A phase transition in block-weighted random maps

We consider the model of random planar maps of size $n$ biased by a weight $u>0$ per $2$-connected block, and the closely related model of random planar quadrangulations of size $n$ biased by a weight $u>0$ per simple component. We exhibit a phase transition at the critical value $u_C=9/5$. If $u<u_C$, a condensation phenomenon occurs: the largest block is of size $\Theta(n)$. Moreover, for quadrangulations we show that the diameter is of order $n^{1/4}$, and the scaling limit is the Brownian sphere. When $u>u_C$, the largest block is of size $\Theta(\log(n))$, the scaling order for distances is $n^{1/2}$, and the scaling limit is the Brownian tree. Finally, for $u=u_C$, the largest block is of size $\Theta(n^{2/3})$, the scaling order for distances is $n^{1/3}$, and the scaling limit is the stable tree of parameter $3/2$.


Introduction
Models of planar maps exhibit a form of universality: many "natural" classes of random maps exhibit a similar behaviour when the size grows to infinity. This can be made precise by considering scaling limits: when taking an object M n uniformly among all objects of size n in some class, then, after an appropriate rescaling, the sequence converges towards a certain metric space. This was first proved for uniform quandrangulations by Miermont [Mie13] and Le Gall [LG13], following a sequence of results on this subject [MM03,CS04,LG07,LG10,LM11]. Since then, these results have been extended to other families of maps: the sequence converges towards the Brownian sphere M e (also called Brownian map, see Fig. 1), always with a rescaling by cn 1/4 for some model-dependent c > 0. Gromov-Hausdorff's topology allows to give meaning to the convergence of a sequence of maps to a certain limit, considering them as (isometry classes of) compact metric spaces. In particular, uniform planar maps also converge towards the Brownian sphere [BJM14], as well as other families such as uniform triangulations and uniform 2q-angulations (q 2) [LG13], uniform simple triangulations and uniform simple quadrangulations [ABA17], bipartite planar maps with a prescribed face-degree sequence [Mar18], (2q + 1)-angulations [ABA21] and Eulerian triangulations [Car21].
On the other hand, "degenerate" classes of maps that "look like" trees also exhibit similar behaviours. In particular, up to rescaling by cn 1/2 , there is a convergence to the Brownian tree T (2) (see Fig. 2), the scaling limit of uniform critical Galton-Watson trees [Ald93,LG06]. This is the case for classes of maps with a tree-decomposition such as stack triangulations [AM08]; classes of maps with some particular boundary conditions, such as quadrangulations of a polygon [Bet15], outerplanar maps [Car16]; or, more generally for "subcritical" classes [Stu20] (see [PSW16] for the case of graphs).  Models interpolating between the Brownian tree and the circle can be obtained by using looptrees [CK13b]. Curien and Kortchemski considered the boundary of large percolation clusters in the uniform infinite planar triangulation (which is the local limit of large triangulations) where each vertex is coloured (independently) white with probability a ∈ (0, 1) and black otherwise. They showed that if a ∈ (0, 1/2), the scaling limit is the Brownian tree, if a ∈ (1/2, 1) it is the unit circle and if a = 1/2 it is the stable looptree of parameter 3/2 [CK13a], which correspond to the stable tree of parameter 3/2 (see Fig. 3) where each branching point is replaced by a circle. Richier [Ric18] also showed that the boundary of critical Boltzmann planar maps with degrees of faces in the domain of attraction of a stable distribution with parameter α ∈ (1, 2] exhibit a similar phase transition: if α ∈ (1, 3/2), the scaling limit is the stable looptree of parameter (α − 1/2) −1 , and, with Kortchemski, Richier showed that it is the circle of unit length if α ∈ (3/2, 2] and conjectured that this holds also for α = 3/2 [KR20]. In both cases, the parameter of the model allows the number of cut vertices appearing on the boundary to be adjusted, thus changing from a "round" to a "tree" phase. Some natural models also interpolate between the Brownian sphere and the Brownian tree. For example, consider random quadrangulations with n faces and a boundary of length , where / √ n → σ. When σ = 0, the scaling limit is the Brownian sphere, when σ = ∞ it is the Brownian tree, and for all σ ∈ (0, ∞) it is the Brownian disk with boundary length σ [Bet15].
Model. The purpose of this paper is to propose yet another model interpolating between the Brownian sphere and the Brownian tree, but where the transition does not appear through the boundary. It relies on a parameter tuning the density of separating elements. In this model, a map m is sampled with a probability which depends on its number b(m) of maximal 2-connected 3 Figure 3: Approximation of the stable tree 3/2 by a tree of size approximately 150 000. 4 components, or "blocks", for which a precise definition will be given later in Section 2.
In fact, we will consider two probability distributions on maps, both indexed by a parameter u > 0. The first one is a fixed size model: for any n ∈ Z 0 , we define where M n is the set of maps with n edges, M (z, u) = m∈M u b(m) z |m| = n∈Z 0 ([z n ]M (z, u)) z n and |m| is the number of edges of m. The second one is a Boltzmann-type distribution which samples maps with random sizes. More precisely, write ρ(u) for the radius of convergence of z → M (z, u). We define: ρ(u) |m| for any m ∈ M. (2) The qualitative properties of maps sampled according to these measures change drastically when u varies, and we will see that it gives rise to different regimes with a phase transition. In this paper, blocks will be either maximal 2-connected components of maps, or maximal simple components of quadrangulations. Indeed, both models have the same underlying structure, so one study gives results for both (see Sections 2.4 and 2.5), except for some of the scaling limit results, where some convergence results for 2-connected maps are missing. However, our approach could be generalised to many other models with an underlying tree structure (see Table 3), such as the ones described in [BFSS01]. In particular, the case u = 1 corresponds to sampling a uniform map and u → 0 to sampling a uniform block.
Block decompositions have already been used in the context of scaling limits, and some joint convergences are known: a quadrangulation, its largest 2-connected block, and its largest simple block jointly converge to the same Brownian sphere [ABW17].
The model with a weight per 2-connected blocks was already analysed with a combinatorics point of view by Bonzom [Bon16,§8] with physical applications in mind (see [DS09] for a thorough discussion). The so-called quadric model studied in his work can be specialized to our model. Bonzom obtains rational parametrisations for the generating series, and exhibits the possible singular behaviours, which suggest the existence of three different regimes: a "map behaviour", a "treebehaviour", and in-between a "proliferation of baby universes". Since his focus is much broader, he does not go into details to study this particular model from a probabilistic point of view, and this is the main topic of the present article. For u = 1, which corresponds to sampling maps uniformly, this model has also been studied with the point of view of block decomposition in [BFSS01] and [AB19].
Results. Our results are summarized in Table 1. In Section 4, we show that, with high probability, when u < 9/5, there is condensation with one block of size Θ(n) and all others of size O(n 2/3 ) (Theorem 2); when u > 9/5, the largest block has size Θ(log(n)) (Theorem 3) and when u = 9/5 the largest block is of size Θ(n 2/3 ) (Theorem 4).
In Section 5, we give a unified proof of the convergence towards T (2) , after renormalising distances by n 1/2 , in the supercritical case u > 9/5; and towards T (3/2) , after renormalising distances by n 2/3 , in the critical case u = u C (Theorem 5). For u > 9/5, we retrieve a previous result by Stufler for more general weighted models [Stu20]. All these results hold for both maps and their 2-connected cores, and quadrangulations and their simple cores. Finally, when u < 9/5, we show in Theorem 6 that quadrangulations converge towards the Brownian sphere when renormalising distances by n 1/4 . In the case of quadrangulations, these results are consistent with existing literature for the case Largest block Scaling Scaling limit u < 9/5 Θ(n) n 1/4 Brownian sphere M e 1 u = 9/5 Θ(n 2/3 ) n 1/3 Stable tree T (3/2) u > 9/5 Θ(log(n)) n 1/2 Brownian tree T (2)  LG13,BJM14], as well as when u → 0 [ABA17]. We rely crucially on the convergence of uniform simple quadrangulations with the same normalisation, which is proven in [ABA17], and recalled in Proposition 22 below. A similar convergence result for uniform 2-connected maps would be needed in order to prove a version of Theorem 6 for maps, see the discussion after the statement of Proposition 22. Such a convergence is expected to hold and hinted at for instance by Lehéricy's results [Leh22], which show that graph distances on a uniform map of size n and on its quadrangulation via Tutte's bijection behave similarly when n → ∞. Section 2 and Theorem 1 introduce tools to prove these theorems. We show that maps and quadrangulations can be decomposed into blocks with an underlying tree structure. We show that the law of such trees can be described by a Galton-Watson model (as in several papers cited above). From there, we exhibit in Section 3 a phase diagram going from a condensation phenomenon (u < 9/5) to a critical "generic" regime (u > 9/5) going through a "non-generic" critical point (u = 9/5).

Maps and their enumeration
A planar map m is the proper embedding into the two-dimensional sphere of a connected planar finite multigraph, considered up to homeomorphisms. Let V (m) be the set of its vertices, E(m) the set of its edges and F (m) the set of its faces. The size of a planar map m -denoted by |m| -is defined as its number of edges.
A half-edge e is an oriented edge from u to v (with possibly v = u) and is represented as half of an edge starting from u. Its starting vertex u is denoted by e − and its end vertex v is denoted e + . Let − → E (m) be the set of half-edges of m.
A corner is the angular sector between two consecutive edges in counterclockwise order around a vertex. Each half-edge is canonically associated to the corner that follows it in counterclockwise order around its starting vertex. The degree of a face is the number of corners incident to it.
All the maps considered in this paper are rooted, meaning that one of their half-edges (or one of their corners) is distinguished. The set of rooted planar maps -simply called maps in the following -is denoted by M. For n in Z 0 , let m n be the number of maps of size n and M (z) = n∈Z 0 m n z n be the associated generating series. By convention, we set m 0 = 1 which corresponds to the vertex map: the map reduced to a single vertex. Similarly, define the edge map as the map reduced to a single edge between two vertices.
Rooting simplifies the study by avoiding symmetry problems, however we expect our results remain true in the non-rooted setting due to the general results of [RW95]. The enumerative study  of rooted planar maps was initiated by Tutte in the 60s. In particular, he obtained the following result: Proposition 1 ( [Tut63]). The number m n of maps of size n is equal to This implies in particular that ρ M = 1/12 and M (ρ M ) < ∞, where ρ M denotes the radius of convergence of M (z).

2-connected maps and block decomposition
Definition 1. Let m ∈ M and v ∈ V (m). Then, v is said to be a cut vertex if m \ {v} is not connected. A map m is said to be separable if it has at least one cut vertex. It is said to be 2-connected otherwise, see Fig. 4.
For n ∈ Z 0 , we write B n for the set of 2-connected maps of size n, and b n = |B n |. From Fig. 5, we see that b 0 = 1, b 1 = 2 and b 2 = 1. Notice in particular that the only 2-connected map with a loop is the map reduced to a loop-edge. In the 60's, Tutte introduced the so-called "block decomposition of maps" [Tut63], which roughly speaking corresponds to cutting the map at all cut-vertices, and is illustrated on Fig. 6 (this is known for graphs as well and called block-cut tree, see e.g. [Har69]).
We describe here this decomposition drawing inspiration from Addario-Berry's presentation [AB19,§2]. Let m be a map and let b be the block containing its root. For each half-edge e of b, we define the pendant submap m e of e as the maximal submap of m disjoint from b except at e − and located to the left of e (it is possibly reduced to the vertex map). If m e has at least one edge, we root it at the half-edge of m following e in counterclockwise order around e − (see Fig. 7). From b and the 2|E(b)| pendant submaps {m e , e ∈ − → E (b)}, it is possible to reconstruct the map m: for each m e rooted at the half-edge ρ, insert m e in the corner associated to e in such a way that ρ is the first edge after e in counterclockwise order and merge ρ − and e − . Thus, a map can be encoded as a block where each edge is decorated by two maps. This decomposition induces an identity of generating series, thanks to the symbolic method [FS09,Ch1]. Letting B(y) = n∈Z 0 b n y n , Tutte's block decomposition translates into the following equality of generating series: Thanks to (4) and an explicit expansion for M (z) obtained in [Tut62], Tutte obtained the following enumerative results for 2-connected maps.
Moreover, writing ρ B for the radius of convergence of the series B, we have In the following, we consider maps enumerated by both their number of edges and their numbers of blocks. Namely, we consider the following bivariate series: M (z, u) = m∈M z |m| u b(m) (recall that b(m) is the number of blocks of m and |m| is its number of edges). Tutte's decomposition of a map into blocks translates in the following refined version of (4): where the term 1 − u accounts for the fact that the vertex map has no block by Definition 2 (even if it is 2-connected). For u > 0, denote by ρ(u) the radius of convergence of z → M (z, u). Since for z 0 and u 1 M (z, u) m∈M z |m| u |m| = M (uz), if |uz| < ρ M = 1/12, then M (z, u) is a converging sum. Hence, for u 1, ρ(u) 1 12u > 0. On the other hand, since ρ(u) is decreasing, for u 1 we have ρ(u) ρ(1) = ρ M = 1/12 (and ρ(u) ρ(0) = ρ B = 4/27).
In view of the form of the equation (7) and in particular that it is non-linear, it holds that M (ρ(u), u) < ∞. Indeed, since B(y) 1 + 2y for all y 0, we get M (z, u) 1 + 2uzM (z, u) 2 . This shows that it is impossible that

Block tree of a map and its applications
Tutte's block decomposition can also be applied recursively, i.e. we consider first the root block and then apply the block decomposition to each of the pendant submaps. By doing so, for any map m we can obtain a decomposition tree T m , which was first explicity described by Addario-Berry in [AB19, §2]. More precisely: 1. Let b = (b, ρ) be the maximal 2-connected submap containing the root ρ. The root v ρ of T m represents b, and has 2|E(b)| children (in particular, if b is of size 0, v ρ is a leaf); 2. List the half-edges of b as a 1 , . . . , a 2|E(b)| according to an arbitrarily fixed deterministic order on half-edges (e.g. the order in a left-to-right depth first search). Let m i be the pendant submap in the corner corresponding to the half-edge a i in b. The i-th pendant subtree of T m is the subtree encoding T m i .
By abuse of language, we might refer to (b v , v ∈ T m ) as the family of blocks (even if blocks necessarily have positive size). A direct consequence of this proposition is that to study the block sizes of a map m, it is sufficient to study the degree distribution of T m . This is precisely the strategy developed by Addario-Berry in [AB19]. This allows him to study the block sizes of a uniform random map M n of size n, by describing T Mn as a Galton-Watson tree with an explicit degree distribution conditioned to have 2n edges, and one of our contributions is to extend his result to our model.

Block tree of a quadrangulation
We describe in this section how a quadrangulation can be decomposed into maximum simple quadrangular components, in the same way that a map can be decomposed into maximum 2connected components.
Definition 3. A quadrangulation is a map with all faces of degree 4.
Planar quadrangulations are bipartite, i.e. their vertices can be properly bicolored in black and white. In the following, we always assume that they are endowed with the unique such coloring having a black root vertex. Although quadrangulations are maps, when an object is explicitly defined as a quadrangulation, its size will be its number of faces. Thus, a quadrangulation of size n has 2n edges.
Definition 4. A quadrangulation of the 2-gon is a map where the root face -the face containing the corner associated to the root -has degree 2 and all other faces have degree 4.
A quadrangulation of the 2-gon with at least two faces can be identified with a quadrangulation of the sphere by simply gluing together both edges of the root face.
Definition 5. A quadrangulation is called simple if it has neither loops nor multiple edges. Definition 6. Let e 1 e 2 be a 2-cycle of a quadrangulation q, its interior is the submap of q between e 1 and e 2 (both included) which does not contain the root corner of q. A 2-cycle is maximal when it does not belong to the interior of another 2-cycle.
Definition 7. Let e 1 e 2 be a maximal 2-cycle of a quadrangulation q, its pendant subquadrangulation is defined as its interior, which is turned into a quadrangulation of the 2-gon by rooting it at the corner incident to the unique black vertex of e 1 e 2 .
Let e be a half-edge of a quadrangulation q. If e is oriented from black to white and there exists a half-edge f such that ef is a maximal 2-cycle of q, then the pendant subquadrangulation of e is the pendant subquadrangulation of ef . Else, it is the edge map (which is also a quadrangulation of the 2-gon).
For q a quadrangulation, its simple core q s -the simple block containing the root -is obtained by collapsing the interior of every maximal 2-cycle of q. Similarly as for maps, a decomposition tree T (q) q can be associated to a quadrangulation q, by recursively decomposing the pendant subquadrangulations at the simple core, see Fig. 9. Simple blocks are recursively defined as the simple cores e m e Figure 10: Reconstructing a quadrangulation from its simple core and the pendant subquadrangulations appearing in the underlying arborescent decomposition. We then have an exact parallel with the situation of maps and their 2-connected components.
Given a simple quadrangulation q s and a collection of |E(q s )| = 2|q s | quadrangulations of the 2-gon {m e , e ∈ E(q s )}, it is possible to construct a quadrangulation: for each m e of root ρ e replace e by m e such that ρ e has the orientation e. See Fig. 10 for an illustration. This transformation is invertible. Thus, a quadrangulation can be encoded as a simple quadrangulation where each edge is decorated by one quadrangulation of the 2-gon, i.e. each face is decorated by two quadrangulations of the 2-gon: where Q is the generating series for quadrangulations (with a weight z for faces, and u for simple blocks) and S is the generating series for simple quadrangulations (with a weight z for faces). Note that this equation is isomorphic to (7). This decomposition and the former one presented for general maps are in fact two sides of the m φ(m) 2. The half-edge e created in the corner of ρ is now the root, oriented from black to white; 3. Remove the original edges.
Proposition 4. For n ∈ Z >0 , the function ϕ is a bijection between maps of size n and quadrangulations of size n. Moreover, it maps bijectively 2-connected maps of size n 1 to simple quadrangulations of size n.
The construction ϕ is due to Tutte [Tut63, §5] (he defines the notion of derived map, from which the angular map is extracted by deleting one of the 3 classes of vertices, as explained in [Bro65,§7]). The specialization to 2-connected maps is explained e.g. in [Bro65]. In particular, it implies that S(y) = B(y). Moreover, given Equations (7) and (8), this gives M (z, u) = Q(z, u) + 1.
Finally, when constructing the decomposition tree T

Probabilistic consequences
Recall the model defined in Equations (1) and (2) for general maps. As promised, we now define its analogue on quadrangulations, and show their equivalence. To that end, we set for all m ∈ M n , and for all q ∈ Q n , and consider for all m ∈ M and q ∈ Q the singular Boltzmann laws (remember that, as explained in Section 2.2, M (ρ(u), u) = Q(ρ(u), u) < ∞) By Proposition 5, one has: Proposition 6. For all q ∈ Q and n ∈ Z 0 , P quad n,u (q) = P map n,u ϕ −1 (q) and P quad so, denoting by * the pushforward, for all n ∈ Z 0 , P quad n,u = ϕ * P map n,u and P quad u = ϕ * P map u .

A word on the probabilistic setting
We denote by M : M → M the canonical random variable on the space of maps, and let Q = ϕ(M). We denote by T the block tree associated to M (and also to Q by Proposition 5). In this way, under P u (resp. P n,u ), M has law P map u (resp. P map n,u ), and, by Proposition 6, Q has law P quad u , (resp. P quad n,u ). Therefore, we will simply use P n,u and P u as a shorthand notation for P map n,u and P map u . Maximal simple components of quadrangulations will also be called "blocks" because everything that has been said about blocks (in the sense of maximum 2-connected components) can also be said about the maximum simple quadrangular components of quadrangulations; and likewise in everything that follows. As a consequence, every result about the size of the blocks of a map of size n is valid for blocks of quadrangulations of size n as well. For These random variables will be studied under probability measures P u and (P n,u ) n 1 , which were introduced in Section 2.5. We write accordingly E u [. . . ] and E n,u [. . . ] the expectations with respect to these probability measures. Unless mentioned otherwise or if it is clear from context, other random variables shall be viewed as defined on some probability space (Ω, P ), and the according expectations will be written as E [. . .]. In particular we will use the following random variables defined on (Ω, P ):

13
• For each u 0, the triplet (T n,u , M n,u , Q n,u ) is (T, M, Q) under the law P n,u .
• For each k 1, the pair (B map k , B quad k ) consists of a 2-connected map B map k with k edges sampled uniformly, together with B quad k = ϕ(B map k ) its image by Tutte's bijection. By Proposition 4, the latter is a simple quadrangulation with k faces sampled uniformly.

Phase diagram
For µ a probability distribution on Z 0 and n ∈ Z 0 , we denote by GW (µ, n) the law of a Galton-Watson tree with offspring distribution µ and conditioned to have n edges. Following [AB19], for u > 0 we aim at finding a measure µ u such that T under P n,u has law GW (µ u , 2n). To that end, for any y ∈ [0, ρ B ] we introduce the following probability distribution where b j and B are defined in Proposition 2. Moreover (see Remark 1 for a discussion), we set y(u) := ρ(u)M 2 (ρ(u), u) and µ u := µ y(u),u for any u > 0, where we recall that ρ(u) is the radius of convergence of z → M (z, u). On Fig. 12, the value of y(u) is represented, using an explicit expression (see Remark 2) . Notice that in view of (7), y(u) ρ B for all u > 0 and 1 + u(B(y(u)) − 1) = M (ρ(u), u).
Then, by (5), for all u > 0, we have: so that by setting it holds that The following proposition extends [AB19, Proposition 3.1] to our setting.
For every u > 0, under P u , the law of tree of blocks (T, (B v , v ∈ T)) can be described as follows.
• T follows the law GW (µ u ); • Conditionally given T = t, the blocks (B v , v ∈ t) are independent random variables, and, for v ∈ t, B v follows a uniform distribution on the set of blocks of size is the number of children of v in t. For every n 1, the same statements hold under P n,u , only replacing GW (µ u ) with GW (µ u , 2n).
Proof. It suffices to prove the first part of the statement.
Let t be a tree with 2n edges, where each vertex has an even number of children, and let (b v , v ∈ t) be a family of (2-connected, or simple) blocks, with 2|b v | = k v (t) for any v ∈ t. Let m be the map (or quadrangulation) with block decomposition given by (t, (b v , v ∈ t)).
Then, we have This concludes the proof.
Notice that the case u = 1, which corresponds to uniform planar maps, as studied by Addario-Berry [AB19], falls in the subcritical regime.

A singularity of
is obtained as an immediate consequence of Equations (6) and (15), and the asymptotic behaviour of µ u (2j) comes from Equations (12) and (13). This happens iff u 9/5.
Notice that at u = u C , both types of singularity are reached.
Remark 1. The proof of Theorem 1 highlights the reasons behind our choice of y(u). When u 9/5, we choose y(u) such that E(u) = 1. When u < 9/5, this is not possible, and we choose the value of y(u) maximising E(u) so that, when conditioning the trees to be of size 2n, the conditioning is as little degenerated as possible.
Remark 2. Using (16), we obtain an explicit expression for y in terms of u for u u C . By [Tut63], the series B is algebraic and for all y ∈ [0, 4/27], This gives an expression of B in terms of B, and taking the resultant between this new equation and (16) allows to eliminate B. Initial conditions then give 4 Study of the size of the largest blocks

Subcritical case
To investigate the distribution of the size of the largest blocks, in the subcritical case, we follow the approach developped in [AB19], which consists in studying the degrees in the block tree of a map.
To that end, we rely on results of condensation in Galton-Watson trees: exactly one of the nodes has a degree linear in the size. To that end, we rely on Janson's survey [Jan12], in which there is a refinement of the study of the largest degree of a subcritical Galton-Watson tree with condensation by Jonsson and Stefánsson [JS11]. The condensation phenomenon is visible in the following result where, denoting by d T V the total variation distance, we write X n Proposition 8 ([Jan12, Theorem 19.34]). Let µ be a probability distribution on Z 0 such that µ(0) > 0, E [µ] < 1 and there exists c satisfying µ(k) ∼ k→∞ ck −5/2 . Let D n,1 D n,2 . . . D n,n be the ranked list of the number of children of a µ-Galton-Watson tree conditioned to have n edges. Then, letting ξ 1 , . . . , ξ n−1 be a family of n − 1 independent random variables of law µ and ξ n−1 their decreasing reordering, it holds that: We combine this proposition with the fact that T is a Galton-Watson tree under P u to get the following generalization of [AB19, Theorem 3.3] to every value of u ∈ (0, 9/5) 2 . This is a rephrasing of the results for trees of [Jan12], to which we add the proof of the joint convergence. For m a map of size n, denote by LB 1 (m) . . . LB b(m) (m) the sizes of its blocks in decreasing order. By Theorem 2. Let u ∈ (0, 9/5). Recall that E(u) and c(u) are defined in Equations (12) and (14).
When u → 0, 1 − E(u) → 1: as expected, if the map has only one block, its size is n.
is a Stable process of parameter 3/2 satisfying E e −sL 1 = e Γ(−3/2)s 3/2 for s such that Re(s) 0; then, it is known that (see [Ber96, Theorem 1] and its proof ): Proof. Recall that the subcritical case corresponds to u ∈ (0, 9/5), for which we have We follow essentially the same lines of proof as in [AB19], but refining the arguments so as to establish the joint convergence stated in (21). Theorem 1 shows that the hypotheses of Proposition 8 are satisfied in the subcritical case.
Let (ξ i ) i 1 be a family of iid random variables of law µ u and let ξ be the decreasing reordering of its first n variables (take the convention ξ Let us consider the following cumulative process: It is standard [Fel71, Theorem XVII.5.2] [JS87, Chapter VII, Corollary 3.6] that there exists a Lévy process (L t ) t∈[0,1] with Lévy measure π(dx) = x −5/2 dx1 {x>0} so that for s such that Re(s) 0, and such that the following convergence holds in the Skorokhod topology By definition of the process L In particular, denoting by ∆P t the jump of the process (P t ) at time t (which may equal 0),

By [JS87, Chapter VI, Proposition 2.4], (22) gives
By construction of a Lévy process, (∆L (j) ) j 1 has same law as the decreasing rearrangement of the atoms of a Poisson random measure with intensity π on R + (see e.g. [Ber96, Theorem 1]). By denoting t So, applying again [JS87, Chapter VI, Proposition 2.4], one gets, denoting by t 1 the time of the largest jump of (L 1 ): It is again possible to iterate by subtracting the largest jump: for all k 1, However, by Proposition 8 and (20), one has (recall that a map of size n has 2n + 1 components, some of which might be empty): Therefore, for all k 2 fixed This allows to conclude since k is arbitrary.

Supercritical case
The supercritical case corresponds to u ∈ (9/5, +∞) and y(u) = ρ(u)M 2 (ρ(u), u) ∈ (0, 4/27). Recall that in this case T is distributed under P u as a critical Galton-Watson tree with exponential moments by Proposition 7 and Theorem 1. Properties of the maximum degree of critical Galton-Watson trees have been extensively studied by Janson [Jan12], building on work by Meir and Moon [MM90]. For the case where the offspring distribution admits exponential moments, Janson shows the following result.
Proposition 9 ([Jan12, Theorem 19.16]). Let µ be a probability distribution on Z 0 such that µ(0) > 0, and µ(k + 1)/µ(k) converges to a finite limit as k → ∞. Let D n,i be the i-th maximal number of children of nodes in a µ-Galton-Watson tree conditioned to have n edges. Denote by ρ the radius of convergence of Φ : In our case, the asymptotic of k(n) can be computed thanks to results about the Lambert W function, which is the compositional inverse of x ∈ R → xe x ∈ [−e −1 , +∞). This gives the following theorem.
Theorem 3. Let u > u C . For all fixed j 1, it holds as n → ∞ that Proof. The probability µ u ({2k}) is decreasing with k. So, by (13), for n large enough, to study k(n) it is sufficient to study for which k one has For sake of compactness, set w(u) = 27 Consequently, the previous inequality is equivalent to Notice that this is equivalent to Therefore, k(n) is the largest integer such that: where W denotes the Lambert W function. It is known that W satisfies, for x → ∞, which concludes the proof.

Critical case
The critical case corresponds to u = 9/5 and ρ(u)M 2 (ρ(u), u) = 4/27. As shown in Theorem 1, the offspring distribution has a power law tail in cj −α−1 , where α = 3/2 ∈ (1, 2). In this case, the variance is infinite, so that the method of Section 4.2 cannot be used. However, this case is directly treated in Janson's survey [Jan12, Example 19.27 and Remark 19.28].
Theorem 4. The following convergence holds: where the E (j) are the ordered atoms of a Point Process E on [0, ∞], satisfying that the random variable E a,b = # (E ∩ [a, b]) has a probability generating function convergent for all z ∈ C with The intensity measure π of E satisfies, for x > 0, and, for all j 1, E (j) > 0 almost surely.

Scaling limits
The preceding sections exhibited, via a study of the block-tree, a phase transition of a combinatorial nature, in terms of the size of the largest blocks, when the parameter u reaches u C = 9/5, both for the model on general maps and the one on quadrangulations. The goal of the present section is to expand on this phase transition by considering metric properties of the models in each phase, in the sense of taking scaling limits, see Section 5.1 for definitions. Because Tutte's bijection commutes with the block decomposition of both models under consideration, as stated in Proposition 5, the combinatorial picture of Section 4 is the same for both models. However, obtaining global metric properties under either model requires a good understanding of the metric behaviour of the underlying blocks. As of now, the required results exist only for simple quadrangulations. Consequently, our scaling limit results are complete only for the quadrangulation model.
In Section 5.1, we introduce the relevant formalism to state our scaling limit results, as well as a deviation estimate for the diameters of blocks, which will be useful for all values of u.
In Section 5.2, we prove Theorem 5, which identifies scaling limits simultaneously when u > u C and u = u C . For both models, there is convergence after suitable rescaling to a random continuous tree, namely a Brownian tree when u > u C and a 3/2 stable tree when u = u C . This convergence holds in the Gromov-Hausdorff (GH) sense, between metric spaces. In the quadrangulation model, this convergence is strengthened to a Gromov-Hausdorff-Prokhorov (GHP) convergence between measured metric spaces when quadrangulations are equipped with their mass measure on vertices. Doing the same with the model on maps is harder, so we only state a GHP convergence for maps equipped with an ad hoc measure, the degree-biased measure on their vertices.
Finally in Section 5.3, we prove Theorem 6 which deals with the GHP scaling limit when u < u C . In this phase, the one-big-block identified in Theorem 2 converges after rescaling to a scalar multiple of the Brownian sphere, and the contribution of all other blocks is negligible. This result is proved only for the quadrangulation model since it relies crucially on the scaling limit result for uniform simple quadrangulations obtained in [ABA17]. No such result is available yet for uniform 2-connected general maps, although one expects that it should hold.

The Gromov-Hausdorff and Gromov-Hausdorff-Prokhorov topologies
Originating from the ideas of Gromov, the following notions of metric geometry have become widely used in probability theory to state scaling limit results. We refer the interested reader to [BBI01] for general background on metric geometry and [Mie07, Section 6] for an exposition of the main properties of the Gromov-Hausdorff and Gromov-Hausdorff-Prokhorov topologies, and especially their definition via correspondences and couplings that we use here.
Define a correspondence between two sets X and Y as a subset C of X ×Y such that for all x ∈ X, there exists y ∈ Y such that (x, y) ∈ C, and vice versa. The set of correspondences between X and Y is denoted as is a correspondence, one may define its distortion: This allows to define the Gromov-Hausdorff distance between (isometry classes of) compact metric spaces One can modify this notion of distance in order to get a distance between compact measured metric spaces. For measured spaces (X, ν X ) and (Y, ν Y ) such that ν X and ν Y are probability measures, let us denote by Coupl(ν X , ν Y ) the set of couplings between ν X and ν Y , i.e. the set of measures γ on X × Y with respective marginals ν X and ν Y . Then the Gromov-Hausdorff-Prokhorov distance is defined as When (X, d X ) and (Y, d Y ) are the same metric space, one can bound this distance by the Prokhorov distance between the measures ν X and ν Y . This distance is defined for ν 1 and ν 2 two Borel measures on the same metric space (X, d) by where A ε is the set of points x ∈ X such that d(x, A) < ε. The bound mentioned above then corresponds to the inequality which is a consequence of Strassen's Theorem, see [Dud02, Section 11.6]. Finally we will use the following fact, the proof of which is left to the reader. For a ∈ [0, 1) and Borel probability measures µ, ν and ν on some metric space (X, d), it holds that

Formulation of the GHP-scaling limit problem
Let us begin by setting the notations for the measured metric spaces that one can canonically associate to the combinatorial objects under consideration. We associate to a tree (resp. map or quadrangulation) the following measured metric spaces: • For t a tree with at least one edge, denote by V + (t) the set of its non-root vertices, d t the distance that the graph distance induces on V + (t), ν t the uniform probability measure on V + (t) and t the measured • For m a map, recall that V (m) is its vertex set, and denote by d m the graph distance on V (m), ν m the uniform probability measure on V (m) and m the measured metric space • For q a quadrangulation, denote by V (q) its vertex set, d q the graph distance on V (q), ν q the uniform probability measure on V (q) and q the measured metric space The problem of finding a GHP-scaling limit consists in finding a suitable rescaling of a sequence of random compact measured metric spaces so that it admits a non-trivial limit in distribution for the GHP-topology. Let us introduce a convenient notation for the rescaling operation on a measured metric space. For X = (X, d, ν) a measured metric space and λ > 0, we denote by λ · X the measured metric space (X, λd, ν).

A useful deviation estimate
We shall now prove a deviation estimate for the diameters of the blocks of M and Q. It will prove useful for all values of u > 0. We recall the definition of stretched-exponential quantities, as this notion provides a concise way to deal with the probabilities of exceptional events.
Definition 8. A sequence (p n ) of real numbers is said to be stretched-exponential as n → ∞ if there exist constants γ, C, c > 0 such that As is evident from the definition, if (p n ) n and (q n ) n are stretched-exponential sequences, then so are the sequences (p n + q n ) n , (p n q n ) n , (n α p n ) n and (n α sup k n β p k ) n with arbitrary α, β > 0.
The input we shall rely on to derive our estimate is a deviation estimate for the diameter of one block, in both the case of 2-connected blocks of maps and simple blocks of quadrangulations.
Proof. The estimate for uniform 2-connected maps (B map k ) k 0 is obtained from [CFGN15, Theorem 3.7, specialized to x = 1]. To obtain the estimate for uniform simple blocks of quadrangulations (B quad k ) k 0 , one easily checks that for any path of length l 0 in a map m, there exists a path with same endpoints and length at most 2l in ϕ(m), its image by Tutte's bijection. Therefore for every map m one has diam(ϕ(m)) 2 diam(m). In particular diam(B quad k ) 2 diam(B map k ), and the conclusion follows from the estimate for (B map k ) k 0 This deviation estimate for the diameter of one block allows to control the deviations of the diameter of every block of M n,u and Q n,u , in the sense of the following corollary.
Proof. Let b be either a 2-connected map, or a simple quadrangulation. Then diam(b) is bounded by its number of edges, which is |b| if b is a map, and 2|b| if it is a quadrangulation. In particular, recalling that the outdegrees in the block-tree are twice the sizes of the respective blocks, we get for all u > 0 and n 1, that Denote by A(M n,u ) the "bad" subset of T n,u made of the vertices v such that both k v (T n,u )/2 n 1/6 and diam(b k v (T n,u ) (1+δ)/4 . By the above trivial bound on diameters, to show that the probabilities (26) are stretched-exponential as n → ∞, it suffices to see that the probability of the event {A(M n,u ) = ∅} is stretched-exponential as n → ∞.
By Proposition 7, conditionally on T n,u , each block b Mn,u v is sampled uniformly from 2-connected maps with size k v (T n,u )/2 respectively. Therefore, conditionally on T n,u , for each vertex v in T n,u we have Since T n,u has 2n + 1 vertices, this yields by a union bound, which is stretched-exponential as n → ∞ by Proposition 10, as announced. A similar use of Proposition 10 proves that the probabilities (27) are stretched-exponential as n → ∞.

Statement of the result
For 1 < θ 2, let us denote by T (θ) a θ-stable Lévy tree equipped with its mass measure. There are several equivalent constructions of these objects. A common way is to define them via excursions of θ-stable Lévy processes. Namely, T (θ) is the real tree encoded by the height process of an excursion of length one of a θ-stable Lévy process, see [Duq03]. To fix a normalization for T (θ) , we consider in the construction an excursion obtained by a cyclic shift from a θ-stable Lévy Bridge with Laplace exponent λ → λ θ . Note that the measured metric space T (2) corresponds to √ 2 times the Brownian Continuum Random Tree, which is encoded by an excursion of length 1 of the standard Brownian motion. The precise definition via excursions is not important for our statement and one can take Proposition 14 below as an alternative definition.
Theorem 5. There exist positive constants (κ map u , κ quad u ) u u C such that we have the following joint convergences in distribution, in the Gromov-Hausdorff-Prokhorov sense: where we set 2. If u = u C = 9/5, we have Additionally, the constants (κ map u , κ quad u ) u u C can be expressed as follows.
is the expectation of the distance, in a uniform 2-connected map with j edges (resp. simple quadrangulation with j faces) of the distance of the root vertex to the base vertex of a uniform corner (resp. to the closest endpoint of a uniform edge).
Remark 4. We believe that the same statement holds with M instead of M B , the choice of the latter is for convenience.
Differentiating the algebraic equation satisfied by B (18) with respect to y and taking (16) as a fourth equation gives a polynomial system from which can get B(y(u)) and B (y(u)) as functions of u only (using the resultant or a Gröbner basis algorithm). Replacing in the expression of σ(u) allows to conclude.
Remark 6. The quantity κ quad u could in principle be obtained via the explicit formula obtained in [BG10] for the generating function g of simple edge-rooted quadrangulations with a distinguished edge at prescribed distance from the root vertex.

Discussion and overview of the proof
Let u u C .
Consider a geodesic in either M n,u or Q n,u between two distant blocks b and b, respectively indexed by v and v in the block-tree. This geodesic must go through all the blocks whose index w in the block-tree T n,u is on the path from v to v, in the order induced by this path in the tree.
We have seen in Proposition 7 that under the law of M n,u or Q n,u , the blocks are independant conditionally on the block-tree, and when u u C they tend to all have non-macroscopic o(n) size by Theorems 3 and 4. One therefore expects that when n is large, the distance between two distant blocks b and b falls into a law of large numbers behaviour and is of the same order as d Tn,u (v, v). According to this heuristic, the macroscopic distances in M n,u and Q n,u should be concentrated around a deterministic scalar multiple of the distances in T n,u . But T n,u is a critical Galton-Watson tree conditioned to have 2n + 1 vertices, with explicit tail asymptotic given by Theorem 1 for its offspring distribution, yielding that its scaling limit is a stable tree.
To make this heuristic work, one needs to understand the typical distribution of degrees on a typical path in the tree. It turns out that on a typical path in a size-n critical Galton-Watson tree, the degrees are asymptotically independent and identically distributed, with distribution the size-biased version of the offspring distribution. This will be obtained by a spine decomposition for trees, adapted to our context.
We bring the attention of the reader to the fact that a proof similar in spirit has been done for the Gromov-Hausdorff metric in the general abstract setup of enriched trees by Stufler [Stu20, Theorem 6.60], and we could readily apply this result to deal with the case u > u C , modulo a technical complication regarding the additivity of distances in the quadrangulation case. When u = u C however, the distances within blocks have fat tails, so we fall outside the scope of Stufler's result. To deal with this, our last technical ingredient is a suitable large deviation estimate: we show that after an adequate truncation of the variables depending on n, large (and moderate) deviation events still have very small probability.
We now proceed with the proof.

Additivity of the distances along consecutive blocks
In Lemmas 12 and 13, we justify that a macroscopic distance is indeed a sum of distances on "in-between" blocks, in the case of blocks lying on the same branch in the block-tree. Lemma 12. For 0 i h , we have Recalling that x h = x, we get by the triangle inequality that the left-hand-side is at most the right-hand side. Therefore it suffices to show that any geodesic path in m from x = x h to x i visits each of the points (x j ) i<j<h , in decreasing order of j.
Let j be such that i < j < h . Denote by t j the tree of descendants of v j in t and also m j and m j the submaps of m made of the blocks (b m v ) v∈t j and (b m v ) v∈t\t j respectively. By the recursive description of the block-tree, the submaps m j and m j share only the vertex x j . But x is a vertex of m j since v is a descendant of v j , and x i is a vertex of m j since v i is an ancestor of v j . Hence any injective path between x and x i must visit x j in decreasing order of j; and in particular for a geodesic path.
Notice that it does not require the (x j ) to be mutually distinct. This concludes the proof.
The quadrangulation case. A slight complication arises for quadrangulations because the "interface" between two blocks is a double edge, containing two vertices instead of a single vertex in the map case. At first sight it is thus unclear through which of these vertices a geodesic should go. We show that there is a canonical choice: the vertex between those two which is closest to the root vertex. This relies crucially on the fact that quadrangulations are bipartite. Fix a quadragulation q. For b a simple block of q, and l an integer in {1, . . . , 2|b|}, let us denote by D q (b, l) the graph distance in b between the endpoint of the l-th edge of b in the ordering described thereafter, and the endpoint of the root edge of b which is closest to the root vertex of q. The order on the edges of b that we use is the image of the lexicographic order on vertices of t via the block-tree decomposition. This is consistent with the ordering of corners in the map case.
Fix a vertex x on q and let e be any edge incident to x, and v be the vertex corresponding to e in the block-tree t. Define the height h and the ancestor line (v i ) 0 i h as in the map case. Let also (x i ) 0 i h be the respective root vertex of b v i . In particular x h is either x or the other endpoint of e . Finally, let (l i ) 1 i h be the respective breadth-first index of the edge in b q v i−1 to which the root edge of b q v i is attached. The situation is illustrated on Fig. 14.
Proof. The idea is quite similar in principle as in the preceding lemma, except that consecutive blocks share two vertices in the quadrangulation case, instead of one.
For 0 j h , denote by y j the endpoint of the root-edge of b q v j which is closest to the root vertex of q. In particular y 0 is the root vertex of q. Let 0 i h . Then by construction, y i and x i are adjacent to the root edge of b q v i , and similarly y := y h and x are adjacent to the root edge of b q v h . Therefore, there exists some δ x,x i ∈ {0, ±1, ±2} such that We shall prove the following, which are sufficient to conclude: 1. for 0 i h , the identity d q (y, y i ) = i j<h d q (y j+1 , y j ), Let us argue that it is sufficient to show the following Assuming this is true, we directly get by applying it iteratively that d q (y, y i ) = i j<h d q (y j+1 , y j ).
To verify the second set of identities, recall that y j is defined as the endpoint of the root edge of b q v j which is closest to the root vertex y 0 of q. Denote by y j the other endpoint. Then, for 0 j < h we have The first equality is equivalent to the definition of D q (b q v j , l j ). The second one comes from the fact that within a block b of q, the graph distance respective to q and the graph distance respective to b coincide. But assuming (29), it holds that d q (y j+1 , y j ) = d q (y j+1 , y 0 ) − d q (y j , y 0 ) d q (y j+1 , y 0 ) − d q (y j , y 0 ) d q (y j+1 , y j ), where the first inequality comes from the definition of y j+1 , and the second inequality from triangle inequality. In particular, the above minimum is d q (y j+1 , y j ) and we have as needed d q (y j+1 , y j ) = D q (b q v j , l j ).
We still have to prove (29). Let us first prove the case i = 0 and deduce the general case. Let 0 j k h , and let γ be a geodesic path from y 0 to y k . If γ visits y j , we readily have d q (y 0 , y k ) = d q (y 0 , y j ) + d q (y j , y k ).
Otherwise it visits y j , and denote by γ 1 , γ 2 the portions of γ form y 0 to y j , and from y j to y k respectively. By definition of y j , we have d q (y 0 , y j ) d q (y 0 , y j ). But since q is a quadrangulation, it is bipartite and the inequality is strict d q (y 0 , y j ) < d q (y 0 , y j ). Form γ 1 the concatenation of a geodesic path from y 0 to y j and of the oriented edge (y j , y j ). Then from the strict inequality we mentioned, len( γ 1 ) len(γ 1 ), and in particular the concatenation of γ 1 and γ 2 is a geodesic path from y 0 to y k which visits y j . Therefore we also have in this case the identity (30). Finally, let us deduce the case i = 0. Let 0 i j k h . We have This proves (29) and concludes the proof of the lemma.

Scaling limit and largest degree of critical Galton-Watson trees
A slight technical complication that arises in our setting is that the block-tree has a lattice offspring distribution with span 2, in the sense of the following definition Definition 9. A measure µ on Z is called lattice if its support is included in a subset b + dZ of Z, with d 2. The largest such d is called its span. If d = 1, µ is called non-lattice.
The results that we need [Kor13, Theorem 3] are stated for non-lattice offspring distributions. This turns out to be purely for convenience and we state the following more general result that is suited to our needs.
We recall that a probability distribution µ with mean m µ is said to be in the domain of attraction of a stable law of index θ ∈ (1, 2] if there exist positive constants (C n ) n 0 such that we have the following convergence in distribution where (U 1 , . . . U n ) are i.i.d. samples of the law µ, and X (θ) is a random variable with Laplace transform E exp(−λX (θ) ) = exp(λ θ ).
Let µ be a probability distribution on Z 0 , with µ(1) = 1, and which is assumed to be critical. Assume additionally that it is in the domain of attraction of a stable law of index θ ∈ (1, 2]. Let d 1 be the span of the measure µ. Then under those assumptions, we have 1. For all m large enough, the GW µ (dT )-probability that T has dm edges is positive. This probability is equivalent to c θ /(C dm dm) for some constant c θ > 0.
2. If we denote by T n a GW µ -tree conditioned to have n edges, then in the Gromov-Hausdorff-Prokhorov sense, with (C n ) n 0 the sequence in (31).
3. The largest degree in T dm is of order at most C dm , in the sense that for any ε > 0 Proof. The first statement can be obtained by a straightforward adaptation of the proof of [Kor13, Lemma 1], which relies on a local limit theorem and the cycle lemma. We specify below how this local limit theorem should be adapted. The cycle lemma adapts straightforwardly. For the second statement, let us justify that [Kor13, Theorem 3] still applies when the non-lattice (or aperiodic) assumption is dropped, but with the number of vertices n + 1 taken only along the subsequence (dm + 1) m 0 . This will prove functional convergence of the contour functions of the trees (T dm ) m when properly rescaled, to the contour function of T (θ) . This convergence of contour functions is sufficient to get the announced Gromov-Hausdorff-Prokhorov convergence.
The local limit theorem [Kor13, Theorem 2, (ii)] changes as follows lim n→∞ sup k∈Z a n d P (Y n = k) − p 1 k a n = 0.
See for instance [Ibr71, Theorem 4.2.1]. Notice that the only difference with the non-lattice (d = 1) local limit theorem is the factor 1/d in the last display. Examining the details of Kortchemski's arguments, this extra 1/d factor would appear only in the discrete absolute continuity relations which are used in the proof. But in each instance, it would appear in both the numerator and denominator of some fraction. Hence the fraction simplifies and this factor has no impact on the proof, which carries without change, except that the integer n, which in the paper is the number of vertices, should now only be taken in dZ + 1. Finally, in order to get the third statement, one can take as a basis the local limit theorem above. From this, one can get the functional convergence of the Łucasiewicz path of T dm , when it is rescaled by dm in time and C dm in space. In particular, (C dm ) −1 times the largest degree in T dm is tight, and one obtains the claimed probabilistic bound. One could for instance use the same arguments as in the proof of [KM21, Proposition 3.4] Corollary 15 then just identifies the explicit scaling constants in specific instances of the abovementioned scaling limit theorem.
Corollary 15. Let µ be a critical probability distribution on Z 0 with span d 1, and with µ(1) = 1. Denote by T n a GW µ -tree conditioned to have n edges, for n ∈ dZ large enough. Then the following holds.
Proof. Note that in the case where ν has exponential moments, [MM03] treats the case of lattice distributions. That would suffice for our applications when u > u C . We still need the second statement to treat the case u = u C . Let us apply the preceding proposition and identify the right constants, in these two cases. Statement 1. If µ has finite variance σ 2 , then by the Central Limit Theorem, for i.i.d. samples (U i ) i of the law µ, we have the convergence in distribution U 1 + · · · + U n − n σ · n 1/2 where G is a standard normal variable. In particular, G has the same law as 1 √ 2 · X (2) . Therefore, the hypotheses of Proposition 14 are satisfied, with C n = σ √ 2 · n 1/2 , and the conclusion follows from this proposition.
Statement 2. We consider the case where µ ([x, +∞)) ∼ x→∞ cx −θ with θ ∈ (1, 2] and c > 0. Let U := U 1 and let us also introduce the notations The function M 3 is non-decreasing and using the assumed tail asymptotic of µ, one has the asymptotic M 3 (x) ∼ cx 2−θ /(2 − θ)(θ − 1). We may therefore use the Karamata Tauberian theorem [BGT89, Theorem 1.7.1] to get where M 3 is the Laplace-Stieljes transform of M 3 [BGT89, Paragraph 1.7.0b]. Then, if we integrate by parts three times, we obtain This, together with the fact that M 2 (0) = 1 since it is the expectation of µ, yields the following expansion when h → 0 + , Now, if we set and plug h = λ/C n into (32), we get for all λ 0, Hence there is convergence in distribution of U 1 +···+Un−n Cn to X (θ) , as required in Proposition 14. So this proposition applies with the above-chosen sequence (C n ) n , and the conclusion follows.

The spine decomposition and size-biased laws
In this section we present a size-biasing relation for the block-tree, in the sense of [LPP95]. Actually, we extend in a straightforward way this size-biasing relation to our setting, where we have a Galton-Watson tree and some decorations, namely the blocks. More precisely, consider the following measure on maps with a distinguished corner (m, c ) where δ c is the Dirac measure A → δ c (A) = 1 {c∈A} . Then this σ-finite measure can be decomposed as a sum of probability measures h 1 P u,h (dM, dB ), where under P h u there are exactly h blocks on the path from the root corner to the distinguished corner, those blocks having a size-biased law as defined below. The present section makes that precise.

Description of P u,h .
Definition 10. Let ν be a probability distribution on Z 0 with finite expectation m ν . Then the size-biased distribution ν is defined by When ν is a (sub-)critical offspring distribution with ν(0), ν(1) = 0, denote by GW ν,h h 0 the following family of laws, on the sets of discrete trees with a distinguished vertex at height h respectively. It may be described algorithmically: • Each vertex will either be mutant or normal, and their number of offspring are sampled independently from each other; • Normal vertices have only normal children, whose number is sampled according to ν; • Mutant vertices of height less than h have a number of children sampled according to the size-biased distribution ν, all of which are normal except one, chosen uniformly, which is mutant; • The only mutant vertex at height h reproduces like a normal vertex and is the distinguished vertex V .
This yields a pair (T, V ), where T is a discrete tree and V is a distinguished vertex of T with height h. We denote by (V i ) 0 i h T (V ) the ancestor line of V , and L i the order of V i+1 in the children of V i respectively. Observe that the construction gives that (k V i (T)) i are i.i.d. with law ν, and conditionally on those variables, the variables (L i ) i are independent with uniform law on {1, . . . , k V i (T)} respectively. We may now define the family of probability measures ( P u,h ) h 0 as follows. Let h 0.
• Sample (T, V ) according to the law GW µ u ,h .
• For each v ∈ T, sample independently and uniformly a 2-connected map b M v with k v (T)/2 edges.
• Build the map M whose block decomposition is (b M v ) v∈T , and Q = ϕ(M) its image by Tutte's bijection.
• Denote by B the corner of M, resp. E the edge of Q, that the block-tree decomposition associates to the vertex V of T.
We are now equipped to state the size-biasing relation.
Proposition 16. For u u C , the σ-finite measure P u (dM) c corner of M δ c (dB ) on maps with a distinguished corner decomposes as the following sum of probability measures, Proof. The standard size-biasing relation for (sub-)critical Galton-Watson trees reads When u u C , the offspring distribution µ u is critical, so m µ u = 1. Specializing the last display to ν = µ u and to the value of (t, v ) associated to some map with a distinguished corner (m, c ), this gives for all such (m, c ), Therefore, if we multiply both sides by v∈t 1 b kv (t)/2 , we get the following by Proposition 7: Since c corner of m δ c (c ) = 1, the last display expresses the measure P u (dM) c corner of M δ c (dB ) as a sum of the probability measures ( P u,h ) h 0 .
Probabilistic properties of P u,h . Since we need metric information on blocks whose size follows the size-biased law µ u , let us introduce adequate notations. Let u u C . Denote by ξ u a sample of the distribution µ u on some probability space (Ω, P ). Then jointly define the random variables Lemma 17. For all h 1, we have the identity in law Proof. Recall that under P u,h , the pair (T, V ) has law GW µ u ,h . By definition of the law GW µ u ,h , the ancestor line of the distinguished vertex V in T is made of mutant vertices. This means that the family (k v i (T)) 0, i<h is i.i.d. sampled from the size-biased distribution µ u , which is the law of ξ u , and that independently of each other, each V i+1 has uniform rank L i among the k V i (T) children of V i . Hence we have the identity in law Now under P u,h the conditional law of the blocks (b M v ) v∈T with respect to T is that of independent blocks, sampled uniformly from blocks with size (k V (T)/2) v∈T respectively. In particular, the blocks (b M V i ) 0 i<h are sampled independently, uniformly from blocks with size (k V i (T)/2) 0 i<h respectively. Therefore the preceding identity in law extends to the following one Lemma 18. Denote by D(b, l) the distance in a simple quadrangulation b between its root vertex and the closest endpoint of the l-th edge in the order induced by the block-tree decomposition, the same order as the one introduced before Lemma 13. Then for all h 1, there is the identity in law Proof. Recall from the notations introduced for Lemma 13 that for b a simple block of a quadrangulation q, and l an integer in {1, . . . , 2|b|}, D q (b, l) is the graph distance in b between the endpoints of the l-th edge of b in breadth-first order, and the endpoint of the root edge of b which is closest to the root vertex of q. Denote by b → F (b) the mapping which reverses the rooted oriented edge of a simple quadrangulation. Introduce also for b a simple quadrangulation, f b the permutation of {1, . . . , 2|b|} which maps the breadth-first order on b to the breadth-first-order on F (b). Finally, define the event E i that the root vertex of b Q V i is closer to the root vertex of Q than the other endpoint of the root edge of b Q V i . Then by definition, for all 0 i < h we have that is independent of F i , and has the same law as ξ u , B map u , B quad u , U ). Now the crucial point is that the event E i is F i -measurable, since it can be decided whether or not it holds by looking only at the first i blocks on the spine. In particular it is independent of (k This implies the following The proposition is therefore proved if we justify the identity in law To check this, first notice that F is a bijection since it is involutive, so that in particular the uniform law on simple quadrangulations with k edges is invariant under F . By definition, for b a simple quadrangulation, f b is also a bijection so that the uniform measure on {1, . . . , 2|b|} is invariant under it. Denoting U k a uniform random variable on {1, . . . , 2k}, this gives for each k 1 the identity in law Since the pair ( B quad u , U ) is the ξ u /2-mixture of the laws (B k , U k ) k 1 , the identity in law (33) also holds and this conludes the proof.
Moments of typical distances in a size-biased block. We may now examine how fat are the tails of this i.i.d. family of distances along the spine, which we wish to sum.
Proposition 19. Let D be either the variable D( B map u , U ) or D( B quad u , U ). Then for u > u C , there exists ε > 0 such that E[exp(tD)] < ∞ for all real t < ε. And for u = u C , we have E D β < ∞ for all 0 < β < 2.
Hence it suffices to prove that the above moments are finite when we replace D by diam( B u ).
Let u > u C . Then diam( B u ) ξ u , and the latter variable has exponential moments since P ( ξ u x) = 2j x 2jµ u ({2j}), where µ u has a tail decaying exponentially fast by Theorem 1. Now take u = u C = 9/5 and let δ ∈ (0, 2). Also let ε > 0 to be chosen later depending on δ. Using the notation B k for B map k or B quad k , set p ε (k) = P diam(B k ) k 1/4+ε . By Proposition 10, we have that p ε (k) decays stretched exponentially as k → ∞. Therefore we get a constant C > 0 such that k 2 p ε (k) C for all k. Recall that we have diam( B u ) ξ u . Distinguishing upon whether diam( B u ) ( ξ u ) 1/4+ε or diam( B u ) > ( ξ u ) 1/4+ε and taking a conditional expectation with respect to ξ u , we get If ε is small enough so that (1/4 + ε)(2 − δ) < 1/2, then the last sum is finite since by Theorem 1 Let us make a brief commentary, and justify that when u = u C , Proposition 19 is optimal, in the sense that D( B quad u C , U ) does not have moments of order β for β 2. Firstly, one easily checks that functionals on pointed measured metric spaces of the form are continuous with respect to the Gromov-Hausdorff-Prokhorov topology. Addario-Berry and Albenque [ABA21] prove the GHP convergence of size k uniform simple quadrangulations, rescaled by cst · k −1/4 , to the measured Brownian sphere (S, D * , λ). This holds when putting either the uniform measure on vertices of B quad k or the size-biased one by [ABW17]. In particular, by the abovementioned continuity, we have the convergence in distribution where U k is uniform on {1, . . . , 2k} and x 0 is the distinguished point on the Brownian sphere. Since the variable S λ(dx) D * (x 0 , x) β is almost surely positive, the left-hand-side forms a tight sequence of (0, ∞)-valued random variables. Therefore it is bounded away from 0 with uniform positive probability. This implies a lower bound E D(B quad k , U k ) β c(k 1/4 ) β , for some c = c(β) > 0. In particular, The latter sum is infinite when β 2, which proves that D( B quad u C , U ) does not have moments of order β for β 2. The same argument would hold for D( B map u C , U ), but we lack at the moment the GHP convergence of size-k uniform 2-connected maps.

Moderate deviations estimate.
When increments of a random walk possess only a polynomial moment of order β > 1, as is the case of D( B map u , U ) and D( B quad u , U ) when u = u C , moderate and large deviation events can possibly have probabilities which decay slowly, that is polynomially with n. In the case of heavy-tailed increments, this indeed happens since those moderate and large deviation events can be realised by taking one large increment. This one-big-jump behaviour is actually precisely how these large deviations events are realised. This phenomenon, which we have already encountered in Section 3 for u < u C , is known as condensation. For a more precise statement, see [Jan12, AL09, AB19].
One could hope that if we prevent the variables from condensating, we could still get stretchedexponentially small probabilities for large deviation events. We make this precise in the following proposition, by stating that this is the case when we suitably truncate the increments. We were not able to find an instance of such an estimate in the literature, although it has certainly been encountered in some form. We thus include a short proof, which as usual relies on a Chernoff bound.
Notice that since β ∈ (1, 2], for all M > 0 the following inequality holds for t near 0 or −∞ Therefore, if one takes M large enough it holds for all t ∈ (−∞, 1]. Fix such a constant M . Given λ, s 0, distinguishing upon whether λx ∈ (−∞, 1] or not and using that x1 {x s} x and x1 {λx 1} x (even when x < 0), we get for all x ∈ R, Applying this inequality with x = X, λ = n −θ , s = n γ and taking expectations we obtain Recall that E[X] = 0 by hypothesis, that γ − θ < 0 by choice of θ, and use Markov's inequality. This yields Since by hypothesis E |X| β < ∞, we have where the last inequality comes from the choice of θ, which is greater than 1/β. Therefore (34) is satisfied and the proposition is proved.

A lemma to compare m, q and t
Let us state a lemma which elaborates on the additivity of distances on consecutive blocks, so that we can bound the GHP-distance between a map (resp. a quadrangulation) and its block-tree scaled by some constant. Let κ 1 and κ 2 be positive constants. Let m be a map, q its associated quadrangulation by Tutte's bijection, and t their block-tree.
For c a corner of m, denote by e the associated edge in q and v the associated vertex in t. is attached, which is also the index of the edge in b q v i to which the root edge of b q v i+1 is attached. Finally, denote by ∆(m) (resp. ∆(q)) the largest diameter of the blocks of m (resp. q), set the quantities Lemma 21. With this notation, we have for all ε > 0, and d GHP ε · q , εκ 2 · t 3ε + ε 2 ∆(q) + ε max e edge of q R(q, e , κ 2 ) + max(ε, 1/|V (q)|).
Proof. Recall the notation introduced in Section 2.6. Let us treat the inequality involving q, which is a bit more involved. Consider the correspondence C between V (q) and V + (t) defined as follows.
A vertex x of q is set in correspondence with a non-root vertex v of t if and only if it belongs to the edge of q that the block-tree decomposition associates to v. Let γ be the uniform measure on the set C ⊂ V (q) × V + (t). The projection on V + (t) is 2-to-1, so that the image measure of γ is uniform on V + (t), i.e. it is ν t . The projection on V (q) of the measure γ is biased on the other hand, as the weight each vertex receives is proportional to its degree. Therefore, γ tautologically defines a coupling between ν B q and ν t supported on C, i.e. γ (V (q) × V (t)) \ C = 0. By the triangle inequality and the preceding observations, we have The last inequality uses (24). Now, [ABW17, Lemma 5.1] bounds the Prokhorov distance between the uniform and degree-biased measures on (V (q), εd q ). Namely, d (V (q),εdq) P (ν q , ν B q ) max(ε, 1/|V (q)|).
It remains to bound the distortion dis(C; d q , κ 2 d t ). Take x, x vertices of q, and let v, v be any vertices of t such that (x, v) and ( x, v) are in correspondence in C. We shall bound |d q (x, x) − κ 2 d t (v, v)|. Denote by e and e the respective edges that the block-tree decomposition associates to v and v respectively. By definition of the correspondence C, e is incident to x and e to x. Define accordingly as in the statement of Lemma 13, h , Then, let i be such that v i = v i is the last common ancestor of v and v in t. First remark that there exists δ 0 ∈ {0, ±1, ±2} such that Indeed, similarly as in the proof of Lemma 13, a geodesic from x to x must visit, once and in that order, • the vertex x, • either x i+1 , or x i+1 the other endpoint of the root-edge of b q v i+1 , • either x i+1 , or x i+1 the other endpoint of the root-edge of b q v i+1 , • the vertex x.
Since x i+1 and x i+1 , and also x i+1 and x i+1 are at distance 1 respectively, and since a geodesic between points in b v i must stay in b v i , we get that (36) holds, for some δ 0 ∈ {0, 1, 2}. Then, Lemma 13 applies to decompose the distances d q (x, x i+1 ) and d q ( x i+1 , x), with some δ, δ in {0, 1}. Combining this with (36), this gives The sum of the first three terms has absolute value at most 6, the fourth one at most ∆(q), and the two sums each have absolute value at most R(q, e , κ 2 ). Therefore by the triangle inequality, Since this holds for every (x, v) ∈ C and ( x, v) ∈ C, the right-hand side is actually a bound on the distortion dis(C; d q , κ 2 d t ), which is precisely what we needed to conclude. For the inequality involving m, the reasoning is quite similar, except we keep the degree-biased measure and do not compare it to the uniform measure. Take C the correspondence such that x ∈ V (m) is in correspondence with v ∈ V + (t) if and only if the vertex x is the origin of the corner associated to the non-root vertex v. Similarly, the uniform measure γ on C defines a coupling between the measure ν t and the degree-biased measure ν B m on vertices of m, and this coupling is tautologically supported on C. The distortion of C is bounded with a very similar argument as above involving Lemma 12 instead of Lemma 13, except that there are no terms δ 0 , δ, δ to introduce. We leave the details to the reader. All in all we get R(m, c , κ 1 ).

Proof of Theorem 5
Let u u C . Let us first prove the claimed scaling limit for the block-tree T n,u . By Proposition 7, T n,u has law GW(µ u , 2n), where the offspring distribution µ u has span 2.
Scaling limit of T n,u for u > u C . If u > u C , then by the third statement of Theorem 1, µ u is critical and admits a variance σ(u) 2 < ∞. Corollary 15 thus gives the announced scaling limit for T n,u , The expression for σ(u) given in the statement comes from a straightforward computation from the generating function of µ u , which by (9) is k 0 x k µ u (k) = uB(x 2 y(u)) + 1 − u uB(y(u)) + 1 − u .
We also obtain from this bound on diameters the following Bounding the height of T n,u . We have identified above the scaling limit of T n,u and the appropriate normalization of distances. In particular, n (α−1)/α · T n,u is tight in the GHP-topology. An immediate consequence is that n −(α−1)/α H(T n,u ) is tight, where H(T n,u ) is the height of T n,u . In particular, our problem reduces once more to showing that for sufficiently small δ > 0, Using the spine decomposition. Fix δ > 0, as small as necessary. Let us only treat the term involving Q in (39), as the expression for R(q, e , κ 2 ) we used to define the event A Q 1;n,η carries more dependence than that of R(m, c , κ 1 ). Indeed the summands D q (b q v i , l i ) involve in their definition a global metric property of q. The case of the term involving M is similar and simpler.
Recall that by definition, the law P n,u is the law P u , conditioned on the event {|T| = n}. Since P u (|T| = n) decays polynomially by Corollary 15, we may get rid of the conditioning if the unconditional version of the probabilities we wish to bound decays sufficiently fast. Namely, it suffices to prove that for all η > 0, the following (unconditional) probability is stretched-exponential in n P u A Q 1;n,η ∩ A Q 2;n,δ ∩ {H(T) η −1 n By a union bound and then by Proposition 16, using the notations introduced above it, one can bound this by The last inequality may require some explanations. First we apply a union bound with respect to the sign of the expression under the absolute value. Then we use the control that A Q 2;n,δ offers on ∆(Q) the maximum diameter of blocks of Q, and the positivity of the distances D Q (b Q v j ), to insert an indicator function. Hence the appearance of ψ n,δ .
Reducing to a large deviations event with truncated variables. We let ξ u,j , B quad u,j , U j j 0 be an i.i.d. sequence of copies of the triple ξ u , B quad u , U . We also let X j = D B quad u,j , U j − κ 2 . Then by Lemma 18, the arguments of the function ψ n,δ that appear in the last upper bound we obtained, are actually i.i.d. and have joint law under P u,h the law of (X j ) 0 j<h . Therefore this last upper bound is equal to Since the sequence (X j ) 0 j<h is i.i.d., we re-order the terms of the two sums which appear inside the probabilities in the last display, so that they run on indices j = 1, . . . , i. Hence, if we set h n = n (α−1)/α , then we can bound the last display by Using the moderate deviations estimate. Let γ = γ(δ) = (1 + δ) 2 /2. Then (h n ) γ κ 2 for n large, so By definition of κ 2 , the variables (X j ) are centered. And by Proposition 19, they possess moments of order β for all 1 β < 2. But for δ sufficiently small, we have γ < 1. Therefore Proposition 20 yields that is stretched-exponential as n → ∞, and the same holds when replacing (X j ) by (−X j ). This proves that for each η > 0, the probability (40) is indeed stretched-exponential in n, and concludes the proof.

Scaling limit of the quadrangulations in the subcritical case
Let us finally identify the scaling limit of the quadrangulation Q n,u when u < u C .

Statement of the result
Denote by S = (S, D * , λ) the Brownian sphere, also known as the Brownian Map. One may take Proposition 22 below as a definition.
Theorem 6. Assume u < u C = 9/5. We have the following convergence in distribution for the Gromov-Hausdorff-Prokhorov topology 9(3 + u) 8(9 − 5u) 1/4 n −1/4 · Q n,u In the case u = 1, one recovers the Brownian sphere as the scaling limit of uniform quadrangulations with n faces, which has been proven in [LG13] and [Mie13]. It is also the scaling limit of uniform simple quadrangulations with n faces, which was proven in [ABA17]. The latter corresponds informally to the case u → 0.
We emphasize that those results, and especially the one of [ABA17], serve as an input in our proof and we do not provide a new proof of them. Accordingly, let us precisely state the latter result, so that we can use it in the subsequent proof.

Proposition 22 ([ABA17]
). Uniform simple qudrangulations with k faces admit the Brownian sphere as scaling limit, with the following normalization 3 8k This is precisely the result [ABA17, Theorem 1.1], restricted to the case of simple quadrangulations. Notice that in their result, the scaling limit is stated in terms of M n , a uniform simple quadrangulation with n vertices, not faces. This is not a problem since by Euler's formula, a quadrangulation has n vertices if and only if it has n − 2 faces. Therefore B quad k has the same law as M k+2 .
• To obtain a Gromov-Hausdorff scaling limit, the missing ingredient is the equivalent for 2-connected maps of the result of [ABA17], that is to say GH(P) convergence of uniform 2-connected maps with n edges, rescaled by a constant times n −1/4 , to the Brownian sphere.
• In order to strengthen this to GHP convergence when the map is equipped with the uniform measure on vertices, one would need the above mentioned convergence of 2-connected maps, but in the GHP sense. It would also require a way to compare, in the Prokhorov sense, the degree-biased measure on vertices of M n,u , and the uniform measure. For quadrangulations on the other hand, this comparison can be done using [ABW17, Lemma 5.1].
The paper [ABW17] makes precise the relationship between the convergence of uniform quadrangulations with n faces [LG13,Mie13], and the convergence of simple uniform quadrangulations with n faces [ABA17]. It is shown that a quadrangulation sampled uniformly among those which have size n and whose biggest block has size k(n) ∼ cn with an adequate c > 0, converges jointly with said biggest block to the Brownian sphere, in the GHP sense.
The proof of Gromov-Hausdorff convergence for these quadrangulations amounts to showing that pendant submaps that are grafted on the macroscopic block have negligible diameter, that is o n 1/4 , which is done by [ABW17, Proposition 1.12]. The strategy of proof is not directly applicable here, since it uses an a priori diameter bound on the pendant submaps, which we do not have for general u. As explained in what follows, it is sufficient to have an a priori diameter bound on single blocks themselves, which is why we need Proposition 10. To strengthen GH convergence to GHP convergence however, we use the same arguments as those exposed in [ABW17] modulo some technical details.

Sketch of the proof
On the combinatorics side, Theorem 2 characterizes the phase u < u C by a condensation phenomenon: when n is large, there is precisely one block of linear size, while others have size O(n 2/3 ). This theorem is stated for a map with law P n,u , that is the law of M n,u , but by Section 2.5, Tutte's bijection commutes with the block decomposition, so that the same happens for Q n,u .
On the metric side, there is not much more going on. The block-tree is subcritical in this phase by Theorem 1 and therefore has small height. Combining this with the O(n 2/3 ) bound on the size of non-macroscopic blocks, and the deviation estimate of Proposition 10 on diameters of blocks, we get that Q n,u is approximately equal to its largest block, in the Gromov-Hausdorff sense in the scale n 1/4 . This argument is rather general and should be easy to adapt to other models of graphs or maps with a block-tree decomposition under a condensation regime.
In order to strengthen this convergence to one in the Gromov-Hausdorff-Prokhorov sense, we use the rather general result [ABW17, Corollary 7.2], by comparing the mass measure on vertices with a projection on the macroscopic block, which is modulo some technical details an exchangeable vector on the edges where the pendant submaps are attached. This corollary tells that this random measure is well-approximated by its expectation, which is uniform on the edges of the macroscopic block, or equivalently that it is degree-biased on its vertices. The last part of the argument is specific to quadrangulations, for which we can compare the degree-biased and the uniform measure on vertices by [ABW17, Lemma 6.1].

Comparison of a quadrangulation and its biggest block
Let us introduce some notation. Denote by t[v] the subtree of descendants of a node v in t, rooted at v. For an edge e of q, and v the vertex of t that the block-decomposition associates to e, we will denote by q[e] the quadrangulation whose block-tree decomposition is (b q w ) w∈t [v] . Recall that by convention, if v is a leaf then q[e] is the edge map, with 2 vertices and 1 edge, the edge e.
Let v • be the vertex of t with largest outdegree, choosing one arbitrarily if there are several, and let q • = b q v • . Write also q + for the quadrangulation whose block decomposition is In particular, q • is the simple core of q + , and its other blocks are the blocks of the pendant subquadrangulations (q[e]) e∈E(q • ) .
Finally, let π q + q • be the probability measure on vertices of q • obtained by projection of the contribution to ν q of each pendant map (q[e]) e∈E(q • ) to the biggest block q • . More formally, for each edge e of q • , let {e + , e − } be its extremities. Then, Observe that since q • shares exactly 2 vertices with each pendant map (q[e]) e∈E(q • ) , the last display indeed defines a probability measure.
Lemma 23. It holds that Proof. There are successive comparisons to be made for the GHP distance.
Metric comparison. The term r GH bounds how distant the spaces ε · q, ε · q + and ε · q • are, from a metric point of view, i.e. in the GH sense. Recall that we can see q and q + as their biggest block q • , together with some maps attached to it. Therefore one needs to bound the maximal diameter of the attached maps. We use a brutal bound on the diameter of the non-macroscopic blocks by their maximal diameter, together with a bound on the number of consecutive blocks in the attached maps. This number is bounded by diam(t) 2H(t). Therefore the maximal diameter of attached maps in ε · q or ε · q + is bounded by In particular, take the correspondence B 1 on V (q) × V (q + ) such that x ∈ V (q) is in correspondence with only itself if it belongs to V (q + ), or otherwise with both endpoints of the root-edge of q + if it belongs to V (q) \ V (q + ). The uniform measure on B 1 is a coupling between ν q and some measure µ + on V (q + ). One therefore gets, using triangle inequality and (24), Similarly, take the correspondence B 2 on V (q + ) × V (q • ) such that x ∈ V (q + ) is in correspondence with only itself if it belongs to V (q • ), or otherwise with both endpoints {e + , e − } of the root-edge of q[e] if x belongs to V (q[e]) \ {e + , e − } for some edge e ∈ E(q • ). Then the uniform measure on B 2 is a coupling between ν q + and some measure µ • on V (q + ). We get as above Comparing the uniform measures on vertices of q and q + . Observe that ν q + is the counting measure on V (q + ) renormalized to a probability distribution, while µ + is the renormalized version of the same counting measure but with additional mass the latter being split equally on the endpoints of the root-edge of q + . Elementarily, this yields a total variation bound, as follows Since the Prokhorov distance is bounded by the total variation distance, we have Comparing the uniform measures on vertices of q + and q • . From the definition of π q + q • and from the partitioning observe that the measure µ • obtained from the correspondence B 2 above decomposes as follows In particular, we obtain from (25) that Concluding the proof. By the triangle inequality, we have Using (41) and (43) to bound the first term, and (42) and (44) to bound the second one, we get the claimed inequality.

Exchangeable decorations
We aim to use Addario-Berry & Wen's argument for [ABW17, Lemma 6.2] which tells that for exchangeable attachments of mass on edges of Q n , a quadrangulation with n faces sampled uniformly, the resulting measure on Q n is asymptotically close to the uniform measure on vertices, in the sense of the Prokhorov distance on n −1/4 · Q n . They use the following ingredients: 3. GHP convergence of n −1/4 · Q n to the Brownian sphere.
4. Properties of the Brownian sphere such as compacity and re-rooting invariance.
The first ingredient is rather general and actually stated for any graph in [ABW17, Lemma 5.3]. We will ever-so-slightly adapt its proof since there is a double edge in their setting which we do not have, and the mass is not projected on vertices in the exact same way. The second ingredient is specific to quadrangulations and one may need different arguments to compare the degree-biased and uniform measures for other classes of maps.
Let us state which result we extract for our purpose from Addario-Berry & Wen's paper. For n = (n(e)) e∈E(G) a family of nonnegative numbers indexed by edges of a graph G, we denote its p-norm for p 1 by n(e) 1 2 δ e + + 1 2 δ e − , with {e + , e − } the set of endpoints of the edge e. Notice that this definition is slightly different from that of ν n G in [ABW17, Section 5], because the mass of an edge is projected uniformly and independently on either of its enpoints in their case, while we deterministically split this mass on both endpoints. This does not change much except that we find it easier to work with. One of their results translates as the following.
Proposition 24 ([ABW17, Corollary 6.2]). Let Q k = B quad k , which is a simple quadrangulation with k faces, sampled uniformly. Consider for each k 1, a random family n k = (n k (e)) e∈E(Q k ) of nonnegative numbers, such that conditionally on Q k it is an exchangeable family. Assume that |n k | 2 /|n k | 1 → 0 in probability as k → ∞. Then there holds the convergence in probability where ν Q k is the uniform measure on vertices of Q k and ε k = k −1/4 . This is the statement of [ABW17, Corollary 6.2], adapted to our setting. The proof goes mutatis mutandi, except for an adjustement in the concentration inequality [ABW17, Lemma 5.3], which we adapt below in Lemma 25.
Lemma 25 ([ABW17, Lemma 5.3]). Let G be a graph and n = (n(e)) e∈G a random and exchangeable family of nonnegative numbers with |n| 2 > 0 almost surely. Then for any V ⊂ V (G), and any t > 0, The proof goes the same way as that of [ABW17, Lemma 5.3], except that we do not have a double edge here, and the mass on edges is projected deterministically on vertices in our case, instead of randomly. The reader may notice that there is an extra term inside the probability in their lemma. This term accounts for the double edge, which we do not have here. The same line of arguments still works though. Indeed, we have The last equality holds because the degree biased-measure counts twice each edge of G[V ], since this edge appears in the degree of both its endpoints, while the edges of ∂ e V are only counted once, in the degree on the only one of its endpoints which is in V . Then one concludes as in the proof of [ABW17, Lemma 5.3], by a Hoeffding-type bound for exchangeable vectors.

Proof of Theorem 6
Scaling limit of the biggest block. By Proposition 3 and Proposition 4, the biggest block of Q n,u , whose size we denote C(n, u), is a simple quadrangulation sampled uniformly with size C(n, u). Also by Theorem 2, this size is asymptotically in probability, C(n, u) = (1 − E(u))n + O P (n 2 /3) = 9 − 5u 3(3 + u) n + O P (n 2 /3).
By conditioning on C(n, u) and using Proposition 22, we therefore get the following GHP scaling limit for the biggest block GHP comparison of Q n,u with its biggest block. By the preceding scaling limit, and the use of Lemma 23 with q = Q n,u and ε = n −1/4 , the proof of the theorem reduces to showing the convergence to 0 in probability of the following quantities r GH := 2 n 1/4 H(T n,u ) max v =v • diam(b Qn,u v ) r P := 2 V (Q n,u ) \ V (Q + n,u ) |V (Q n,u )| where ε n = n −1/4 .
Bounding r GH . By Theorem 2, the second-biggest block of Q n,u has size O(n 2/3 ) in probability. Combining this with Corollary 11, one gets for all δ > 0 the bound in probability Also, by Theorem 1, T n,u is a non-generic subcritical Galton-Watson tree conditioned to have 2n + 1 vertices, in the terminology of [Kor15]. We may therefore use [Kor15, Theorem 4] to get for all δ > 0 the bound in probability H(T n,u ) = o n δ .
Bounding r P . First, notice that since Q n,u is a quadrangulation we have |V (Q n,u )| = |E(Q n,u )| = 2n, and by the block-tree decomposition which puts in correspondence edges of Q n,u and edges of T n,u , we also have V (Q n,u ) \ V (Q + n,u ) = E(Q n,u ) − E(Q + n,u ) = E(T n,u ) − E(T + n,u ) = E(T n,u \ T + n,u ) .
Therefore we have to bound the size of the subtree T n,u \ T + n,u . A moment of thought shows that it is bounded by U → (T n,u ) + U ← (T n,u ), where U → (t) is the index in lexicographical order of the vertex with largest degree of the tree t, and U ← (t) is the index in reverse lexicographical order of that same vertex. Now, [Kor15,Theorem 2] shows that (U → (T n,u )) n 1 is a tight sequence. Since U ← (T n,u ) has the same law as U → (T n,u ), the respective sequence is also tight. All in all, we get in probability Let us argue that conditionally on Q • n,u , this family n is exchangeable. Recall that Q n,u has the law of Q under P u , conditioned to the event {|Q| = n}. By the symmetries of the Galton-Watson law and Proposition 7, the family is i.i.d. conditionally on Q • , where v e is the child of v • that the block-tree decomposition associates to e. In particular, this family is exchangeable. Since the event {|Q| = n} is invariant by each permutation of the subtrees attached to the node v • with their respective blocks, the above family stays exchangeable when conditioning by this event. Therefore n is indeed exchangeable. All the hypotheses of Proposition 24 have been checked, so that we may apply it, after conditioning by the size of Q • n,u , since conditionally on its size k it is a uniform simple quadrangulation of size k. We obtain in probability Hence, d P also tends to 0 in probability and this concludes the proof.

Concluding remarks and perspectives
We have exhibited a phase transition phenomenon for two closely related models of random maps with a weight u > 0 per block. The phase transition occurs at u = 9/5, and we have established the existence of three regimes, regarding the size of the largest block, and regarding the scaling limit (and the order of magnitude of distances).
Extension to other models. Our method can be generalised to other models which can be decomposed into appropriate blocks with an underlying tree structure, for example the models described in [BFSS01, Table 3], which is partially reproduced in Table 2. A triangulation is a map where all faces have degree 3. It is irreducible if every 3-cycle defines a face. In this section, we use the same notation for the various models as in the rest of the article. Models described in [BFSS01, Table 3] where maps are decomposed into blocks weighted with a weight u > 0 undergo a phase transition at the critical value u C written down in Table 3. More precisely, Theorems 1 to 4 hold for these models with the constants of Table 3. Notice that for the decomposition of general maps into 2-connected maps (i.e. the schema linking M 1 and M 4 ) -which is the case studied in this paper -we get results consistent with Theorem 1. Moreover, the values of u C and E(u) are consistent since it always holds that E(u C ) = 1. Furthermore, for u = 1, we retrieve the results of [BFSS01, Table 4]: indeed, our 1 − E(1) equals their α 0 3 . Models from [BFSS01, Table 3] are amenable to computations similar to this article's in order to get the values above. We show in Table 3 the most obvious results and models requiring more care will be described in a separate note. In the cases of Table 2, there is d ∈ Z >0 such that H(z) = z(1 + M ) d , and the corresponding law µ u (except for triangulations) comes naturally as: µ u (dm) = 1 m =0 ub m y(u) m + 1 m=0 uB(y(u)) + 1 − u , µ u (m) = 0 when d m.
The cases dealing with triangulations require more care as the series are counted by vertices but the substitution is done on edges in one case, and on internal faces in the other; but keeping this in mind, the same methods can be applied. For all models, we expect to get similar regimes as in Table 1 (assuming the convergence of the family of blocks is known, as well as diameter estimates). However, conditioning is more difficult for some models, as the size of the map is not always immediately deduced from the size of the Galton Watson tree (e.g. for simple triangulations (T 2 ) decomposed into irreducible triangulations (T 3 ), the size is the number of leaves of the Galton-Watson tree).
Perspectives. We plan to study similar models in the context of decorated planar maps (e.g. treerooted maps or Schnyder woods), where the generating series exhibit different singular behaviours. In future work, we also want to investigate more closely the rate of the phase transition at the critical value u = 9/5, in analogy to the study of the largest component for the Erdös-Rényi random graph [CFGK20].
Finally, as mentioned in the introduction, the model of maps with a weight u per 2-connected block has been studied as encoding certain discrete spaces of dimension larger than 2, with motivations from theoretical physics [Bon16,Lio17]. The metric properties are however modified via the correspondence, and it would be interesting to determine if the scaling limits remain the same.