Cyril Marzouk

ON SCALING LIMITS OF PLANAR MAPS WITH STABLE FACE-DEGREES by

On scaling limits of planar maps with stable face-degrees

. Introduction and main result

This work deals with scaling limits of large random planar maps viewed as metric measured spaces. We assume that the reader is already acquainted with this theory; let us describe the precise model that we consider before stating our main results.

We study rooted planar maps, which are nite (multi-)graphs embedded in the two-dimensional sphere, viewed up to homeomorphisms, and equipped with a distinguished oriented edge called the root-edge. For technical reasons, we restrict ourselves to bipartite maps which are those maps in which all faces have even degree. Given a sequence q = (q k ) k ≥1 of non-negative numbers such that q k 0 for at least one k ≥ 3 (in order to discard trivial cases), we de ne a Boltzmann measure w q on the set M of all nite bipartite maps by assigning a weight:

w q (M) = f ∈Faces(M ) q deg(f )/2 ,
to each such map M. We shall also consider rooted and pointed maps in which we distinguish a vertex in a map M; we then de ne a pointed Boltzmann measure on the set M • of pointed maps by setting w q,• (M, ) = w q (M). Let W q = w q (M) and W q,• = w q,• (M • ) be the total mass of these measures; obviously the latter is greater than the former, but Bernardi, Curien and Miermont [BCM ] have shown that W q < ∞ if and only if W q,• < ∞.

When this holds, we say that the sequence q is admissible and we normalise our measures into probability measures P q and P q,• respectively. We assume further that q is critical, which means that the number of vertices of a map has in nite variance under P q , or equivalently in nite mean under P q,• . Such models of random maps have been rst considered by Marckert and Miermont [MM ] who gave analytic admissibility and criticality criteria, recast in [Mar b], and which we shall recall later. Following the terminology introduced in the very recent work of Curien and Richier [CR ], we further assume that there exists α ∈ (1, 2] such that our distributions are discrete stable with index α, which we de ne as follows. Let µ q be the law of the half-degree of the face adjacent to the right of the root-edge (called the root-face) under P q,• . Then we assume that µ q belongs to the domain of attraction of a stable law with index α ∈ (1, 2], which means that either it has nite variance: ∞ k =0 k 2 µ q (k) < ∞ and then α = 2, or the tail can be written as ∞ k=j µ q (k) = j -α l(j), where l is a slowly varying function at in nity which means that for every c > 0, it holds that lim x →∞ l(cx)/l(x) = 1.

It can be checked that the degree of the root-face under the non-pointed law P q is more regular, and under this assumption has nite variance for every α ∈ (1, 2]. We shall interpret µ q as the law of the half-degree of a typical face in a large pointed or non-pointed Boltzmann random map. Such an assumption was rst formalised by Richier [Ric ] (except that the case α = 2 was restricted to nite variance) and is more general than the one used e.g. in [LGM , Mar b].

For every integer n ≥ 2, let M E=n , M V =n and M F =n be the subsets of M of those maps with respectively n -1 edges, n + 1 vertices (these shifts by one will simplify later statements) and n faces. For every S = {E, V , F } and every n ≥ 2, we de ne

P q S =n (M) = P q (M | M ∈ M S =n ), M ∈ M S =n ,
the law of a rooted Boltzmann map conditioned to have 'size' n. We de ne similarly pointed laws P q,• S =n . Let us denote by ζ (M n ) the number of edges of the map M n sampled from such a law; note that it equals n -1 if S = E but it is random otherwise. We shall implicitly assume that the support of q generates the whole group Z, not just a strict subgroup, so these laws are well-de ned for every n large enough; the general case only requires mild modi cations. We consider limits of large random maps in the following sense: given a nite map M, we endow its vertex-set (which we still denote by M) with the graph distance d gr and the uniform probability measure p gr ; the topology we use is then that given by the so-called Gromov-Hausdor -Prokhorov distance which makes the space of compact metric measured spaces (viewed up to isometries) a Polish space, see e.g. Miermont [Mie ].

T

. . -There exists an increasing sequence (B n ) n ≥1 such that the following holds. Fix S ∈ {E, V , F } and for every n ≥ 2, sample M n from P q S =n or from P q,• S =n , then: (i) If α = 2, then we have the convergence in distribution in the sense of Gromov-Hausdor -Prokhorov This result is reminiscent of the work of Duquesne [Duq ] and Kortchemski [Kor ] on sizeconditioned Bienaymé-Galton-Watson trees (see Equation ( ) below) and indeed, the sequence (B n ) n ≥1 is the same as there; it is of order n 1/α up to a slowly varying function, and in the nite-variance regime, it takes the form B n = (nσ 2 /2) 1/2 for some σ 2 ∈ (0, ∞).

R

. . -We shall see in Remark . that under P q S =n or P q,• S =n we have for some constant

Z q > 1 n -1 ζ (M n ) P -→ n→∞ Z q if S = V and n -1 ζ (M n ) P -→ n→∞ (1 -Z -1 q ) -1 if S = F ,
so the factor B -1/2 ζ (M n ) may be replaced by Z -1/(2α ) q B -1/2 n and (1 -Z -1 q ) 1/(2α ) B -1/2 n respectively.

In the Gaussian case α = 2, tightness in the sense of Gromov-Hausdor of rescaled uniform random 2κ-angulations (all faces have degree 2κ xed) with n faces was rst obtained by Le Gall [LG ]. The Brownian map was then characterised independently by Le Gall [LG ] and Miermont [Mie ] which yields the convergence of these maps; building upon the pioneer work of Marckert and Miermont [MM ], Le Gall [LG ] also includes Boltzmann planar maps conditioned by the number of vertices, assuming exponential moments. This assumption was then reduced to a second moment in [Mar b], as a corollary of a more general model of random maps 'with a prescribed degree sequence'. Let D * = ( 9 8 ) 1/4 D, then in this nite variance regime, Theorem . reads thanks to the preceding remark:

M n , 9 4σ 2 ζ (M n ) 1/4 d gr , p gr (d ) -→ n→∞ (M, D * , m),
which recovers [Mar b, Theorem ].

In the case α < 2, Theorem . extends a result due to Le Gall and Miermont [LGM ] who studied the Gromov-Hausdor convergence of such maps conditioned by the number of vertices in the particular case where µ q (k) = Ck -α -1 (1 + o(1)) for some constant C > 0. Because the conjectured 'stable maps'

have not yet been characterised, the extraction of a subsequence is needed in Theorem . . Nonetheless, as in [LGM ], we derive some scaling limits which do not necessitate such an extraction: in Theorem . below, we give the limit of the maximal distance to the distinguished vertex in a pointed map, or to a uniformly chosen vertex in a non-pointed map, as well as the pro le of the map, given by the number of vertices at distance k to such a vertex, for every k ≥ 0. Let us nally mention the work of Richier [Ric ] and more recently with Kortchemski [KR ] who analyse the geometric behaviour of the boundary of the root-face when conditioned to be large, and so, roughly speaking, the geometric behaviour of macroscopic faces of the map.

R

. . -(i) As in [Mar b], the proof of Theorem . actually shows that we can also take as notion of size of a map the number of faces whose degree belongs to a xed subset A ⊂ 2N, at least when either A or its complement is nite. (ii) As observed in [Mar b, Theorem ], the conditioning by the number of edges is special since M E=n is a nite set for every n xed so we may de ne the law P q E=n even when q is not admissible and our results still holds under appropriate assumptions. (iii) As in [LGM ], Theorem . and the other main results below hold when conditioning the maps to have 'size' at least n, the references cover this case and the proofs only require mild modi cations.

The proof of convergences as in Theorem . in [LG , LGM ] relied on a bijection due to Bouttier, Di Francesco and Guitter [BDFG ] which shows that a pointed map is encoded by a 'two-type' labelled tree and one of the key steps was to prove that this labelled tree, suitably rescaled, converges in distribution towards a 'continuous' limit which similarly describes the limit (M, D, m). In [Mar b] we studied this two-type tree by further relying on a more recent work of Janson and Stefánsson [JS ] who established a bijection between such 'two-type' trees and 'one-type' trees which are much easier to control. The scheme of the proof of the analogous statement in [Mar b] was rst to prove that this 'one-type' labelled tree converges towards a continuous object, then transporting this convergence to the two-type tree and nally conclude from the arguments developed in [LG , LGM ].

In this paper, we bypass the bijection [BDFG ] and only work with the one-type tree from [JS ]; we prove the convergence of this object in Theorem . and deduce Theorem . by recasting the arguments from [LG , LGM ]. On the one-hand, the advantage of the bijection from [BDFG ] is that it also applies to non-bipartite maps (but it yields a 'three-type' tree even more complicated to study) so in principle, one may use it to prove the convergence of such maps, whereas the bijection from [JS ] only applies to bipartite maps. On the other hand, the latter bijection reduces the technical analysis of the tree, which opens the possibility to study more general models of random bipartite maps, such as those from [Mar b] in more complicated 'large faces' regimes. In particular, our proof does not necessitate a tight control on the geometry of the tree, since it mostly relies on its Łukasiewicz path which is rather simple to study.

The rest of this paper is organised as follows: In Section , we recall the key bijection with labelled trees which in our case are randomly labelled size-conditioned Bienaymé-Galton-Watson trees. We recall their continuous analogues in Section , then we state and prove their convergence in Theorem . in Section which contains most of the technical parts and novelties of this work. Finally, in Section , we state and prove Theorem . on the pro le of distances and then prove Theorem . .

Acknowledgement

I wish to thank Nicolas Curien for helping providing the simulations in Figure and for a discussion on what 'discrete stable map' could mean. Thanks are also due to the anonymous referee for a swift reading.

This work was supported by a public grant as part of the Fondation Mathématique Jacques Hadamard.

. Maps and labelled trees

In this rst section we brie y recall the notion of labelled (plane) trees and introduce some useful notation. We also describe the bijection between a pointed planar map and such a tree.

. . Plane trees. -Following the notation of Neveu [Nev ], we view discrete trees as words. Let N = {1, 2, . . . } be the set of all positive integers and set N 0 = { }. Then a (plane) tree is a non-empty subset T ⊂ n ≥0 N n such that:

(i) ∈ T , it is called the root of T ; (ii) if = ( 1 , . . . , n ) ∈ T , then pr ( ) = ( 1 , . . . , n-1 ) ∈ T ; (iii) if = ( 1 , . . . , n ) ∈ T , then there exists an integer k ≥ 0 such that i = ( 1 , . . . , n , i) ∈ T if and only if 1 ≤ i ≤ k .
We shall view each vertex of a tree T as an individual of a population for which T is the genealogical tree. The vertex is the ancestor and for every = ( 1 , . . . , n ) ∈ T , the vertex pr ( ) is its parent, k is the number of children of (if k = 0, then is called a leaf, otherwise, is called an internal vertex), and 1, . . . , k are these children from left to right, χ = n is the relative position of among its siblings, and | | = n is its generation. We shall denote by u, the unique non-crossing path between u and .

Fix a tree T with N + 1 vertices, throughout this work, we shall list the vertices in the lexicographical

order as = u 0 < u 1 < • • • < u N .
We describe two discrete paths which each encode T , depicted in Figure . First, its Łukasiewicz path W = (W (j); 0 ≤ j ≤ N + 1) is de ned by W (0) = 0 and for every 0

≤ j ≤ N , W (j + 1) = W (j) + k u j -1.
One easily checks that W (j) ≥ 0 for every 0 ≤ j ≤ N but W (N + 1) = -1. Next, we de ne the height process H = (H (j); 0 ≤ j ≤ N ) by setting for every 0 ≤ j ≤ N ,

H (j) = |u j |.
-F . A tree on the le , with its vertices listed in lexicographical order, and on the right, its Łukasiewicz path W on top and its height process H below.

The next lemma, whose proof is left as an exercise, gathers some deterministic results that we shall need (we refer to e.g. Le Gall [LG ] for a thorough discussion of such results). In order to simplify the notation, we identify the vertices of a tree with their index in the lexicographic order.

L

. . -Let T be a plane tree and W be its Łukasiewicz path. Fix a vertex u ∈ T , then

W (uk u ) = W (u), W (uj ) = inf [u j,u j ] W and j -j = W (uj) -W (uj )
for every 1 ≤ j ≤ j ≤ k u .

Note that W (u) -W (pr (u)) equals the number of siblings of u which lie to its right, so W (u) equals the total number of individuals branching o to the right of the ancestral line , u .

. . Labelled trees. -For every k ≥ 1, let us consider the following set of discrete bridges ( )

B + k = {(x 1 , . . . , x k ) : x 1 , x 2 -x 1 , . . . , x k -x k-1 ∈ {-1, 0, 1, 2, . . . } and x k = 0} .
A labelling of a plane tree T is a function de ned on its set of vertices to Z such that (i) the root of T has label ( ) = 0, (ii) for every vertex u, with k u ≥ 1 children, the sequence of increments (l(u1)l(u), . . . , l(uk u )l(u))

belongs to B + k u . We stress that the last child of every internal vertex carries the same label as its parent, for example, the right-most branch in the tree only contains zeros. De ne the label process L(k) = (u k ), where (u 0 , . . . , u N ) is the sequence of vertices of T in lexicographical order; the labelled tree is encoded by the pair (H, L), see Figure . Without further notice, throughout this work, every Łukasiewicz path shall be viewed as a step function, jumping at integer times, whereas height and label processes shall be viewed as continuous functions after interpolating linearly between integer times.

. . Labelled trees and pointed maps. -Bouttier, Di Francesco and Guitter [BDFG ] proved that pointed maps are in two-to-one correspondence with some labelled trees, di erent from the preceding section; in the bipartite case, Janson and Stefánsson [JS ] then related these trees to trees as in the preceding section. Let us describe a direct construction of this correspondence between our labelled trees and pointed bipartite maps and leave to the reader as an exercise to verify that it indeeds corresponds to the two previous bijections (one may compare the gures here and those in [Mar b]).

Let us start with the construction of a pointed map from a labelled tree (T , ), depicted in Figure ; the construction contains two steps. Let (u 0 , . . . , u N ) be the vertices of T listed in lexicographical order. For every 0 ≤ i ≤ N , set u N +1+i = u i . We add an extra vertex labelled min u ∈T (u) -1 outside of the tree T and construct a rst planar graph G on the vertex-set of T and by drawing edges as follows: for every 0 ≤ i ≤ N -1, -if (u i ) > min 0≤k ≤N (u k ), then we draw an edge between u i and u j where j = min{k > i :

(u k ) = (u i ) -1}, -if (u i ) = min 0≤k ≤N (u k )
, then we draw an edge between u i and .

We stress that we exclude the last vertex u N in this construction; it indeed yields a planar graph G. In a second step, we merge every internal vertex of the tree T with their last child; then G becomes a map M with labelled vertices. We shift all labels by subtracting min u ∈T (u) -1; it can be checked that these new labels are just the graph distance to in the map M. We also distinguish the image after the merging operation of the rst edge that we drew, for i = 0. The latter is non-oriented; let e + and e -be its extremities so that d gr (e -, ) = d gr (e + , ) -1, then we orient the edge either from e -to e + or from e + to e -; these maps are called positive and negative respectively in [MM ].

0 -1 -2 1 0 0 -1 -2 0 -1 1 0 0 0 -1 -2 -1 -3 0 -1 -2 1 0 0 -1 -2 0 -1 1 0 0 0 -1 -2 -1 -3 -1 -2 0 -2 0 -1 1 0 -1 -2 -1

F

. The negative pointed map associated with a labelled tree. Labels indicate, up a to a shi , the graph distance to the distinguished vertex, which is the one carrying the smallest label.

Let us next construct a labelled tree from a negative pointed map (M, ), as depicted in Figure . First, label all vertices by their graph distance to . In every face of M, place a new unlabelled vertex and mark each corner when the next vertex of M in clockwise order (1) has a smaller label. Then start with the root-face, adjacent to the right of the root-edge. Link the new vertex in this face to every marked corner if it is the only marked corner of this vertex, otherwise, erase the mark and link the new vertex to the one in the face which contains the next marked corner of this vertex in clockwise order. Proceed recursively with the new vertices attached to the one in the root-face: link each of them to the marked corners in their face if they are the only remaining ones around their vertex, otherwise, remove the mark and link the new vertex to the next one in clockwise order around the vertex. Once all faces have been considered we obtain a planar tree that we root at the new vertex in the root-face. Since the map is negative, the corner of the origin e + of the root-edge in the root-face is marked, then the last child of the root of the tree is either e + or the new vertex in the face which contains the next marked corner of this (1) When drawn in the plane instead of the sphere, the clockwise order in the exterior face appears as counter-clockwise.

0 2 1 3 1 3 2 4 3 2 1 2 0 2 1 3 1 3 2 4 3 2 1 2 -1 -2 0 -2 0 -1 1 0 -1 -2 -1

F

. The labelled tree associated with a negative map. In the top le , marked corners are indicated in fat red. The oriented edge in the tree goes from the root to its last child; unlabelled black vertices then receive each the label of its their child.

vertex in clockwise order around it if any. Then assign to each new vertex the label of its last child and nally shift all labels so the root of the tree has label 0 to get a labelled tree as in the preceding section. We claim that these constructions are the inverse of one another and yield a bijection between labelled trees and negative maps (the construction is very close to [BDFG ], one can thus follow their detailed proof). Recall that the root-face of a map is the face adjacent to the right of the root-edge. This bijection enjoys the following properties:

(i) The leaves of the tree are in one-to-one correspondence with the vertices di erent from the distinguished one in the map, and the label of a leaf minus the in mum over all labels, plus one, equals the graph distance between the corresponding vertex of the map and the distinguished vertex. (ii) The internal vertices of the tree are in one to one correspondence with the faces of the map, and the number of children of the vertex is half the degree of the face. (iii) The root-face of the map corresponds to the root-vertex of the tree. (iv) The number of edges of the map and the tree are equal.

In order to have a bijection between labelled trees and positive maps, one just reverse the root-edge in order to get a negative map. Note that Property (iii) above does not hold anymore and it does not seem clear which internal vertex of the tree corresponds to the original root-face. Nonetheless, by 'mirror symmetry' of the map (which preserves positivity or negativity of the map), the degree of the faces on both sides of the root-edge have the same distribution, so in both cases of positive or negative maps, the half-degree distribution of the root-face is the o spring distribution of the root of the tree. Property (i) above explains how to partially translate the metric properties of the map to the labelled tree, whereas Property (ii) is important because it gives us the distribution of the tree when the map is a random Boltzmann map, as described below.

. . Random labelled trees. -Let us introduce the law of the labelled tree associated with a pointed map sampled from P q,• . Let q 0 = 1 and de ne the power series

q (x) = k ≥0 x k 2k -1 k -1 q k , x ≥ 0.
Then q is convex, strictly increasing and continuous until its radius of convergence, and q (0) = 1. In particular, it has at most two xed points, and if it has exactly one, then at that point, the graph of q either crosses the line = x, or is tangent to it. It was argued in [Mar b, Section . ], recasting the discussion from [ MM , Section . ] in the present context, that the sequence q is admissible and critical exactly when q falls into the last case, and we denote by Z q the only xed point, which satis es q (Z q ) = 1. Let us mention that Z q equals (W q,• + 1)/2 > 1. Such a sequence q thus induces a probability measure on Z + = {0, 1, 2, . . . } with mean one, given by:

( ) µ q (k) = Z k-1 q 2k -1 k -1 q k , k ≥ 0.
We shall consider random labelled trees, sampled as follows. First, let T be a Bienaymé-Galton-Watson tree with o spring distribution µ q , which means that the probability that T equals a given nite tree τ is u ∈τ µ q (k u ). For every subset A ⊂ Z + such that µ q (A) 0 and for every n ≥ 1, we let T A,n be such a tree conditioned to have exactly n vertices with o spring in A; scaling limits of such trees have been investigated by Kortchemski [Kor ] and Rizzolo [Riz ] (who also considers a di erent model of trees satisfying a 'Markov branching' property). We shall be particularly interested in the sets A = Z + , A = {0}, and A = N corresponding respectively to conditioning on the total progeny, on the number of leaves, and on the number of internal vertices. We let ζ (T A,n ) be the number of edges of T A,n .

Next, conditional on the tree T (or T A,n ), we sample uniformly random labels ( (u)) u ∈T satisfying the conditions described in Section . : the root has label ( ) = 0 and the sequences ( (ui) -(u)) 1≤i ≤k u are independent when u ranges over all internal vertices of T and are distributed respectively uniformly at random in B + k u . Let us observe that the cardinal of B + k is precisely the binomial factor 2k -1 k-1 in the de nition of µ q in ( ). Also, it is well-known and easy to check that a uniform random bridge in B + k has the law of the rst k steps of a random walk conditioned to end at 0, with step distribution i ≥-1 2 -i-2 δ i , which is centred and with variance 2.

One easily checks (see e.g. [Mar b, Proposition ]) that this labelled tree (T , ( (u)) u ∈T ) is the one associated, in the bijection described previously, with a pointed Boltzmann map sampled from P q,• . Finally, the tree is conditioned to have n vertices, or n internal vertices, or n leaves, when the map is conditioned to have n -1 edges, n faces, and n + 1 vertices respectively. Thanks to Property (iii) of the bijection in the preceding section, µ q is the law of the half-degree of the root-face under P q,• . For the rest of this paper, we further assume that it belongs to the domain of attraction of a stable law with index α ∈ (1, 2]. We refer the reader to [CR , Proposition ] for three equivalent assumptions. Let us remark, using the notation from this reference, that the root-face under P q has degree 2k with probability proportional to q k W (k ) q which, under our assumption, behaves as q k r -k q k -α -1/2 . Using the fact that q k r -k q is almost ν q (k) and that ν q has regularly varying tails with index α -1/2, we see that, informally, ν q (k) behaves as k -α -1/2 , so nally q k W (k ) q ≈ k -2α -1 and so the root-face under P q has regularly varying tails with index 2α > 2. This can be made rigorous using similar arguments to [CR , Proposition ].

. Continuous labelled trees

In this short section, we brie y describe the continuous limits of labelled size-conditioned Bienaymé-Galton-Watson trees, the statement and proof of the convergence are given in Section .

For the rest of this paper, we x an admissible and critical sequence q such that its support generates the whole group Z and such that µ q de ned by ( ) belongs to the domain of attraction of a stable law with index α ∈ (1, 2]; the dependence in α shall mostly be implicit in what follows. By the standard theory of stable domains of attraction, there exists an increasing sequence α ) whose law is given by the Laplace exponent E[exp(-λX (α ) )] = exp(λ α ) for every λ ≥ 0. Recall that n -1/α B n is slowly varying at in nity and that if µ q has variance σ 2 q ∈ (0, ∞), then this falls in the case α = 2 and we may take B n = (nσ 2 q /2) 1/2 . We stress that with this normalisation, X (2) has the centred Gaussian law with variance 2.

(B n ) n ≥1 such that if (ξ n ) n ≥1 is a sequence of i.i.d. random variables sampled from µ q , then B -1 n (ξ 1 + • • • + ξ n -n) converges in distribution to a random variable X (
. . The stable trees. -The continuous objects analogous to size-conditioned Bienaymé-Galton-Watson trees are the so-called stable Lévy trees. Let X = (X t ; t ∈ [0, 1]) denote the normalised excursion of the α-stable Lévy process with no negative jump whose value at time 1 has the law of X (α ) , and let further H = (H t ; t ∈ [0, 1]) be the associated height function; we refer to e.g. [Duq ] for the de nition of this object. In the case α = 2, the two processes X and H are equal, both to √ 2 times the standard Brownian excursion. In any case, H is a non-negative, continuous function, which vanishes only at 0 and 1. As any such function, it encodes a 'continuous tree' called the α-stable Lévy tree T of Duquesne, Le Gall and Le Jan [Duq , LGLJ ], which generalises the celebrated Brownian tree of Aldous [Ald ] in the case α = 2. Precisely, for every s, t

∈ [0, 1], set d(s, t) = H s + H t -2 min r ∈[min(s,t ),max(s,t )] H r .
One easily checks that d is a random pseudo-distance on [0, 1], we then de ne an equivalence relation on [0, 1] by setting s ∼ t whenever d(s, t) = 0. Consider the quotient space T = [0, 1]/∼, we let π be the canonical projection [0, 1] → T; then d induces a distance on T that we still denote by d. The space (T, d) is a so-called compact real-tree, naturally rooted at π (0) = π (1).

. . The continuous distance process. -We construct next another process L = (L t ; t ∈ [0, 1]) called the continuous distance process on the same probability space as H which is intrinsically di erent according as wether α = 2 or α < 2. Let us start with the latter case which is analogous to the discrete setting. Indeed, in the discrete setting, the label increment between a vertex and is parent was given by the value of a random discrete bridge of length equal to the o spring of the parent, at a time given by the position of the child, and these last two quantities are encoded by the Łukasiewicz path (recall Lemma . ). Loosely speaking, we do the same when α < 2, by taking random Brownian bridges.

Precisely, suppose that α < 2 and let (b i ) i ≥1 be i.d.d. standard Brownian bridges of duration 1 from 0 to 0 de ned on the same probability space as X and independent of the latter; by the scaling property, for every x > 0, the process (x 1/2 b 1 (t/x); t ∈ [0, x]) is a standard Brownian bridge of duration x. For every 0 ≤ s ≤ t ≤ 1, put

I s,t = inf r ∈[s,t ] X r .
For every t ∈ (0, 1), let ∆X t = X t -X t -≥ 0 be the 'jump' of X at time t and let (t i ) i ≥1 be a measurable enumeration of those times t such that ∆X t > 0. We then put for every t ∈ [0, 1]:

( ) L t = √ 2 i ≥1 ∆X 1/2 t i b i I t i ,t -X t i - ∆X t i 1 {I t i , t ≥X t i -} 1 {t i ≤t } .
According to Le Gall and Miermont [LGM , Proposition and ], this series converges in L 2 and the process L admits a continuous modi cation, even Hölder continuous for any index smaller than 1/(2α). The factor √ 2 is added here in the de nition of L in order to have statements without constants. When α = 2, the process X is √ 2 times the Brownian excursion so it has continuous paths. To understand the de nition, imagine that in the discrete setting, the tree T n is binary: internal vertices always have two children, then the label increment between such an internal vertex and its rst child equals -1, 0, or 1 with probability 1/3 each, and given a 'typical' vertex, each of its ancestor is either the rst or the second child of its parent, with probability roughly 1/2 each, so the sequence of increments along an ancestral line resembles a centred random walk with step -1 or 1 with probability 1/6 each and 0 with probability 2/3. In the continuous setting of the Brownian tree, we de ne the process L conditional on H as a centred Gaussian process satisfying for every s, t ∈ [0, 1],

E |L s -L t | 2 H = 2 3 • d(s, t) or, equivalently, E [L s L t | H] = 2 3 min r ∈[min(s,t ),max(s,t )] H r .
Again, the factor 2/3 removes the constants in our statements and will be explained below. This process is called the head of Brownian snake driven by H [ LG , DLG ]; it is known, see, e.g. [LG , Chapter IV. ] that it admits a continuous version. In all cases α ∈ (1, 2], without further notice, we shall work throughout this paper with the continuous version of L. Observe that, almost surely, L 0 = 0 and L s = L t whenever s ∼ t so L can be seen as a random motion indexed by T by setting L π (t ) = L t for every t ∈ [0, 1]. We interpret L x as the label of an element x ∈ T so the pair (T, (L x ; x ∈ T)) is analogous to labelled plane trees.

R

. . -We point out that, when α = 2, the process L is loosely speaking a Brownian motion indexed by the Brownian tree, which is denoted by S in [Mar a], but it is not a Brownian motion indexed by the stable tree in the case α < 2 (along an ancestral line, it may jump at the branch-points); such a Brownian motion is studied in [Mar a].

. Scaling limits of labelled trees Throughout this section, we x A ⊂ Z + such that either A or Z + \ A is nite and µ q (A) 0 and for every n ≥ 1, we let T A,n be a Bienaymé-Galton-Watson tree with o spring distribution µ q conditioned to have exactly n vertices with o spring in A; recall that ζ (T A,n ) denotes the number of edges of T A,n . Then conditional on T A,n , we sample uniformly random labels ( (u)) u ∈T A, n as in Section . .

Duquesne [Duq ] in the case A = Z + (so ζ (T A,n ) = n -1) and then Kortchemski [Kor ] in the general case, proved the convergence of the Łukasiewicz path and height process:

( ) 1 B ζ (T A, n ) W n (ζ (T A,n )t), B ζ (T A, n ) ζ (T A,n ) H n (ζ (T A,n )t) t ∈[0,1] (d ) -→ n→∞ (X t , H t ) t ∈[0,1] , in D([0, 1], R) ⊗ C([0, 1], R).
Let us point out that the work [Kor ] focuses on the case A = {0} and most of the results we shall need are developed in this case, but as explained in Section there, the arguments extend to the general case, at least as long as either A or its complement is nite.

R

. . -As observed by Kortchemski [Kor ], see e.g. Corollary . there for a stronger result, it holds that

n -1 ζ (T A,n ) P -→ n→∞ µ q (A) -1 .
Therefore, we may replace ζ (T A,n ) by µ q (A) -1 n in ( ) above and Theorem . below; moreover, since (n -1/α B n ) n ≥1 is slowly varying, we may replace B ζ (T A, n ) by µ q (A) 1/α B n . In the cases A = {0} and A = N, recall that T A,n is related to a Boltzmann map conditioned to have n + 1 vertices and n faces respectively. Since µ q (0) = 1µ q (N) = Z -1 q , this explains Remark . .

As alluded in the introduction, the key to prove Theorem . is the following result.

T . . -The convergence in distribution

B -1/2 ζ (T A, n ) L n (ζ (T A,n )t) t ∈[0,1] (d ) -→ n→∞ (L t ) t ∈[0,1] ,
holds in C([0, 1], R) jointly with ( ).

The proof of the convergence of L n occupies the rest of this section. We rst prove that it is tight and then we characterise the nite dimensional marginals, using two di erent arguments for the non-Gaussian case α < 2 and the Gaussian case α = 2, since the limit process L is de ned in two di erent ways. Indeed, let us comment on this statement and on the constants in the de nition of L. We assume A = Z + to ease the notation in this informal discussion.

For a vertex u ∈ T A,n , the label increments between consecutive ancestors are independent and distributed as X k, j when an ancestor has k ≥ 1 children and the one on the path to u is the j-th one, where (X k,1 , . . . , X k,k ) is uniformly distributed in B + k , as de ned in ( ). Since the latter has the law of a random walk conditioned to be at 0 at time k, with step distribution i ≥-1 2 -i-2 δ i which is centred and with variance 2, then a conditional version of Donsker's invariance principle for random bridges (see e.g. [Bet , Lemma ] for a detailed proof of the latter) yields ( )

(2k) -1/2 X k,kt t ∈[0,1] (d ) -→ n→∞ (b t ) t ∈[0,1] ,
where, as usual, on the left we have linearly interpolated, and b is the standard Brownian bridge. The factor √ 2 is the same as in the de nition of L for α < 2 in ( ) and one must check that the k's and j's converge towards the ∆X t i 's and the X t i -I t i ,t 's.

In the case α = 2, suppose furthermore that the variance σ 2 q of µ q is nite, so B n = (nσ 2 q /2) 1/2 . Then one can calculate the variance of X k, j , which equals 2j(kj)/(k + 1), and as we will see, there is typically a proportion about µ q (k) of corresponding ancestors, so (u) has variance about

k ≥1 k j=1 |u |µ q (k) 2j(k -j) k + 1 = |u| k ≥1 µ q (k) k(k -1) 3 ≈ |u| σ 2 q 3 .
If u is the vertex visited at time nt in lexicographical order, then, by ( ) we have |u | ≈ (n/B n )H t so we expect L n (nt), once divided by B 1/2 n , to be asymptotically Gaussian with variance

1 B n n B n H t σ 2 p 3 = 2 3 H t ,
which exactly corresponds to L t . The case α = 2 but µ q has in nite variance is more involved, but this sketch can be adapted by taking the truncated variance.

. . Tightness of the label process. -The rst step towards the proof of Theorem . is to show that the sequence of processes

B -1/2 ζ (T A, n ) L n (ζ (T A,n )t) t ∈[0,1]
is tight. This was proved in [Mar b, Proposition ] in a slightly di erent context of trees 'with a prescribed degree sequence' in a nite variance regime but the argument are easily adapted to our case. The main point is to apply Kolmogorov's tightness criterion; thanks to the properties of uniform random bridges in B + k , we can see that the increment of labels between a vertex u and one of its ancestors is about the square-root of the numbers of vertices branching o of the path u, , which can be described in terms of the Łukasiewicz path. We thus shall need later the following tail bounds.

L

. . -Fix any θ ∈ (0, 1/α). There exists c 1 , c 2 > 0 such that for every n large enough, for every 0 ≤ s ≤ t ≤ 1, every x ≥ 0, and every δ ∈ (0, α/(α -1)), we have

P W n (ζ (T A,n )s) -min s ≤r ≤t W n (ζ (T A,n )r ) > B ζ (T A,n ) |t -s | θ x ≤ c 1 exp(-c 2 x δ ).
Consequently, the moments of

B -1 ζ (T A,n ) |t -s | -θ (W n (ζ (T A,n )s) -min s ≤r ≤t W n (ζ (T A,n )r )) are uniformly bounded.
Proof. -First note that we may restrict ourselves to times |ts | ≤ 1/2. Let us start with the more familiar case A = Z + . It is well-known that W n is an excursion of a random walk S with i.i.d. steps distributed as k ≥-1 µ q (k + 1)δ k in the sense that we condition the path to hit -1 for the rst time at time n + 1. Moreover, such an excursion can be obtained by cyclicly shifting a bridge S n of this walk (i.e. conditioning the walk to be at -1 at time n + 1, but without the positivity constraint) at the rst time it realises its overall minimum, see e.g. Figure in [Mar a]; this operation is called a discrete Vervaat transform, see e.g. Pitman [Pit , Chapter . ] for details. Our claim holds when W n is replaced by S, in which case we may take s = 0; indeed, according to Kortchemski [Kor , Proposition ], it holds that P S(ns)min

s ≤r ≤t S(nr ) > xB n |t -s | ≤ c 1 exp(-c 2 x δ ).
Since n -1/α B n is slowly varying at in nity, the so-called Potter bounds (see e.g. [ BS , Lemma . ] or [Kor , Equation ]) assert that for every ε > 0, there exists a constant c depending only on ε such that for every n large enough,

(n|t -s |) -1/α B n |t -s | n -1/α B n ≤ c • |t -s | -ε ,
and so P S(ns)min

s ≤r ≤t S(nr ) > c • x • |t -s | -ε +1/α • B n ≤ c 1 exp(-c 2 x δ ).
One can then transfer this bound to S n ; an argument based on the Markov property applied to S indeed results in an absolute continuity between the rst n/2 steps of S and of S n , see e.g. [Kor ], near the end of the proof of Theorem there. Finally, we can transfer this bound from S n to W n using the preceding construction from a cyclic shift, see e.g. the end of the proof of Equation in [Mar a].

In the case A = {0}, the construction of W n from a bridge S n is discussed by Kortchemski [Kor , Section . ], and as explained in Section there, and it extends mutatis mutandis to the general case A either nite or co-nite. Here, the bridge S n is obtained by conditioning the walk S the be at -1 after its n-th jump in the set A -1. Therefore, it su ces again to prove our claim when W n is replaced by S n and |ts | < 1/2. Again, we may cut the path of S n at the time it realises its (n/2)-th jump in the set A -1 and this path is absolutely continuous with respect to that of the unconditioned walk S cut at the analogous stopping time. This follows from the same argument as alluded above, appealing to the strong Markov property. So nally, we have reduced our claim to showing that it holds when W n is replaced by S, when s = 0, and ζ (T A,n ) is replaced by the time of the n-th jump of S in the set A -1. This random time, divided by n converges almost surely towards µ q (A), see e.g. [ Kor , Lemma . ] so we may replace it by (1 ± γ )µ q (A)n with a xed γ ∈ (0, 1) arbitrarily small and conclude from the previous bound on S (it only a ects the constants).

We next turn to the proof of tightness of the label process. Recall that we may replace B ζ (T A, n ) by B n . Our argument closely follows the proof of Proposition in [Mar b].

Proof of the tightness in Theorem . . -Fix q > 2α α -1 and β ∈ (1, q(α -1) 2α ). We aim at showing that for every n large enough, for every pair 0 ≤ s ≤ t ≤ 1, it holds that ( )

E |L n (ζ (T A,n )s) -L n (ζ (T A,n )t)| q ≤ C(q) • B q/2 n • |t -s | β ,
where here and in all this proof, C(q) stands for some constant, which will vary from one equation to the other, which depends on q, β, and the o spring distribution, but not on n nor s nor t. Tightness follows from ( ) appealing to the standard Kolmogorov's criterion. Without loss of generality, we may, and do, restrict to those times s and t such that |t -s | ≤ 1/2 and both ζ (T A,n )s and ζ (T A,n )t are integers. Let us then denote by u and the vertices corresponding to the times ζ (T A,n )s and ζ (T A,n )t respectively in lexicographical order, so

L n (ζ (T A,n )s) -L n (ζ (T A,n )t) = (u) -( ).
Let u ∧ , be the most recent common ancestor of u and and further û and ˆ be the children of u ∧ which are respectively ancestor of u and . We stress that u and correspond to deterministic times, whereas u ∧ , û and ˆ correspond to random times which are measurable with respect to T A,n . We write:

(u) -( ) = w ∈ û,u (w) -(pr (w)) + ( ( û) -( ˆ )) + w ∈ ˆ , (pr (w)) -(w) .
Recall the notation 1 ≤ χ û ≤ χ ˆ ≤ k u∧ for the relative position of û and ˆ among the children of u ∧ . By construction of the labels on T A,n , conditional on the tree, the di erence ( û) -( ˆ ) is distributed as X p,i -X p, j with p = k u∧ , i = χ û and j = χ ˆ and where X p has the uniform distribution on the set of bridges B + p . According to Le Gall and Miermont [LGM , Lemma ], we thus have

E | ( û) -( ˆ )| q T A,n ≤ C(q) • (χ ˆ -χ û ) q/2 .
Next, x w ∈ û, u ; since (pr (w)) = (pr (w)k pr (w ) ), similarly, we have

E | (w) -(pr (w))| q T A,n ≤ C(q) • (k pr (w ) -χ w ) q/2 ,
and, for every w ∈ ˆ , ,

E | (pr (w)) -(w)| q T A,n ≤ C(q) • χ q/2 w .
It was argued in [Mar b, Equation ], appealing to the so-called Marcinkiewicz-Zygmund inequality, that if Y 1 , . . . , Y m are independent and centred random variables which admit a nite q-th moment, then

E m i=1 Y i q ≤ C(q) • m i=1 E [|Y i | q ] 2/q q/2 .
In our context, this reads

E | (u) -( )| q T A,n ≤ C(q) • w ∈ û,u (k pr (w ) -χ w ) + (χ ˆ -χ û ) + w ∈ ˆ , χ w q/2 ≤ C(q) • w ∈ û,u (k pr (w ) -χ w ) + (χ ˆ -χ û ) q/2 + w ∈ ˆ , χ w q/2 . ( )
Let us rst consider the rst term in ( ). Appealing to Lemma . , we have

χ ˆ -χ û = W n ( û) -W n ( ˆ ),
and similarly, for every w ∈ û, u ,

k pr (w ) -χ w = W n (w) -W n (pr (w)k pr (w ) ) = W n (wk w ) -W n (pr (w)k pr (w ) ), so w ∈ û,u (k pr (w ) -χ w ) + (χ ˆ -χ û ) = W n (u) -W n ( ˆ ) = W n (u) -inf [u, ]
W n .

ON SCALING LIMITS OF PLANAR MAPS WITH STABLE FACE-DEGREES Then Lemma . applied with θ = 2β/q < 1/α yields

E       w ∈ û,u (k pr (w ) -χ w ) + (χ ˆ -χ û ) q/2       ≤ C(q) • B q/2 n • |t -s | β .
We next focus on the second term in ( ). We would like to proceed symmetrically but there is a technical issue: on the branch û, u , we relied on the fact that (wk w ) = (w) in order to only count the number of vertices branching o of this path strictly to the right, but this is not the case on ˆ , : we do not have (w1) = (w) in general so we must also count the vertices on this path. Let T - A,n be the 'mirror image' of T A,n , i.e. the tree obtained from T A,n by ipping the order of the children of every vertex; let us write w -∈ T - A,n for the mirror image of a vertex w ∈ T A,n ; make the following observations: (i) T - A,n has the same law as T A,n , so in particular, their Łukasiewicz paths have the same law; (ii) for every w ∈ ˆ , , the quantity χ w -1 in T A,n corresponds to the quantity k pr (w -)χ w -in T - A,n ; (iii) the lexicographical distance between the last descendant in T - A,n of respectively ˆ -and -is smaller than the lexicographical distance between ˆ and in T A,n (the elements of ˆ , = ˆ -, -are missing).

With theses observations, the previous argument used to control the branch û, u shows that

E       w ∈ ˆ , (χ w -1) q/2       ≤ C(q) • B q/2 n • |t -s | β .
Finally, as proved recently (for the conditioning A = Z + but the general case follows similarly) in [Mar a]: for every γ < (α -1)/α, ( )

E # ˆ , q/2 ≤ C(q) • B ζ (T A, n ) ζ (T A,n ) q/2 • |t -s | γ q/2 ,
which is smaller than the bound we are looking for; indeed, since we assume that both ζ (T A,n )s and ζ (T A,n )t are integers, then ζ (T A,n ) -q/2 ≤ |ts | q/2 ≤ 1 and γ can be chosen close enough to (α -1)/α to ensure that |ts | (γ +1)q/2 ≤ |ts | β .

Let us mention that we have hidden the technical di culties in ( ). Nevertheless, there is a di erent argument which does not necessitate any control on the length of the branches. Indeed the bound ( ) answers in this context of size-conditioned Bienaymé-Galton-Watson trees Remark in [Mar b] on trees with a prescribed degree sequence. We could have argued instead as in the proof of Proposition there that, if χ w ≥ 2, then χ w ≤ 2(χ w -1), so in order to control the moments of w ∈ ˆ , χ w , it su ces to bound those of #{w ∈ ˆ , : χ w = 1}. But according to [Mar a, Lemma ] (which recasts [Mar b, Corollary ] in the context of size-conditioned Bienaymé-Galton-Watson trees), with high probability, uniformly for all pair of vertices ˆ , such that ˆ is an ancestor of , (2) there is a proportion at most 1µ q (0)/2 < 1 of individuals w ∈ ˆ , such that χ w = 1. Then the bound ( ) holds under the conditional expectation with respect to this event, so tightness of the label process holds conditional on this event, and so also unconditionally.

. . Finite dimensional marginals in the non-Gaussian case. -In this subsection, we assume that α < 2, and prove the following result which, together with the tightness obtained in the preceding subsection, concludes the proof of Theorem . in this case.

(2) And the path ˆ , has length at least of order ln n, but shorter paths do not cause any issue.

P

. . -For every k ≥ 1 and every 0

≤ r 1 < • • • < r k ≤ 1, it holds that B -1/2 ζ (T A, n ) L n ( ζ (T A,n )r i ) 1≤i ≤k (d ) -→ n→∞ (L r i ) 1≤i ≤k ,
jointly with ( ).

Our argument follows closely that of Le Gall and Miermont [LGM , proof of Proposition ] who considered the two-type tree associated with the maps via the Bouttier-Di Francesco-Guitter bijection, whereas we use the Janson-Stefánsson bijection which eliminates several technicalities. The argument relies of the convergence of the Łukasiewicz path in ( ) which we assume for the rest of this subsection to hold almost surely, appealing to Skorokhod's representation Theorem. To ease the exposition, we start with the one-dimensional marginals.

Proof in the case k = 1. -Fix t ∈ [0, 1] and recall the notation

I s,t = inf r ∈[s,t ] X r for every s ∈ [0, t]. Let (s i ) i ≥1 be those times s ∈ [0, t] such that X s-< I s,t ,
which are ranked in decreasing order of the values of the jumps of X:

∆X s 1 > ∆X s 2 > . . . Similarly, let κ n be the number of integers k ∈ {0, . . . , ζ (T A,n )t -1} such that W n (k) = min r ∈[k, ζ (T A, n )t ] W n (r ),
and let us denote by a n,1 , . . . , a n,κ n these integers, ranked so that

W n (a n,1 + 1) -W n (a n,1 ) ≥ • • • ≥ W n (a n,κ n + 1) -W n (a n,κ n ).
It follows from the almost sure version of ( ) that almost surely, for every i ≥ 1, we have ( )

1 ζ (T A,n ) a n,i -→ n→∞ s i , 1 B ζ (T A,n ) W n (a n,i + 1) -W n (a n,i ) -→ n→∞ ∆X s i , 1 B ζ (T A,n ) min k ∈[a n, i +1, ζ (T A, n )t ] W n (k) -W n (a n,i ) -→ n→∞ I s i ,t -X s i -, Let u 0 , u 1 , . . . , u ζ (T A,n
) be the vertices of T A,n listed in lexicographical order. Observe that the a n,i 's are exactly the indices of the strict ancestors of u ζ (T A,n )t . We may then write

L n ( ζ (T A,n )t ) = (u ζ (T A, n )t ) = κ n i=1 ( (u ψ (a n,i ) ) -(u a n, i )),
where u ψ (a n, i ) is the only child of u a n,i which is an ancestor of u ζ (T A,n )t . We claim that only the rst values of i matters. Indeed, by classical results on uctuation theory, it is well known that

X t = i ≥1 (I s i ,t -X s i -),
whence, for every ε > 0, there exists an integer N ≥ 1 such that with probability at least 1ε, it holds that

X t - i ≤N (I s i ,t -X s i -) ≤ ε/2.
Then ( ) and ( ) imply that for every n su ciently large, with probability at least 1 -2ε, it holds that

1 B ζ (T A, n ) W n ( ζ (T A,n )t ) - N ∧κ n i=1 min k ∈[a n, i +1, ζ (T A, n )t ] W n (k) -W n (a n,i ) < ε.
Observe that the left-hand side equals

1 B ζ (T A, n ) κ n i=N +1 min k ∈[a n,i +1, ζ (T A,n )t ] W n (k) -W n (a n,i ),
which is therefore arbitrarily small when xing N large enough. Now recall that, conditional on T A,n , the label increments (u ψ (a n,i ) ) -(u a n,i ) for 1 ≤ i ≤ κ n are independent and distributed as X k i , χ i where

(X k i ,1 , . . . , X k i ,k i ) is a uniform random bridge in B + k i de ned in ( ),
where

k i = W n (a n,i + 1) -W n (a n,i
) + 1 is the number of children of a n,i , and where

χ i = W n (a n,i + 1) -W n (ψ (a n,i )) + 1 is the position of ψ (a n,i ) amongst its siblings; note that W n (ψ (a n,i )) = min j ∈[a n,i +1, ζ (T A,n )t ] W n (j).
As in the preceding section, according to Le Gall and Miermont [LGM , Equation ], there exists some universal constant K > 0 such that

E | (u ψ (a n,i ) ) -(u a n, i )| 2 T A,n = K χ i (k i -χ i ) k i ≤ K min j ∈[a n, i +1, ζ (T A, n )t ]
W n (j) -W n (a n,i ) .

Since, conditional on T A n , these increments are centred and independent, we conclude that

E B -1/2 ζ (T A,n ) κ n i=N +1 ( (u ψ (a n, i ) ) -(u a n, i )) 2 T A,n ≤ K B ζ (T A,n ) κ n i=N +1 min j ∈[a n,i +1, ζ (T A,n )t ] W n (j) -W n (a n,i ),
which, on a set of probability at least 1 -2ε for every n large enough, is bounded by Kε according to the preceding discussion.

We next focus on a n,1 , . . . , a n, N . Conditional on X, let (γ i ) 1≤i ≤N be independent Brownian bridges of length ∆X s i respectively; it follows from ( ) and ( ) together with Donsker's invariance principle for random bridges ( ) that

B -1/2 ζ (T A, n ) N i=1 ( (u ψ (a n, i ) ) -(u a n,i )) (d ) -→ n→∞ √ 2 N i=1 γ i (X s i -I s i ,t ),
and the right-hand side further converges in distribution as N → ∞ towards

√ 2 i ≥1 γ i (X s i -I s i ,t
) which has the same law as L t = √ 2 i ≥1 γ i (I s i ,t -X s i -) by the time-reversal property of the Brownian bridge.

We next brie y sketch the argument for the multi-dimensional marginals.

Proof in the case k ≥ 2. -To ease the notation, we only treat the case k = 2, but the arguments are valid in the more general case. Let us x 0 < s < t; let us denote by 0

= a n,0 < • • • < a n,κ n the indices of the strict ancestors of u ζ (T A,n )s , and let similarly 0 = a n,0 < • • • < a n,κ n be the indices corresponding to the ancestors of u ζ (T A, n )t . Let j(n) ∈ {0, . . . , ζ (T A,n )s } be the index such that u j(n) is the last common ancestor of u ζ (T A, n )s and u ζ (T A, n )t ; we implicitly assume that j(n) < ζ (T A,n )s but this case is treated similarly. Let i(n) ∈ {0, . . . κ n ∧ κ n } be the index such that j(n) = a n,i(n) = a n,i(n) . Note that j(n) is the unique time such that W n (j(n)) ≤ min k ∈[ ζ (T A,n )s , ζ (T A, n )t ] W n (k) < min k ∈[j(n)+1, ζ (T A, n )s ] W n (k).
By analogy with the discrete setting, we interpret the times r ∈ [0, s] such that X r -< I r,s as the times of visit of the ancestors of the vertex visited at time s, and similarly for t. Introduce then the unique time r 0 ∈ [0, s] such that X r 0 -< I s,t < I r 0 ,s , which intuitively corresponds to the time of visit of the last common ancestor of the vertices visited at time s and t, and indeed, from the almost sure version of ( ), r 0 is the almost sure limit of ζ (T A,n ) -1 j(n).

Let us consider the label increments at the branch-point: conditional on X, let γ be a Brownian bridge of length ∆X r 0 , then similar arguments as in the one-dimensional case show that the pair

B -1/2 ζ (T A, n ) L n (a n,i(n)+1 ) -L n (j(n)), L n (a n,i(n)+1 ) -L n (j(n)) converges in distribution as n → ∞ towards √ 2 • (γ (X r 0 -I r 0 ,s ), γ (X r 0 -I r 0 ,t )).
If one removes the branch-point from the subtree of T A,n spanned by its root and the vertices u ζ (T A,n )s and u ζ (T A, n )t , then one gets three branches and the label increments between their root and their leaf are independent; we may apply the arguments of the previous proof to prove that these three increments, divided by

B 1/2 ζ (T A,n
) converge in distribution towards the 'label increments' given by L. Details are left to the reader, we refer to the end of the proof of Proposition in [LGM ].

. . Finite dimensional marginals in the Gaussian case. -We now focus on the Gaussian regime α = 2. As opposed to the other regimes, we consider random marginals. Precisely, we prove the following result.

P

. . -For every k ≥ 1, sample U 1 , . . . , U k i.i.d. uniform random variables in [0, 1] independently of the labelled trees, then the convergence

B 1/2 ζ (T A,n ) L n ( ζ (T A,n )U i ) 1≤i ≤k (d ) -→ n→∞ (L U i ) 1≤i ≤k ,
holds jointly with ( ), where the process L is independent of U 1 , . . . , U k .

Since we know that the sequence of continuous processes

B -1/2 ζ (T A, n ) L n (ζ (T A,n ) •
) is tight, this su ces to characterise the subsequential limits as L. Indeed, given any nite collection of xed times in [0, 1], one can approximate them by sampling su ciently many i.i.d. uniform random variables in [0, 1]; then the equicontinuity given by the tightness shows that the images of L n at these random times approximate well the values at the deterministic times, and the same holds for the uniformly continuous limit L, see e.g. Addario-Berry and Albenque [ ABA , proof of Proposition . ] for a detailed argument.

As previously, we rst treat the one-dimensional case.

Proof in the case k = 1. -The approach was described earlier in this section. Sample U uniformly at random in [0, 1] independently of the rest and note that the vertex u n visited at the time ζ (T A,n )U in lexicographical order has the uniform distribution in T n \ { } (we shall omit the fact that is excluded for the sake of clarity). Let us write

1 B ζ (T A,n ) 1/2 (u n ) = B ζ (T A, n ) ζ (T A,n ) |u n | 1/2 • ζ (T A,n ) B 2 ζ (T A, n ) |u n | 1/2 (u n ).
It follows from ( ) that the rst term on the right converges in distribution towards H U , it is therefore equivalent to show that, jointly with ( ), we have the convergence in distribution

( ) 3ζ (T A,n ) 2B 2 ζ (T A, n ) |u n | 1/2 (u n ) (d ) -→ n→∞ G
where G has the standard Gaussian distribution. Recall that according to Remark . , we may, and do, replace

ζ (T A,n ) by µ q (A) -1 n and B ζ (T A,n ) by µ q (A) -1/2 B n .
For every k ≥ j ≥ 1, let us denote by A k, j (u n ) the number of strict ancestors of u n with k children, among which the j-th one is again an ancestor of u n :

A k, j (u n ) = # { ∈ , u n : k = k and j ∈ , u n } .

ON SCALING LIMITS OF PLANAR MAPS WITH STABLE FACE-DEGREES

The idea is to decompose (u n ) as the sum of the label increments between two consecutive ancestors w and pr (w); conditionally on T n and u n , these random variables are independent and, whenever k pr (w ) = k and w = pr (w)j, the label increment has the law of j-th marginal of a uniform random bridge in B + k , which is centred and has variance, say, σ 2 k, j . This variance is known explicitly, see e.g. [ MM , page (3) ]: we have ( )

σ 2 k, j = 2j(k -j) k + 1 , so k j=1 σ 2 k, j = k(k -1) 3 .
Let ∆(T A,n ) denote the largest o spring of a vertex of T A,n . As in the classical proof of the central limit theorem, we may write for every z ∈ R,

E exp iz 3n 2B 2 n |u n | 1/2 (u n ) T n , u n = ∆(T A, n ) k =1 k j=1 1 - z 2 2 3nσ 2 k, j 2B 2 n |u n | + o nσ 2 k, j B 2 n |u n | 2 A k, j (u n ) = exp - z 2 2 ∆(T A, n ) k =1 k j=1 A k, j (u n ) 3nσ 2 k, j 2B 2 n |u n | + O nσ 2 k, j B 2 n |u n | 2 .
We claim that ( )

∆(T A, n ) k=1 k j=1 3nσ 2 k, j 2B 2 n |u n | A k, j (u n ) P -→ n→∞ 1, and
∆(T A,n ) k=1 k j=1 nσ 2 k, j B 2 n |u n | 2 A k, j (u n ) P -→ n→∞ 0.
Then an application of Lebesgue's Theorem yields our claim. First, since ∆(T A,n ) is the largest jump plus one of the Łukasiewicz path, from ( ) in which X is a Brownian excursion, it su ces to work conditional on the event ∆(T A,n ) ≤ ηB n for any η > 0 xed. A short argument uses the following very recent spinal decomposition due to Kortchemski and Richier [KR ]. For every 0 ≤ i < |u n |, let a i (u n ) be the ancestor of u n with height

|a i (u n )| = i, let k i (u n ) = k a i (u n )
be its number of children, and χ i (u n ) = χ a i +1 (u n ) be the position of its only child which is an ancestor of u n . Then according to [KR , Theorem ], the vector

((k i (u n ), χ i (u n )); 0 ≤ i < |u n |)
approaches in total variation distance the vector

((K i , i )); 0 ≤ i < RnB -1 n ),
where (K i ) i ≥0 is a sequence of i.i.d. random variables sampled from the size-biased law µ * q = k ≥1 kµ q (k)δ k , conditional on this sequence, the i 's are independent and sampled uniformly at random in {1, . . . , K i } respectively, and R is independent and has the Rayleigh distribution 2xe -x 2 dx on (0, ∞). For every h ≥ 1 and k ≥ j ≥ 1, let us set

N k, j (h) #{1 ≤ i ≤ h : (K i , i ) = (k, j)}.
Then we may reduce ( ) to proving that for any ε > 0, we have

( ) lim η ↓0 sup η 1/2 nB -1 n ≤h ≤η -1/2 nB -1 n lim sup n→∞ P ηB n k =1 k j=1 3nσ 2 k, j 2B 2 n h N k, j (h) -1 > ε = 0, and
( ) lim η ↓0 sup η 1/2 nB -1 n ≤h ≤η -1/2 nB -1 n lim sup n→∞ P ηB n k =1 k j=1 nσ 2 k, j B 2 n h 2 N k, j (h) > ε = 0,
(3) Note that they consider uniform random bridges in B + k +1 ! which we do by simple moment calculations. Indeed, we have rst

E ηB n k =1 k j=1 3nσ 2 k, j 2B 2 n h N k, j (h) = 3n 2B 2 n ηB n k=1 k j=1 σ 2 k, j P ((K 1 , 1 ) = (k, j)) = 3n 2B 2 n ηB n k =1 k j=1 σ 2 k, j µ q (k).
Let ξ have the law µ q , with unit expectation, then according to ( ), the preceding expectation equals

3n 2B 2 n ηB n k =1 k(k -1) 3 µ q (k) = n 2B 2 n Var ξ 1 {ξ ≤ηB n } + o(1) .
Recall that the function x → Var(ξ 1 {ξ ≤x } ) is slowly varying so the term η can be removed and it follows from [Fel , Theorem XVII. . ] that

E ηB n k=1 k j=1 3nσ 2 k, j 2B 2 n h N k, j (h) ∼ n→∞ n 2B 2 n Var ξ 1 {ξ ≤B n } -→ n→∞ 1.

Next, by independence

Var

ηB n k =1 k j=1 3nσ 2 k, j 2B 2 n h N k, j (h) = h 3n 2B 2 n h 2 ηB n k =1 k j=1 σ 4 k, j Var 1 {(K 1 , 1 )=(k, j)} .
Let us simply bound the last variance by the second moment, which equals µ q (k); according to ( ), we have σ 2 k, j ≤ k/2 for every pair 1 ≤ j ≤ k, so, from the preceding argument, the right-hand side is bounded by

h 3n 2B 2 n h 2 ηB n 2 ηB n k =1 k j=1 σ 2 k, j µ q (k) ∼ n→∞ h 3n 2B 2 n h 2 ηB n 6 2B 2 n n = 3ηn 4B n h ≤ 3η 1/2 4 ,
uniformly for η 1/2 nB -1 n ≤ h ≤ η -1/2 nB -1 n , which proves the convergence ( ). The preceding calculations also yield

E       ηB n k=1 k j=1 nσ 2 k, j B 2 n h 2 N k, j (h)       = h n B 2 n h 2 ηB n k =1 k j=1 σ 4 k, j µ q (k) ≤ η 1/2 3 (1 + o(1)),
uniformly for η 1/2 nB -1 n ≤ h ≤ η -1/2 nB -1 n and the convergence ( ) follows.

As in the case α < 2, we close this section by sketching the argument for the multi-dimensional marginals. One of the di erences is that now the contribution of the branch-points vanishes.

L

. . -We have the convergence in probability

B -1/2 ζ (T A, n ) max u ∈T n max 1≤i ≤k u (ui) -min 1≤i ≤k u (ui) P -→ n→∞ 0.
Proof. -Note that again, we may, and shall, replace B ζ (T A, n ) by B n . We follow the proof of [Mar b, Proposition ] which dealt with trees 'with a prescribed degree sequence' in the nite-variance regime.

Recall that a uniform random bridge X k in B + k has the same law as the rst k steps of a random walk with step distribution i ≥-1 2 -i-2 δ i conditioned on being at 0 at time k. According to Lemma in [Mar b], there exists two constants c, C > 0 such that for every k ≥ 1 and x ≥ 0, we have

P max 1≤i ≤k X k,i -min 1≤i ≤k X k,i > x ≤ Ce -cx 2 /k . Let ζ k (T A,n
) denote the number of individuals in T A,n with k o springs and let ∆(T A,n ) be the largest o spring in T A,n . Fix ε > 0 and recall the bound ln(1x) ≥ -x 1-x for x < 1. We then have

P max u ∈T n max 1≤i ≤k u (ui) -min 1≤i ≤k u (ui) ≤ εB 1/2 n T A,n = ∆(T A,n ) k=1 P max 1≤i ≤k X k,i -min 1≤i ≤k X k,i ≤ εB 1/2 n T A,n ζ k (T A, n ) ≥ exp -C ∆(T A, n ) k =1 ζ k (T A,n )e -cε 2 B n /k (1 + o(1)) ,
and the claim reduces to showing the convergence in probability

∆(T A,n ) k=1 ζ k (T A,n )e -cε 2 B n /k P -→ n→∞ 0.
As we have seen, the ratio ∆(T A,n )/B n tends to 0 in probability; since x → x 2 e -x is decreasing on [2, ∞), we have on a set of high probability as n → ∞,

∆(T A,n ) k=1 ζ k (T A,n )e -cε 2 B n /k ≤ ∆(T A, n ) k =1 k 2 ζ k (T A,n ) B 2 n × B 2 n ∆(T A,n ) 2 e -cε 2 B n /∆(T A, n ) ,
and the last term

(B n /∆(T A,n )) 2 exp(-cε 2 B n /∆(T A,n
)) tends to 0 in probability so it only remains to prove that the sequence

∆(T A, n ) k =1 k 2 ζ k (T A,n )/B 2 n is bounded in probability in the sense that lim K →∞ lim sup n→∞ P ∆(T A, n ) k =1 k 2 ζ k (T A,n ) B 2 n > K = 0.
Let us intersect the preceding event with {∆(T A,n ) ≤ B n } whose probability tends to one. Let us translate our probability in terms of the Łukasiewicz path: write X n (i) = W n (i +1)-W n (i), then we aim at showing lim

K →∞ lim sup n→∞ P      ζ (T A, n ) i=1 X n (i) 2 B 2 n > K      ∩ max 1≤i ≤ζ (T A, n ) X n (i) ≤ B n = 0.
We then use the same reasoning as in the proof of Lemma . . For a path S and n ≥ 1, let us denote by ς A,n the time such that the n-th jump of S with values in the set A -1 is its ς A,n -th jump in total. Note that our event is shift-invariant so we may replace the excursion W n by the bridge S n obtained by conditioning the random walk S with step distribution k ≥-1 µ q (k + 1)δ k to be at -1 after its ς A,n -th jump. Then, by cutting the time interval in two and using a time-reversibility property for the second half due to Kortchemski [ Kor , Proposition . ], we may in fact only consider the rst half of the bridge, i.e. up to time ς A,n/2 . The latter is absolutely continuous with respect to the unconditioned random walk so, if (ξ i ) i ≥1 are i.i.d. copies of a centred random variable ξ sampled from k ≥-1 µ q (k + 1)δ k and if now ς A,n denotes the least time j ≥ 1 such that #{i ∈ {1, . . . , j} : ξ i ∈ A -1} = n, then there exists C > 0 such that for every n large enough, the probability

P      ζ (T A, n ) i=1 X n (i) 2 B 2 n > K      ∩ max 1≤i ≤ζ (T A, n ) X n (i) ≤ B n is bounded by C times P ς A, n/2 i=1 ξ 2 i B 2 n > K ∩ max 1≤i ≤ς A, n/2 ξ i ≤ B n ≤ P ς A, n/2 i=1 ξ 2 i B 2 n 1 {ξ i ≤B n } > K .
Since ς A,n/2 /n converges almost surely to 1/(2µ q (A)) (see e.g. [ Kor , Lemma . ]), we may replace ς A,n/2 by n/(2µ q (A)). Markov's inequality then yields

P ς A, n/2 i=1 ξ 2 i B 2 n 1 {ξ i ≤B n } > K ≤ n 2K µ q (A)B 2 n E ξ 2 1 {ξ ≤B n } ,
which converges to 1/(K µ q (A)) thanks to [Fel , Theorem XVII. . ] and our claim follows.

Proof of Proposition . in the case k ≥ 2. -Let us only restrict ourselves to the case k = 2 to ease the notation since the general case hides no extra di culty. We sample two independent uniform random vertices of T n , say, u n and n , and we let w n be their most recent ancestor, we denote by ûn and ˆ n the children of w n which are respectively an ancestor of u n and n in order to decompose

(u n ) = (w n ) + ( ( ûn ) -(w n )) + ( (u n ) -( ûn )),
and similarly for n . The important observation is that, conditional on T A,n , u n and n , the random variables (w n ), (u n ) -( ûn ) and ( n ) -( ˆ n ) are independent. According to ( ), we have

B ζ (T A, n ) ζ (T A,n ) (|w n |, |u n | -| ûn |, | n | -| ˆ n |) (d ) -→ n→∞ min r ∈[U ,V ] H r , H U -min r ∈[U ,V ] H r , H V -min r ∈[U ,V ] H r ,
where U and V are i.i.d uniform random variables on [0, 1] independent of H. We claim that, jointly with ( ), we have

( ) 3ζ (T A,n ) 2B 2 ζ (T A, n ) 1/2 (w n ) |w n | , (u n ) -( ûn ) |u n | -| ûn | , ( n ) -( ˆ n ) | n | -| ˆ n | (d ) -→ n→∞ (G 1 , G 2 , G 3 ) ,
where G 1 , G 2 , G 3 are i.i.d. standard Gaussian random variables. This actually follows from the arguments used in the proof of Proposition . in the case k = 1 which show not only the convergence of (u n ), but also that if a n is an ancestor of u n such that the ratio |a n |/|u n | converges in probability to some a ∈ (0, 1) as n → ∞, then we have

3ζ (T A,n ) 2B 2 ζ (T A, n ) 1/2 (a n ) |a n | , (u n ) -(a n ) |u n | -|a n | (d ) -→ n→∞ (G 1 , G 2 ) .
This implies the convergence of the rst two components in ( ). The convergence of the last one also holds since the role of u n and n is symmetric and so ( ) holds by independence. Since Lemma . implies that

B -1/2 ζ (T A, n ) ( ( ûn ) -(w n ), ( ˆ n ) -(w n )) P -→ n→∞ (0, 0) ,
We conclude from ( ) and ( ) that the pair

B -1/2 ζ (T A,n ) ( (u n ), ( n )) converges in distribution as n → ∞ towards 2 3 min r ∈[U ,V ] H r • G 1 + H U -min r ∈[U ,V ] H r • G 2 , min r ∈[U ,V ] H r • G 1 + H V -min r ∈[U ,V ] H r • G 3 , which is indeed distributed as (L U , L V ).
.

Scaling limits of maps

The main goal of this section is to prove Theorem . , we also state and prove scaling limits on the pro le of distances in Theorem . below. We rst prove that large pointed and non-pointed maps are close, in order to focus on pointed maps. Relying of the description of such maps by labelled trees, we then state and prove Theorem . . Finally, we prove Theorem . in the last three subsections. The proof of tightness in Theorem . follows from the functional convergence in Theorem . as in the pioneer work of Le Gall [LG ] who considered maps pointed at the origin of the root-edge, see also [LG , LGM ] for maps pointed as here; all these references rely on a di erent labelled tree obtained by the Bouttier-Di Francesco-Guitter bijection [BDFG ]. We recast their proof using the Janson-Stefánsson bijection [JS ].

Throughout this section, we x S ∈ {V , E, F }, and for every n ≥ 1, we sample a pointed map (M n , ) from P q,• S =n . Recall that ζ (M n ) denotes the number of edges of M n . Recall that we associate with (M n , ) a labelled tree (T n , ) with the same amount of edges, and, as discussed in Section . , T n has the law of T A,n where A = Z + if S = E, and A = 0 if S = V , and A = N if S = F .

. . On the behaviour of leaves in a large Bienaymé-Galton-Watson tree. -Recall that the leaves of the tree are in one-to-one correspondence with the vertices of M n di erent from the distinguished one; we shall need the following two estimates. First, let λ(T n ) be the number of leaves of T n . For every 0 ≤ j ≤ ζ (T n ), let further Λ(T n , j) denote the number of leaves amongst the rst j vertices of T n in lexicographical order, and make Λ a continuous function on [0, ζ (T n )] after linear interpolation.

L

. . -We have the convergence in probability

Λ(T n , ζ (T n )t) λ(T n ) ; t ∈ [0, 1] P -→ n→∞ (t; t ∈ [0, 1]).
Proof. -Kortchemski [ Kor , Corollary . ] (again for trees conditioned by the number of leaves, but it extends to the general case) proved that when we restrict to a time interval [η, 1] with η > 0, the probability that t →

Λ(T n ,ζ (T n )t ) λ(T n )
deviates from the identity decays sub-exponentially fast. We can then extend to the whole segment [0, 1] to get our result by 'mirror symmetry'. Fix t ∈ [0, η] and suppose that ζ (T n )t is an integer to ease the notation. It su ces to observe that the mirror Łukasiewicz path W - n visits approximately the same number of leaves in its last ζ (T n )t steps than the original Łukasiewicz path W n in its rst ζ (T n )t steps. Let T - n be the mirror tree and for every 0

≤ j ≤ ζ (T n ), let Λ -(T - n , j
) denote the number of leaves amongst the last j vertices of T - n in its lexicographical order. The point is that, in order to visit a vertex u on T n in the original lexicographical order, one must rst visits all its ancestors, whereas in the mirror order started from u, these ancestors have already been visited. Nevertheless, no other vertex of T n smaller than u in the original lexicographical has already been visited, so

Λ(T n , ζ (T n )t) = Λ -(T - n , ζ (T n )t -H n (ζ (T n )t)). Our claim then follows from the fact that sup t ∈[0,1] ζ (T n ) -1 H n (ζ (T n )t) converges in probability to 0.
The preceding result states that the leaves of the tree are homogeneously spread. Note that we could replace the leaves by the vertices with o spring in a given set B ⊂ Z + . The next result states that the inverse of the number of leaves, normalised to have expectation 1, converges to 1 in L 1 .

L

. . -We have the convergence in probability

lim n→∞ E 1 λ(T n ) 1 E[ 1 λ(T n ) ]
This convergence is [Mar b, Lemma ] in the nite-variance regime; the proof applies mutatis mutandis in our case since the arguments used there, which are due to Kortchemski [Kor ], hold in the more our general case. Following arguments from [Abr , BJM , BM ], it was then shown in [Mar b, Proposition ] that Lemma . yields the following comparison between pointed and non-pointed maps. S =n be the push-forward measure induced on M by P q,• S =n , then

P q S =n -ϕ * P q, S =n T V -→ n→∞ 0,
where • T V refers to the total variation norm.

Indeed, one can bound this total variation distance by the expectation in Lemma . with λ(T

n ) + 1 instead of λ(T n ). Observe that if (M n , ) is sampled from P q,• S =n , then, conditional on M n , the vertex is uniformly distributed in M n .
. . Radius and pro le. -Although for α ∈ (1, 2), we shall only obtain a convergence along subsequences of the metric spaces, still we do obtain some information about distances in large maps. Recall that we work with pointed maps (M n , ) sampled from P q,• S =n , but according to the preceding section, this pair is close to a non-pointed map sampled from P q S =n , in which we sample a vertex uniformly at random so the next result also holds in this context. Recall that ζ (M n ) denotes the number of edges of M n , let us denote by υ(M n ) its number of vertices.

Let

R(M n ) = max x ∈M n d gr (x, )
be the radius of the map; de ne also a point measure on Z + , called the pro le of distances, by

ρ M n (k) = #{x ∈ M n : d gr (x, ) = k}. k ∈ Z + .
Finally, let ∆(M n ) be the longest distance in M n between and the two extremities of the root-edge (the other extremity is at distance ∆(M n ) -1).

T . . -Let L = sup t ∈[0,1] L t and L = inf t ∈[0,1] L t and observe that L and -L have the same law by symmetry. Then the following convergences in distribution hold as n → ∞:

(i) B -1/2 ζ (M n ) R(M n ) → L -L; (ii) B -1/2 ζ (M n ) ∆(M n ) → L; (iii) For every continuous and bounded function ϕ, 1 υ(M n ) k ≥0 ϕ(B -1/2 ζ (M n ) k)ρ M n (k) (d ) -→ n→∞ ∫ 1 0 ϕ(L t -L)dt .
Proof. -We rely on the bijection with the labelled tree (T n , ). Let us set

L n = min 1≤i ≤ζ (T n ) L n (i) -1; in this bijection, we have R(M n ) = max 0≤i ≤ζ (T n ) L n (i) -L n .
Similarly, the farthest extremity of the root-edge of M n from corresponds to the last leaf of T n , which has label 0 so

∆(M n ) = -L n ,
and the rst two convergences are immediate consequences of Theorem . . We need a little more work for the third assertion. Our argument shall also serve later in Section . and . . Recall the notation λ(T n ) for the number of leaves of T n , which equals υ(M n ) -1, and Λ(T n , j) for the number of leaves amongst the rst j vertices of T n in lexicographical order. For every 1 ≤ i ≤ λ(T n ), let where d gr is the graph distance of M n . We then extend d n to a continuous function on [0, n] 2 by 'bilinear interpolation' on each square of the form [i, i + 1] × [j, j + 1] as in [LG , Section . ] or [LGM , Section ].

De ne for every t ∈ [0, 1]:

H (n) (t) = B ζ (M n ) ζ (M n ) H n (ζ (M n )t), and
L (n) (t) = B -1/2 ζ (M n ) L n (ζ (M n )t),
and for every s, t ∈ [0, 1]: L (n) (r ) .

d (n) (s, t) = B -1/2 ζ (M n ) d n (ζ (M n )s, ζ (M n )t), D L (n) (s, t) = L (n) (s) + L (n) (t
Using the triangle inequality at a vertex where a geodesic from φ(u i ) to and a geodesic from φ(u j ) to in M n merge, Le Gall [LG , Equation ] (see also [LG , Lemma . ] for a detailed proof) obtained the bound ( )

d (n) (s, t) ≤ D L (n) (s, t) + 2B -1/2 ζ (M n ) ,
for every s, t ∈ [0, 1] such that both ζ (M n )s and ζ (M n )t are integers, but then also in the other cases. Let us point out that this bound was obtained using the coding of the Bouttier-Di Francesco-Guitter bijection, where L n is the so-called white label function of the two-type tree in the contour order. Nonetheless, as proved in [Mar b, Lemma ], this process equals (deterministically) our process L n when the trees are related by the Janson-Stefánsson bijection.

Recall from Section . that T n has the law of T A,n where A = Z + if S = E, and A = 0 if S = V , and A = N if S = F . Then Theorem . yields the convergence in distribution of continuous paths L r .

The bound ( ) then easily shows that d (n) is tight. Therefore, from every increasing sequence of integers, we can extract a subsequence along which we have ( ) where (D(s, t)) s,t ∈[0,1] depends a priori on the subsequence and satis es D ≤ D L , see [LG , Proposition . ] for a detailed proof in a similar context. In the next subsections, we implicitly restrict ourselves to a subsequence along which ( ) holds.

H (n) (
. . Tightness of metric spaces. -Appealing to Skorokhod's representation Theorem, let us assume that the convergence ( ) holds almost surely (along the appropriate subsequence). We claim that, deterministically, the convergence ( ) then holds. Let us rst construct the limit space. As limit of the sequence (d (n) ) n ≥1 , the fonction D, which is continuous on [0, 1] 2 , is a pseudo-distance. We then de ne an equivalence relation on [0, 1] by setting s ≈ t if and only if D(s, t) = 0, and we let M be the quotient [0, 1]/≈, equipped with the metric induced by D, which we still denote by D. We let Π be the canonical projection from [0, 1] to M which is continuous (since D is) so (M, D) is a compact metric space, which nally we endow with the Borel probability measure m given by the push-forward by Π of the Lebesgue measure on [0, 1].

Recall our de nition of the Gromov-Hausdor -Prokhorov distance. Recall from Section . that for every 1 ≤ i ≤ λ(T n ), we denote by (i) ∈ {1, . . . , ζ (T n )} the index such that u (i) is the i-th leaf of T n , so the sequence (φ(u (i) )) 1≤i ≤λ(T n ) lists without redundancies the vertices of M n di erent from . The set

R n = φ(u ( λ(T n )t ) ), Π(t) ; t ∈ [0, 1] .
d (n) ( λ(T n )s ) ζ (T n ) , ( λ(T n )t ) ζ (T n ) -D(s, t) ,
which, appealing to ( ), tends to 0 whenever the convergence ( ) holds, which concludes the proof of the tightness.

. . Characterisation of the limit in the Brownian case. -In this last subsection, we assume that α = 2 and we prove that ( ) holds without extracting a subsequence, and then so does ( ), with a limit which we next recall, following Le Gall [LG ] to which we refer for details. As functions on T 2 , we clearly have D * ≤ D L and in fact, D * is the largest pseudo-distance on T satisfying this property. Indeed, if D is another such pseudo-distance, then for every x, ∈ T, for every k ≥ 1 and every a 0 , a 1 , . . . , a k-1 , a k ∈ T with a 0 = x and a k = , by the triangle inequality, D(x, ) ≤ k i=1 D(a i-1 , a i ) ≤ k i=1 D L (a i-1 , a i ) and so D(x, ) ≤ D * (x, ). Furthermore, if we view D * as a function on [0, 1] 2 , then for all s, t ∈ [0, 1] such that π (s) = π (t) we have D * (π (s), π (t)) = 0. We deduce from the previous maximality property that D * is the largest pseudo-distance D on [0, 1] satisfying the following two properties: D ≤ D L and π (s) = π (t) implies D(s, t) = 0.

We point out that since His √ 2 times the standard Brownian excursion, then our process L corresponds to ( 8 9 ) 1/4 Z where Z is used to de ne the Brownian map in [LG ] and in subsequent papers so the standard Brownian map is (M, ( 9 8 ) 1/4 D * , m). Let D be a limit in ( ) and note that it satis es the two preceding properties; we claim that D = D * almost surely.

Our argument is adapted from the work of Bettinelli and Miermont [BM , Lemma ]. According to the maximality property of D * , the bound D ≤ D * holds almost surely so it su ces to prove that if

  4 D, m) is the (standard) Brownian map. (ii) If α < 2, then from every increasing sequence of integers, one can extract a subsequence along which we have the convergence in distribution in the sense of Gromov-Hausdor -Prokhorov, where (M, D, m) is a random compact measured metric space with Hausdor dimension 2α.

  tree on the le , and on the right, its height process on top and its label process below.

  Let ϕ : M • → M : (M, ) → M and let ϕ * P q,•

H

  (n) (t), L (n) (t), D L (n) (s, t) , L t , D L (s, t)) s,t ∈[0,1] ,where, similarly to the discrete setting,D L (s, t) = L s + L t -2 max min r ∈[s∧t,s∨t ] L r ; min r ∈[0,s∧t ]∪[s∨t,1]

  is a correspondence between (M n \ { }, B -1/2 ζ (M n )d gr , p gr ) and (M, D, m). Let further ν be the coupling between p gr and m given by∫ M n \{ }×M ϕ( , x)dν ( , x) = ∫ 1 0 ϕ φ(u ( λ(T n )t ) ), Π(t) dt,for every test function ϕ. Then ν is supported by R n by construction. Finally, the distortion of R n is given by sup s,t ∈[0,1]

  First, we view D L as a function on the tree T by settingD L (x, ) = inf {D L (s, t); s, t ∈ [0, 1], x = π (s) and = π (t)} ,for every x, ∈ T, where we recall the notation π : [0, 1] → T = [0, 1]/∼ for the canonical projection. Then we putD * (x, ) = inf k i=1 D L (a i-1 , a i ); k ≥ 1, (x = a 0 , a 1 , . . . , a k-1 , a k = ) ∈ T .The function D * is a pseudo-distance on T which can be seen as a pseudo-distance on [0, 1] by setting D * (s, t) = D * (π (s), π (t)) for every s, t ∈ [0, 1].

  t), L (n) (t), d (n) (s, t) s,t ∈[0,1]

	(d) -→

n→∞ (H t , L t , D(s, t)) s,t ∈[0,1] ,

(i) ∈ {1, . . . , ζ (T n )} be the index such that u (i) is the i-th leaf ofT n . Since j → Λ(T n , j) is non-decreasing, Lemma . is equivalent to

where as usual, we have linearly interpolated between integer values. Then observe that

which converges in law to ∫ 1 0 ϕ(L t -L)dt according to ( ) and Theorem . .

. . The Gromov-Hausdor -Prokhorov topology. -Let us next brie y de ne this topology used in Theorem . in a way that is tailored for our purpose. Let (X , d x , m x ) and (Y , d Y , m ) be two compact metric spaces equipped with a Borel probability measure. A correspondence between these spaces is a subset R ⊂ X × Y such that for every x ∈ X , there exists ∈ Y such that (x, ) ∈ R and vice-versa. The

Then we de ne the Gromov-Hausdor -Prokhorov distance between these spaces as the in mum of all those ε > 0 such that there exists a coupling ν between m X and m Y and a compact correspondence R between X and Y such that

This de nition is not the usual one and is due to Miermont [Mie , Proposition ]. We refer to Section there for more details on the Gromov-Hausdor -Prokhorov distance. Let us only recall that it makes separable and complete the set of measure-preserving isometry classes of compact metric spaces equipped with a Borel probability measure.

If (M n \ { }, d gr , p gr ) is the metric measured space given by the vertices of M n di erent from , their graph distance in M n and the uniform probability measure, then the Gromov-Hausdor -Prokhorov distance between (M n , d gr , p gr ), and (M n \ { }, d gr , p gr ) is bounded by one so it su ces to prove that from every increasing sequence of integers, one can extract a subsequence along which the convergence in distribution

holds for the Gromov-Hausdor -Prokhorov topology.

. . Tightness of distances. -Recall that the leaves of the labelled tree (T n , ) associated with (M n , ) are in bijection with the vertices of M n di erent from . As for the internal vertices of T n , they are each identi ed with their last child and so to each such internal vertex corresponds a leaf (the end of the right-most ancestral line starting from them) and therefore a vertex of M n \ { }. Let φ : T n → M n \ { } be the map which associates with each vertex of T n its corresponding vertex of M n . Let us list the vertices of T n as u

lexicographical order and for every i, j ∈ {0, . . . , ζ (M n )}, we set

It is known [ LG , Corollary . ] that the right-hand side is distributed as

where s is the (a.s. unique [LGW ]) point at which L attains its minimum. The point is that, in the discrete setting, d n describes the distances in the map between the vertices (φ(u i )) 0≤i ≤ζ (M n ) , and some vertices of M n may appear more often that others in this sequence so if one samples two uniform random times, they do not correspond to two uniform random vertices of the map. Nonetheless, this e ect disappears at the limit, according to ( ). Indeed, x X , Y i.i.d. uniform random variables on [0, 1] such that the pair (X , Y ) is independent of everything else, and set

Note that x and are uniform random vertices of M n \ { }, they can therefore be coupled with two independent uniform random vertices x and of M n in such a way that the conditional probability given M n that (x, ) (x , ) is at most 2/υ(M n ) → 0 as n → ∞; we implicitly assume in the sequel that (x, ) = (x , ). Since is also a uniform random vertex of M n , we obtain that ( )

= d gr ( , ).

By de nition, d gr (x, ) = d n ( ( λ(T n )X ), ( λ(T n )Y )).

Recall that the labels on T n describe the distances from in M n , we therefore have

L n (j) + 1.

We obtain ( ) by letting n → ∞ in ( ) along the same subsequence as in ( ), appealing also to ( ).


Sabrine Jlassi
email: sabrine.jlassi@mines-albi.fr

Florentin Berthet
email: florentin.berthet@mines-albi.fr

Gérard Bernhart
email: gerard.bernhart@mines-albi.fr

INVESTIGATION OF MECHANICAL PROPERTIES OF NONWOVEN SECOND GENERATION COMPOSITE MATERIAL ELABORATED THROUGH A MIXTURE OF CARBON FIBERS AND FILAMENT LENGTHS

Keywords: carbon fiber, recycling, steam thermolysis process, thermoplastic composites, nonwovens

The current study aims to investigate the mechanical properties of ten different carbon fiber nonwoven reinforced thermoplastic composites. A design of experiments was carried out by using a Mixture Design methodology in order to produce nonwoven fabrics by carding and pre-needling. Composites were manufactured by compression molding using film stacking method. Tensile properties such as tensile modulus and tensile strength were investigated as function of proportion of fibers in nonwovens and direction of applied load. It was found that the mechanical properties of the composites were much higher in the Cross direction than in the Machine direction. It was observed that the mechanical properties are mainly influenced by fiber properties and orientation of fibers in the nonwovens. The tensile modulus was then modelled by multilinear regression resulting from the design analysis and Rule of Hybrid Mixtures (RoHMs). A good correlation between the experimental and the predicted values is observed.

Introduction

During the last few years, there has been great deal of interest in recycling carbon-fiber-reinforced polymer composites (CFRP). Different technologies have been proposed to treat CFRP and recover carbon fiber for reuse in 2 nd generation Composites [START_REF] Géraldine | Current status of recycling of fibre reinforced polymers: review of technologies, reuse and resulting properties[END_REF][START_REF] Soraia | Recycling carbon fibre reinforced polymers for structural applications: Technology review and market outlook[END_REF]. The steam thermolysis process is a thermochemical recycling process using superheated steam water to degrade organic matrix and recover carbon fibers from CFRP [START_REF] Boulanghien | Using steam thermolysis to recycle carbon fibres from composite waste[END_REF]. It was shown that the reclaimed carbon fibers (rCF) have retained their original mechanical properties [START_REF] Ye | Parameter optimization of the steam thermolysis: a process to recover carbon fibers from polymer-matrix composites[END_REF].

From an industrial and economical point of view, there is a major challenge related to the sorting of composites by type/grade of fibers before recycling. An optimal solution would be to be able to produce rCF nonwovens with guarantied properties avoiding the initial CFRP sorting step and to make cost effective 2 nd generation composites. Many studies have been carried out on the effect of rCF properties as function of recycling process on the mechanical performance of composites [START_REF] Soraia | Recycling carbon fibre reinforced polymers for structural applications: Technology review and market outlook[END_REF][START_REF] Boulanghien | Using steam thermolysis to recycle carbon fibres from composite waste[END_REF][START_REF] Ye | Parameter optimization of the steam thermolysis: a process to recover carbon fibers from polymer-matrix composites[END_REF][START_REF] Sj Pickering | Applications for carbon fibre recovered from composites[END_REF]. Small number of studies have been reported the influence of variability of sources of rCF on the mechanical properties of composites. Actually, the fibers are mixed during the recycling process, so that reclaimed fibers present a distribution of properties [START_REF] Géraldine | Current status of recycling of fibre reinforced polymers: review of technologies, reuse and resulting properties[END_REF]. In this case, ELG CF explained that blending different products reduces property variation in second generation products [6].

This study aims to investigate the influence of carbon fiber sources variability on the mechanical properties of nonwoven reinforced composites. A design of experiments was carried out by using a Mixture Design methodology in order to produce different nonwoven fabrics [START_REF] Cornell | Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data[END_REF]. Three different virgin carbon fiber tows (T300, T700 and IM7) were selected and cut into lengths up to 110 mm. According to the mixture design, ten nonwoven fabrics, with a weight close to 200g/m², were prepared by carding and needle punching (pre-needling) using the same process parameters. These nonwovens were associated to Polyamide 6 (PA6) matrix films to produce by compression molding process ten composites with a fiber volume fraction of 28 vol%. Composite mechanical properties were evaluated by performing tensile test along the machine direction (MD), which is the direction of nonwoven production, and the cross direction (CD).

Materials and methods

Materials

In this study three virgin carbon fiber tows, T300, T700 (Toray) and IM7 (Hexcel), were selected to produce nonwovens. The basic properties of these fibers are presented in Table 1. They represent high strength and intermediate modulus-high strength carbon fibers. The selection was based on the mechanical properties of the fibers and their use in aircraft and automotive applications (end-of-life composites are generally reinforced by these fibers). The fibers were cut into three lengths 50 mm, 80mm and 110mm.

A design of experiments was carried out by using a Mixture Design methodology in order to produce nonwovens and investigate the mechanical properties of nonwoven reinforced composites (Table 2). The Figure 1 shows a diagram of the design with 10 experiments. Each corner of the diagram consists of 100% of one of the three fibers cut into different lengths. The sides represent binary mixtures. Generally, the first 7 points are used to estimate the regression model, the 3 following to validate it.

The nonwovens were produced via a carding/cross lapping/needle-punching process at STFI (Saxon Textile Research Institute e.V.) in Germany. The fibers were mixed according to the mixture design and carded to be transformed into a veil. The carded veils were then cross lapped and pre-needled (single sided needling, 60 punches/cm², 14mm needle depth), in order to form 100% carbon fiber nonwovens with an areal density of 200g/m².

Composite preparation and testing method

Composite plates with 28% fiber volume content and 2 mm thickness were manufactured by compression molding using film stacking method. In this method, carbon fiber nonwovens and PA6 films were stacked alternately. Before manufacturing, the PA6 films were dried in an oven at 80°C for 24 hours. The obtained composite plates were cut into tensile specimens in accordance with ISO-527 specification with dimensions 250 mm x 25mm x 2mm. Specimen ends were reinforced by gluing ±45°glass fiber/epoxy end tabs to avoid sample failure out of the gage length. Tensile properties were determined according to the ISO 527 standard using 30 KN INSTRON-5200 testing machine at a crosshead speed of 1 mm/min. An extensometer was used to record strain of tested specimens. Composite mechanical properties were evaluated by performing tensile tests along the machine direction (MD) and the cross direction (CD).

Results and discussion

Experimental results

The mechanical properties of the ten composites obtained from tensile tests are summarized in cluster column charts shown in Figure 2 and Figure 3. Each test data chart is the average value from three to five identical specimens and the error bars on the histograms represent the standard deviation. These results show that the tensile modulus and strength values range respectively from 18 GPa to 35 GPa and 200 MPa to 550 MPa in the CD direction. In the MD direction the values range from 15 GPa to 25 GPa for the modulus and 150 MPa to 450 MPa for strength. It is important to note that the results are different depending on the direction of applied load. The composites have higher mechanical properties in the CD direction. This could be attributed to the fiber orientation distribution in nonwovens which could be influenced by the fiber lengths and the process parameters. Indeed, in carded and needle-punched According to the Table 2, the experimental design can be divided into 3 groups. The first group corresponds to the composites C1, C2 and C3. These composites are reinforced by 100% of each fiber cut into different lengths. Due to the best mechanical properties of IM7 fiber, C3 presents the highest modulus and strength in both MD and CD directions. The second group corresponds to the composites C4, C5 and C6. In this case, composites are reinforced by a mixture of two fibers (50/50) cut into one fixed length. The best properties obtained for the C5 in the CD direction. For this composite the cut length fixed at 80 mm and the presence of IM7 fiber in the mixture have increased the modulus as well as the resistance in this direction. The last group corresponds to the rest of the design of experiments (C7, C8, C9 and C10). The composites were reinforced by all the fibers cut into three different lengths. No significant difference was observed between the tensile properties of the composites in both MD and CD directions.

In addition to these investigations, an analysis of fiber orientation in nonwovens was carried out in order to predict the tensile modulus of nonwoven reinforced composites. A regression model was used to predict the contribution of fibers on the tensile modulus. Moreover, a modified Rule of Hybrid Mixtures was developed to predict the composite tensile modulus. This model takes into account the fiber orientation distribution and tensile properties.

Theoretical Analysis: Tensile modulus prediction

In order to study the influence of fiber properties and architecture on the tensile modulus of the composites in MD and CD directions, a regression model and Rule of Hybrid Mixtures were used.

-Regression model [START_REF] Sj Pickering | Applications for carbon fibre recovered from composites[END_REF] A statistical regression model considering fiber proportions as variables and tensile modulus as the response produced the regression line. In this study, tensile modulus be evaluated by the regression equation as follows:

Y =A XT300 + B XT700 + C XIM7. ( 1
)
Where A, B and C are the regression coefficients; Xi is the proportion of each fiber "i" in the composite.

Typically, the coefficients have to be tested as significantly different from "0" or not. This can be done by evaluating the probability values which have to be less than 5% [START_REF] Sj Pickering | Applications for carbon fibre recovered from composites[END_REF].

Modulus in CD and MD can be evaluated by the regression equations ( 2) and (3). In this case the different coefficients are significant and the squared multiple correlation R² is equal to 0.94 for CD regression and 0.95 for MD regression.

Ec (CD) =22863 XT300 +23581 XT700 + 32216 XIM7.

(

) 2
Ec (MD) = 18068 XT300 + 14097 XT700 + 22624 XIM7.

(

-Rule of Hybrid Mixtures RoHMs [8-9-10] In this study, by mixing different carbon fibers, the composites can be assimilated as hybrid materials.

Several studies have been carried out on the effect of carbon fiber hybrid with glass fibers [START_REF] Venkateshwaran | Prediction of tensile properties of hybridnatural fiber composites[END_REF][START_REF]Stiffness Prediction of Hybrid Kenaf/Glass Fiber Reinforced Polypropylene Composites using Rule of Mixtures (ROM) and Rule of Hybrid Mixtures (RoHM)[END_REF]. But, some research has been reported the effect of hybrid composites reinforced with more than two different carbon fibers. In this case the tensile modulus of hybrid nonwoven composite can be predicted using equation 4.

Ehc = ET300 VT300 + ET700 VT700+ EIM7 VIM7. ( 4
)
Where Ehc is the tensile modulus of hybrid composite and VT300, VT700 and VIM7 are the proportion of each fiber in the composite. ET300, ET700 and EIM7 are respectively tensile modulus of T300/PA6, T700/PA6 and IM7/PA6 composites with 28% fiber volume fraction and can be predicted using modified rule of mixture (equation 5) developed by Cox [START_REF] Cox | The elasticity and strength of paper and other fibrous materials[END_REF] and modified by Krenchel [START_REF] Krenchel | Fibre Reinforcement: Theoretical and Practical Investigations of the Elasticity and Strength of Fibre-reinforced Materials[END_REF].

Ei = ηo Ef Vf + Em (1-Vf). ( 5
) ηo = Σ a(i) cos 4 θ(i) . ( 6
)
Where Ef and Em are tensile modulus of fiber and matrix, Vf is the fiber volume content and ηo is an efficiency factor which takes into account the proportion of fibers a(i) oriented according to θ(i) in the nonwoven fabrics. This factor was determined using image analysis method and the technique used is known as Fast Fourier Transform (FFT) [START_REF] Shhs | Manufacturing of Fibre glass Nonwoven Webs Using a Paper Making Method and Study of Fibre Orientation in These Webs[END_REF]. The analyzed images correspond to the SEM observations of nonwoven fabrics as shown in Figure 4. The orientation factor was then determined from the polar plot of fiber orientation distribution using Equation 6. It is found that values of ηo for all the different nonwoven fabrics studied are close to each other. The obtained ηo in MD and CD directions are equal respectively to 0.266 and 0.464. -

Results

According to ηo values, the fiber orientation distribution in nonwovens explains the corresponding anisotropy in the composite properties. The Figure 5 and Figure 6 show the comparison between measured and predicted tensile moduli of composites in CD and MD directions. A good agreement between the experimental and theoretical values of tensile modulus is observed. The regression model tends to be closer to the RoHMs model. Therefore, the regression model could be assimilated as a hybrid model. The models used here take into account the fiber properties and the anisotropy of nonwoven fabrics. These parameters have an important influence on the tensile modulus of nonwoven reinforced composites.

Error Bars: ± Standard Deviation

Conclusions

In this study, the tensile properties of ten different carbon fiber nonwoven composites were investigated. The fiber type and length have great influence on the mechanical properties of the composites. The fiber orientation in nonwoven fabrics leads to the corresponding anisotropy in the composite properties. The tensile modulus and tensile strength of the composites produced are influenced by both tensile properties and architecture of fibers. Theoretical analysis of the tensile modulus of nonwoven reinforced composites are discussed. The tensile moduli of the composites are predicted using regression model and RoHMs equation. Good agreements are observed between the experimental modulus and the theoretical results. Further work is being undertaken to investigate the mechanical performance of recycled carbon fiber nonwoven reinforced composites.

Figure 1 .

 1 Figure 1. A Simplex centroid design for a three different carbon fiber and fiber length mixture.

Figure 2 .Figure 3 .

 23 Figure 2. Tensile modulus of nonwoven reinforced composites in machine and cross directions.

Figure 4 .

 4 Figure 4. Polar plot of fiber orientation distribution in nonwoven fabric.

Figure 5 .Figure 6 .

 56 Figure 5. Measured and predicted tensile modulus in CD direction.

Table 1 .

 1 Structural and mechanical properties of material constituents.

	Fiber/Matrix Density Tensile strength Tensile Modulus Elongation	Length
		(g/cm 3 )	(MPa)	(GPa)	(%)	(mm)
	T300	1.76	3530	230	1.5	50 / 80 / 110
	T700	1.8	4900	230	2.1	50 / 80 / 110
	IM7	1.78	5760	290	2.0	50 / 80 / 110
	PA6	1.14	90	3.2	4.5	---

Table 2 .

 2 Design of mixture table

Acknowledgments

The authors gratefully acknowledge the funding supports of "Direction Générale de l'Armement" and "Alpha Recyclage Composites".