈ T f and every u ∈ t, we denote by (-u1, -u2, . . . , -uk u ) = (uk u , u(k u -1), . . . , u1) the children of u in counterclockwise order. For every t ∈ T f and every R ≥ 0, the left-right ball of radius R in t is the sub-tree B ↔ R (t) defined as follows. First, ∅ ∈ B ↔ R (t). Then, a vertex u ∈ t belongs to B ↔ R (t) iff u ∈ B ↔ R (t), |u| ≤ 2R and u ∈ { u1, . . . , uR} ∪ {-u1, . . . , -uR} (u is among the R first or last children of its parent). We call local- * topology the topology on T f induced by

d * loc (t, t ) := 1/(1 + sup {R ≥ 0 : B ↔ R (t) = B ↔ R (t )})
, t, t ∈ T f . The set T of general trees is the completion of T f for d * loc , while the set T loc of locally finite trees is the completion of T f for d loc .

Local limits of conditioned Galton-Watson trees. We next recall results concerning local limits of Galton-Watson trees conditioned to survive. The critical case. The critical setting was first investigated by Kesten [START_REF] Kesten | Subdiffusive behavior of random walk on a random cluster[END_REF] (see also [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF]) and extended by Stephenson in [START_REF] Stephenson | Local convergence of large critical multi-type Galton-Watson trees and applications to random maps[END_REF]. Let (ρ • , ρ • ) be a critical pair of offspring distributions, and recall that for every probability measure ρ on Z + with mean m ρ ∈ (0, ∞), the size-biased distribution ρ is defined by

ρ(k) := kρ(k) m ρ , k ∈ Z + .
The infinite random tree T

•,• ∞ = T •,• ∞ (ρ • , ρ • ) is
T •,• k (d) -→ k→∞ T •,• ∞ (ρ • , ρ • ).
Here and after, we implicitly work along a subsequence on which GW ρ•,ρ• ({|t| = k}) > 0. The subcritical case. We start with subcritical monotype trees, first considered in [START_REF] Jonsson | Condensation in Nongeneric Trees[END_REF] and studied in full generality in [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF][START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the condensation case[END_REF]. Let ρ be a subcritical offspring distribution (such that ρ(0) ∈ (0, 1)). The infinite random tree T ∞ = T ∞ (ρ) is defined as follows. It has a.s. a unique finite spine of random size L, such that

P (L = k) = (1 -m ρ )m k-1 ρ for k ∈ N.
The last vertex of the spine has infinite degree. The L -1 first vertices of the spine have offspring distribution ρ, and a unique child in the spine chosen uniformly among the offspring. Outside of the spine, vertices have offspring distribution ρ, and all the numbers of offspring are independent. This defines a random element of T . Proposition 5.2. Let ρ be a subcritical offspring distribution with no exponential moment (and ρ(0) ∈ (0, 1)). For every k ≥ 1, let T k be a tree with law GW (k) ρ . Then, in the local- * sense, T k

(d) -→ k→∞ T ∞ (ρ).
Proof. The proof follows from [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF]Theorem 7.1]. However, this result is equivalent to the convergence of left-balls of any radii (see [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF]Lemma 6.3]), which is weaker than our statement. Then, observe that for every t ∈ T f , k ≥ 0 and R ≥ 0 we have

GW (k) ρ (B ↔ R (T ) = t) = GW (k) ρ (B ← 2R (T ) = t) .
Indeed, GW (k) ρ is invariant under the operation consisting in exchanging the descendants of (u(R + 1), . . . u(2R)) and (-u1, . . . -uR) for every u ∈ t such that k u (t) > 2R (which exchanges B ↔ R (t) and B ← 2R (t)). This concludes the argument. We now extend Proposition 5.2 to two-type Galton-Watson trees. Let (ρ • , ρ • ) be a subcritical pair of offspring distributions. We build a two-type version T

•,• ∞ = T •,• ∞ (ρ • , ρ • ) of T ∞ as follows.
It has a.s. a unique spine, with random number of vertices 2L satisfying

P (L = k) = (1 -m ρ• m ρ• )(m ρ• m ρ• ) k-1 , k ∈ N.
In the spine, the topmost (black) vertex has infinite degree, while other vertices have offspring distribution ρ• (if white) and ρ• (if black), with a unique child in the spine chosen uniformly among the offspring. Outside of the spine, white (resp. black) vertices have offspring distribution ρ • (resp. ρ • ), and all the numbers of offspring are independent (see Figure 7).

Proposition 5.3. Let (ρ • , ρ • ) be a subcritical pair of offspring distributions such that ρ • is geometric with parameter 1 -p ∈ (0, 1) (ρ • (k) = (1 -p)p k for k ≥ 0), and ρ • has no exponential moment. For every k ≥ 1, let T •,• k be a tree with law GW (k) ρ•,ρ• . Then, in the local- * sense, T •,• k (d) -→ k→∞ T •,• ∞ (ρ • , ρ • ). Proof. For every k ≥ 1, let T k := Φ JS (T •,• k ). By Proposition 3.1, T k has law GW (k) ρ , where ρ(0) = 1 -p and ρ(k) = p • ρ • (k -1), k ∈ N.
In particular, ρ satisfies the hypothesis of Proposition 5.2. For every N ≥ 1, let u N = u N (T k ) be the first vertex of B ↔ N (T k ) in contour order having 2N offspring (or the root vertex otherwise). For every R ≥ 0, we also let T k u N , R be the collection of subtrees of T k containing all the children of u N different from {±u N 1, . . . ± u N R}, as well as their descendants. Finally, set T k [N, R] := B ↔ N (T k )\T k u N , R , and extend these definitions to T ∞ = T ∞ (ρ). We denote by u ∞ the vertex of infinite degree of T ∞ , and let T ∞ [R] be the subtree of T ∞ in which children of u ∞ other than {u ∞ 1, . . . u ∞ R} and their descendants are discarded. This definition extends to T •,• ∞ . Fix R ≥ 0. By Proposition 5.2 and the definition of T ∞ , we have in the local sense

T k [N, R + 1] (d) -→ k→∞ T ∞ [N, R + 1], and T ∞ [N, R + 1] (d) -→ N →∞ T ∞ [2(R + 1)]. (47)
In particular, the event (measurable with respect to

B ↔ N (T k )) E(R, N, k) := {sup{|u| ∨ k u : u ∈ T k [N, R + 1]} < N }
has probability tending to one when k and then N go to infinity. On the event E(R, N, k), one has

T k \T k [N, R + 1] ⊆ T k u N , R + 1 , which in turn enforces B ↔ R (T •,• k ) = B ↔ R (Φ -1 JS (T k )) ⊆ Φ -1 JS (T k [N, R + 1]). (48)
The critical case.

When (ρ • , ρ • ) is critical, T •,•
∞ is a.s. locally finite. We extend the mapping Loop to t ∈ T loc by defining Loop(t) as the consistent sequence of maps (Loop(B 2R (t)) : R ≥ 0). This mapping is continuous on T loc for the local topology. When t is infinite and one-ended, Loop(t) is an infinite looptree, that is, an edge-outerplanar map whose root face is the unique infinite face. Then, the random infinite looptree 7 for an illustration. Note that similar infinite looptrees also appear in [START_REF] Björnberg | Random Walk on Random Infinite Looptrees[END_REF].

L ∞ = L ∞ (ρ • , ρ • ) is defined by L ∞ := Loop(T •,• ∞ ). See Figure
The subcritical case. When (ρ • , ρ • ) is subcritical, T •,•
∞ has a unique vertex u ∞ with infinite degree. Since u ∞ has odd height, the sequence (B r (Loop(B ↔ R (T •,• ∞ ))) : R ≥ 0) is eventually stationary, for every r ≥ 0. Consequently, we define L ∞ = L ∞ (ρ • , ρ • ) as the local limit

L ∞ := lim R→∞ Loop(B ↔ R (T •,• ∞ )). ( 50
)
Although L ∞ is not a looptree in the aforementioned sense, we keep the notation L ∞ = Loop(T •,• ∞ ). The map L ∞ can also be obtained by gluing onto vertices i ∈ Z the independent looptrees L i := Loop(T i ), with (T i : i ∈ Z) as in Lemma 5.4 (see Figure 7 for an illustration). From the above arguments, we get the following result.

(m ρ • m ρ • < 1) u ∞ T 0 0 1 -1 (m ρ • m ρ • = 1) T 2 T 4 2 3 4 T -2 T 1 T 3 T -1 T -3 T -4 -2 -3 -4 L ∞ (ρ • , ρ • ) L ∞ (ρ • , ρ • ) T •,• ∞ (ρ • , ρ • ) T •,• ∞ (ρ • , ρ • )
L k := Loop(T •,• k ) (d) -→ k→∞ L ∞ (ρ • , ρ • ).
Note that the internal faces of L ∞ = Loop(T •,• ∞ ) are all finite in the critical case, while there is a unique infinite internal face in the subcritical case.

Local limits of Boltzmann maps with a boundary

We now deal with the q-IBHPM introduced in (1). This map, denoted by M ∞ = M ∞ (q), is a.s. one-ended with an infinite boundary (see [START_REF] Curien | Peeling random planar maps[END_REF]Theorem 7]).

The definitions of the scooped-out map and the irreducible components extend to any m ∈ M ∞ . We are now interested in the continuity of Scoop for the local topology. Proof. First, if (#∂m k : k ≥ 1) is bounded, there exists R ≥ 0 such that for every k ≥ 1, ∂m k ⊆ B R (m k ) and the result follows. Thus, we can assume that #∂m k → ∞ as k → ∞.

For every k ∈ N ∪ {∞}, let p(k) := #∂m k /2 and denote by (v k (0), v k (1), . . . , v k (p(k))) the vertices of the root face of m k from the origin, in right contour order (with repetition). We use the notation (v k (0), v k (-1), . . . , v k (-p(k))) for the left contour order.

Let r ≥ 0. We now prove that there exists R ≥ 0 and K ≥ 1 such that for every k ≥ K, As a consequence, in the cycle (-p(k), . . . , p(k)), there exists two distinct sequences of consecutive indices (i, . . . , i + x) and (j, . . . , j + y) such that x, y ≥ N + 2 and

V(B r (m k )) ∩ {v k (l) : i ≤ l ≤ i + x} = {v k (i), v k (i + x)},
and similarly for (j, . . . , j + y). In particular, the sets E 1 := {v k (i + 1), . . . , v k (i + x -1)} and E 2 := {v k (j + 1), . . . , v k (j + y -1)} are disjoint. Indeed, a vertex v ∈ E 1 ∩ E 2 would disconnect Scoop(m k ) in two submaps each containing a vertex at distance less than r from the origin, which is in contradiction with v / ∈ B r (m k ). Now, for every -p(k) ≤ i < p(k), (v k (i), v k (i + 1)) is an edge of Scoop(m k ). Therefore, {((v k (l), v k (l + 1)) : i < l ≤ i + N + 1} and {((v k (l), v k (l + 1)) : j < l ≤ j + N + 1} are disjoint sets of N half-edges contained in B r+N (m k )\B r (m k ). This holds for infinitely many k ≥ 1, thus for m ∞ . Since m ∞ has one end and N is arbitrary, this is a contradiction.

Let us choose R and K such that assertion (51) holds for every k ≥ K (and thus for m ∞ ). For every k ≥ K, let v k (-R), . . . , v k (R) be the sub-map induced by the R first half-edges Remark 6.3. When a = 2, (ν • , ν • ) can either be subcritical or critical. In the critical case (which includes the standard setting of [START_REF] Gall | Scaling limits of random planar maps with large faces[END_REF][START_REF] Borot | A recursive approach to the O(n) model on random maps via nested loops[END_REF] and the sequence q * ), the results of Theorem 1.2 and Proposition 5.7 hold. However, Proposition 6.2 suggests that ν • has a very heavy tail, meaning that ∂M ∞ has very large loops. In the subcritical case, Theorem 1.2 also holds provided that ν • has no exponential moment. This can be proved by using the analogue of Proposition 2.5 to ensure that r q = r q F 2 (r q ) (in this case, the slowly varying correction vanishes at infinity and the expansion is singular). Finally, we expect the scaling limit of the boundary to be a circle, but the normalizing sequence to be negligible compared to the perimeter 2k of the map (typically of order k/ log(k)).

Enumerative results. The sequence q * is convenient because we have an explicit formula for the partition functions (F k : k ≥ 0) by [START_REF] Budd | Geometry of infinite planar maps with high degrees[END_REF]Lemma 14] and [START_REF] Budd | Geometry of infinite planar maps with high degrees[END_REF]Equation (7)]:

F k = 3 4 6 k (k + 3/2)(k + 1/2)
, and

F (x) = 1 4x - 3 4(6x) 3/2 (1 -6x) log 1 + √ 6x 1 - √ 6x .
Consequently, r q = 1/6 and we deduce the asymptotic expansions as x → r - q

F (x) = 3 2 + 3 4 1 - x r q log 1 - x r q + 3 2 (1 -log(2)) 1 - x r q (1 + o(1)), (53)
F (x) = - 9 2 (3 -2 log(2)) - 9 2 log 1 - x r q + o(1). (54)
Unlike the previous cases, an expansion of F is not sufficient; we rather need an expansion of its derivative. The function P (x) = xF 2 (x) is again continuous increasing with inverse P -1 . Moreover, we have as x → r - q P (x) = P (r q )+P (r q ) 1 -x r q log 1 -x r q +P (r q )(2 log(2)-1) 1 -x r q (1+o(1)). [START_REF] Neveu | Arbres et processus de Galton-Watson[END_REF] We put c * := 2 log(2) -1. Let us define the function R and its inverse R -1 both on [0, 1] by R(x) := 1 P (r q ) (P (r q ) -P (r q (1x))) and R -1 (y) = 1 -1 r q P -1 (P (r q )(1y)) . [START_REF] Sheffield | Quantum gravity and inventory accumulation[END_REF] The expansion of R reads R(x) = -x log(x)c * x + o(x), as x → 1 -. We now need the Lambert W function, which is the multivalued inverse of x → xe x . We use the lower branch W -1 , continuous decreasing from [-1/e, 0) onto (-∞, -1], which satisfies

W -1 (-x) = log -x W -1 (-x)
and W -1 (x log(x)) = log(x), x ∈ (0, 1/e].

We also have W -1 (-x) = log(x)log(-log(x)) + o(1) as x → 0 + . We introduce the function

Q(x) := R -x W -1 (-x)
, x ∈ (0, 1/e],

which is continuous increasing. By [START_REF] Stephenson | Local convergence of large critical multi-type Galton-Watson trees and applications to random maps[END_REF], its inverse function Q -1 satisfies Q -1 (y) = -R -1 (y) log R -1 (y) and R -1 (y) = -Q -1 (y) W -1 (-Q -1 (y))

, y ∈ (0, R(1/e)]. [START_REF] Stufler | Limits of random tree-like discrete structures[END_REF]

Figure 2 :

 2 Figure 2: A rigid loop configuration on a quadrangulation with a boundary q.

Figure 5 :

 5 Figure 5: A planar map m and the associated scooped-out map Scoop(m).

  be the law of a Galton-Watson tree with offspring distribution ρ (resp. (ρ • , ρ • )) conditioned to have n vertices, provided this makes sense. We have the following conditioned version of Proposition 3.3 and Corollary 3.4. Corollary 3.7. Let q be a weight sequence of type a ∈ [3/2, 5/2]. Under P (k) q , Tree(M ) has law GW (2k+1) ν•,ν• , and Φ JS (Tree(M )) has law GW (2k+1) ν . Moreover, conditionally on Tree(M ), the maps ( M u : u ∈ Tree(M ) • ) associated to M by Φ TC are independent with law P (deg(u)/2) q .

Lemma 4 . 1 . [ 26 ]

 4126 For every m ∈ M, we have Scoop(m) = Loop(Φ JS (Tree(m))).

  defined as follows. It has a.s. a unique spine, in which white (resp. black) vertices have offspring distribution ρ• (resp. ρ• ), and a unique child in the spine chosen uniformly among their offspring. Outside of the spine, white (resp. black) vertices have offspring distribution ρ • (resp. ρ • ), and all the numbers of offspring are independent. The tree T •,• ∞ is illustrated in Figure 7. Proposition 5.1. [57, Theorem 3.1] Let (ρ • , ρ • ) be a critical pair of offspring distributions. For every k ≥ 1, let T •,• k be a tree with law GW (k) ρ•,ρ• . Then, we have in the local sense

Figure 7 :Lemma 5 . 5 .

 755 Figure 7: The infinite planar map L ∞ and the associated tree T •,• ∞ .

Lemma 5 . 6 .

 56 Let (m k : k ∈ N) be a sequence of maps in M, and m ∞ a one-ended infinite map such that m k → m ∞ in the local sense, as k → ∞. Then, in the local sense, Scoop(m k ) -→ k→∞ Scoop(m ∞ ).

V

  (B r (m k )) ∩ {v k (l) : |l| > R} = ∅. (51)We proceed by contradiction. By local convergence, the sequence(#V(B r (m k )) : k ≥ 0) is bounded. Moreover, for every v ∈ V(B r (m k )) we have #{-p(k) ≤ l ≤ p(k) : v k (l) = v} ≤ deg m k (v) ≤ sup u∈V(Br(m k )) deg m k (u),which is also bounded. Therefore, there exists M ≥ 0 such that for every k ≥ 0,#{-p(k) ≤ l ≤ p(k) : v k (l) ∈ V(B r (m k ))} ≤ M.Let N ≥ 0. By assumption, there exists infinitely many k such that p(k) > 2M (N + 2) and V(B r (m k )) ∩ {v m k (l) : |l| > M (N + 2)} = ∅.

  3.1 Random trees and the Janson-Stefánsson bijectionTrees. A (finite) plane tree t[START_REF] Gall | Random trees and applications[END_REF][START_REF] Neveu | Arbres et processus de Galton-Watson[END_REF] is a finite subset of the sequences of positive integers ∅ ∈ t and is called the root vertex. Then, for every u = (u 1 , . . . , u k ) ∈ t, u := (u 1 , . . . , u k-1 ) ∈ t (and is called the parent of u in t). Finally, for every u = (u 1 , . . . , u k ) ∈ t, there existsk u = k u (t) ∈ Z + (the number of children of u in t) such that uj := (u 1 , . . . , u k , j) ∈ t iff 1 ≤ j ≤ k u . The height |u| of a vertex u = (u 1 , . . . , u k ) ∈ t is |u| = k,and we denote by [∅, u] (resp. [∅, u)) the ancestral line of u in t, u included (resp. excluded). The vertices at even height are called white, and those at odd height are called black. We let t • and t • be the corresponding subsets of vertices of t.

	U :=	n∈Z +	N n
	satisfying the following properties. First,

Acknowledgements. Many thanks to Grégory Miermont for enlightening discussions and for attentively reading this work. I would also like to thank warmly Erich Baur, Jérémie Bouttier, Timothy Budd, Nicolas Curien and Igor Kortchemski for very useful discussions and comments. Finally, I am deeply indebted to the anonymous referees for their careful rereading, crucial corrections and remarks.

This work was partially accomplished at UMPA, École Normale Supérieure de Lyon and supported by the grant ANR-14-CE25-0014 (ANR GRAAL).

Indeed, on this event, the images of vertices of T k \T k [N, R + 1] in Φ -1 JS (T k ) are descendants of the children of u N := Φ -1 JS (u N ) that are not in {±u N 1, . . . ± u N R}. (See Figure 6.) Let d ≥ 0, and keep the notation u ∞ for the pointed vertex with d children in T ∞ [d] and T •,• ∞ [d]. We let GW [d] ρ be the law of (T ∞ [d], u ∞ ), and GW [d] ρ•,ρ• be that of (T

We temporarily admit [START_REF] Marckert | Invariance principles for random bipartite planar maps[END_REF] and conclude the proof. Let A be a Borel set for the local- * topology. We have by [START_REF] Lyons | Probability on trees and networks[END_REF] that for every k ≥ 1 and N ≥ 1

Next, for every N ≥ 1, [START_REF] Gall | Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere[END_REF] entails

Then, by [START_REF] Gall | Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere[END_REF] again and the fact that T ∞ [2(R + 1)] is a.s. finite,

Finally, for every

) by definition so that by [START_REF] Marckert | Invariance principles for random bipartite planar maps[END_REF],

As a conclusion, by letting k and then N go to infinity, we have

Let us now prove assertion [START_REF] Marckert | Invariance principles for random bipartite planar maps[END_REF]. Let (t, u * ) be a pointed plane tree such that k u * (t) = d + 1. By definition, (t , v * ) := Φ -1 JS (t, u * ) is a pointed plane tree satisfying k v * (t ) = d, and v * ∈ t • . Then, we have by definition of ρ • and the identity u∈t

Vertices of t • are mapped to leaves of t by Φ JS , while vertices of t • with k children are mapped to vertices of t with k + 1 children. By Proposition 3.1, we get

The image of T k by Φ -1 JS , on the event E(R, N, k). The boxed vertex is the last leaf of T k in contour order, while the crossed vertex is the last leaf among the descendants of u N .

We conclude with a property of T •,• ∞ in the subcritical case. Let u ∞ be the unique vertex with infinite degree of T •,• ∞ , and u ∞ its parent. There exists j ∈ {1, . . . , k u∞ } such that u ∞ = u ∞ j. We define the vertex u ← ∞ as u ∞ (j -1) if j > 1, and u ∞ itself if j = 1. The vertex u ∞ and its incident edges disconnect T •,• ∞ in infinitely many connected components that we denote by (T i : i ∈ Z). For every i = 0, T i is the connected component containing u ∞ i, rooted at the oriented edge going from

∞ , and has the same root edge as T •,• ∞ . Lemma 5.4. The plane trees (T i : i ∈ Z) are independent. For every i = 0, T i has law GW ρ•,ρ• , while T 0 has the size-biased law GW ρ•,ρ• defined by

Moreover, conditionally on T 0 , u ← ∞ has uniform distribution on T 0 . Proof. We focus on T 0 . Let (t, u * ) be a pointed plane tree, and let u

and by definition of ρ • , we obtain

We conclude by Proposition 3.1, which gives

Random infinite looptrees.

We now define infinite planar maps out of the infinite trees T

of Scoop(m k ) in left and right contour order. We denote by H the measurable function such

. By [START_REF] Marzouk | Scaling limits of random bipartite planar maps with a prescribed degree sequence[END_REF], we have for every

which concludes the proof.

Recall that when m ∈ M ∞ has a unique infinite irreducible component, it is called the core of m, and denoted by Core(m). We are now ready to prove Theorem 1.2.

Proof of Theorem 1.2. For every k ≥ 1, let M k be a map with law P (k) q . By Corollary 3.7,

By (1) and Lemmas 5.6 and 5.5, we have Local limits: the subcritical and dense regimes. When q is of type a ∈ [3/2, 2), M ∞ can be built from the looptree L ∞ (ν • , ν • ) and a collection of independent Boltzmann maps. This generalizes [6, Theorem 4] which deals with subcritical quadrangulations. We define a fill-in mapping that associates to a one-ended tree t ∈ T loc and a collection ( m u : u ∈ t • ) of finite maps with a simple boundary of respective perimeter deg(u) the map

obtained from l := Loop(t) by gluing the map m u in the face of l associated to u, for every u ∈ t • . We keep the notation Φ -1 TC by consistency, although we consider infinite trees. This mapping is continuous with respect to the natural topology. Proposition 5.7. Let q be of type a ∈ [3/2, 2), and

• ) be independent maps with a simple boundary and law P

• has the law of the q-IBHPM.

Proof. The proof closely follows that of [START_REF] Baur | Uniform infinite half-planar quadrangulations with skewness[END_REF]Theorem 4]. For every t ∈ T loc and every R ≥ 1, let Cut R (t) be the subtree of t made of vertices u ∈ t such that |u| ≤ 2R. Consistently, if

Let R ≥ 1 and for every k ≥ 0, let M k be a map with law P (k)

q . Let m ∈ M and (t, ( m u : u ∈ t • )) = Φ TC (m). By Proposition 3.3 and 5.1, we have

Local limits: the dilute and generic regimes. When q is of type a ∈ (2, 5/2], M ∞ cannot be fully described using finite maps. We believe that the finite irreducible components of M ∞ are independent Boltzmann maps with a simple boundary (conditionally on Scoop(M ∞ )). Moreover, we conjecture that there exists a distribution P (∞) q on one-ended maps with an infinite simple boundary such that

q . This result is proved for quadrangulations in [START_REF] Curien | Uniform infinite planar quadrangulations with a boundary[END_REF]Proposition 6], but relies on enumeration results for quadrangulations with a simple boundary that are unknown for general maps.

6 The non-generic critical case with parameter α = 3/2

We now deal with the parameter α = 3/2 (a = 2) that has been excluded so far. The results of Section 2 still hold by considering a = 2 as part of the dense regime if F (r q ) = ∞, and of the dilute regime if F (r q ) < ∞. For instance, the proofs of Propositions 2.4 and 2.5 can be slightly adapted. For the former, we use [10, Proposition 1.5.9 a-b] to check that the assumption [10, Equation (8.1.11 a-c)] of Karamata's theorem [10, Theorem 8.1.6] is satisfied. For the latter, when F (r q ) = ∞, we use the so-called de Bruijn conjugate of a slowly varying function [10, Theorem 1.5.13] instead of Lemma 2.7. The issue comes from Proposition 3.6, because in this case Karamata's theorem merely provides information on the tail of the size-biased version of ν, see [START_REF] Bingham | Regular Variation[END_REF]]. We now bypass this difficulty by using a special weight sequence introduced in [START_REF] Ambjørn | Generalized multicritical one-matrix models[END_REF], and by calling on de Haan theory [START_REF] Bingham | Regular Variation[END_REF]Chapter 3]. Let us start with a general statement regarding the criticality of the tree of components that is a consequence of ( 15) and [START_REF] Gwynne | Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times[END_REF]. Lemma 6.1. Let q be a weight sequence of type a = 2. Then, ν is critical if and only if

The special weight sequence q * = (q * k : k ∈ N) introduced in [START_REF] Ambjørn | Generalized multicritical one-matrix models[END_REF] (see also [START_REF] Budd | Geometry of infinite planar maps with high degrees[END_REF]Section 5]) is defined by

The sequence q * is admissible, critical, and of type a = 2. We will prove the following. Proposition 6.2. Let q * be the sequence defined by [START_REF] Miermont | An invariance principle for random planar maps[END_REF]. Then, (ν

, and ν

In particular, ν and ν • are in the domain of attraction of a Cauchy distribution.

Using the above expansions, we get as x, y → 0

Together with [START_REF] Stufler | Limits of random tree-like discrete structures[END_REF] and the expansion of W -1 , this yields

Finally, by ( 56) we obtain

This proves that r q = P (r q ). By differentiating both sides in the equation of Lemma 2.6 and using ( 53), ( 54) and ( 61) we obtain the wanted expansion of F : as y → P (r q ) -,

log 2 1 -y P (rq)

The tree of components. We are now interested in properties of the tails of ν and ν • . To do so, we need estimates on the derivative of L ν . Recalling [START_REF] Flajolet | Analytic Combinatorics[END_REF] and ( 62