Random cubic planar graphs converge to the Brownian sphere

In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.


Introduction
In recent years, a lot of progress has been achieved in the understanding of the scaling limit of random planar maps (planar graphs embedded in the sphere). Miermont [28] and Le Gall [24] established the first results of convergence of random planar maps towards the Brownian sphere. Since these major results, many other families of maps have also been proved to admit the Brownian sphere as their scaling limit. Moreover, the properties of the Brownian sphere have been thoroughly investigated, we refer to the following surveys and references therein [25,29,30] for nice entry points to this field.
However, much less is known about the scaling limit of random planar graphs, which are graphs that can be embedded in the sphere but for which the embedding is not fixed. In this article, we obtain the first result of convergence of a family of random planar graphs towards the Brownian sphere. Our main result is the following (a graph or multigraph is called cubic if all its vertices have degree 3): Theorem 1. Let C n (resp. M n ) be a uniformly random connected cubic labeled planar graph (resp. multigraph) with 2n vertices. Let µ Cn and µ Mn denote respectively the uniform distribution on V (C n ) and V (M n ). Let d gr denote the graph distance. Then, there exist constants c, c m ∈ (0, ∞) such that the following convergence holds: and where the convergence holds in distribution for the Gromov-Hausdorff-Prokhorov topology for measured metric spaces, and m ∞ denotes the Brownian sphere.
While finishing this project, we became aware that Benedikt Stufler obtained simultaneously and independently the scaling limit of random connected cubic planar graphs [38,39], using the same general strategy but with different techniques to resolve the main difficulties.
Our motivation to consider not only cubic planar graphs but also cubic planar multigraphs comes from the fact that they appear naturally as the kernel of random connected planar graphs in the sparse regime (i.e. when the excess of the graph is o(n)). The scaling limit of random planar maps in the sparse regime has recently been investigated in [12], via the study of the scaling limit of their cubic kernel. To extend this result to the case of random sparse planar graphs, the scaling limit of random connected cubic planar multigraphs is thus a decisive step 1 .
Before giving the main ideas behind the proof of this result, let us place it in the context of the existing literature on the enumeration of planar graphs and on the study of their random properties. Unlike planar maps, for which closed enumerative formulas are often available [41], the enumeration of planar graphs is more involved. Building on a canonical decomposition of graphs into their 3-connected components [42], a (differential) system of equations for their generating series can be obtained. Major achievements have consisted in deriving precise asymptotic estimates from these equations. A non-exhaustive list of contributions includes the case of 2connected planar graphs [6], the case of cubic planar graphs [7] and cubic planar multigraphs [22] and the case of planar graphs [18]. We also refer to the survey [32] and references therein. These enumerative results have made possible the study of random properties of planar graphs, and to establish limit laws for various parameters, such as the maximum degree [14]. They have also been a key ingredient in the recent proofs of the convergence of random cubic planar graphs [40] and random planar graphs [37] for the local topology.
However, only few results are available about the metric properties of random planar graphs. It has only been shown that the order of magnitude of the diameter of random planar graphs with n vertices is concentrated around n 1/4 [11]. Let us also mention that the scaling limit of some families of planar graphs have been previously established. However, such results only hold for so-called subcritical families of graphs (which include in particular outerplanar graphs and series-parallel graphs) and their scaling limit is Aldous' Brownian Continuum Tree [34,36].
A key ingredient in the proof of Theorem 1 is the following result, which is of independent interest: Theorem 2. Let t n be a uniformly random simple triangulation with n + 2 vertices. Let µ tn and µ t † n denote respectively the uniform distribution on V (t n ) and on V (t † n ). Assume that the support of ν is included in [η 0 , ∞) for η 0 > 0, and that ν has exponential tails.
Then, there exists a constant c ν > 0 such that the following convergence holds jointly in distribution for the Gromov-Hausdorff-Prokhorov topology: where m ∞ is the Brownian sphere.
Note that the convergence of the first component was established by Addario-Berry and the first author in [1]. By applying this theorem to the case where ν = δ 1 is the Dirac measure on {1}, and by applying Whitney's theorem, we obtain as an immediate corollary: Theorem 3. Let K n be a uniformly random 3-connected cubic planar graphs with 2n vertices. Denote by µ Kn the uniform distribution on V (K n ). Then, there exists a constant c 3 > 0 such that the following convergence holds in distribution for the Gromov-Hausdorff-Prokhorov topology: where m ∞ is the Brownian sphere.
When adapting the framework of Curien and Le Gall to our setting, the main difficulty comes from the fact that we consider simple triangulations rather than (unconstrained) triangulations. The article [13] relies extensively on the so-called skeleton decomposition of triangulations, introduced by Krikun in [23]. However, the skeleton decomposition does not behave as nicely on simple triangulations as on general triangulations. To circumvent this issue, we apply the skeleton decomposition to the model of quasi-simple triangulations (defined in Section 5). The very nice property of quasi-simple triangulations is that their skeleton decomposition can be encoded via exactly the same branching process as general triangulations. Therefore, many results of [13] extend to our setting effortlessly. This comes at the price that quasi-simple triangulations are not invariant under re-rooting (its root plays indeed a special role), so that some arguments need to be adapted. This is the purpose of Section 6. On the other hand, in his independent preprint [38], Benedikt Stufler deals directly with a skeleton decomposition for simple triangulations, where some forbidden configurations have to be handled.
From i.i.d edge weights to the scaling limit of random cubic planar graphs. Given Theorem 2, there remain two last steps to prove Theorem 1. The first one deals with the metric property: indeed, we need to relax the hypothesis that the edge lengths are independent. The second one deals with the measure on the vertices. Roughly speaking, we have to prove that the uniform measure on the vertices of a random connected cubic planar graph on one hand, and on the vertices of its 3-connected core on the other hand, do not differ asymptotically.
More precisely, to prove Theorem 1, we establish in fact the following joint convergence result: Theorem 4. Let C n be a uniformly random connected cubic labeled planar graph (resp. multigraph) with 2n vertices. Write K(C n ) for its 3-connected core and ∆(C n ) for the (unlabeled, rooted) dual of K(C n ). We denote respectively by µ Cn , µ K(Cn) and µ ∆(Cn) the uniform measures on V (C n ), V (K(C n )) and V (∆(C n )). Moreover, we write dist Cn for the distance on K(C n ) induced by the graph distance on C n . Then, there exist two positive constants c 1 and c 2 (with c 2 explicit) such that: C n , 1 c 1 n 1/4 d gr , µ Cn , K(C n ),

Definitions on graphs and maps
In this article, we consider both multigraphs, and graphs (i.e., multigraphs with no loop nor multiple edges). A multigraph is called vertex-labeled if its n vertices carry distinct labels in [1..n], and is called half-edge-labeled if its 2m half-edges carry distinct labels in [1..2m]. When dealing with random cubic planar multigraphs (resp. graphs) we will usually consider that they are half-edge-labeled (resp. vertex-labeled). A connected multigraph is rooted if it comes with a distinguished oriented edge called its root edge. The tail vertex of the root edge is called the root vertex.
Note that a connected graph or multigraph can be seen as a discrete metric space, where the distance between two vertices is the minimal length over the paths connecting them. More generally, we will consider so-called metric graphs, i.e., graphs where every edge e carries a length (e) ∈ R * + . In such a graph seen as a metric space, the length of a path γ = (e 1 , . . . , e m ) is defined as L(γ) = (e 1 ) + · · · + (e m ), and the distance between two vertices is the length of a shortest path connecting them (on the other hand, in the context of metric graphs, we will use the notation |γ| for the number of edges in γ).
A planar map (shortly, a map) m is a connected multigraph properly embedded on the sphere, up to orientation-preserving homeomorphism. A face of m is a connected component of the sphere cut by the embedding, a corner of m is a sector between two consecutive edges around a vertex (a corner thus lies in a unique face of m). The degree of a vertex (resp. face) of m is the number of corners that are incident to it. We write E(m), V (m) and F (m) for the set of edges, vertices and faces of m respectively.
A rooted map is a map that comes with a distinguished oriented edge called its root edge, i.e., the underlying multigraph is rooted. The face on the right of the root edge is called the root face (also called the outer face, since it is usually taken as the unbounded face when the embedding of the map is represented on the plane), and the corner incident to the root vertex just after the root edge in clockwise order is called the root corner. Similarly, vertices incident to the root face are called outer vertices, and inner vertices otherwise.
In general, maps are considered as unlabeled (whether on vertices or on half-edges). Indeed, in contrast to the situation for multigraphs, the rooting operation is sufficient to suppress the possibility of having non-trivial automorphisms. A map is simple if the underlying multigraph is a graph, i.e. if there is no double edge and no loop.
For m a map, the dual map m † is constructed as follows (see Figure 1): we add a "type F" vertex in every face of m. Then for every edge e of m, we draw an edge e † (the dual edge of e) between the type F vertices of the two faces that are incident to e, in such a way that e † crosses e and no other edge (including the new ones) -if both sides of e are incident to the same face, then e † is a loop. Then m † has vertex set the set of type F vertices, and edge set the set of dual edges. If m is rooted, the root edge of m † is taken as the dual of the root edge e of m, oriented in such a way that it crosses e in the counterclockwise direction around the root vertex.
One can see from the definition that (m † ) † is the map obtained from m by reversing the direction of its root edge. Furthermore, the one-to-one correspondence between the faces of m and the vertices of m † preserves the degree: if f is a face of m, and f † is the associated vertex of m † , then deg m (f ) = deg m † (f † ). The same holds for vertices of m: every face of m † contains a unique vertex of m, and if v is a vertex of m and v † is the face of m † that contains it, then deg m (v) = deg m † (v † ).
A multigraph or map is called cubic if every vertex has degree 3. The size of such a multigraph or map is the integer n 1 such that the number of vertices is 2n and the number of edges is 3n. By Euler's formula, a cubic map of size n has n + 2 faces. On the other hand, a triangulation is a map where all faces have degree 3. Since taking the dual of a map sends vertices of the primal to faces of the dual and vice-versa, it follows that the dual of triangulations are cubic maps; and conversely, the dual of cubic maps are triangulations. A triangulation with n + 2 vertices thus has 2n faces and 3n edges.
A multigraph is called 3-connected it is simple, has at least 4 vertices, and at least 3 vertices need to be deleted to disconnect it. By Whitney's theorem [43], such a graph has exactly two embeddings on the sphere, which differ by a mirror. A 3-connected map is a map whose underlying graph is 3-connected. It is known that duality preserves 3-connectivity [31], and that a triangulation is 3-connected if and only if it is simple.
Remark 1. For a 3-connected graph k that is vertex-labeled and rooted, we can canonically choose one of the two embeddings R, R (which differ by a mirror) as follows. In R we let f 1 , f 2 be the two faces incident to the root edge e. Let v 0 be the root-vertex, and let v 1 (resp. v 2 ) be the unique vertex of f 1 (resp. f 2 ) adjacent to v 0 and not incident to the root edge. By 3-connectivity, we have v 1 = v 2 . If the label of v 1 is smaller than the label of v 2 , then we take R as the embedding associated to k, otherwise we take R . The outer edges of k are those incident to the outer face in the canonical embedding. Note that each 3-connected (unlabeled) rooted map with n vertices has n!/2 preimages under this correspondence. . . . . . . . . . . . . . . . . . . . . . . . . . .
Let (X , d) be a metric space. For every K ⊂ X and every ε > 0 let K ε = {x ∈ X : d(x, K) < ε}. The Hausdorff distance between two compact sets K 1 and K 2 of X is The Gromov-Hausdorff distance allows us to compare two compact metric spaces by isometrically embedding them in a common, larger metric space, and comparing the Hausdorff distance of their embedding. More formally for every pair of compact metric spaces (X , d X ) and (Y, d Y ), we define their Gromov-Hausdorff distance as: where the infimum is over every metric space Z and isometries Φ X : X → Z and Φ Y : Y → Z. Since isometric spaces are at Gromov-Hausdorff "distance" zero, d GH only defines a pseudodistance on compact metric spaces, but it becomes a true distance when considered on M the set of compact metric spaces seen up to isometry. Furthermore, (M, d GH ) is a Polish space.
If (Z, d) is a metric space with Borel σ-algebra B(Z), we define the Lévy-Prokhorov distance between two Borel probability measures µ, µ on Z as follows: If Z is a Polish space then the Lévy-Prokhorov distance makes the set of probability measures on Z a Polish space, and d LP metrizes the weak convergence of measures. There is also a characterization of d Z LP (µ, µ ) in terms of coupling, see for instance [20]. A coupling between µ and µ is a probability measure ν on Z × Z such that π 1 * ν = µ and π 2 * ν = µ (where π 1 and π 2 denote the canonical projections with respect to the first and second coordinate respectively). We denote by M(µ, µ ) the set of couplings between µ and µ . Then, if (Z × Z) denotes the set if and only if there exists a coupling ν ∈ M(µ, µ ) such that ν((Z × Z) ) 1 − .
A weighted metric space is a metric space equipped with a probability Borel measure. The Gromov-Hausdorff-Prokhorov distance d GHP is a metric on isometry classes of compact weighted metric spaces, defined as follows. For any measured spaces X , Z, measurable function Φ : X → Z and measure µ on X , write Φ * µ = µ(Φ −1 (·)) for the pushforward of µ under Φ. Let (X , d X ) and (Y, d Y ) be two compact metric spaces equipped respectively with the probability measures µ X and µ Y ; define where (as in the definition of d GH ) the infimum is over every metric space (Z, d Z ) and isometries Φ X : X → Z and Φ Y : Y → Z. Because isometric weighted metric spaces are at Gromov-Hausdorff-Prokhorov "distance" zero, the function d GHP only defines a pseudo-metric; to make it a true metric, we consider K the set of isometry classes of compact weighted metric spaces. Then d GHP defines a true metric on K, and (K, d GHP ) is a Polish space. There exists also an equivalent description of d GHP based on couplings which is better suited for our purposes. Let us introduce the concept of correspondence. A correspondence between two sets X , X is a subset R ⊂ X × X such that π X (R) = X and π X (R) = X , where π X , resp. π X is the canonical projection on X , resp. X . We write C(X , X ) for the set of correspondences between X and X . Given a correspondence, define its distortion The Gromov-Hausdorff distance can be expressed in terms of distortion: and this infimum is attained. Let X = (X , d X , µ X ) and Y = (Y, d Y , µ Y ) be two compact weighted metric spaces. Recall the notation M(µ X , µ Y ) for the set of couplings between µ X and µ Y , then we have the following alternate characterization of the Gromov-Hausdorff-Prokhorov distance: 3 The 3-connected core of cubic planar graphs

Decomposition into 3-connected components
We recall here a decomposition [7,22,33] of so-called cubic networks (closely related to rooted cubic planar graphs) and of cubic planar (multi-)graphs into 3-connected components.
A cubic network is a connected planar multigraph N with two marked vertices v − , v + called the poles, such that the poles have degree 1 and the other vertices have degree 3. Note that the number of non-pole vertices has to be a positive even number 2n, then n 0 is called the size of N. The two edges adjacent to the poles are called the legs of N, and the other edges are called plain. The trivial network is the one of size 0, with a single edge connecting the two poles. Note that a cubic network N of positive size identifies to a rooted cubic planar multigraph N, which is obtained by merging the two poles, smoothing out the merged vertex, and orienting the resulting edge from the neighbor of v − to the neighbor of v + . Then N is called polyhedral if N is 3-connected, in which case the outer edges of N are the (plain) edges of N corresponding to the outer edges (without including the root edge) of N. We also define the double-edge network as the cubic network of size 1 where the neighbors of the poles are connected by a double edge.
Given two non-trivial cubic networks N 1 , N 2 , we let N := N 1 + N 2 be the cubic network obtained by merging the plus-pole of N 1 with the minus-pole of N 2 and smoothing out the merged vertex, so that the plus-pole of N is the plus-pole of N 2 , and the minus-pole of N is the minus-pole of N 1 . A cubic network N that can be obtained as N = N 1 + N 2 is called an S-network, in which case N has a unique decomposition as N = N 1 + N 2 where N 1 is a (non-trivial) non-S-network, and by unfolding it also has a unique decomposition as N = N 1 + · · · + N k where each component N i is a non-trivial non-S-network.
In this article, edge-substitution in a graph G is the operation consisting, for each edge e = (u − , u + ) of G (canonically endowed with a minus-extremity and a plus-extremity), of substituting it by a cubic network N e , so that the plus-pole and minus-pole of N e are identified respectively with u − and u + (note that substitution by the trivial network leaves the edge unchanged). If G is itself a cubic network, only the plain edges are substituted. A cubic network obtained by edge-substitution from the double-edge network (resp. from a polyhedral network) is called an P -network (resp. a H-network).
Note that S-networks, P -networks and H-networks are such that the two poles are at distance greater than 2. An L-network is a cubic network N whose poles are at distance 2. In an L-network, let u be the common neighbor of the poles, and let v be its other neighbor (note that the edge {u, v} is an isthmus). Let l 1 be the L-network of size 1 where v carries a loop. And let l 2 be the L-network of size 2 where v has two neighbors (apart from u) each carrying a loop. As shown in [7,22], an L-network N is either obtained from l 1 where only the loop-edge is allowed to be substituted, or is obtained from l 2 where only the two loop-edges are allowed to be substituted (the first and second case correspond respectively to N \v having two or three connected components). The two cases are called type 1 and type 2, respectively.
We have the following decomposition result for cubic networks, shown in [22, Lem.1] (the version for graphs appeared previously in [7, Lem.1]): Lemma 5. Any non-trivial cubic network is in exactly one of the following classes: {L-network, S-network, P -network, H-network }.
As detailed in [33] and illustrated in Figure 2, Lemma 5 yields a tree-decomposition of cubic networks, i.e., a rooted (unembedded) tree τ (N) can be associated to a non-trivial cubic network N, whose inner nodes have labels in {L, R, M, T }. Precisely, if N is an S-network, of the form N 1 + · · · + N k , then the root node of τ (N) is labeled R and the hanging subtrees are τ (N 1 ), . . . , τ (N k ) (these are not S-network hence do not have root label R). If N is an P -network or H-network, then the root node is labeled M (resp. T ) and the hanging subtrees are the trees associated to the substituted networks. Finally, for L-networks, the root node has label L. In type 1, it has one child also labeled L and of arity in {0, 1}, depending on whether the loop-edge Left: a connected (simple) cubic graph g, realized as a cubic network upon rooting at a (non-isthmus) edge. Right: the associated decomposition-tree. The tree is considered as unrooted: it is intrinsic to g (i.e., it does not depend on the root-choice), which is also reflected by the difference in the choice of letters (a series-decomposition in the rooted setting corresponds to a "ring" of components in the unrooted setting, giving a node of type R). The left part also shows the partition of the vertices of g by components (tree-nodes) of type in {L, M, T }. of l 1 is unchanged or substituted by a non-trivial network N (in which case the hanging subtree at the child corresponds to τ (N )). In type 2, it has two children labeled L, each corresponding to one of the two loops of l 2 . Each child again has arity in {0, 1}, depending on whether the corresponding loop is left unchanged or substituted by a non-trivial network.
Each T -node of the tree τ (N) corresponds to a certain rooted 3-connected planar graph k whose vertex-set is a subset of the vertex-set of N. Then k is called a 3-connected component of N. The hanging subtrees of the node correspond to the (non-trivial) cubic networks substituted at its edges. As detailed in [33], the tree-decomposition and 3-connected components are intrinsic to the underlying unrooted cubic planar graph (essentially it means that rerooting the cubic multigraph at another non-isthmus edge amounts to rerooting the same underlying unrooted tree, a fact established by Tutte [42] for the decomposition of graphs into 3-connected components, which can then be specialized to the cubic case). Accordingly, to any connected cubic planar graph g, one can associate an unrooted decomposition tree, denoted τ (g). Let us mention the following subtlety: when building the tree in the rooted case, every R-node has arity at least 2, hence degree at least 3 if not at the root and degree at least 2 if at the root. In case the root node is of type R of arity 2, then when unrooting the tree the corresponding node of degree 2 is smoothed out, so that all R-nodes in the unrooted tree are of degree at least 3. The (unrooted) decomposition-tree satisfies the following properties, which can be visualized in Figure 2: For g a connected cubic planar multigraph and τ (g) the associated decompositiontree, the vertices of g are partitioned among the L-components, the M -components, and the T -components (3-connected components) of τ (g) (the R-components have no contribution).
The L-components are in one-to-one correspondence with the separating vertices of g, the M -components are in one-to-one correspondence with vertex-pairs {u, v} such that g is obtained by edge-substitution in the triple-bond connecting u and v.
The other vertices belong to the 3-connected components. Precisely, for V a vertex-subset of g with |V | 4, and k a 3-connected planar graph (unrooted) with vertex-set V , k is a 3-connected component of g if and only if g is k with networks substituted at the edges.
We focus our study first on simple cubic planar graphs; accordingly the cubic networks are also simple (extension of the arguments to multigraphs will be given in Section 3.4). In the network-decomposition, this yields the constraint that loops (when dealing with an L-network at the current node) are to be substituted by non-trivial non-L-networks, whereas when substituting in the double-edge network (when dealing with a P -network at the current node), at least one of the two edges is to be substituted by a non-trivial network. On the decomposition-tree for unrooted graphs, being simple is equivalent to the fact that every leaf has label T , every L-node of degree 2 has exactly one neighbor of type L, and every M -node has degree in {2, 3}; for networks the same constraints hold at non-root nodes, while the root node is allowed to be a leaf (i.e., to have arity 1) of type L or type M .
We define families F = ∪ n F n of cubic simple networks (with n the size-parameter), where F is replaced by L, S, P, H according to the type; we also define D as the family of all simple cubic networks and K as the family of polyhedral networks. Each of these families is vertex-labeled, in the sense that the 2n non-pole vertices have distinct labels in [1..2n] (on the other hand the two poles are unlabeled). We denote by F (z) = n 1 (2n)! |F n |z n the associated generating function, replacing F by the corresponding letter for each family.
We can express K in the following way. Let T n be the set of rooted simple planar triangulations with n+2 (unlabeled) vertices (and 2n faces), and T (x) = n 2 |T n |x n the corresponding ordinary generating function, which is known [41] to have the following exact expression: where ξ is the unique formal power series in x with constant term equal to 0 and defined by x = ξ(1 − ξ) 3 . By duality, simple planar triangulations correspond to 3-connected cubic planar maps. Thus, applying Whitney's theorem and taking vertex-labeling into account, we have (2n)!|T n | = 2|K n |, hence K(x) = T (x)/2.
Then a decomposition at the root of the decomposition tree of cubic networks yields the following system (from which the coefficients can be extracted): In the second line the two terms correspond respectively to the root L-node having arity 2 or 1.
The last line (with T (x) given in (7)) relies on the fact that a cubic network of size n has 3n − 1 plain edges. Let C = ∪ n C n be the family of vertex-labeled connected cubic planar graphs of size n, i.e., the 2n vertices carry distinct labels in [1..2n]. Let C(z) = n 1 1 (2n)! |C n |z n be the associated exponential generating function. Using Claim 6, one obtains the following expression for the exponential generating function C • (z) = 2zC (z) of pointed (i.e., with a marked vertex) cubic planar graphs [33]: where we define G(z) := K(zD(z) 3 ). The three groups correspond to the marked vertex being involved in an L-component (with the two terms corresponding to the marked vertex yielding two or three connected components upon its removal), an M -component, and a T -component respectively (the factor 1/3 in front of G(z) is due to having a marked vertex instead of a marked directed edge).

Largest 3-connected component
We now use singularity analysis of the involved generating functions, in order to show (in this subsection and the next one) that a random cubic connected planar graph almost surely has a unique giant 3-connected component (of linear size), and to establish related properties. This analysis follows from a well-established methodology [5,17]. Similar results appear in the recent preprint [40] (in particular Theorem 1.2). We provide here our own analysis (thereby explaining the mild differences with the statements and proof of [40]) for the sake of completeness, and since we will extend these results to cubic planar multigraphs in Section 3.4. It follows from [33,Sec.3.1] that the generating functions F (z) with F ∈ {L, D, G, C • } have the same radius of convergence ρ ≈ 0.101905 and these series have singular expansion (in a slit neighborhood) of the form 2 : Moreover, these series are analytically continuable to a domain {|z| ρ + η, z − ρ / ∈ R + } for some η > 0. By transfer theorems of analytic combinatorics [16,VI.3 We say that a pointed connected cubic planar graph is of type T if the marked vertex is incident to a T -component. According to (9), this occurs with probability Let E(z) = zD(z) 3 (note that E(z) also has a singular expansion of the form E(z) = E 0 − E 2 Z 2 + E 3 Z 3 + O(Z 4 )), so that the generating function of cubic connected planar graphs of type T is 1 3 As shown in [33,Sec.3.1], this composition scheme is critical, in the sense that the radius of convergence of y → K(y) is y 0 = ρD 3 0 , and K(y) has a singular expansion of the form We can now use the results from [5]. Let A(x) be the density function defined by For c > 0, the probability distribution of density cA(cx) is called Airy-map distribution of parameter c.
Proposition 7 (Apply Theorem 5 in [5]). Define the constants For C • n the random pointed cubic 3-connected planar graph of type T , let X n be the size of the 3-connected component incident to the marked vertex. Then, for any x ∈ R (and uniformly in any compact set of R)  The 3-connected core of a cubic connected planar multigraph is its largest 3-connected component (if there is a tie, we break the tie by choosing the one whose minimal label is smaller). We now obtain: Theorem 8. For C n the random connected cubic planar graph with 2n vertices, let X * n be the size of the 3-connected core of C n . Then, for any x ∈ R (and uniformly in any compact set of R), we have, with the constants α, c defined in Proposition 7: Note that Theorem 8 ensures that X * n is concentrated around αn with fluctuations of order n 2/3 , and that (X * n − αn)/n 2/3 converges in law to the Airy-map distribution of parameter c.
Proof. Let C • n be the set of pointed connected cubic planar graphs with 2n vertices, and let C • n be uniformly random in C • n . In view of (11), if we define and define X n to be 0 if C • n is not of type T , and X n to be the size of the 3-connected core otherwise, then by (11) and Proposition 7, we have for any x ∈ R (and uniformly in any compact set of R), P (X n = αn + xn 2/3 ) = n −2/3 p cA(cx) · (1 + o(1)).
To show (12) we rely on a re-rooting argument given in [5,Appendix D]. The graphs with a giant 3-connected component of size ≈ αn incident to the marked vertex are in proportion ∼ p among C • n , and they admit ∼ 1/α re-rootings at an arbitrary vertex. In addition, they produce different objects in C • n , because, as observed in [40], the fact that α > 1/2 ensures that there is at most one such component within the graph, for n large enough.
Thus p /α 1, and to conclude that almost all graphs in C • n are of that form it remains to show that p = α. Here we proceed a bit differently from [40]. Rather than computing the precise evaluations of α and p , we will show that each of α, p has a rational expression in terms of ρ, L 0 , L 2 and that these expressions are the same. Thereby our calculations do not require any evaluations of the involved quantities, treated as formal symbols, nor the exact expression of K(x) = T (x)/2 given in (7); we just use the fact that the series in z have a singular expansion of the form (10). (Thus the coincidence of p and α can be seen as a "generic" property of the decomposition grammar.) The second line of (8) yields for each i ∈ {0, 2, 3} a rational expression for D i in terms of {ρ, L 0 , L i }, precisely Since E(z) = zD(z) 3 , this directly yields for each i ∈ {0, 2, 3} a rational expression for E i in terms of {ρ, L 0 , L i }, e.g.
We can then inject the expressions of E 0 and E 2 into α = E 0 E 2 , and find It remains to find a similar rational expression for p = 1 from which we extract for each i ∈ {0, 2, 3} a (quite larger) rational expression for G i in terms of {ρ, L 0 , L i }. From (9) we can then extract a rational expression for C • 3 in terms of {L 0 , L 3 , ρ}. From the expressions of E 2 , E 3 , G 2 , G 3 , we obtain a rational expression of p that exactly matches the one of α given above. Thus p = α.

Cubic planar graphs with a marked 3-connected component
For two integers n, q, let C (q) n be the set of connected cubic planar graphs of size n having a marked 3-connected component of size q, and denote by C n . It will also prove convenient for us to see C (q) n as closely related to a model with more independence. The critical Boltzmann distribution on D is the probability distribution µ (D) defined as where we recall that ρ is the radius of convergence of D(z). Let C (q) be a random connected cubic planar graph with a marked 3-connected component of size q constructed in the following way. Letting K be a uniformly random 3-connected graph of size q, we replace (independently) each edge of K by a cubic network sampled from µ (D) . The resulting graph is C (q) . Note that, for any n, q 1, C (q) n is distributed as C (q) conditioned to have size n.
As stated next, the conditioning probability is polynomial when q = αn + O(n 2/3 ):
With the notation (10), we have A local limit theorem (for a stable law of parameter 3/2) holds [5, Theo.11(ii)]: for n = 1 α q − xq 2/3 , uniformly in x on any compact interval, we have Let C be a connected cubic planar graph with a marked 3-connected component K. Note that C is obtained from K by substituting every edge of K by a cubic network; such a network is called an attached network of C. n have size at most n 2/3+ . Proof. Let = /2. We argue via C (q) and conditioning on the size. The 3q edges of the marked 3-connected component can be canonically ordered (e.g. lexicographic ordering on the pair of labels of the incident vertices), which yields a canonical ordering M 1 , . . . , M 3q of the 3q attached networks. In C (q) these are independently drawn under µ (D) , and the size of (a large deviation result, obtained by a saddle-point estimate).
) (we apply the same argument to (3q − 1) attached networks instead of 3q). By the union bound, the probability that Hence, when q, n are related by q = αn + O(n 2/3 ), Lemma 9 ensures that the probability that |M i | > n − 1 (1). Moreover, for n large enough, this event is implied by the event that |M i | > n 2/3+ for some i ∈ [1..3q].

Extension of the results to cubic planar multigraphs
In the context of multigraphs, it is somehow more natural (e.g. for kernel extraction relations) to consider that the half-edges are labeled instead of the vertices 3 . Thus, let M n denote the set of half-edge-labeled connected cubic planar multigraphs of size n (i.e., with 2n vertices), and M n denotes an element of M n taken uniformly at random. Regarding cubic networks (allowing for loops and multiedges), we consider that, in size n (i.e., with 2n non-pole vertices), the 6n − 2 half-edges on plain edges carry distinct labels in [1..6n − 2]. For each family F of cubic networks, with F ∈ {D, L, S, P, H}, we let F (z) = n 1 (6n−2)! |F n |z n be the associated generating function. In this context, the equation-system (8) (again obtained from a decomposition at the root of the associated tree) becomes We warn the reader of our abuse of notations, as D, L, S, P and H all denote different objects in the context of graphs and multigraphs; we do so because of the extreme similarity of the proofs and results. Which object we consider should be clear from context. Now, with M (z) = where is the same as before, since 3-connected graphs are simple).
We can now rely on the singularity analysis performed in [15,22] to obtain the analogue of Theorem 8. The generating functions F (z) with F ∈ {L, D, E, G, M • } have the same radius of convergence ρ = 54/(79) 3/2 (we still use the letter ρ even if the radius of convergence for multigraphs is different than for simple graphs), and these series have singular expansion (in a slit neighborhood) of the form As in the case of graphs, the composition scheme in (15) is critical, so that we can apply Theorem 5 from [5], exactly in the same way as in Proposition 7. Defining the constants , we obtain that, for M • n the random pointed connected cubic planar multigraph of size n, conditioned to have the marked vertex on a 3-connected component, the size X n of the 3connected component at the marked vertex satisfies for any x ∈ R (and uniformly in any compact set of R) The constants α, c are again explicitly computable, and we find the following values, much simpler than in the case of graphs: We can now obtain the analogue of Theorem 8.
Theorem 11. For M n a uniformly random connected cubic planar multigraph of size n, let X * n be the size of its 3-connected core. Then, for any x ∈ R, and uniformly in any compact set of R, where α, c are given in (18).
Again, this implies that X * n is concentrated around α n, and (X * n − α n)/n 2/3 converges to the Airy-map distribution of parameter c.
Proof. Similarly as in Theorem 8, we have to check that the constant p : p is equal to α (again we do not need the precise algebraic equations of the involved series, the proof is "generic" and does not require the expression of K(x)). The second line of (15) yields, for each i ∈ {0, 2, 3}, a rational expression of D i in terms of {ρ, L 0 , L i }: . It remains to find a similar rational expression for p = 1 . From (15) (note that the third line yields S(z) = (D(z) − 1) 2 /D(z)), we extract the equation which yields, for i ∈ {0, 2, 3}, a rational expression for G i in terms of {ρ, L 0 , L i }. Finally, we can extract from (16) a rational expression for we obtain a rational expression for p that exactly matches the one of α given above. Hence p = α.
Similarly as for cubic graphs, for two integers n, q, we let M n . And we denote by M (q) the random connected cubic planar multigraph with a marked 3-connected component K of size q, each edge of K being independently replaced by a cubic network drawn under the critical Boltzmann distribution on D, i.e., the probability distribution µ (D) (D denoting here the family of cubic networks allowing for double edges and loops) defined as where ρ is the radius of convergence of D(z). As before, M (q) n is M (q) conditioned to have size n. By a very similar analysis as in Section 3.3, when n, q are related by q = αn + O(n 2/3 ), the probability that M (q) has size n is Θ(n −2/3 ); and for > 0, a.a.s. every cubic network attached at the marked 3-connected component has size smaller than n 2/3+ . Moreover, since α > 1/2, for q = αn + O(n 2/3 ) and n large enough, M (q) n coincides with the random cubic connected planar multigraph of size n conditioned to have its 3-connected core of size q.
4 First results about the metric properties of random cubic planar graphs

Pole-distance in cubic networks under the critical Boltzmann distribution
For N a cubic network, the pole-distance of N is the distance between its two poles, it is denoted δ(N ). We use the notation D for the class of networks in both contexts (simple cubic networks, labeled at non-pole vertices, or cubic networks allowing for loops and multiedges, labeled at half-edges on plain edges). Let ν be the distribution of δ(N ), where N is drawn under µ (D) (critical Boltzmann distribution on D). Our aim here is to show that ν has an exponential tail, both in the case of graphs and multigraphs (the reason to show this is that ν will be the law of edge-lengths to be considered on the 3-connected core, and our subsequent analysis of modified distances in 3-connected cores assumes the law of edge-lengths to have exponential tails).

Lemma 12.
There exist constants a , b > 0 such that, for X drawn under ν , Proof. Similarly as in [11,Lem.5.6], the approach is to define another parameter χ on cubic networks that dominates δ and is easier to track along the network decomposition, and to show via generating functions that χ has an exponential tail. For N ∈ D, χ(N ) is defined as follows: • if N is of type M , with N 1 and N 2 the two networks substituted at each edge, then χ(N ) = χ(N 1 ) + χ(N 2 ) + 2, • if N has a 3-connected core K, and if N 1 , . . . , N k are the cubic networks substituted at the outer edges of K, then χ(N ) = 2 + χ(N 1 ) + · · · + χ(N k ). It is easy to check recursively along the decomposition that χ(N ) δ(N ) for any N ∈ D.
For each family F of networks, we let F (z, u) be the bivariate generating function (with z for the size and u for the parameter χ). We also write T (z, v) for the bivariate generating function of rooted simple triangulations, with v for the root degree minus 1. By Remark 1, T (z, v)/2 is the bivariate (exponential) generating function of 3-connected edge-rooted cubic planar graphs, with v conjugate to the number of outer edges minus 1. For the case of graphs, the network decomposition gives Hence, noting that S(z, u) where ).
We are now going to prove the property that there exists u 0 > 1 such that D(ρ, u 0 ) converges; this will prove the lemma, since for N drawn under µ (D) one has What we will obtain more precisely is that, in a complex neighborhood of (ρ, 1) (with ρ the radius of convergence of D(z)), we have a singular expansion for D(z, u) of the form where D(Z, U ) is analytic around (0, 0) and satisfies [Z 1 ] D(Z, U ) = 0, and [Z 2 ] D(Z, 0) = 0; and moreover we will check that ρ is the radius of convergence of z → D(z, u) for u ∈ R around 1.
To prove that, we first obtain a similar singular expansion for T (x, v). An explicit expression is given in [8]: where η ≡ η(x) is the algebraic series given by x = η 3 (1 − η), which has an expansion of the form where T (X, V ) is analytic around (0, 0) and satisfies [X 1 ] T (X, V ) = 0, and [X 2 ] T (X, 0) = 0. Next, we have (from [7,33] Taking Z = 0, this ensures that the univariate function u → D(ρ, u) is analytic at 1. Letting a n,m := [z n u m ]D(z, u), we have D(ρ, u) = n,m a n,m ρ n u m for u 1 (because n,m a n,m ρ n = D(ρ) < ∞). The analyticity of u → D(ρ, u) at u = 1 then ensures that the convergence of the above sum extends for u u 0 for some u 0 > 1. Hence, there exists u 0 > 1 such that D(ρ, u 0 ) converges.
The proof for the case of multigraphs is completely similar (the equation-system is the same, except for P (z, u) = 1 2 u 2 D(z, u) 2 ).

A first bound on the diameter
For later purpose, we need to establish a first bound on the diameter of random connected cubic planar graphs (resp. multigraphs), which is the analogue of the one for random planar graphs obtained in [11]. We say that a property A n, related to a random object of size n, and also formulated in terms of a real parameter > 0, holds a.a.s. with exponential rate if there are positive constants a, b, c such that P (not A n, ) a exp(−b n c ) for all n 0 and > 0 small enough. The diameter of a graph g, denoted Diam(g), is the maximum of the pairwise distances between vertices in g. The purpose of this section is to prove the following result: Proposition 13. Let C n (resp. M n ) be the random connected cubic planar graph (resp. multigraph) on 2n vertices. Then Diam(C n ) As a first step, we establish the bound in the 3-connected case (starting in the dual setting of random simple triangulations).

Lemma 14.
Let t n be the random rooted simple triangulation with n + 2 vertices. Then Diam(t n ) n 1/4+ a.a.s. with exponential rate.
Proof. Let T n be the set of rooted triangulations with a marked vertex (distinct from the root vertex), having n + 2 vertices. We use the fact (to be detailed in Section 5.2, and illustrated in Figure 4) that T n−1 injects into the set Q n,1 of so-called quasi-simple triangulations of the 1-gon having n inner vertices, in such a way that the underlying metric space is preserved; and that both |Q n,1 | and |T n | are Θ((256/27) n n −3/2 ). Let G(x) be the counting series of quasi-simple triangulations of the 1-gon with respect to the number of vertices minus 2. For i 1, let G (i) (x) be the counting series of quasi-simple triangulations of the 1-gon where the root vertex and the marked vertex are at distance i, counted with respect to the number of vertices minus 2. Let q (1) n be a uniformly random map in Q n,1 . Denoting by L n the distance between the root vertex and the marked vertex in q (1) n−1 , we have An explicit expression 4 for G (i) (x) is given in [3,Prop.5.3]. Precisely, there exists an (algebraic) counting series Y (x), with a quartic dominant singularity at 27/256, is an explicit rational expression that is continuous and positive at ( , where b is any positive constant smaller than 24 1/4 (e.g. b = 24 1/4 /2). Given the expression of uniformly over i n 1/4+ , and thus so that L n n 1/4+ a.a.s. with exponential rate. Let t n be the uniform random rooted simple triangulation with n + 2 vertices. LetL n be the distance between the root vertex and a random vertex (distinct from the root vertex) in t n . Note thatL n is distributed as L n conditioned on the event that the triangulation associated with q n−1 is simple (via the mapping described in Section 5.2). Since that event has probability Θ(1), we conclude thatL n n 1/4+ a.a.s. with exponential rate. Since the number of choices for a root and a marked vertex is O(n 2 ), the union-bound ensures that Diam(t n ) n 1/4+ a.a.s. with exponential rate.

Lemma 15.
Let K n be the random 3-connected cubic planar graph on 2n vertices. Then Diam(K n ) n 1/4+ a.a.s. with exponential rate.
Proof. Up to choosing one of the two embeddings, and choosing a root edge, Diam(K n ) is distributed as the diameter of the random rooted 3-connected cubic planar map with 2n vertices. Let m be a planar map, and let m † be its dual. Hence, Diam(K n ) is stochastically dominated by (1 + Diam(t n )) · maxDeg(t n ). Given Lemma 14, it remains to prove that maxDeg(t n ) n a.a.s. with exponential rate. Letting rootDeg(t n ) be the root degree in t n , we have (with T (x, v) defined in Section 4.1) On the other hand, a standard re-rooting argument ensures that ) for k n . Hence, maxDeg(t n ) n a.a.s. with exponential rate.
Let g be a (non-rooted) cubic connected planar multigraph, and let τ (g) be the associated decomposition-tree. We call R-attached network within g one of the cubic networks assembled around an R-cycle of components (node of label R in τ (g)). Note that such a component must be a non-trivial non-S-network. It corresponds to an edge e with a R extremity in τ (g), precisely it is the network associated to the subtree hanging from the non-R side of e. We let δ R (g) be the maximal value of δ(N ) over all R-attached networks within g. And we let ξ R (g) be the maximal degree over all R-nodes of τ (g). Similarly, we define a T -attached network as a network substituted at one of the edges of a T -component of g. It corresponds to an edge e with a T extremity in τ (g), precisely it is the network associated to the subtree hanging from the other side of e. We let δ T (g) be the maximal value of δ(N ) over all T -attached networks within g. And we let ξ T (g) be the maximal diameter over all 3-connected components of g.
Similarly as in [11,Eq.(6)], the following bound on the diameter of g holds: Proof. A diametral path γ in g induces a path γ in τ (g). Some edges on γ correspond to edges of γ (where γ leaves a component and enters another component). The other edges of γ are "consumed" within the nodes on γ . For a T -node w visited by γ, corresponding to a 3-connected component K, the length of γ consumed within w is bounded by where att w is the set of T -attached networks at K. Similarly, for an R-node w, of degree k, visited by γ, the length of γ consumed within w is bounded by k · max N ∈attw δ(N ), where att w is the set of R-attached networks at w. Finally, for an M -node w, at most one edge of γ is consumed by w (this happens if one of the three edges of the M -component for w is not substituted, and this edge is traversed by γ).
In view of Claim 16, in order to conclude the proof of Proposition 13, it just remains to show the following.

Lemma 17.
For C n the random connected cubic planar graph with 2n vertices, we have a.a.s. with exponential rate.
The same holds when replacing C n by M n .
Proof. We give the proof details for graphs (the arguments for multigraphs are completely similar). We start with ξ T (C n ). Consider a 3-connected component K in C n , and let n n be its size. Then K is uniformly distributed over cubic 3-connected planar graphs with 2n vertices. If n n 1/4 /2, then obviously Diam(K) n 1/4+ . If n ∈ [n 1/4 /2, n], then by Lemma 15 there are positive constants a, b, c such that, for > 0 small enough, we have (where Lemma 15 yields the middle inequality): Hence, uniformly over n ∈ [0, n], the diameter of a random 3-connected cubic planar graph of size n is smaller than n 1/4+ a.a.s. with exponential rate. Since the number of 3-connected components in C n is O(n), by the union-bound we conclude that ξ T (C n ) n 1/4+ a.a.s. with exponential rate.
We now deal with ξ R (C n ). LetS(z) := L(z) + P (z) + H(z) be the generating function for cubic networks that are eligible to be R-attached (i.e., non-trivial non-R cubic networks). From the 1st and 3rd line in (8) (or directly, from the decomposition of S-networks), we get S(z) =S 2 1−S(z)) . Since S(z) converges at ρ, we must haveS(ρ) ∈ (0, 1). For k 3, the generating function of connected cubic planar graphs with a marked R-component of degree k is equal toS(z) k /k. Hence, for k 3, the number of connected cubic planar graphs of size n having a marked R-component of degree k is bounded by (2n)![z n ]S(z) k , which itself is bounded by (2n)!β k ρ −n , where β =S(ρ) ∈ (0, 1). The number of connected cubic planar graphs of size n having a marked R-component of degree larger than k is thus bounded by (2n)! β k 1−β ρ −n . Since the number of R-component is O(n), the number of connected cubic planar graphs of size n having at least one R-component of degree larger than k is O((2n)!n β k 1−β ρ −n ) On the other hand, the number of connected cubic planar graphs of size n is Θ((2n)!ρ −n n −7/2 ). Hence, the probability that C n has an R-component of degree larger than k is O(n 9/2 β k ), which is O(β n /2 ) for k n , ensuring that ξ R (C n ) n a.a.s. with exponential rate.
We now consider the parameter δ T (C n ). Again, we use the fact that a T -attached network of size n in C n is uniformly distributed over cubic networks of size n . This distribution is also µ (D) conditioned to have size n , an event that occurs with probability Θ(n −5/2 ) (as follows from the asymptotic form of the counting coefficients for cubic networks). Hence, letting D n be the random cubic network of size n , Lemma 12 ensures that there are positive constants a , b such that P(δ(D n ) > k) a n 5/2 exp(−b k).
Hence, δ(D n ) n a.a.s. with exponential rate, uniformly over n n. Since the number of T -attached networks in C n is O(n), the union bound ensures that δ T (C n ) n a.a.s. with exponential rate. The same argument is easily adapted to δ R (C n ) as well (an R-attached network of size n is a network drawn under µ (D) conditioned to have size n and to be a non-trival non-S-network, an event that holds with probability Θ(n −5/2 )).
We now consider Diam(τ (C n )). We first rewrite the system (8) in a positive form, where t n is the number of rooted simple triangulations with 2n faces.
This system, of the form y = F(z, y), is irreducible. It is also critical in the sense of [11,Sec.5], as the quantities L (z), S (z), P (z), H (z) converge when z ∈ (0, ρ) tend to ρ.
, H h (z) be the counting series gathering the contributions of L(z), S(z), P (z), H(z) where the associated decomposition-tree has height at most h, and let H h (z)). We clearly have y h+1 = F(z, y h ) for h 0. Thus, the height of the decomposition-tree is a height-parameter for the system, with the terminology of [11,Sec.5]. We can then rely on [11,Lem.5.3.], which ensures that for a random cubic network in each type ∈ {L, S, P, H}, the height of the decomposition-tree is bounded by n a.a.s. with exponential rate. Since the diameter of a rooted tree is at most twice its height, we easily conclude (via the decomposition (9) for pointed cubic connected planar graphs in terms of cubic networks) that Diam(τ (C n )) n a.a.s. with exponential rate.

Skeleton decomposition for quasi-simple triangulations
We introduce in this section a modification of simple triangulations, which will prove to be more convenient to deal with, when studying Krikun's skeleton decomposition. We will indeed prove in Sections 5.4 and 5.5 that the skeleton of these so-called quasi-simple triangulations admit exactly the same encoding by a branching process as general triangulations.

Simple triangulations
For every n 1, we write T n be the set of rooted simple triangulations with n + 2 vertices. Then, for every p 3 and n 0, let T n,p be the set of rooted simple triangulations of the p-gon with n inner vertices (that is, simple triangulations whose root face is a simple cycle of degree p, with n vertices that are not incident to the root face). Observe in particular that T n = T n−1,3 . Enumerative formulas for T n,p have been obtained by Brown in [8] and read: We deduce from this expression and from Stirling's formula that: with Because of the polynomial correction in n −5/2 (usual for planar maps), the generating series T p (x) := n 0 |T n,p |x n evaluated at x = 27/256 converge for every value of p. Following [8, (4.5)], we get: From which we deduce: The critical Boltzmann distribution on simple triangulations of the p-gon is the one assigning probability (27/256) n /Z(p) to the elements in T n,p , for n 0.

Quasi-simple triangulations: definition and enumeration
Definition 18. For p 1, let q be a triangulation of the p-gon with a marked inner vertex v . We call (q, v ) a quasi-simple triangulation if all its 1-cycles and 2-cycles separate the p-gon and the pointed vertex, see Figure 3.
Remark 4. Observe that in a quasi-simple triangulation, each vertex can carry at most one loop, otherwise the two loops would form a 2-cycle that does not contain the pointed vertex.
Therefore, we can turn a quasi-simple triangulation of the 1-gon into a triangulation of the sphere by merging the two edges that form a triangular face with the root edge, rooting the The asymptotic behavior of |Q n,p | as n goes to infinity can be deduced directly from this expression. It is given in [8, (9.1)] and reads: with Note the unusual polynomial correction in n −3/2 , which comes from the fact that quasi-simple triangulations are inherently rooted and pointed. We hence retrieve the classical behavior of (non-pointed) planar maps with a polynomial correction in n −5/2 . Following the same approach as in the proof of [13, Lemma 1], we obtain from (28) the following bounds for |Q n,p |: There exists a constant c > 0 (independent of p and n) such that, for every n 1 and p 1, we have:

Decomposition of quasi-simple triangulations of the 1-gon into simple components
Let Q (1) be the family of quasi-simple triangulations of the 1-gon, where the size-parameter is the number of inner vertices. Let T , T − , T • − , T •−• be the families of rooted simple triangulations with respectively a marked inner vertex, a marked inner edge, and a marked inner edge incident (resp. not incident) to the root vertex, where the size-parameter is the number of vertices minus 2 (which is also half the number of faces). We use a decomposition of Q (1) described in [3,Sec.3.2] (decomposition along the nested sequence of 1-cycles and 2-cycles) that yields Q (1) = X · U, where the families U, V admit the following decomposition-grammar 5 : with X an atom that accounts for 1 in the size. This system can easily be turned into a regular expression for U = Q (1) involving only the "terminal families" T , T − , T • − , T •−• (as well as the atom-family X ). Note that the second line gives V = T + 2 · X + 2 · X · U · Seq(T − ).
Injecting it into the first line, we obtain where Proof. We rely on classical results of analytic combinatorics regarding the size of the largest component in combinatorial decompositions [19]. For α ∈ R\N, a combinatorial class C is said to be of singularity-type α if the radius of convergence ρ of the counting series C(z) = n |C n |z n is positive, and C(z) can be analytically continued to a domain {|z| ρ + η, z − ρ / ∈ R + } (for some η > 0), where it admits a singular expansion around ρ of the form for some polynomial P (z) of degree smaller than α, and some non-zero constant d. Transfer theorems of analytic combinatorics [16,VI.3] then guarantee that |C n | ∼ d Γ(−α) ρ −n n −α−1 . The following dictionary can then be obtained regarding the size of the greatest component (we also include a sum-rule to track which families along the decomposition have asymptotically non-zero probability of having the largest component): Sum rule. Let C = A+B, where A, B have the same radius of convergence ρ, and have respective singularity types α, α / ∈ N, both positive. Let C n be the random structure in C n . Then C n is almost surely in A (resp. B) if α < α (resp. α < α).
Product rule. Let C = A · B, where A, B have the same radius of convergence ρ, and have respective singularity types α, α / ∈ N, both positive. Let C n be the random structure in C n . Let X n be the size of its largest component (either in A or in B, in A if there is a tie). Then n − X n converges in law. In addition, the largest component is a.a.s. in A (resp. in B) if α < α (resp. α < α).
Sequence rule. Let C = Seq(A) (with no object of size 0 in A), where C, A have same radius of convergence and same positive singularity type α / ∈ N (non-critical case in [19]). Let C n be the random structure in C n , and let X n be the size of the largest component (in A, if there is a tie the leftmost component of largest size is chosen). Then n − X n converges in law.
We can then use these rules repeatedly in the regular expression (31) to prove the statement, using also the fact [3] that the counting series of T , T − , T • − , T •−• all have radius of convergence 27/256, with singularity type 1/2 for T , T − , T •−• , and singularity type 3/2 for T • − (the fact that the largest component is a.a.s. not in T • − is due to the larger singularity type for this family).
We will also need the following statement, ensuring that, for the decomposition (31), the largest simple component of q Proof. The statement for T n (resp. T − n ) is obvious, since the number of allowed markings on an object in T n is n − 1 (resp. 3n − 3). For T •−• n , the number of allowed markings of an object in T n , with root degree d, is 3n − 1 − d. As we have seen in the proof of Lemma 15, the root degree of the random triangulation in T n is a.a.s. at most n for any > 0 (even more is true, it converges in law), hence the number of allowed markings is concentrated around 3n, ensuring that the distribution induced by T •−• n on T n is at total-variation distance o(1) from the uniform distribution on T n .

Quasi-simple triangulations of the cylinder and their skeleton decomposition
In this section, we describe the structure of the neighborhood of the root in a quasi-simple triangulation, and how it can be encoded via the so-called skeleton decomposition introduced by Krikun in [23]. We follow here the presentation given by Curien and Le Gall in [13, Section 2.2] for general triangulations, and we emphasize the necessary adjustments to deal with quasi-simple triangulations. We first introduce quasi-simple triangulations of the cylinder, which will appear naturally as the hulls of quasi-simple triangulations.

Definition 22.
Fix r ∈ Z >0 . A quasi-simple triangulation of the cylinder of height r is a rooted planar map such that:

All its faces are triangles except for two distinguished faces: its root face (also called the bottom face) and the top face. 2. The boundaries of the bottom and of the top face are disjoint simple cycles. 3. Every vertex incident to the top face is at graph distance exactly r from the boundary of the bottom face, and every edge incident to the top face is also incident to a triangle whose third vertex is at distance r − 1 from the bottom face. 4. Cycles of length 1 and 2 necessarily separate the bottom face and the top face. If ∆ is a quasi-simple triangulation of the cylinder of height r, the boundary of its root face (bottom cycle) is denoted by ∂∆ and its top cycle (boundary of the top face) is denoted by ∂ * ∆.
We call ∆ a quasi-simple triangulation of the (|∂∆|, |∂ * ∆|)-cylinder.
Moreover, we denote by Z p,r the set of quasi-simple triangulations of the cylinder of height r with a root face of degree p.
Note that this is the same definition as [13, Definition 1], except for item 4. To describe the encoding of a triangulation of the cylinder via the skeleton decomposition, we need additional definitions.

Definition 23.
Let ∆ be a fixed quasi-simple triangulation of the cylinder of height r. For 1 j r, the ball of radius j of ∆ -denoted B j (∆) -is the submap of ∆, which is spanned by its faces which are incident to at least one vertex at distance strictly smaller than j from ∂∆.
Moreover, the hull of radius j of ∆ -denoted B • j (∆) -is defined as the reunion of B j (∆) with the connected components of ∆\B j (∆) that do not contain ∂ * ∆.
Note that for any j ∈ {1, . . . , r}, B • j (∆) is itself a quasi-triangulation of the cylinder (of height j), and we set ∂ j (∆) := ∂ * (B • j (∆)). We extend this notation to j = 0 and set ∂ 0 (∆) := ∂∆. Every edge of ∂ j (∆) is incident to exactly one triangle whose third vertex belongs to ∂ j−1 (∆). Such triangles are called the downward triangles at height j, and we write F j (∆) for the set of all downward triangles at height j. LetĒ(∆) be the collection of all edges that belong to one of the cycles ∂ j (∆), for 0 j r. The planar embedding of ∆ enables to encode the edges ofĒ(∆) as the vertices of a forest f, with q trees where q := |∂ |. This encoding is depicted in Figure 5 and we refer to [13, page 9] for its formal definition.  Given the forest f := (τ 1 , . . . , τ q ), to reconstruct the full quasi-triangulation ∆, we need to fill the "slots" that lie between successive downward triangles. More precisely, for e ∈ ∂ j (∆) with 1 j r, we associate to e a (possibly empty) slot in the following way: the slot is included in , and is bounded by the edges of ∂ j−1 (∆) that correspond to children of e and by the two "vertical" edges that connect the initial vertex of e (when ∂ j (∆) is oriented so that B • j (∆) lies on its left) with vertices of ∂ j−1 (∆). The definition should be clear from Figure 5. Writing k f (e) for the number of children of e in the forest f, the slot associated to e has to be filled with a simple triangulation of the k f (e) + 2-gon. Note that, when k f (e) = 0, the slot is filled by the edge-triangulation, or in other words the two vertical edges defining the slot are identified (whereas in the setting of [13] of general triangulations, non-trivial slots with a boundary of length 2 may occur).
To characterize the forests that can appear via the encoding, we need the following definition:

Definition 24. A forest f with a marked vertex is said to be (p, q, r)-admissible if:
1. The forest f := (τ 1 , . . . , τ q ) is made of q rooted plane trees, 2. The maximal height of these trees is r.
3. The total number of vertices of the forest at generation r is p, 4. The distinguished vertex has height r, 5. The distinguished vertex is in τ 1 . We write f for the set of vertices of f with height strictly smaller than r.
Moreover, the set of (p, q, r)-admissible forests is denoted F p,q,r and we define F p,r := ∪ q 1 F p,q,r . Proposition 25. The skeleton decomposition described above is a bijection between, quasi-simple triangulations ∆ of the (p, q)-cylinder of height r, and pairs (f, (t v ) v∈f ), where f is a (p, q, r)admissible forest and for any v ∈ f , with e ∈Ē(∆) the corresponding edge, t v is a simple triangulation of the (k f (e) + 2)-gon.
We write (F(∆), (t v ) v∈F(∆) ) for the image of ∆ via this bijection.  Proof. This proposition is very similar to the skeleton decomposition introduced originally by Krikun in [23]. However, in this paper, he dealt with loopless triangulations rather than quasisimple triangulations. In other words, loops were always forbidden (whereas they are allowed in our setting if they separate ∂∆ and ∂∆ ) and multiple edges were always authorized (whereas we allow them only if they separate ∂∆ and ∂∆ ). This is the only differences between his decomposition and ours. Hence, we only need to prove that we indeed obtain a quasi-simple triangulation. In other words, we only need to check that 1-cycle and 2-cycle separate ∂∆ and ∂∆ . One can argue via the universal cover of the cylinder, which is the periodic plane (R/Z) × R. Indeed, the constraint of being quasi-simple on the cylinder is equivalent to being simple in the universal cover representation. Thus, one just has to check that, on (R/Z) × R, the obtained (periodic) triangulation is simple. Since, in the universal cover representation, each boundary ∂ j (∆) can conveniently be drawn as the horizontal line {y = j}, with the next added layer in the horizontal band {j y j + 1}, the result follows directly.
For sake of illustration, some configurations in which a loop or a pair of multiple edges can appear are represented on Figure 6 (note that it is only possible to have a loop or a pair of edges if F(∆) has only one or two vertices at some generation j with 0 < j < r).

Hulls and encoding via a branching process
In this section, we show how the probability to have a fixed neighborhood of the root can be expressed in terms of a branching process via the skeleton decomposition.

Definition 26.
Let (q, v * ) be a quasi-simple triangulation of the p-gon. For R 1, the ball of radius R of q -denoted B R (q) -is the submap of q, which is spanned by its faces which are incident to at least one vertex at distance strictly smaller than R from its root face, see Figure 7.
Moreover, the hull of radius R of (q is the submap of q, that consists of the reunion of B R (q) with the connected components of q\B R (q) that do not contain v .
It follows directly from the definition of the hull, that B • R (q, v ) is either a quasi-simple triangulation (with a marked vertex) if B • R (q, v ) = (q, v ), or a quasi-simple triangulation of the cylinder ∆, with |∂∆| = p, otherwise.

Proposition 27. Let p, q, r be positive integers. Let ∆ be a quasi-simple triangulation of the (p, q)-cylinder with height r. Then, for q (p)
n a uniformly random element of Q n,p , we have: where κ and θ are defined by: Remark 5. It follows from the definition of θ that: Hence our definition of θ matches the definition given in [23, (22)] and in [13, (15)]. This is not surprising and can be explained via classical substitution relations between general, loopless and simple triangulations.
Note that Proposition 27 is fully analogous to [13, Lemma 2], except that here Z and ρ denote respectively the generating series and the radius of convergence associated to simple triangulations, whereas in [13], they correspond to general

Proof.
We set ρ = 27/256 to be the radius of convergence of the generating series of quasi-simple triangulations and β = 9/64 to be inverse of the growth rate of Z(p) defined in (26). We denote by |∆| the total number of vertices of ∆. Recall that Q n,p is the set of quasi-simple triangulations of the p-gon with n inner vertices. Then: where the asymptotic equivalent is obtained from (29).
To lighten notation, in this proof we set f := F(∆), and recall that f * is the set of its vertices with height strictly smaller than r. According to the skeleton decomposition, the vertices of ∆ can be partitioned into two sets. They either belong to one of the cycles δ j ∆, for 0 j r, or they are inner vertices of the simple triangulations that are inserted in the slots. Hence |∆| = |f | + v∈f Inn(S v ). Since |f | = q + v∈f k v , we get from (34) that: Following similar computations performed in [23] and [13], and in order to interpret this quantity in terms of a branching process, we perform a change of variables.
we can multiply the former expression by (ρ/β) p−q− v∈f (kv−1) to get: Now, informally, we would like to decompose each term of the product into two contributions. On the one hand, a term representing the probability for a vertex to have k children in the skeleton decomposition and, on the other hand, a term for the probability (under the critical Boltzmann distribution) that a given simple triangulation of the (k + 2)-gon appears in the corresponding slot. To do so, we set θ(k) := ρβ k−1 Z(k + 2), which yields: In view of the definition of β and ρ and the expression of Z(k) given in (26), this concludes the proof.

Quasi-simple UIPT
In exactly the same manner as in Section 2.4 of [13], we can define a probability measure P p,r on F p,r by setting for every forest f ∈ F p,q,r : We then define a probability measure P p,r on the set Z p,r , by saying that under P p,r the law of the skeleton on the triangulation is given by P p,r and, conditionally given the skeleton, the triangulations filling the slots are independent critical Boltzmann simple triangulations with a simple boundary (whose boundary lengths are determined by the skeleton). Proposition 27 can then be restated as: for any ∆ ∈ Z p,r . It follows from the preceding convergence, that we can define a random infinite quasi-simple triangulation of the plane with a simple boundary of length p -denoted q ∞ ) is given by P p,r , for every integer r 1. The map q (p) ∞ is called the uniform infinite quasi simple triangulation of the p-gon and it follows directly from (36) that q (p) ∞ is the local limit of q (p) n as n → ∞. By the correspondence Ψ (illustrated in Figure 4) between quasi-simple triangulations of the 1-gon and quasi-simple triangulations of the sphere, from q (1) ∞ we can define the so-called quasi-simple triangulation of the sphere, denoted q ∞ .

Half-plane simple models
Recall from Remark 5 that the branching process encoding the skeleton decomposition of quasisimple triangulations has exactly the same offspring distribution as the one obtained in [13] for general triangulations.
In this work, two half-plane models of random geometry are introduced, the upper half-plane triangulation and the lower half-plane triangulation and we refer to [13,Section 3] for their construction. This allows us to define readily the upper half-plane simple triangulation U s and the lower half-plane simple triangulation L s , by only requiring that in their construction, the slots are filled with simple triangulations rather than general triangulations.
All results stated in Sections 3 and 4 of [13] can be extended verbatim to our setting. Indeed, in most of the proofs they do not need to know how the slots are filled in, and only study the distribution of the skeleton, which is the same in our setting and in [13]; the only part that we need to change is in [13,Lemma 10], where (to reflect the different constants in the enumeration of simple triangulations) ρ = 12 √ 3 should be replaced by ρ = 256/27 and α = 12 by α = 64/9. Notice that triangulations of the half-plane that we obtain are simple triangulations and not quasi-simple triangulations, as can be directly observed from the construction of U s and L s .

Coupling between simple and general models
It follows from the preceding section that q (1) ∞ and L s can be constructed via exactly the same branching process as the (general) UIPT of the 1-gon and L respectively. The only modification between these pairs of models is that the slots are filled in the first case by simple critical Boltzmann triangulations with a simple boundary and in the second case by general critical Boltzmann triangulations with a simple boundary.
The simple core of a critical Boltzmann general triangulation with a simple boundary of length 3 (obtained by collapsing all 2-cycles into edges) is itself a critical Boltzmann simple triangulation with a boundary. So that, by taking the simple core of each triangulation grafted in the slots, this induces a coupling between the (general) UIPT of the 1-gon and q (1) ∞ on one hand and between L and L s on the other hand. This remark will be useful to derive bounds in our model directly from the bounds obtained in [13].

Modification of distances in simple triangulations
The goal of this section is to give the proof of Theorem 2. To do that, we follow closely the work of Curien and Le Gall [13] and rely heavily on the skeleton decomposition of quasi-simple triangulations presented in the previous section.
Throughout this section, we assume that ν is a probability distribution with support included in [η 0 , ∞) for a given η 0 > 0, and that there exist constants A, λ > 0 such that, for X a random variable with law ν: Moreover, we assume that any planar map m (random or not) considered in this section is equipped with a collection (w e ) e∈E(m † ) of random weights, such that (w e ) are i.i.d random variables sampled from ν. Recall from the introduction that we defined the first-passage percolation distance d on m † by setting: where the infimum runs over all paths γ going from u to v in m † . For the sake of simplicity, we write d without emphasizing its dependency in ν, it should not bear any confusion.
Since the skeleton decomposition of quasi-simple triangulations gives the exact same branching process as the skeleton decomposition of general triangulations, many results of [13] can be applied verbatim to our setting. When this is the case, we do not duplicate the proofs. Rather, we only state the results for which the proofs need to be adapted. Note also that in [13], the authors focus first on the case of modifications of distances in triangulations (and not on their dual). They then treat the dual case without giving as many details as for the primal case. Remark 6. Theorem 2 and its proof could be extended verbatim to deal with modification of distances on simple triangulations rather than on their dual as was initially done in [13]. However, for sake of conciseness, we only give the proof of this result in the dual setting. . . . . . . . . . . . . . .

Upper bounds in the infinite models
The purpose of this section is to derive statements similar to Lemma 24 and Proposition 25 of [13] for the lower half-plane simple triangulation L s . For every j 0, let L s j for the vertices of L s that are the vertices of depth −j of its skeleton; or rather, L s j is the set (forming an infinite simple path) of all edges of L s that connect these vertices -it is the analogue of ∂ r+j (∆) when ∆ is a quasi-simple triangulation of the cylinder of height r. We also denote by L s, † j the set of all downward triangles incident to an edge of L s j , and by f 0,0 the face in L s, † 0 that is incident to the root edge of L s .
For every downward triangle f in L s, † j , the downward path from f is an infinite path in the dual map, defined as follows (see [13,Section 7.2] for a precise definition): starting from f , and if v is the vertex of f that is not an endpoint of an edge of L s j , turn around v in counterclockwise order until crossing an edge of L s j−1 . The face we reach is itself in L s, † j−1 : we continue inductively with the downward path starting from this latter face.

Lemma 28.
For r 1, let γ r be the downward path in L s connecting f 0,0 to a downward triangle incident to L s −r , and write |γ r | for the length of γ r . There exist two constants µ > 0 and K < ∞ such that for every integer r 1: Proof. The result follows from Lemma 24 of [13], and from the coupling described in Section 5.6.3, under which the length of the downward path can only decrease.
For every j 0, let L s, † j be the set of downward triangles incident to an edge of L s j in the lower half-plane model. We assume that L s is equipped with the f.p.p distance d defined above (the dual map is infinite here, its vertices corresponding to the triangular faces of L s , and its edges corresponding to the edges in L s with a triangular face on each side). Then, Proposition 25 of [13] can be readily adapted into the following result: [η 0 , ∞). There exists a constant c ν 2η 0 such that:

Proposition 29. Recall that the support of ν is included in
As in the original article, the result follows directly from a subadditive argument together with the fact that E[d (f 0,0 , L s, † −r )] < ∞ by Lemma 28 and d (f 0,0 , L s, † −r ) 2η 0 r by construction.
Thanks to Lemma 28, the proof of Lemma 26 and Corollary 27 of [13] can directly be extended to our setting. We obtain:

Lemma 30. Consider the quasi-simple UIPT of the 1-gon q
(1) ∞ , with i.i.d weights on its edges sampled from ν. There exist positive constants K , α 0 , β 0 such that, for any integer 0 r < s: where f s is a downward triangle at height s chosen uniformly at random, and where B • 0 (q (1)

∞ ) should be interpreted as the bottom face.
Moreover, for any fixed δ > 0, define A R (δ) to be the event where the bound holds for every 0 r < s R with s − r δR, and for every downward triangle f at height s. Then, there exists another positive constantβ such that for any sufficiently large R,

Upper bound in finite simple triangulations
We finally turn our attention to finite triangulations. The purpose of this section is to obtain a result equivalent to [13,Lemma 28] but for uniform simple triangulations rather than uniform general triangulations and for the d distance rather than the graph distance or the Eden model. For x ∈ V (t n ) and f ∈ F (t n ), write x f to mean that the vertex x is incident to the face f . Then we have: Proposition 31. Let α 0 be as in Lemma 30. Let t n be a uniform rooted simple triangulation with n + 2 vertices. For ε ∈ (0, 1/4), define E n as the event where the bound holds for every x, y ∈ V (t n ) and f, g ∈ F (t n ) such that x f and y g. Then: The same result holds when replacing t n by q n is a uniform quasi-simple triangulation of the 1-gon with n inner vertices.
Proof. The main difficulty that arises in our setting and does not exist in [13] is that we have to derive results for simple triangulations, whereas we established (in particular in the previous section) results about the local limit of quasi-simple triangulations. Fortunately, these two models are similar enough to obtain the results we need.
In this proof, we always consider that the random planar maps are endowed with random weights on their edges, sampled independently under a law ν with support included in [η 0 , ∞) for some η 0 > 0, and with exponential tails. Let q (1) n be a uniform quasi-simple triangulation of the 1-gon with n inner vertices. Following the proof of [13,Lemma 28], we write ρ n for its root vertex, o n for its pointed vertex and set d n := d gr (ρ n , o n ). Then, we write Θ for the set of quasi-simple triangulations t that belong to Z 1,r for some r > 1 and are such that there exists a face f incident to ∂ * t whose d from the bottom face is greater than αr. By the exact same chain of arguments as in the proof of [13,Lemma 28] that leads to their equation (73), we obtain that there exist two constants c > 0 and a > 0, such that: By, again, the same line of arguments as in [13] and given the definition of Θ, it is then easy to see that on the event {d n > n ; B • dn−1 (q , for any f, g ∈ q (1) n , with ρ n f and o n g (where we denote by maxDeg the maximum vertex degree of a map, and by w * the maximal dual weight of an edge of t n ).
The original proof continues with a rerooting argument, which cannot apply here since quasi-simple triangulation are not invariant under rerooting. However, we can transfer the bound obtained in (43) from quasi-simple triangulations to simple triangulations (up to changing the value of the constant c) in the following way. The asymptotic enumerative results for simple triangulations and quasi-simple triangulations of the 1-gon given respectively in (24) and (29) imply that there exists κ > 0, such that: where we recall that T n denotes the set of rooted simple triangulations with n + 2 vertices and a marked inner vertex. Moreover, conditionally on the fact that Ψ(q (1) n ) ∈ T n−1 , it is clear from the construction that q (1) n has the same distribution as a uniformly random element of T n−1 (as a random measured metric space) 6 .
Let t n be uniformly random in T n . Writeρ n for its root vertex,õ n for its pointed vertex and d n for d gr (ρ n ,õ n ). It follows from the discussion above and from the discussion following (43) that there exist two constantsc > 0 andã > 0 such that: P d n > n ; d † gr (f, g) > α 0dn + 3w · maxDeg(t n ) wheneverρ n f andõ n g c exp(−ãn ).
(45) Now, by rerooting t n at another oriented edge e n chosen uniformly and independently from o n , we obtain a pointed and rooted simple triangulation with the same distribution. Hence, the bound obtained above remains valid upon replacingρ n by ρ n , where ρ n is the initial vertex of e n andd n by d gr (õ n , ρ n ).
We can then proceed as in the original proof to establish that: Hence, asymptotically almost surely, the following bound holds: whenever u, v ∈ V (t n ), d gr (u, v) > n and u f and v g.
Since ν is assumed to have exponential tails, w can be bounded by n outside a set of probability tending to 0 as n → ∞. Besides, the geometric bound obtained for the degree of the root vertex in [4,Lemma 4.1] gives the existence of a constant A > 0 for which maxDeg(t n ) A log n with probability tending to 1 as n → ∞. This gives the desired result, except for the requirement that d gr (u, v) > n . This case can be dealt with by hand using the fact that d (f, g) w · maxDeg(t n )(d gr (u, v) + 1), which concludes the proof for simple triangulations.
To finish the proof for quasi-simple triangulations, we rely on Lemmas 20 and 21. Fix > 0 and recall that we write T (q) for the largest simple component of a quasi-simple triangulation q. By Lemma 20, there exists K ∈ N such that: We fix k ∈ [0, K] and |T (q n ) with a uniform simple triangulation t n−k of size n − k such that P(T (q (1) n ) = t n−k ) < . We extend this coupling to the model with weights on the edges and make sure that the weights also coincide when the triangulations are equal.
By applying the result of the proposition for t n−k , we know that outside a set of probability smaller than , the bound (41) holds. This gives the result when u, v ∈ T (q (1) n ), we can use the easy fact that there exists u ∈ T (q (1) n ) with d gr (u, w) k, which concludes the proof of the proposition.

Two-point function
Thanks to the bounds we obtained in the two previous sections, we are now ready to prove the equivalent of Proposition 21 of [13], stated below as Proposition 34. Roughly speaking, this proposition establishes that the difference between the 2-point function for the graph distance and the d 2-point function is o(n 1/4 ).
The proof of Proposition 34 relies on the following similar result obtained for the infinite quasi-simple triangulation. Given the upper bounds established in Lemma 30, we can apply verbatim the proof presented in Proposition 19 (in the primal case) and in Proposition 29 (in the dual case) of [13] (see the remark at the end of Section 5.6.2) to readily obtain: Proposition 32. Let > 0 and δ > 0. We can find η ∈ (0, 1/2) such that, for every sufficiently large n, we have: where we recall that F n (q To transfer this result to finite simple triangulations, we rely on the following absolute continuity relation between finite and infinite quasi-simple triangulations (analogous to [13,Lemma 22]). Recall that Z 1,r denotes the set of quasi-simple triangulation of the cylinder of height r and rooted on a loop. For ∆ ∈ Z 1,r , we denote by |∆| the number of inner vertices of ∆. Recall that q (1) n denotes the random quasi-simple triangulation of the 1-gon with n inner vertices. Then, we have: There exists a constantc, such that for every n 1, for every r 1 and every ∆ ∈ Z 1,r such that n > |∆|, Proof. The result follows directly from the uniform upper bound obtained in Lemma 19, from a similar lower bound on |Q n,1 | that we deduce from (28), and from the limit of (34) together with the fact that q (1) ∞ is the local limit of q (1) n as n → ∞.
We can now state the main result of this section: (52) The same result holds also if we replace q (1) n by a random rooted simple triangulation having n + 2 vertices and a random marked vertex o n , and f * by its root face.
Proof. Again, we follow closely the proof of [13,Proposition 21]. Only two modifications must be made. First, if we follow verbatim their proof, then at the very end, we need to adapt the upper bound they obtain. Indeed, at this point they use the fact that the first-passage distance on the primal graph (i.e on the triangulation) is upper-bounded by the graph distance on the primal. In particular, they fix three close values α j < β j < γ j and restrict their attention to the event {β j n 1/4 d gr (ρ n , o n ) γ j n 1/4 }. Then, they can give a deterministic upper bound for the first-passage percolation distance between o n and the hull of radius α j n 1/4 . This last step does not apply directly anymore when working on the dual graph, but can be made valid if we replace the deterministic bound by the probabilistic bound obtained in Proposition 31.
Secondly, write B r (q (1) n , o n ) for the ball of radius r centered at o n in q (1) n . We have to establish the following analogous statement of [13, (62)], and prove that for any y < 1, there exists b ∈ (0, 1) such that: Given the convergence of simple triangulations to the Brownian sphere in the Gromov-Hausdorff-Prokhorov topology [1], this statement is known to be true for uniform simple triangulations. It can then be extended to quasi-simple triangulations, upon extracting their largest simple component and by applying Lemmas 20 and 21. This concludes the proof for quasi-simple triangulations.
To extend the result to simple triangulations, we rely on the same line of arguments as in the proof of Proposition 31, and particularly on (44). It states that a positive proportion of quasi-simple triangulations are in fact simple triangulations (after applying the mapping Ψ, as depicted in Figure 4) Moreover, when Ψ(q (1) n ) is a simple triangulation, it yields a direct coupling between a random inner vertex of q (1) n and of Ψ(q (1) n ). Since we have just shown that (52) holds for quasi-simple triangulations, and since Ψ preserves the underlying metric space, it has to hold for simple triangulations as well.

Coupling between µ n and µ † n and proof of Theorem 2
Theorem 35. Let t n be uniformly random in T n . Consider the two following weighted metric spaces T n = (t n , d gr , µ n ) and K n = (t † n , d , µ † n ), where µ n and µ † n denote respectively the uniform measures on V (t n ) and on V (t † n ). Then, there exists an explicit correspondence C n between (t n , d gr ) and (t † n , d ) and an explicit coupling π n between µ n and µ † n , such that: Before giving the proof of this theorem, notice that in view of the known convergence of simple triangulations towards the Brownian sphere established in [1], it concludes the proof of Theorem 2.
Proof. The correspondence C n is defined as the set of pairs: Next, the coupling we construct between µ n and µ † n is based on the theory of Schnyder woods for simple triangulations [35], that we now briefly recall. A simple triangulation t can be canonically endowed by an orientation of its edges, such that for each v ∈ V (t), writing out(v) for the outdegree of v, we have: Note that to ensure that this orientation is canonical, it is enough to require that directed counterclockwise cycles are not allowed, but this is not relevant for our purpose. However, this canonical orientation enables to construct an explicit coupling between µ n and µ † n . Letting o n be a random vertex of t n , we associate to o n a uniform face of t n in the following way. If o n is not incident to the root face of t n , then let e n be an edge chosen uniformly among one of the three outgoing edges from o n . With probability 1/2, we pick f n to be the edge on the left of e n , and otherwise f n is the edge on the right of e n . Now, assume o n is incident to the root face of t n . Then, with probability 1/3, we let e n to be the unique outgoing edge from o n , and with probability 2/3, we let e n to be a uniform edge of t n . In both cases, we proceed as above.
It is clear from the construction, that e n is a uniform edge of t n . Since each face has degree 3, f n is then a uniform face of t n . Moreover, apart on an event of probability at most 2/(n + 2) (corresponding to the case where o n is incident to the root face and e n is sampled uniformly in E(t n )), we have o n f n .
To conclude the proof of the theorem, it is then enough to prove that dis(C n ) n 1/4 , or in other words that: To prove this last statement, we only need to adapt slightly the framework of [13, Theorem 1, p.680], and thus we only sketch the proof. We start by presenting the general strategy of the proof. First, thanks to a rerooting argument, we can extend the result for the 2-point function stated in Proposition 34 to control the pairwise distance between any finite number of random points. Next, we rely on the known convergence of simple triangulations to the Brownian sphere in the Gromov-Hausdorff-Prokhorov sense (established in [1]) to transfer the compactness of the Brownian sphere from the continuum world to the discrete world. This allows us to approximate the metric structure of a large simple random triangulation by restricting our attention to a (large) fixed number of uniform points. Lastly (and this is the only difference with the proof of [13, Theorem 1]), Proposition 31 extends this approximation result from a simple triangulation to its dual.
Let us now state more formally those main steps. Let t n be a uniform element of T n . Let Φ n : V (t n ) → F (t n ) be a deterministic function (given t n ) which associates to each vertex v of t n a face f with v Φ n (v). Let o n and o n be two independent uniform vertices of t n . By the same chain of arguments as in [13, p.680], it follows from Proposition 34 that for any > 0: Now let (o i n ) i 1 be a sequence of independent uniform vertices of t n . For any δ > 0, there exists N 1 such that for every n ∈ N: By the triangle inequality, we can then write: We take N large enough so that (57) holds. By Proposition 31, outside a set of probability at most δ and provided that n is large enough, the last term in the upper bound can be bounded by 2 sup x∈V (tn) inf 1 j N d gr (x, o j n ) + n 1/8 . In view of (57) and (56) (applied simultaneously to all pairs (o i n , o j n )), this concludes the proof.

Scaling limit of the 3-connected core with distances induced by the connected graph
Informally, the purpose of this section is to establish a result analogous to Proposition 34 (and then a result analogous to (55)), but where we relax the independent hypothesis for the weights on the edges. Recall from Section 3.3 that C (q) n is the set of cubic connected planar graphs of size n having a marked 3-connected component of size q, that C (q) n is a uniformly random element of C  n (for q > n/2 it coincides with the random connected cubic planar multigraph of size n conditioned to have its 3-connected core of size q). Recall also that for g ∈ C n , we write K(g) for the 3-connected core of g.
We aim at proving the following result (analogous to (55), with a dependence on the edgelengths): Theorem 36. Let (q(n), n 1) be such that q(n) = αn + O(n 2/3 ) (where α ≈ 0.85 is defined in Proposition 7 and with the informal notation introduced below Lemma 9). Write respectively dist C (q(n)) n and d gr for the distance on K(C (q(n)) n ) induced by the graph distance on C (q(n)) n and for the graph distance on K(C (q(n)) n ). Then, there exists a constant c > 0 such that, for any > 0: A similar result holds for the random multigraph M (q(n)) n , when q(n) = 199 316 n + O(n 2/3 ).
We note that, given Theorem 3, this result ensures that the scaling limit of K(C (q(n)) n ) with distances (rescaled by c 1 q 1/4 for some c 1 ∈ (0, ∞)) induced by the distances in C (q(n)) n is the Brownian sphere. Moreover, combined with Theorem 2, it gives the joint convergence of the two last coordinates in Theorem 4.
To prove this theorem, we first establish the following similar result for the 2-point function (analogous to Proposition 34, with a dependence on the edge-lengths).

Proposition 37.
Consider the same setting as in the previous theorem. Let o, o be two random vertices of K(C (q(n)) n ). Then, there exists a constant c > 0 (the same as in Theorem 36) such that, for any > 0: A similar result holds for the random multigraph M (q(n)) n , when q(n) = 199 316 n + O(n 2/3 ). Sections 7.1 to 7.5 will be devoted to the proof of this proposition. The proof of Theorem 36 will then follow along the same lines as the part proving (55) in the proof of Theorem 35 and is given in Section 7.6.
In the rest of Section 7, we always assume that n and q(n) are related as in Proposition 37. To simplify notations, we drop the index n and simply write q for q(n). Since q(n) = αn + O(n 2/3 ), it makes sense to write q → ∞ in place of n → ∞. Similarly, with a slight abuse of notation, in all this section, we write K q for K(C (q(n)) n ) and K q := (K q , d) for the metric space (K(C (q(n)) n ), dist C (q(n)) n ). On the other hand, recall that ν is the law of the pole-distance of a critical Boltzmann cubic network as defined in Section 4.1. We define K q := (K q , d) to be the random metric graph K(C (q(n)) n ), equipped with the first-passage percolation distance obtained from ( (e)) e ) where ( e ) is a collection of i.i.d random variables sampled from ν . It follows from Lemma 12 that ν has exponential tails. Since the support of ν is by definition included in [1, ∞), all the results obtained in the previous section apply to K q . In particular, by applying Proposition 34 twice (once with ν equals to the Dirac measure on {1} and once with ν := ν ), there exists a constant c > 0 such that, for o, o two random vertices in K q and for every > 0, we have We emphasize that in fact this statement serves at the very definition of the constant c > 0 that appears both in Theorem 36 and in Proposition 37.
To prove Proposition 37, we are going to construct a coupling between K q and K q , so as to minimize the the number of edges whose edge-lengths differ. We will then show that, under this coupling, the weighted distance between two random vertices o, o differs by o(q 1/4 ) in probability.

Definition of the coupling between K q and K q
To define the coupling between K q and K q , we first introduce a handful of notations.
Similarly, let ( δ 1 , . . . , δ 3q ) be the random vector giving the edge-lengths in K q inherited from C q n . Then, we denote by V q the push-forward distribution on Σ 3q via the mapping Mult(.) of the distribution of ( δ 1 , . . . , δ 3q ).
.3q], (e j ) = δ j and (e j ) = δ j . It is clear from the construction that K q and K q defined in this manner follow the required distribution. In the rest of this section, they are always assumed to be sampled jointly in this manner.
Remark 7. In the following, it will be useful to be able to "choose an edge in K q using some information about K q but no information from K q ", or symmetrically. To make rigorous sense of this, we can slightly adapt the sampling procedure as follows to better differentiate that K q is first drawn, and K q subsequently (we use the notation E =i for the set of edges of length i in a metric graph with integer edge-lengths): 1. Sample K q , N and N.
2. Consider a random permutation of E(K q ), and list its edges as (e 1 , . . . , e 3q ) in this random ordering.

For each
4. For each i ∈ Z 1 , reorder uniformly at random the edges of E =i ( K q ) and apply the canonical construction of Figure 9 to construct K q following N. In this way, for every i Of course, the roles played by K q and K q are perfectly symmetric, and we can proceed by first sampling K q and then K q if needs be. .

First properties of the coupling
To quantify the number of edges which do not have the same length in both models, we start by proving the following estimate: N = ( N 1 , N 2 , . . .) and N = ( N 1 , N 2 , . . .) be independently sampled from V q and from V q respectively. Then there exist positive constants a and A such that a.a.s. as q → ∞, the following event E q holds:

Lemma 38. Let
• for every positive integer i a log(q), we have min( N i , N i ) q 3/4 and | N i − N i | q 2/3 .
• for all i A log(q), we have N i = N i = 0.
Proof. Let p i := ν (i). We can obtain a lower bound for p i by noticing that any cubic network that is a series of i L-networks has pole-distance i + 1. Hence: and L(ρ) is the generating series of L-networks evaluated at its radius of convergence. Let a = 1 5κ . Then, for i a log(q), we have p i 1 D(ρ) q −1/5 . On the other hand, by definition, N i follows a Binomial(3q, p i ) distribution, so that E( N i ) = 3q p i 3 D(ρ) q 4/5 . By Hoeffding's inequality, we have that, for any x > 0: ). By Lemma 9, N i has the same law as N i conditioned on an event of probability Θ(q −2/3 ), which implies that Hence, which gives the first property (using the lower bound on 3q p i ).
Regarding the second property, we proved in Lemma 12 that ν has exponential tail. Hence, if ( δ 1 , . . . , δ 3q ) is a sequence of i.i.d. random variables sampled from ν , then there exists b > 0, such that P(max( δ 1 , . . . , δ 3q ) i) = O(q exp(−bi)). This is equivalent to the fact that P( N j = 0 for some j i) = O(q exp(−bi)). Using again Lemma 9, we can derive from this property, the fact that P( N j = 0 for some j i) = O(q 5/2 exp(−bi)). Hence, setting A = 3/b, this gives the desired result.

Corollary 39.
Conditionally on the event E q defined in the previous lemma, for any e ∈ E(K q ) such that (e) a log(q), and chosen without using any information on K q , we have: P( (e) = (e)) q −1/12 .
Similarly, for any edge e ∈ E(K q ), such that (e) a log(q), and chosen without using any information on K q , we have P( (e) = (e)) q −1/12 .
Proof. By construction of the coupling (or rather of its variant described in Remark 7), in the first case, we have that: Summing over the values of i in {1, .., a log(q) }, and using the bound for N i and N i , this proves the result. The other case is treated in exactly the same manner.
Let us apply this notion to K q and K q . For an edge e of K q , we denote by (k) (e) (resp. (k) (e)) its truncated edge-length in K q (resp. K q ). For γ a path in K q , its truncated edge-length is denoted by L (k) (γ) (resp. L (k) (γ)) in K q (resp. K q ). Furthermore, for two vertices o, o ∈ K q , we denote by d (k) (o, o ) (resp. d (k) (o, o )) the minimum of L (k) (γ) (resp. of L (k) (γ)) over all paths connecting o and o .
Finally, we denote by K q ) the metric graphs (K q , d (k) ) (resp. (K q , d (k) )). Applying the results of Section 6 to K (k) q instead of K q , we obtain that there exists a constant c (k) such that, for every > 0, we have: where o and o are two random vertices of K q (picked independently from anything else). Note that c (k) is increasing with k and is bounded by c.
The purpose of this section is now to prove the analogous statement on K where o and o are two random vertices of K q (picked independently from anything else).
Proof. Set N k := i k N i and N k := i k N i , and define E q , by the same reasoning as in Corollary 39, we have: On the other hand, let N k be the number of edges having length at least k in K q and smaller than k in K q . By construction of the coupling between K q and K q , N k holds, N k k q 2/3 . Therefore, conditionally on E (k) q , we have: Now, we define γ (k) as a canonically chosen path among the paths γ connecting o to o such that L (k) (γ) = d (k) (o, o ) (e.g. γ (k) is the lexicographically smallest, for the sequence of labels of the visited vertices). For any edge of γ (k) , we can apply (66) or (67). Moreover, by definition of truncation, for any edge e, we clearly have | (k) (e) − (k) (e)| k. Hence, conditionally on E (k) q , and in view of (66) and (67), we have Therefore, conditionally on E (k) q , and by Markov's inequality, It follows directly from Theorem 3 that a.a.s. d gr (o, o ) we also have a.a.s. | γ (k) | k q 1/4 log(q), so that: For any > 0, the event d (k) (o, o ) c (k) d gr (o, o ) + q 1/4 holds a.a.s. according to (63). Since ), we conclude that, for any > 0, Symmetrically, applying the argument to γ (k) , we obtain that, for every > 0, we have a.a.s.
which concludes the proof.

Lengths of truncated geodesic paths in both models
We now state and prove additional estimates in view of completing the proof of Proposition 37.

Lemma 41.
Let k 1. Then, for every > 0, we have where o and o are two random vertices for K q (picked independently from anything else), and where we recall that γ (k) is a canonical geodesic path in K (k) q between them.
Proof. In all this proof, we work conditionally on the a.a.s. event E q of Lemma 38, and consider q large enough such that a log(q) > k. For e an edge chosen using solely information given by K (k) q and for i k, we have: Hence, by Lemma 12, letting κ = a /ν ([k, ∞)), we have P( (e) a log(q)) κ q −ab . Since E | (e) − (e)| A log(q) P (e) = (e)| (e) < a log q + P (e) a log q , by Corollary 39, we get that: E | (e) − (e)| (q −1/12 + κ q −ab )A log(q).
We further work conditionally on the a.a.s. event E q under which where ξ = min(1/12, ab )/2. Hence, by Markov's inequality we have

Lemma 42.
Let X be a random variable drawn under ν . There exist positive constants a , b , and an infinite subset N ⊂ N such that, for every k ∈ N and every i 0, we have Proof. Let p i = P(X = i). We have seen in the proof of Lemma 38 that p i > 0 for any i 0 and that p i = O(exp(−bi)) for some b > 0. Set p i := p i exp(bi/2). Then p i → 0 as i → ∞. Hence, we can define a subset N of integers such that, for every k ∈ N , p k = max(p i , i k) holds. Hence, for k ∈ N and i k, we have p i p k exp(−b(i − k)/2). This gives

Lemma 43.
There exists a positive constantâ such that, for every k ∈ N , where o and o are two random vertices for K q (picked independently from anything else), and where we recall that γ (k) is a canonical geodesic path in K (k) q between them. Moreover, we have lim k→∞ c (k) = c.
copies of the random variable X (k) , whose distribution is given by: By (63) and Theorem 3, we know that P(H q ) → 1 as q → ∞. If H q holds, we have |E k | 2c k d gr (o, o ), so that: so that the first statement is proved (withâ = 2a  (o, o ). This implies that c (k) +â k c. Since c (k) c, this gives lim k→∞ c (k) = c.

Proof of Proposition 37
We start by establishing the lower bound for d(o, o ) given in (59) of Proposition 37. Fix > 0 and η > 0. By Theorem 3, we know that there exists b > 0 such that for q large enough: On the other hand, by Lemma 40, we have for q large enough: Then, Lemma 43 grants the existence of k such that c (k) c − /(2b). Combined with the two preceding inequalities and since d (k) (o, o ) d(o, o )) it yields for q large enough: Regarding the upper bound, let k ∈ N be large enough such thatâ k 4b . By Lemma 43, there exists q 1 such that, for q q 1 , we have By Lemma 41, there exists q 2 such that, for q q 2 , Hence, for q max(q 1 , q 2 ), we have

Proof of Theorem 36
Given Proposition 37, the proof of Theorem 36 follows the exact same lines as in the proof of (55). The only missing point is a result analogous to Proposition 31, which we now state and prove: Lemma 44. Let K be any constant larger than E(ν ). For ∈ (0, 1/4), defineẼ q as the event where the bound d(x, y) K d gr (x, y) + q holds for every pair of vertices x, y in K q . Then P(Ẽ q ) → 1 as q → ∞.
Proof. We start by proving a similar result but for K q rather than K q . LetÊ q be the event where the bound d(x, y) K d gr (x, y) + q holds for every pair of vertices x, y in K q . Let x, y be two fixed vertices in K q . Since ν has exponential tail, the Chernoff bound ensures that there exist positive constants a , b (not depending on x, y) such that In particular, if d gr (x, y) > q /2 , this implies that P d(x, y) > K d gr (x, y) a exp(−b q /2 ).
We now deal with the case where d gr (x, y) q /2 . Let w be the maximal weight over the edges in K q . The union bound and the fact that ν has exponential tail ensure that (with a , b the constants in Lemma 12) Hence, if d gr (x, y) q /2 , we have P( d(x, y) > q ) 3q · a · exp(−b q /2 ) Letting b be a positive constant smaller than b and b , this guarantees the existence of a constant a such that, for every given pair x, y of vertices in K q , By the union bound, this gives P(¬Ê q ) q 2 a exp(−b q /2 ). Finally, by Lemma 9, we obtain that: Given the preceding discussion, this concludes the proof of Theorem 36.

Gromov-Hausdorff-Prokhorov distance between a random connected cubic planar graph and its 3-connected core
The aim of this section is to establish the following result: Proposition 45. Let (q(n), n 1) be such that q(n) = αn + O(n 2/3 ) (where α ≈ 0.85 is defined in Proposition 7). Let C (q(n)) n be a uniformly random element of C (q(n)) n . Write respectively dist C (q(n)) n and d gr for the distance on K(C (q(n)) n ) induced by the graph distance on C (q(n)) n and for the graph distance on K(C (q(n)) n ). Then d GHP C (q(n)) n , 1 n 1/4 d gr , µ C (q(n)) n , K(C (q(n)) n ), 1 n 1/4 dist C (q(n)) n , µ K(C (q(n)) where, as usual, µ C (q(n)) n and µ K(C (q(n)) n ) denote respectively the uniform distribution on V (C (q(n)) n ) and on V (K(C (q(n)) n )). A similar results holds for M (q(n)) n , with q(n) = 199 316 n + O(n 2/3 ). By Theorem 8 of Section 3.2, the size of the 3-connected core of a random cubic planar graph is αn + O(n 2/3 ) with high probability. We established in Section 7 (see the discussion below Theorem 36) that (K(C (q) n ), 1 q 1/4 dist C (q) n , µ K(C (q) n ) ) converges to the Brownian sphere. Hence Proposition 45 directly implies the joint convergence of (K(C n ), 1 n 1/4 dist Cn , µ K(Cn) ) and of (C n , 1 n 1/4 d gr , µ Cn ), which corresponds to the joint convergence of the first and second coordinates stated in Theorem 4.
As in the previous section, in all this section q, n are related by q = αn + O(n 2/3 ). To lighten notation, we drop the explicit dependency of q in n and write q instead of q(n).
Roughly speaking, this result relies on two main steps. First, we have to prove that the Gromov-Hausdorff distance between these two measured metric spaces is small, or in other terms, that any vertex of C (q) n is at distance o(n 1/4 ) from a vertex of K(C (q) n ). Then, we need to prove that the Levy-Prokhorov distance between both spaces is small. Roughly speaking, it means that the uniform measure on K(C (q) n ) and the projection of the masses of the uniform measure on C (q) n into its 3-connected core are close for the d LP distance. Our approach follows closely the strategy developed in an article by Addario-Berry and Wen [2], dealing with the joint convergence of a quadrangulation and its simple core. We give the arguments only for graphs, but the exact same approach can be developed for multigraphs.
Following [2], we first define a measure on V (K(C (q) n )) that reflects a "projection of masses". Let v 1 , . . . , v 2q be the vertices of K(C (q) n ), ordered by increasing label. For i ∈ [1..2q], let s i be the total number of vertices of (non-pole) vertices in the attached networks at the 3 edges incident to v i , and let n i = 1 + s i /2. Then the measure µ Proof. Let n = (n 1 , . . . , n 2q ) be defined from C (q) n as above. Lemma 5.3 in [2] 7 implies that for any vertex-subset W ⊂ V (K(C (q) n )), and for any t > 0, we have Here, |n| 1 = 2n, and by Lemma 10 we know that max(n 1 , . . . , n 2q ) n 3/4 a.a.s., which implies |n| 2 2 n 7/8 (since 2q i=1 n 2 i max(n 1 , . . . , n 2q ) · |n| 1 ). Taking t = n 1/16 and letting q = 2n −1/16 + 1 2q+1 = o(1), we thus have a.a.s. |µ K(C (q) n ) (W ) − µ n )). We can then proceed with the exact same arguments as in Corollary 6.2 in [2] (compactness of the Brownian sphere m ∞ , and the fact that (K(C (q) n ), 1 q 1/4 d gr , µ K(C (q) n ) ) converges toward m ∞ for the GHP topology, as we have already established in Theorem 3). Proof. We use the characterization of the Lévy-Prokhorov distance in terms of couplings. Let V q = V (K(C (q) n )). Let > 0 and η > 0. Our aim is to show that, for q large enough, with probability at least 1 − η there exists a coupling ν between µ K(C (q) n ) and µ such that, for the distance 1 n 1/4 dist C (q) n , we have, with the notation of Section 2.2, Let β = 2E(ν ), and let¯ = /(β + 1). By Lemma 46 there exists q 1 such that, for q q 1 , with probability at least 1 − η/2, there holds the event F q that there exists a coupling ν between µ K(C (q) n ) and µ such that, for the distance 1 n 1/4 d gr , we have Let E q be the event that dist C (q) n (x, y) βd gr (x, y) + q 1/6 for x, y ∈ V q . Then Lemma 44 guarantees that there exists q 0 such that P(E q ) 1 − η/2 for q q 0 . Let q 2 be large enough so that n −1/4 q 1/6 β+1 for q q 2 . If q q 2 and E q holds, then for every x, y ∈ V q , we have that n −1/4 d gr (x, y) ¯ implies n −1/4 dist C (q) n (x, y) . Hence, if q q 2 and if E q and F q hold, then with the same coupling ν as above, but now for the distance 1 n 1/4 dist C (q) n , we have This concludes the proof, since the event E q ∩ F q occurs with probability at least 1 − η for q max(q 0 , q 1 , q 2 ).
Lemma 48. The GHP distance between (K(C ) and (C (q) is o(1) in probability.