Universality for random surfaces in unconstrained genus

Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov and Pittel (arXiv:1503.01816) have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson--Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov and Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.


Introduction . Gluings of polygons and a conjecture
Suppose we are given a set of k 1 polygons whose perimeters are prescribed by P = {p 1 , p 2 , . . . , p k } where p i ∈ {1, 2, 3, . . . }. We can then form a random surface by gluing their sides two-by-two in a uniform manner, see In this work, we release the constraint that the polygon's perimeters are larger than 3 but will only consider con gurations which do not contain too many 1 or 2-gons. If P = {p 1 , . . . , p k } is a con guration of polygons, we write #P = k, |P| = 1 2 k i=1 p i , L(P) = #{i : p i = 1}, B(P) = #{i : p i = 2} respectively for half of its total perimeter, the number of 1-gon (loops) and the number of 2-gons (bigons) in P. A sequence (P n ) n 1 of con gurations of polygons so that |P n | = n is said to be good if L(P n ) √ n → 0 and B(P n ) n → 0 as n → ∞.

( )
We focus on good (sequences of) con gurations because in this case the surface obtained after the uniform gluing is with high probability connected (see Proposition ). Speci cally, given P, we randomly label the sides of the polygons from 1, 2, . . . , 2|P| in a uniform way and then glue them two by two using an independent pairing of {1, 2, . . . , 2|P|}, that is, an involution without xed points. In all that follows, we only consider oriented surfaces and when we glue two edges we always assume we make sure to preserve the orientation of each polygon . When the gluing is connected, the images of the edges of the polygons form a map M P with |P| edges drawn on the surface created. By abuse of notation we speak of M P as our "random surface". The labeling of the sides of the polygons yields a labeling of the oriented edges of the map by 1, 2, . . . , 2|P|: the map M P is labeled . If we forget the labelings and the orientation of the surface, we get a random multi-graph G P , which is the object of study in this work.
To motivate our results, we start with a conjecture which roughly says that given |P| and provided the random graph G P is connected, its law is always the same (in a strong sense), regardless of the con guration of polygons we started with and is close to the law of the random graph obtained from a uniform random map with |P| edges. However, it is easy to see from Euler's formula that the number of vertices of G P has the same parity as |P| + #P and so the proper conjecture needs to deal with this parity constraint. Let G n be the random graph structure of a uniform random labeled map on n edges, and denote G odd n (resp. G even n ) the random graph G n conditioned respectively on having an odd (resp. even) number of vertices.
Conjecture (Universality for G P ). Let (P n ) n 1 be a good sequence of con gurations. We denote by ϵ n ∈ {even, odd} the parity of n + #P n . Then we have d TV G P n , G ϵ n n → 0, as n → ∞.
Here and later we write d TV (X , Y ) for the total variation distance between the laws of two random variables X and Y . Compelling evidence for the above conjecture is the result of Chmutov & Pittel [ ] (generalizing work by Gamburd [ ] in the case when the polygons have the same perimeter), which asserts that when all the polygon's perimeters are larger than 3 then up to an error of O(1/n) in total variation distance, the degree distribution of G P n is the same as that of G ϵ n n . The proof of [ ] is based on representation theory of the symmetric group. One of the goal of this work is to give a probabilistic proof of a weak version (Theorem ) of the above conjecture. We also take this work as a pretext to gather a few results (some of which may belong to the folklore) on the geometry of a uniform random map with n edges.
In the rest of the paper, all the maps considered are labeled.

. Geometry of random maps
For n 1, we denote by M n a random (labeled) map chosen uniformly among all (labeled) maps with n edges. Recall that its underlying graph structure is G n . It is well known that the distribution of degrees of a random map is closely related to the cycle structure of a uniform permutation, we make this precise in Theorem .
Permutations and Poisson-Dirichlet distribution. For n 0 we denote by U n = {u (n) 1 u (n) 2 . . . u (n) k } the cycle lengths in decreasing order of a uniform permutation σ n ∈ S n . This distribution is well-known and in particular where the (standard) Poisson-Dirichlet distribution PD(1) is a probability measure on partitions of 1, i.e. on sequences x 1 > x 2 > x 3 > · · · such that x i = 1, which is obtained by reordering in decreasing Their result is expressed in terms of the cycle structure of a uniform permutation over A 2n if k and n have the same parity (resp. A c 2n if k and n have di erent parity) where A 2n ⊂ S 2n is the group of alternate permutations over {1, 2, . . . , 2n}. But given our Theorem this can be rephrased as the degree distribution of G odd/even n . order the lengths U 1 , U 2 (1 − U 1 ), U 3 (1 − U 1 )(1 − U 2 ), . . . where (U i : i 1) is a sequence of i.i.d. uniform variables on [0, 1]. We refer e.g. to [ ] for details.
Recall our notation M P for the map obtained by the uniform labeling and gluing of the sides of oriented polygons whose perimeters are prescribed by P . If P is itself random, one can still consider M P by rst sampling the perimeters in P and then performing our random gluing.
Theorem (Random map as a random gluing with discrete Poisson-Dirichlet perimeters). For some constant C > 0 we have for all n 1 d TV M n ; M U 2n C n .
In words, a random uniform (labeled) map can be obtained, up to a small error in total variation distance, by a random gluing of random polygons whose sides follow the discrete Poisson-Dirichlet law U 2n .
We can deduce several consequences of the above result. First of all, the faces degrees in M n have the same law as U 2n up to a small error in total variation distance. By well-known results on the distribution of U 2n (see [ ]) this shows that the number of faces of degree 1, 2, 3, . . . in M n converge jointly towards independent Poisson random variables of means 1, 1/2, 1/3, . . . . On the other side, by ( ) the large face degrees, once rescaled by 1/(2n), converge towards PD(1). Finally by ( ), the number of faces #F(M n ) in M n is close in total variation distance to #U 2n and thus of Poisson(log n).
Since M n is self-dual, the same results hold for M † n the dual map of M n . We shall prove in Proposition that #F(M n ), #V(M n ) the number of faces and vertices of M n and its genus Genus(M n ) obey where N 1 and N 2 are independent standard Gaussian random variables. The convergences of the rst and second components alone follow from the discussion above. The perhaps surprising phenomenon is that the number of vertices and faces of M n are asymptotically independent: this is a consequence of our forthcoming Theorem . This result was also proved recently by Carrance [ , Theorem . and its proof] by using the techniques of [ ].
Con guration model. We will use Theorem in conjunction with the self-duality property of M n to deduce an approximate construction of its graph structure. More precisely, if Graph(m) denotes the (multi)graph obtained from a map m by forgetting the labeling and the cyclic orientation of edges then, up to an error of O(1/n) in total variation distance, we have The point is that Graph(M † U 2n ) has a particularly simple probabilistic construction: it is obtained as a con guration model with degrees prescribed by U 2n . Recall that the con guration model with vertices of degrees d 1 , . . . , d k is the random graph obtained by starting with k vertices having d 1 , . . . , d k "legs" and pairing those legs two by two in a uniform manner. This model was introduced by Bender & Can eld [ ] and Bollobás [ ] and was later studied in depth, see e.g. [ ]. This remark is used to prove the following striking property (which is new to the best of our knowledge): Corollary (The diameter of a random map is 2 or 3.). There exists a constant ξ ∈ (0, 1) such that The proof of Corollary gives an expression of ξ in terms of a rather simple random process involving independent Poisson random variables and a PD(1) partition. Unfortunately, we have not been able to transform this expression into a close formula. A numerical approximation shows that ξ ≈ 0.3.
As mentioned above, the literature concerning the con guration model is abundant. Notice however that the conditions we impose on our perimeters are very di erent from the usual "critical" conditions that can be found e.g. in [ , ]. We also mention the works [ , ] which study respectively random plane trees and random planar maps with prescribed degrees. .

Poisson-Dirichlet universality for random surfaces
We now turn to random maps obtained by gluing polygons of prescribed perimeters. We will prove (Proposition ) that the uniform gluing of polygons yields a connected surface with high probability provided that we control the number of loops and bigons.
Number of vertices. Recall the notation M P and G P respectively for the random map and the corresponding graph created by the uniform labeling/gluing of sides of polygons of P. We also write #V P for the number of vertices of M P . Finally, for α > 0, let Poisson odd α (resp. Poisson even α ) be a Poisson variable of parameter α conditioned on being odd (resp. even).
Theorem (Universality for the number of vertices). Let (P n ) n 1 be a good sequence of con gurations and ϵ n be the parity of n + #P n . Then we have d TV #V P n , Poisson ϵ n log n → 0, as n → ∞.
This is supporting our Conjecture . : indeed recall from ( ) that d TV (#U 2n , Poisson(log n)) → 0 as n → ∞. Hence, using our Theorem and duality, we see that Poisson odd log n is close in total variation to #V(G odd n ) (and similarly in the even case). In the case when all the perimeters of P n are larger than 3, the last result is a trivial consequence of [ ] (although the idea of the proof is very di erent). In essence, the above theorem says that up to parity considerations, the number of vertices of M P n is asymptotically independent of its number of faces, which is key in proving ( ).
Connectivity of the edges. On the way towards Conjecture . we describe the connectivity properties of G P , at least for the vast majority of its edges. We start with a de nition. Given a (random) graph g n with n edges, denote by 1 , 2 , . . . the vertices of g n ordered by decreasing degrees. For i, j 1, we write [i, j] g n for the number of edges between i and j with the convention that [i, i] g n is twice the number of self-loops attached to i . We say that a sequence (g n ) n 1 of random graphs satis es the Poisson-Dirichlet universality if Remark . The above Poisson-Dirichlet universality convergence can be rephrased in the theory of edge exchangeable random graphs as the convergence towards the rank multigraph driven by the Poisson-Dirichlet partition, see [ , Example . and . ] and [ ].
Using the approximate construction of G n as a con guration model based on U 2n , it follows from easy concentration arguments that the graphs G n satisfy the Poisson-Dirichlet universality ( ). On the other hand, an intuitive way to formulate ( ) is that the graphs g n look like con guration models for a large proportion of the edges. We show that this phenomenon actually holds true for more general polygonal gluings: Theorem . For any good sequence (P n ) n 1 of con gurations, the graphs G P n satisfy the Poisson-Dirichlet universality ( ).
Let us draw a few consequences of the last theorem. The total number of loops (each loop is counted twice) in the graph G P n can be written as i [i, i] G Pn and hence satis es The last convergence (without identi cation of the limit law) was recently established in [ ] in the case of uniform maps using the method of moments . They also studied the degree and the number of edges incident to the root vertex (without counting loops twice) in a random map. Since the root vertex is a degree-biased vertex, if we introduce a random index I 0 chosen proportionally to the degree of I in G P n , then these variables can respectively be written as and thus converge once rescaled by 1/(2n) towards X and X − 1 2 X 2 where X is a size-bias pick in a Poisson-Dirichlet partition, for which it is well known that X = Unif([0, 1]) in distribution. This extends the results of [ ] to a much broader class of random maps. The above result shows that the distribution of large degrees is universal among random maps obtained by gluing of good con gurations. Actually, the distribution of small degrees, namely the fact that they converge in law towards independent Poisson random variables of means 1, 1/2, 1/3, ... should also be universal among this class of random maps (and this would indeed be implied by our Conjecture . ). An approach using the method of moments might be possible but would not t the general scope of this paper and so we leave this problem for future works. Finally, let us mention that our proof of Theorem is robust and also allows to obtain results about the location of small faces. For example, if P n has a positive proportion of triangles, then the proportion of these triangles whose three vertices are i , j and k (in this order) is asymptotically X i X j X k (see Remark ).
The paper is organized as follows.
Section is devoted to the study of uniform maps with a xed number of edges and no restriction on the genus. Using the classical coding of maps by permutations, we prove that random maps can approximately be seen as a gluing of polygons whose perimeters follow the cycle length of a uniform permutation. By duality, this enables us to see the graph structure of a uniform map as a con guration model which is key in our proof of Corollary .
We then focus on the more general model M P . We prove Theorem and Theorem using "dynamical" explorations of the random surfaces M P . Although most of the ideas presented here are already underlying papers in the eld (see e.g. [ , ]), we draw a direct link with the circles of ideas used in the theory of random planar maps and in particular with the peeling process, see [ ]. We then use two speci c algorithms to explore the surface M P either by peeling the minimal hole or by discovering the vertices one by one, which yield Theorem and Theorem respectively.

Uniform random maps
Most of what follows in this section is probably known to many specialists in the eld, but we were not able to nd precise references and thus took the opportunity to ll a gap in the literature.
. Uniform maps as a con guration model Let m be a (connected) labeled map with n edges. The combinatorics of the map is then encoded by two permutations α, ϕ ∈ S 2n : the permutation α is an involution without xed points coming from the pairing of the oriented edges into edges of the map and ϕ is the permutation whose cycles are the oriented edges arranged clockwise around each face of the map, see e.g. [ , Chapter . . ].
We denote by I 2n ⊂ S 2n the subset of involutions without xed points (i.e. product of n non overlapping transpositions). Clearly, we have #I 2n = (2n − 1)!!. Remark that a pair (α, ϕ) ∈ I 2n × S 2n is not necessarily associated to a (connected labeled) map: for n = 2, the "map" associated with the permutations α = (12)(34) and ϕ = (1)(2)(3)(4) consists of two disjoint loops. However, this situation is marginal: Proposition (Uniform maps are almost uniform permutations). Let C 2n = {(α, ϕ) ∈ I 2n × S 2n : (α, ϕ) encodes a connected labeled map}, so that C 2n is in bijection with labeled maps with n edges and the number of rooted maps with n edges is 1 (2n−1)! #C 2n . Then we have the asymptotic expansion In particular, if (A n , F n ) ∈ I 2n × S 2n is the pair of permutations associated with a uniform labeled map with n edges and if (α n , ϕ n ) is uniformly distributed over I 2n × S 2n , then Proof. Let (α n , ϕ n ) ∈ I 2n × S 2n be uniformly distributed. If (α n , ϕ n ) does not yield a connected map, that means that the subgroup generated by α n and ϕ n does not act transitively on {1, 2, . . . , 2n}, or equivalently that {1, 2, . . . , 2n} can be partitioned into two non-empty subsets I and such that both I and are stable by α n and ϕ n . By partitioning according to the smallest stable subset containing 1, we obtain the following recursive relation : Writing c n = #C 2n /(2n − 1)! for the number of rooted maps with n edges and ξ n = (2n − 1)!!, the above recursive equation is equivalent to c n = 2n × ξ n − n−1 =1 c ξ n− . Iterating this yields The terms 2nξ n and −2(n − 1)ξ n−1 ξ 1 give (2n − 1)!!(2n − 1 + O(1/n)) while the total sum of the absolute values of the other terms is easily seen to be of order O(1/n) × (2n − 1)!!. This proves the rst claim of We also note that a very similar (but di erent) formula appears in [ ].
the theorem. The second one is a trivial consequence of the de nition of total variation distance since (A n , F n ) is uniformly distributed over C 2n .
Proof of Theorem . Theorem is an easy consequence of the last proposition. Indeed, if we rst sample the polygon's perimeter U 2n and then assign in a uniform way labels {1, 2, . . . , 2n} to the edges of the polygons, this represents a uniform permutation ϕ n ∈ S 2n whose cycles are the labeled polygons. The independent involution α n without xed points then plays the role of the gluing operation which identi es the edges of the polygons by . Provided the resulting surface is connected, the random labeled map M U 2n it creates is plainly associated to the pair of permutation (α n , ϕ n ) ∈ I 2n × S 2n .
As we alluded to in the introduction, we will use Theorem in conjunction with the self-duality property of M n to deduce an approximate construction of its graph structure. Indeed, recall that up to an error of O(1/n) in total variation distance, we have . Euler's relation in the limit We prove ( ) which we recall below.
Proposition (Euler's relation in the limit). If #F(M n ), #V(M n ) and Genus(M n ) are respectively the number of faces, vertices and the genus of M n then we have where N 1 and N 2 are independent standard Gaussian random variables.
Proof. We will rely on Theorem which is proved later in the paper. We beg the reader's pardon for this inelegant "back to the future" construction. Let us rst focus on the second component. By Theorem together with ( ) we deduce that It is then standard that λ −1/2 (Poisson(λ) − λ) converges in law towards a standard Gaussian as λ → ∞. This proves the convergence of the second coordinate, the rst one being obtained by self-duality. It remains to show that the (rescaled) number of faces and vertices are asymptotically independent. Once this is done, the convergence of the third component follows from Euler's relation: To prove the asymptotic independence, we rely on Theorem . Indeed, if (U 2n ) n 1 is the cycle structure of a uniform random permutation of S 2n , then as recalled in the introduction, the numbers of loops and bigons in U 2n satisfy with independent Poisson variables. In particular, for any ε > 0, the probability that either B(U 2n ) n 1−ε or L(U 2n ) n 1 2 −ε tends to 0 as n → ∞. We can then apply Theorem (using its notation) to deduce that where ϵ n is the parity of n + #U 2n . But since λ −1/2 (Poisson ϵ λ − λ) → N regardless of the parity ϵ we indeed deduce that the rescaled number of faces and vertices in M U 2n are asymptotically independent. The same is true in M n by Theorem . .
The diameter is 2 or 3 In this section we prove Corollary , showing that the diameter of a random map with n edges converges in law towards a random variable whose support is {2, 3}. Before going into the proof, let us sketch the main idea (a quick glance at Figure may help to get convinced). We rst prove that with high probability all vertices have a neighbor among the vertices of degree comparable to n. Since all these vertices are linked to each other, this proves that the diameter of the map is at most 3. The diameter of the map can even be equal to 2 if all pairs of vertices of low degree share a neighbor among the vertices of high degree.
Let us recall a simple calculation that we will use several times below. If U 2n is the cycle structure of a uniform random permutation of S 2n , then for any positive function ψ : simply because the law of the length of a typical cycle in a uniform permutation of S 2n (i.e. a cycle sampled proportionally to its length or equivalently the cycle containing a given point) is uniformly distributed over {1, 2, . . . , 2n}.
Proof of Corollary . According to ( ), it su ces to prove the result for the random graph obtained from a con guration model whose vertices have degrees prescribed by U 2n , which we denote below by ConfigModel(U 2n ). We write C (n) 1 , . . . , C (n) 2n for the number of occurrences of 1, 2, . . . , 2n in U 2n , i.e. the number of vertices of degree 1, 2, . . . , 2n in ConfigModel(U 2n ). For δ > 0 we denote by V (n) δ n the set of vertices of ConfigModel(U 2n ) whose degree is larger than or equal to δn. We rst state two lemmas: Lemma . For any ε > 0, we can nd δ > 0 and A > 0 such that for all n su ciently large, in the random graph ConfigModel(U 2n ), we have P all vertices have a neighbor in V (n) ( ) The next lemma is a decoupling result between the small degrees and the large degrees in U 2n : Lemma . If C (n) 1 , C (n) 2 , . . . denote the number of occurences of 1, 2, . . . in U 2n and if M (n) 1 > M (n) 2 > . . . are the degrees of U 2n ranked in decreasing order, then we have the following convergence in distribution in the sense of nite-dimensional marginals: where P 1 , P 2 , . . . are independent Poisson random variables of mean 1, 1 2 , 1 3 , . . . and PD (1) is an independent Poisson-Dirichlet partition.
Proof of Corollary given Lemmas and . Given the last two lemmas, the proof of Corollary is rather straightforward. Indeed, combining ( ) and ( ) we deduce that with probability at least 1 − 2ε all vertices of ConfigModel(U 2n ) are linked to V (n) δ n , and if δn A then all the vertices of V (n) δ n are linked to each other. Hence the diameter of the graph is less than 3 and we even have d gr (u, ) 2 as soon as u or has degree larger than A. On the other hand, we clearly have Diameter(ConfigModel(U 2n )) 2 with high probability. To see this, just consider a vertex u with degree of order O(1): this vertex cannot be linked to all the vertices (there are ≈ log n vertices) and thus must be at distance at least 2 from another vertex in ConfigModel(U 2n ). By this reasoning, up to an event of probability 2ε, to decide whether the diameter of ConfigModel(U 2n ) is 2 or 3, one must know whether or not we can nd two vertices u, such that and u, do not share a neighbor in V (n) δ n . ( ) But clearly, by Lemma and the de nition of the con guration model, one can describe the limit in distribution of the connections of all vertices of low degree as follows. Consider P 1 , P 2 , . . . independent Poisson random variables of means 1, 1 2 , 1 3 , . . . . When P i > 0, we imagine that we have P i vertices Independently of this, consider X = (X 1 > X 2 > · · · ) a partition of unity distributed according to PD(1). The vertices (i) k describe the low degree vertices, X a describe the large degree vertices and U ( (i) k ) describe the connections of those low degree vertices to the large degree vertices. Hence, for two distinct "small vertices" (i) k and (j) , we say that they share a neighbor if we can nd 1 a i and 1 b j such that U a ( (i) k ) and U b ( (j) ) fall into the same component of [0, 1] induced by X. If we put ξ = P(there exist two vertices which do not share a neighbor), then we leave the reader verify that ξ is the limit of the probability of the event in ( ). But since the event in ( ) is, up to an error of probability at most 2ε, the same event as {Diameter(M n ) = 3} when n is large, the corollary is proved. Finally, elementary computations show that in the limit model, almost surely, the number of pairs of small vertices with no common neighbour is nite and that 0 < ξ < 1.
See also the proof of Lemma below.
We now prove the two lemmas.
Proof of Lemma . We start with the rst point. Let δ > 0 and let us write E(δ, n)).
By the convergence of the renormalized degrees towards the Poisson-Dirichlet partition (actually, we only use the fact that this is a partition of unity), for any ε ∈ (0, 1/2) we can nd δ > 0 such that for all n large enough we have P(E(δ, n) ε) ε.
By the de nition of the con guration model, conditionally on U 2n , the probability that a given vertex of degree d 1 is not linked to V (n) δ n is bounded above by E(n, δ ) d . Hence we deduce that the probability of the event in ( ) is bounded as follows: The second point is similar. Conditionally on U 2n , if w is a vertex of degree larger than δn then the probability that any xed vertex of degree d is not linked to w is bounded above by (1 − δ 2 ) d . Since there are deterministically less than 2 δ vertices in V (n) δ n , the complement probability of ( ) is bounded above by For xed δ > 0, the right-hand side can be made arbitrary small by taking A large and this proves the second point of the lemma.
Proof of Lemma . This is essentially a corollary of the very general results presented in [ ]. More precisely, the convergence in law of the small cycle lengths count is a corollary of their Theorem . , whereas the convergence of the large cycle lengths is the convergence towards PD(1) recalled in the introduction. To see that they are asymptotically independent, just notice that for xed k 0 , conditionally on C (n) 1 = c 1 , . . . , C (n) k 0 = c k 0 for xed values c 1 , . . . c k 0 0, the law of C (n) k 0 +1 , . . . , C (n) 2n is distributed as P k 0 +1 , . . . , P 2n conditioned on 2n i=k 0 +1 = 2n − k 0 i=1 ic i which again falls in the general "conditioning relation and logarithmic condition" of [ ], for which we can thus apply their Theorem . giving in particular the convergence towards Poisson-Dirichlet.

Dynamical exploration of random surfaces
In this section we describe the "dynamical" exploration of the random surface M P . Our approach is close to the peeling process used in the theory of random planar maps, see [ , ].

. Exploration of surfaces and topology changes
We x a con guration P of (oriented) polygons with |P| = n whose sides have been labeled from 1 to 2n, and ω ∈ I 2n a pairing of its edges. Since everything we describe below is deterministic, we do not need to assume here that ω is random.
We will construct step by step the discrete surface M P that is created by matching the edges by according to ω. More precisely, we will create a sequence S 0 → S 1 → · · · → S n = M P of "combinatorial surfaces" where S 0 is made of the set of labeled polygons whose perimeters are speci ed by P and where we move on from S i to S i+1 by identifying two edges of the pairing ω. More speci cally, S i will be a union of labeled maps with distinguished faces called the holes (they are in light green in the gures below). The holes are made of the edges which are not yet paired. The set of these edges will be called the boundary of the surface and be denoted by etc. etc.

Figure :
Starting configuration (on the le ) and a typical state of the exploration (on the right). Here and later the labeling of the oriented edges does not appear for the sake of visibility. The final vertices of the graph are black dots whereas "temporary" vertices are in white. Notice on the right-hand side that S i contains a closed surface without boundary: if this happens, the final surface S n = M P is disconnected.
To go from S i to S i+1 we select an edge on ∂S i which we call the edge to peel (in red in the gures below) and identify it with its partner edge in ω (in green in the gures below), also belonging to ∂S i . We now describe the possible outcomes of the peeling of one edge. The reader should keep in mind that our surfaces are always labeled and oriented and that when identifying two edges we glue them in a way compatible with the orientation. We shall also pay particular attention to the process of creation of vertices. Indeed, the initial vertices of the polygons are not all vertices in the nal map M P : in the gures below those "temporary" vertices belonging to holes are denoted by white dots, whereas actual vertices of M P = S n are denoted by black dots and called "true" vertices.  Figure : If we identify two edges on the same hole, then this hole splits into two holes of perimeters p 1 and p 1 such that p 1 + p 1 + 2 is the perimeter of the initial boundary. The two other ways to create true vertices: if we identify (le ) two loops (holes of perimeter ), then we create one true vertex. If we identify (right) the two edges of a hole of perimeter 2, then we create two true vertices.

. Peeling exploration
We now move on to our random setting and suppose that ω ∈ I 2n is independent of the starting con guration of labeled polygons. On top of ω, the sequence S 0 → S 1 → · · · → S n depends on an algorithm called the peeling algorithm which is simply a way to pick the next edge to peel A(S i ) ∈ ∂S i . The function A(S i ) can be deterministic or may use another source of randomness, as long as it is independent of ω: we call such an algorithm Markovian. Highlighting the dependence in A, we can thus form the random exploration sequence S A 0 → S A 1 → · · · → S A n = M P by starting with S A 0 , the initial con guration made of the labeled polygons whose perimeters are prescribed by P. To go from S A i to S A i+1 , we perform the identi cation of the edge A(S i ) together with its partner in the pairing ω. When ω is uniform and A is Markovian, the sequence (S A i : 0 i n) is a simple (inhomogeneous) Markov chain: Proposition . If the gluing ω is uniformly distributed and independent of the labeled polygons of P, then for any Markovian algorithm A, the exploration (S A i ) 0 i n is an inhomogeneous Markov chain whose probability transitions are described as follows. Conditionally on S A i and on A(S A i ), we pick E i uniformly at random among the 2(n Proof. It su ces to notice by induction that at each step i 0 of the exploration, conditionally on S i , the pairingω of the (unexplored) edges of ∂S A i is uniform. Hence, if an edge A(S A i ) is picked independently ofω then its partner is a uniform edge on ∂S A i \{A(S A i )}.
Remark (Uniform peeling and split/merge dynamics). Probably the most obvious Markovian peeling algorithm is the following. For i 0, given the discrete surface S i , we pick the next edge to peel uniformly at random on ∂S i (independently of the past operations and of the gluing of the edges). Although very natural, we shall not use this peeling algorithm in this paper. We note however that the Markov chain this algorithm induces on the set of perimeters of the holes of S i is very appealing. If {p 1 , . . . , p k } is the con guration of the perimeters at time i, the next state is obtained by rst sampling independently two indices I , ∈ {1, 2, . . . , k } proportionally to p 1 , . . . , p k . If I , then we replace the two numbers p I and p by p I + p − 2. If I = , we replace p I by a uniform splitting of p I − 2 into {{0, p I − 2}, {1, p I − 2}, . . . , {p I − 2, 0}}. This is a discrete version of the split-merge dynamic, which preserves the Poisson-Dirichlet law, considered in [ , ], except that we have a deterministic "erosion" of −2 at each step in the above dynamic.
The strength of the above proposition is that, as for planar maps [ ], we can use di erent algorithms A to explore the same random surface M P and then to get di erent types of information. We will see this motto in practice in the following sections. When exploring our random surface with a given peeling algorithm, we will always write (F i ) 0 i n for the canonical ltration generated by the exploration. .

Controlling loops and bigons for good con gurations
In this subsection, we give rough bounds on the number of holes of perimeter 1 and 2 that appear during Markovian explorations of good sequences of con gurations. We will call loop (resp. bigon) a hole of perimeter 1 (resp. 2). The purpose of such estimates will be to bound the number of steps at which the cases of Figure occur during the exploration, which will be useful several times in the next sections. Recall from the introduction the de nition of a good sequence (P n ) n 1 of con gurations and assume we explore M P n using a Markovian algorithm. With an implicit dependence on P n and n, denote by (L i : 0 i n) and (B i : 0 i n) the number of loops and bigons in ∂S i during the exploration. We also write π (i) for the perimeter of the hole to which the peeled edge at time i belongs.

Proposition (Bounds on loops and bigons). For any con guration and any Markovian peeling algorithm we have the stochastic dominations
Bern 4 i , with independent Bernoulli random variables. In particular, if (P n ) n 1 is a good sequence of con gurations then we have sup L i = o( √ n), also sup i B i = o(n) and n−1 i=0 1 π (i)=2 = o(n) with very high probability.
Finally, for any good sequence of con gurations, for any ε > 0, we have Proof. Let us describe rst the variation of the number of bigons and loops during a peeling step using Section . . One can create loops or bigons if at step i 0 we are peeling on a p-gon with p 3 and identify the peeled edge with the second edge on its right or left along the same hole (creation of one loop) or the third edge on its right or left along the same hole (creation of a bigon). When p = 4, peeling the second edge on the right or left are the same event but it yields to creation of 2 loops, and similarly when p = 6 for bigons. In any of these cases we have ∆L i 2 and ∆B i 2, and the conditional (on F i ) probability of those events is bounded above by In the case p = 2, recall that the identi cation of both sides of a bigon only results in the disappearance of the bigon and the creation of 2 true vertices. Otherwise, the i-th peeling step identi es the peeled hole of perimeter π (i) with another hole of perimeter we denote by ξ (i). This results in the creation of a hole of perimeter π (i) + ξ (i) − 2 (in the case π (i) = ξ (i) = 1, i.e. when we identify two loops, we just create a true vertex). F . By the above description, appart from the rst kind of events where we identify two edges on the same hole, we notice that we always have ∆L i 0. Hence, the distribution of ∆L i conditionally on F i is stochastically dominated by where Bern(p) is a Bernoulli variable of parameter p. Clearly sup i <n (L i − L 0 ) is stochastically dominated by Z = 2 n−1 i=0 Bern 2 2(n−i)−1 with independent Bernoulli variables, and this proves the rst claim. S . We now focus on ∆(L i + B i ) = L i+1 − L i + B i+1 − B i . Again, appart from the cases where we identify two edges on the same hole which result in ∆(L i + B i ) 2, we always have and the same reasoning as above yields the second point of the Proposition. T . With a closer look at the cases above, we see that the conditional distribution of 1 π (i)=2 + ∆(L i + B i ) given F i is again stochastically dominated by 2 · Bern( 4 2(n−i)−1 ). Summing over 0 i n − 1 yields the result.
L . For the last convergence of the proposition, remark that where the rst term in the right hand side comes from splitting a hole of perimeter p 3, and the second term comes from the case where a loop is peeled and identi ed with a bigon to produce a new loop. Taking expectation, after summing over 0 i (1 − ε)n, we get for n large By our second domination sup i B i B 0 +L 0 +X n and our goodness assumption L 0 +B 0 = o( √ n) +o(n) = o(n), we easily deduce that P(sup 0 j n B j ε 2 n) decreases faster than n −2 . Therefore, after re-arranging the last display, we obtain

Peeling the minimal hole and number of vertices
In this section we study the number of vertices of M P and prove Theorem . The corner stone of the proof is to explore our surface using the following algorithm.
Algorithm H or peeling the minimal hole: Given S i for 0 i < n, the next edge to peel H(S i ) is one of the edges which belong to a hole of minimal perimeter. If there are multiple choices, we pick the edge having the minimal label.
It is clear that the above algorithm is Markovian and so the transitions of the Markov chain (S i ) i 0 are described by Proposition . In the rest of this section, we always explore our random surfaces using the above algorithm.
. Towards a single hole Fix a con guration P with |P| = n and let us explore the random surface M P using the peeling algorithm H. All our notations below depend implicitly on P. A rst observation concerns the number of holes during this exploration. If H i denotes the number of holes of S i then H 0 = #P and it is easy to see from the possible topology changes (Section . ) that 1}, then we have identi ed two edges of the same hole during the i-th peeling step. Since the minimal perimeter of a hole is at most |∂S i |/H i , if F i is the σ -eld of the past exploration up to time i, we get from Proposition that In other words, as long as H i is large, the process H undergoes a strong negative drift (it decreases by 1 or 2 with a high probability at each step). If τ = inf {i 0 : H i = 1} is the rst time at which S i has a single hole, then the above inequality together with easy probabilistic estimates show that H hits the value 1 "almost as soon as possible".
Lemma (τ ≈ #P). For every ε > 0 we can nd C ε > 0 such that uniformly in P, we have Proof. Fix ε > 0 and pick A 1 large so that A A−2 (1 + ε). Let us de ne τ A = inf {i 0 : H i A}. Clearly, by the Markov property of the exploration and the above calculation, as long as i τ A , the increments of H are stochastically dominated by independent random variables ξ i with law P(ξ = 1) = 1 A and P(ξ = −1) = 1 − 1 A . As a result, τ A is stochastically dominated by α(#P) the hitting time of A by a random walk with i.i.d. increments of law ξ started from #P. Now, as soon as H i drops below A and until time τ , we can stochastically bound its increments by those of a simple symmetric random walk on Z. If β denotes the hitting time of 1 by a simple symmetric random walk on Z started from A, we can thus write the stochastic inequality τ α(#P) + β, where the last two variables are independent (but we shall not use it). Notice that by the law of large numbers α(#P) ∼ #P A A−2 as #P → ∞. By our choice of A, this implies that P(α(#P) (1 + ε)#P) ε/2 for all #P larger than some p 0 1. We then x C ε large so that P(β C ε /2) ε/4 and P(sup p p 0 α(p) C ε /2) ε/4. When doing so, the statement of the lemma holds true.
Back to one hole and unicellular maps. Performing the exploration of the surface with algorithm H until time τ is particularly convenient. Indeed, S τ has a single hole of perimeter 2(n − τ ) and by the Markov property of the exploration, to get the nal surface M P , one just needs to glue the edges of this hole using an independent uniform pairing of its edges. This is obviously a particular case of our construction of random surface for the case where P = {2(n − τ )} is made of a single polygon. The (obviously connected!) surface M {2k } is known as a unicellular map with k edges since it is a map having a single face. Using this observation, we can draw two conclusions: rst, the surface M P is connected if and only if S τ is connected. Second, if we focus on the number of vertices V P of the underlying surface we can write where X τ is the number of "true" vertices of S τ (i.e. actual vertices of M P ) and V {2(n−τ )} is the number of vertices of M {2(n−τ )} a unicellular map with n − τ edges, which is conditionally on τ , independent of S τ . Let us use these two remarks to study the connectedness and the number of vertices of M P .

. Connectedness
Let us focus rst on the connectivity of our surface. We x a good sequence (P n ) n 1 of polygonal con gurations and for each n, perform the exploration of M P n using algorithm H. We write τ ≡ τ n to highlight the dependence in P n and n. Remark that if (P n ) is a good sequence of con gurations, then we have few loops and bigons and so #P n n 2 3 ( ) asymptotically as n → ∞. In particular, by Lemma , when performing the peeling using algorithm H, we reach a con guration having a single hole typically before 2n/3 out of the n peelings steps.
By the last discussion, M P n is connected if and only if we managed to reach time τ n without having disconnected a piece of our surface en route. We prove that for a good sequence of con gurations, this situation is very likely: Proposition . If (P n ) n 1 is a good sequence of con gurations, then M P n is connected with high probability as n → ∞.
Remark . It is an exercise to prove that if P either contains more that ε |P| loops or more than ε |P| bigons, then there exists a constant c ε > 0 such that P(M P is not connected) c ε uniformly for all con gurations. Our assumptions for connectedness are thus optimal.
Proof. We recall that S 0 → S 1 → · · · → S n = M P n is the exploration of M P n using the peeling algorithm H. By the list of the possible outcomes of a peeling step, we see that if S τ n is not connected, then at some time 0 i < τ n , we have performed a peeling step identifying either the two sides of a bigon or two loops together. We shall see that such operations are unlikely to happen before τ n if we start with few loops and bigons. Recall that L i (resp. B i ) is the number of loops (resp. bigons) in ∂S i and that π (i) is the perimeter of the hole peeled at time i. For each 0 i < n, conditionally on the past of the exploration, the probability of either closing the two sides of a bigon or identify two loops together is Summing for all 0 i 3n/4 and taking expectations, we deduce that there is a constant C > 0 such that, for every ε > 0, we have But by Proposition again, Cn × P(sup j n L j ε √ n) goes to 0 as n → ∞ since by our goodness assumption we have L 0 = o( √ n). Hence the probability to perform an event which may yield to disconnection of the surface before time 3n/4 is going to 0. Since τ n 3n/4 with high probability, by Lemma , the result is proved.
It follows from the above proof that it is very unlikely that two loops are glued together before time τ n . If so, the number of holes cannot decrease by more than 1 at each step and so with high probability we have τ n #P n − 1, ( ) which complements the upper bound of Lemma . The above proof of connectedness of M P n is probabilistic in essence and should be compared with the analytical proof of Chmutov & Pittel [ , Theorem . ] in the case when all perimeters are larger than 3. The strategy followed by their proof is closer to that of Proposition , with more involved calculations. .

Number of vertices and Theorem
We now turn to the proof of Theorem , which is obviously based on ( ). We shall estimate separately the two contributions of ( ) and start by controlling X τ n , the number of vertices created during the exploration of the surface M P n until time τ n : Lemma . If (P n ) n 1 is a good sequence of con gurations, then Remark . Notice that by ( ) the parameter in the Poisson law above asymptotically belongs to [0, log 3], which implies that (X τ n ) n 1 is tight. This is the only fact we shall use to prove Theorem .
Proof. Fix n 1 and perform the exploration of the surface M P n using algorithm H and stop at τ n . Using the description of the possible topology changes (Section . ), for each 0 i < n, conditionally on F i , the number of vertices created by the next peeling step is The proof of Proposition shows that the expected number of vertices created before time 3n/4 by the last two possibilities is negligible as n → ∞ and τ n 3n/4 with high probability. Furthermore, by Proposition , all but o(n) peeling step take place on p-gons with p 3. Since the sum of o(n) Bernoulli variables with parameter bounded by 3 n is 0 with high probability, we deduce that the law of X τ n is well approximated by Once X τ n is controlled, we need to get our hands on the other part of ( ), namely V {2n } , which is the number of vertices of a unicellular map with n edges. We prove the analog of our target result Theorem for those random maps: Lemma . Let ϵ n ∈ {odd, even} be the opposite parity of n. Then we have Proof. First proof. This result is a straightforward consequence of the general theorem of Chmutov & Pittel [ ], once we recalled that the number of cycles of a random permutation of S n is close in total variation to a Poisson random variable of parameter log n. Second proof. The well-known Harer-Zagier formula [ ] precisely gives access to the generating function of the number of vertices of unicellular maps. This formula has been exploited to give a local limit theorem for V {2n } in [ , Theorem . ]. Combining this local limit theorem with the explicit distribution of a Poisson random variable of parameter log n yields the result (we leave the straightforward calculations to the courageous reader).
Sketch of a third proof. The above lines may disappoint the reader who expected that our results are "self-contained" and do not rely on any algebraic method. Let us explain how to get the lemma without relying on [ ] nor [ ]. The idea is to directly show that the number of vertices of a uniform unicellular map with n edges is close (in total variation distance) to the number of vertices of a uniform map with n edges, provided that it has the same parity as n + 1. To x ideas, let us suppose that n is even. using algorithm H until time τ n . Since the number of polygons of U odd 2n is typically of order log n, we get from Proposition and ( ) that τ n ≈ log n and from Lemma that with high probability, no vertex has been created by the exploration until time τ n . Hence, we can write We are almost there. Notice rst that, with high probability, since we have not created vertices up to time τ n , then τ n is even and so is n −τ n . The above line shows that the number of vertices of a unicellular map with a random number of edges n − τ n is close in total variation to our goal Poisson odd log n . To nish the proof, it remains to see that if n > n have the same parity and n − n = O(log n), then V {2n } ≈ V {2n } in total variation distance. To see this, we will couple the two discrete surfaces M {2n } and M {2n } so that they have the same number of vertices with high probability. The idea will be to couple their explorations using algorithm H in such a way that they are independent until some stopping time ξ , and coincide afterwards. The key is that during these explorations, the number of holes is a process which spends most of its time on small values. More precisely, let us explore independently M {2n } and M {2n } using algorithm H and denote by H and H the processes of the number of holes in each exploration. Notice that as long as we do not create vertices (which happens only after ≈ n steps by Lemma ), the numbers H i and H i have the same parity. We will be interested in the stopping time By ( ) (see also the proof of Lemma ), the processes (H i ) and (H n −n+i ) are dominated (up to time ≈ n) by two independent copies X and X of a positive recurrent process, where X and X have the same parity. In particular, there is a small i such that X i = X i = 1. It follows that ξ happens quickly in the sense that P(ξ C log n and no vertex has been created by then in either surfaces) − −−− → n→∞ 1.
Therefore, the surfaces S ξ and S n −n+ξ both have exactly one hole, with the same size 2(n − ξ ). Hence, on the event described above, we can couple M {2n } and M {2n } by identifying their explorations from time ξ and ξ + n − n respectively on. In particular, when this coupling occurs, we have V {2n } = V {2n } as desired. We leave the details to the interested reader.
The second-to-last equality uses the fact that Poisson log n and Poisson log(n−i) are close for d TV uniformly over 0 i 3n 4 . The last one uses the fact that X τ n is tight and for any xed i, the variables Poisson log n and i + Poisson log n are close for d TV as n → +∞. Finally, if M P n is connected, by applying the Euler formula to S τ n , it is easy to check that X τ n + n − τ n + 1 has the same parity as n + #P n . We just landed on the desired Poisson variable having the correct parity.

Peeling vertices and the Poisson-Dirichlet universality
In this section we prove the Poisson-Dirichlet universality, that is our Theorem . Again, the idea is to explore our random surface via a peeling algorithm tailored to our objective. Since we are interested in the vertex degrees, our algorithm will explore the 1-neighborhood of a given vertex. Once all the edges adjacent to this vertex have been discovered, we choose a new vertex on the boundary of the current surface and iterate. More precisely: Algorithm R or peeling vertices: Given the initial con guration of labeled polygons S 0 ≡ P we pick a "red" vertex R 0 ∈ ∂S 0 uniformly at random. Inductively, given the discrete surface S i with a distinguished "red" point R i ∈ ∂S i , we peel the edge lying immediately on the left of R i to get S i+1 . If during this peeling step the red vertex has been swallowed by the process, then we resample R i+1 ∈ ∂S i+1 uniformly at random (independently of the past and of the gluing). Otherwise R i+1 canonically results of R i .
It is easy to see that the above algorithm is again Markovian and so we can apply Proposition . We shall always use the above algorithm when exploring our surfaces in this section.

. Closure times and targeting distinguished vertices
If S i is a discrete surface with a red vertex R i ∈ ∂S i , then the peeling of the edge immediately on the left of R i leaves the red vertex on the boundary of S i+1 except in two cases: • if the peeled edge is glued to the edge immediately on the right of R i , see Figure . We say that time i is then a strong closure time; • or if the red vertex R i belongs to a hole of perimeter 1, which is glued to another hole of perimeter 1, see Figure left. In this case, we say that i is a loop closure time.
Notice that in both cases the peeling step yields the creation of at least 1 true vertex of M P (this can be 2 if the peeling step closes the two sides of a bigon). There is actually another scenario which yields to Proof. The second convergence is a straightforward consequence of the fourth convergence of Proposition . To prove the rst one, conditionally on the past exploration up to time i, the probability to perform a loop closure time is equal to 1 π (i)=1 L i −1 2(n−i)−1 . Hence, summing over 0 i (1 − ε)n, taking expectation and splitting according to whether sup L i ε √ n, we get for every ε > 0: By our goodness assumption and Proposition , this tends to 0.
We will need to rule out a few other annoying situations. Assume we track a distinguished label during the exploration, say the label 1. Note that the only case where this label is never glued to the red vertex is if it is swallowed at some point by a weak closure time. Part of Proposition is that this situation does not occur (this is important since it ensures that the edges of M P n are concentrated on the vertices closed at strong closure times). We will also prove that with high probability, two xed distinguished labels do not coalesce before being glued to the red vertex (this is important to rule out strong correlations between the times σ (j) n ).
Lemma . Assume that (P n ) n 1 is a sequence of good con gurations and perform the exploration of M P n using algorithm R after having labeled the vertices of S 0 by {1, 2, . . . , 2n} arbitrarily (independently of the matching of the edges). Then for every ε > 0, with high probability as n → ∞, none of the following events occur before time (1 − ε)n: . the label 1 disappears before being glued to the red vertex; . the labels 1 and 2 coalesce before being glued to the red vertex; . the red vertex is moved to the vertex carrying the label 1 after some strong closure time.
Of course, once the lemma is proved for labels 1 and 2, it easily extends to the labels 1, 2, . . . , k for any xed k.
Proof. We start with the rst item. Fix ε > 0. In the event A i where the label 1 is swallowed at time i by a weak closure time, this label is necessarily carried by the vertex immediately to the left of R i and a weak closure time happens at time i. Hence P(A i ) 1 2(n − i) − 1 P(1 is carried by the vertex on the left of R i ).

( )
We write α i = P(1 is carried by the vertex on the left of R i ). Then we can estimate α i for i 1 by looking at the peeling step i − 1 as follows: • Either the (i − 1)-th peeling step swallows the red vertex and the new one is sampled uniformly on the boundary. In this case, the probability that R i is on the right of the label 1 is 1 2(n − i) .
• Either the (i − 1)-th peeling step glues a bigon on the left of R i−1 and, if the label 1 was already on the left of R i−1 , it stays on the left of R i . The conditional probability of this scenario is thus bounded above by 2B i−1 2(n − i) + 1 1 1 is carried by the vertex on the left of R i −1 .
• Or the (i − 1)-th peeling step glues a loop on the left of R i−1 and if the label was on the second vertex on the left of R i−1 , then it becomes immediately on the left of R i . We can crudely bound the conditional probability of this event by L i 2(n − i) + 1 .
• In all other situations, in order for the label 1 to be on the left of R i , the (i − 1)-th peeling step should identify the peeled edge with the second edge on the right of the label 1, which occurs with probability 1 2(n − i) + 1 .
In total, for any i (1 − ε)n, taking expectation and then splitting according to whether sup 0 j n L j ε 2 √ n and sup 0 j n B j ε 2 n, we have for large n and 0 i (1 − ε)n α i 2 2(n − i) + E 2B i−1 2(n − i) + 1 1 1 is carried by the vertex on the left of R i −1 + E L i 2(n − i) + 1 C n + 2ε · α i−1 + P(sup B j ε 2 n) + ε √ n + P(sup L j ε 2 √ n).
Using Proposition , and by our goodness assumption, we see that the two probabilities in the right-hand side are negligible compared to ε √ n for large n and so we get When 2ε < 1 this easily implies that α i C ε / √ n uniformly in i (1 − ε)n as n → ∞ for some constant C > 0 depending on ε. Plugging this back in ( ), we obtain P(A i ) C ε n 3/2 for i (1 − ε)n, so P (1−ε )n i=0 A i goes to 0 as n → ∞. For the second item, if we want the labels 1 and 2 to merge in the same vertex, then one of the two, say 1, must be immediately on the left of R i at time i and then the peeling step should identify the edge on the left of R i with the edge on the left of the label 2. This has probability α i 1 2(n−i)−1 and the last calculation shows that after summing over i (1 − ε)n we get a negligible contribution.
The third item is the most obvious: the probability that i is a strong closure time and that the red vertex is moved to the label 1 is at most [ ] Timothy Budd, The peeling process of in nite boltzmann planar maps, Electronic Journal of Combinatorics ( ).
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