On the sphericity of scaling limits of random planar quadrangulations

We give a new proof of a theorem by Le Gall&Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces.


Introduction
A planar map is a combinatorial embedding of a connected graph into the 2-dimensional sphere.Random planar map have drawn much attention in the recent probability literature due to mathematical physics motivations [2] and a powerful encoding of planar maps in terms of labeled trees due to Schaeffer [15,5].In turn, scaling limits of labeled trees are well-understood thanks to the works of Aldous, Le Gall and others [1,8,9].Using this line of reasoning, many results have been obtained on the geometric aspects of large random quadrangulations (where faces all have degree 4), and other families of maps.Le Gall [10] showed in particular that scaling limits of random quadrangulations are homeomorphic to the Brownian map introduced by Marckert & Mokkadem [13], and Le Gall & Paulin [11] showed that the topology of the latter is that of the 2-dimensional sphere, hence giving a mathematical content to the claim made by physicists that summing over large random quadrangulations amount to integrating with respect to a (still ill-defined) measure over surfaces.
The aim of this note is to give an alternative proof of Le Gall & Paulin's result.We still strongly rely on the results established by Le Gall [10], but use very different methods from those of [11], where the reasoning uses geodesic laminations and a theorem due to Moore on the topology of quotients of the sphere.We feel that our approach is somewhat more economic, as it only needs certain estimates from [10] and not the technical statements [11,Lemmas 3.1,3.2]that are necessary to apply Moore's theorem.On the other hand, this is at the cost of checking that quadrangulations are close to being path metric spaces, which is quite intuitive but needs justification (see definitions below).Our main geometric tool is a reinforcement of Hausdorff convergence, called 1-regular convergence and introduced by Whyburn, and which has the property of conserving the topology of surfaces.We will see that random planar quadrangulations converge 1-regularly, therefore entailing that their limits are of the same topological nature.In the case, considered in this paper, of surfaces with the topology of the sphere, the 1-regularity property is equivalent to [11,Corollary 1], stating that there are no small loops separating large random quadrangulations into two large parts.We prove this by a direct argument rather than obtaining it as a consequence of the theorem.
The basic notations are the following.We let Q n be the set of rooted1 quadrangulations of the sphere with n faces, which is a finite set of cardinality 2 • 3 n (2n)!/(n!(n + 2)!), see [5].We let q n be a random variable picked uniformly in Q n , and endow the set V (q n ) of its vertices with the usual graph distance d gr n , i.e. d gr n (x, y) is the length of a minimal (geodesic) chain of edges going from x to y.
We briefly give the crucial definitions on the Gromov-Hausdorff topology, referring the interested reader to [4] for more details.The isometry class [X, d] of the metric space (X, d) is the collection of all metric spaces isometric to (X, d).We let M be the set of isometry-equivalence classes of compact metric spaces.The latter is endowed with the Gromov-Hausdorff distance d GH , where d GH (X, X ′ ) is defined as the least r > 0 such that there exist a metric space (Z, δ) and subsets X, X ′ ⊂ Z such that [X, δ] = X, [X ′ , δ] = X ′ , and such that the Hausdorff distance between X and X ′ in (Z, δ) is less than or equal to r.This turns M into a complete separable metric space, see [6] (this article focuses on compact R-trees, which form a closed subspace of M, but the proofs apply without change to M).
Remarks.• One of the main open questions in the topic of scaling limits of random quadrangulations is to uniquely characterize the limit, i.e. to get rid of the somewhat annoying "along some subsequence" in the previous statement.
• To be perfectly accurate, Le Gall & Paulin showed the same result for uniform 2kangulations (maps with degree-2k faces) with n faces.Our methods also apply in this setting (and possibly to more general families of maps), but we will restrict ourselves to the case of quadrangulations for the sake of brevity.
• In the work in preparation [14], we provide a generalization of this result to higher genera, in the framework of Boltzmann-Gibbs distributions on quadrangulations rather than uniform laws.
As we are quite strongly relying on Le Gall's results in [10], we will mainly focus on the new aspects of our approach.As a consequence, this paper contains two statements whose proofs will not be detailed (Proposition 2 and Lemma 2), because they are implicit in [10] and follow directly from the arguments therein, and also because their accurate proof would need a space-consuming introduction to continuum tree and snake formalisms.Taking them for granted, the proofs should in a large part be accessible to readers with no particular acquaintance with continuum trees or Schaeffer's bijection.

Gromov-Hausdorff convergence and regularity
We say that a metric space (X, d) is a path metric space if every two points x, y ∈ X can be joined by a path isometric to a real segment (with length d(x, y)).We let PM be the set of isometry classes of compact path metric spaces, and the latter is a closed subspace of (M, d GH ), see [4,Theorem 7.5.1].One of the main tools needed in this article is a notion that reinforces the convergence in the metric space (PM, d GH ), which was introduced by Whyburn in 1935 and was extensively studied in the years 1940's.Our main source is Begle [3].
Definition 1 Let (X n , n ≥ 1) be a sequence of spaces in PM converging to a limit X.We say that X n converges 1-regularly to X if for every ε > 0, one can find δ, N > 0 such that for all n ≥ N, every loop in X n with diameter ≤ δ is homotopic to 0 in its ε-neighborhood.
There are a couple of slight differences between this definition and that in [3].In the latter reference, the setting is that X n are compact subsets of a common compact space, converging in the Hausdorff sense to a limiting set X.This is not restrictive as Gromov-Hausdorff convergence entails Hausdorff convergence of representative spaces in a common compact space, see for instance [7,Lemma A.1].It is also assumed in the definition of 1-regular convergence that for every ε > 0, there exists δ, N > 0 such that any two point that lie at distance ≤ δ are in a connected subset of X n of diameter ≤ ε, but this condition is tautologically satisfied for path metric spaces.Last, the definition in [3] is stated in terms of homology, so our definition in terms of homotopy is in fact stronger.
The following theorem is due to Whyburn, see [3,Theorem 6] and comments before.
Theorem 2 Let (X n , n ≥ 1) be a sequence of elements of PM that are all homeomorphic to S 2 .Assume that X n converges to X for the Gromov-Hausdorff distance, where X is not reduced to a point, and that the convergence is 1-regular.Then X is homeomorphic to S 2 as well.

Quadrangulations
Rooted quadrangulations are maps whose faces all have degree 4, and their set is denoted by Q := n≥1 Q n with the notations of the Introduction.For q ∈ Q we let V (q), E(q), F (q) be the set of vertices, edges and faces of q, and denote by d gr q the graph distance on V (q).

A metric surface representation
One of the issues that must be addressed in order to apply Theorem 2 is that the metric space [V (q), d gr q ] is not a surface, rather, it is a finite metric space.We take care of this by constructing a particular graphical representative of q which is a path metric space whose restriction to the vertices of the graph is isometric to (V (q), d gr q ).Let (X f , d f ), f ∈ F (q) be copies of the emptied unit cube "with bottom removed" endowed with the intrinsic metric d f inherited from the Euclidean metric (i.e. the distance between two points of X f is the minimal Euclidean length of a path in X f ).Obviously each (X f , d f ) is a path metric space homeomorphic to a closed disk of R 2 .For each face f ∈ F (q), we label the four incident half-edges turning counterclockwise as (e 1 , e 2 , e 3 , e 4 ), where the labeling is arbitrary among the 4 possible labelings preserving the cyclic order.Then define In these notations, we keep the subscript f to differentiate points of different spaces X f .In this way, for every e ∈ E(q), we have defined a path c e of length 1 which goes along one of the four edges of the boundary ∂X f = ([0, 1] 2 \ (0, 1) 2 ) × {0}, where f is the face incident to e.
We then define an equivalence relation ∼ on the disjoint union ∐ f ∈F (q) X f , as the coarsest equivalence relation such that for every e ∈ E(q), and every t ∈ [0, 1], we have c e (t) ∼ c e (1 − t).By identifying points of the same class, we glue the boundaries of the spaces X f together in a way that is consistent with the map structure.More precisely, the topological quotient S q := ∐ f ∈F (q) X f / ∼ is a 2-dimensional cell complex whose 1-skeleton E q is a graph representation of q, and where the faces are the interiors of the spaces X f .In particular, S q is homeomorphic to S 2 .We let V q be the 0-skeleton of this complex, i.e. the vertices of the graph.We call the 1-cells and 0-cells of E q and V q the edges and vertices of S q .
We next endow the disjoint union ∐ f ∈F (q) X f with the largest pseudo-metric D q that is compatible with d f , f ∈ F (q) and with ∼, in the sense that D q (x, y) ≤ d f (x, y) for x, y ∈ X f , and D q (x, y) = 0 for x ∼ y.Therefore, the function D q : ∐ f ∈F (q) X f × ∐ f ∈F (q) X f → R + is compatible with the equivalence relation, and its quotient mapping d q defines a pseudo-metric on the quotient space S q .
Proposition 1 The space (S q , d q ) is a path metric space homeomorphic to S 2 .Moreover, the restriction of S q to the set V q is isometric to (V (q), d gr q ), and any geodesic path in S q between two elements of V q is a concatenation of edges of E q .Last, Proof.What we first have to check is that d q is a true metric on S q , i.e. it separates points.To see this, we use the fact [4, Theorem 3.1.27]that D q admits the constructive expression: where we have set d(x, y) = d f (x, y) if x, y ∈ X f for some f , and d(x, y) = ∞ otherwise.
It follows that for a ∈ X f \ ∂X f , and for b = a, D q (a, b) > min(d(a, b), d f (a, ∂X f )) > 0, so a and b are separated.It remains to treat the case a ∈ ∂X f for some f .The crucial observation is that a shortest path in X f between two points of ∂X f is entirely contained in ∂X f .Therefore, the distance D q (a, b) is always larger than the length of a path with values in the edges ∐∂X f / ∼ of S q , where all edges have total length 1.In particular, points in distinct classes are at positive distance.One deduces that d q is a true distance on S q , and by the compactness of the latter, (S q , d q ) is homeomorphic to S 2 [4, Exercise 3.1.14].
From this same observation, we obtain that a shortest path between vertices of S q is a shortest path of edges, i.e. is the geodesic distance for the (combinatorial) graph distance.Thus, (V q , d q ) is indeed isometric to (V (q), d gr q ).The last statement follows immediately from this and the fact that diam (X f , d f ) ≤ 3, entailing that V q is 3-dense in (S q , d q ), i.e. its 3-neighborhood in (S q , d q ) equals S q .

Tree encoding of quadrangulations
We briefly introduce the second main ingredient, the Schaeffer bijection.Let T n be the set of pairs (t, l) where t is a rooted planar tree with n edges, and l is a function from the set of vertices of t to N = {1, 2, . ..}, such that |l(x) −l(y)| ≤ 1 if x and y are neighbors.Then the set Q n is in one-to-one correspondence with T n .More precisely, this correspondence is such that given a graph representation of q ∈ Q n on a surface, the corresponding (t, l) ∈ T n can be realized as a graph whose vertices are V (t) = V (q) \ {x * }, where x * is the origin vertex of the root edge, and l is the restriction to V (t) of the function l(x) = d gr q (x, x * ), x ∈ V (q).Moreover, the edges of t and q only intersect at vertices.The root vertex of t is the tip of the root edge of q, so it lies at d gr q -distance 1 from x * .Let x(0) be the root vertex of t, and given {x(0), . . ., x(i)}, and let x(i + 1) be the first child 2 of x(i) not in {x(0), . . ., x(i)} if there is any, or the parent of x(i) if there is not.This procedure stops at i = 2n, where we are back to the root and have explored all vertices of the tree.We let C i = d gr t (x(i), x(0)), and L i = l(x(i)).Both C and L are extended by linear interpolation between integer times into continuous functions, still called C, L, with duration 2n.The contour process C of t is the usual Harris walk encoding of the rooted tree t, and the pair (C, L) determines (t, l) completely.In the sequel, we will use the fact that x(i) can be identified with a vertex of q.
A simple consequence (see [10,Lemma 3.1]) of the construction is that for i < j,

Estimates on the lengths of geodesics
Our last ingredient is a slight rewriting of the estimates of Le Gall [10] on geodesic paths in quadrangulations in terms of encoding processes.Precisely, let C n , L n be the contour and label process of a uniform random element t n of T n , and let q n be the quadrangulation that is the image of this element by Schaeffer's bijection.In particular, q n is a random uniform element of Q n .Also, recall that a graphical representation T n of t n can be drawn on the representation S qn of Sect.3.1, in such a way that the vertices of T n are V qn \ {x * }, where x * is the root vertex, and T n intersects edges E qn only at vertices.For simplicity we let The main result of [9] says that the convergence in distribution in C([0, 1], R) 2 holds: where ( , Z) is the Brownian snake conditioned to be positive introduced by Le Gall & Weill [12].Moreover, it is shown in [10] that the laws of [V n , n −1/4 d gr n ] form a relatively compact family in the set of probability measures on M endowed with the weak topology.Since V n is 3-dense in S n , the same holds for [S n , n −1/4 d n ].We argue as in [10], and assume by Skorokhod's theorem that the trees t n (hence also the quadrangulations q n ) are defined on a common probability space on which we have, almost-surely , some random limiting space in PM, along some subsequence n k → ∞, and • the convergence (2) holds a.s.along this subsequence.
From this point on, we will always assume that n is taken along this subsequence.In particular, we have that diam S = lim n n −1/4 diam S n ≥ lim n sup n −1/4 L n = sup Z > 0 a.s., so S is not reduced to a point and Theorem 2 may be applied if we check that the convergence is 1-regular.We are going to rely on proposition 4.2 of [10], which can be rephrased as follows.

Proposition 2
The following property is true with probability 1.Let i n , j n be integers such that i n /2n → s, j n /2n → t in [0, 1], where s, t satisfy For n ≥ 1, let γ n be a path in q n between x(i n ) and x(j n ) with the notation of Sect.3.2.Then it holds that lim inf In [10], this proposition was a first step in the proof of the fact that a limit in distribution of (V n , d gr n ) can be expressed as a quotient of the continuum tree with contour function : this lemma says that two points of the latter such that one is an ancestor of the other are not identified.Le Gall completed this study by exactly characterizing which are the points that are identified.

Proof of Theorem 1
Lemma 1 Almost-surely, for every ε > 0, there exists a δ ∈ (0, ε) such that for n large enough, any simple loop γ n made of edges of S n , with diameter ≤ n 1/4 δ, splits S n in two Jordan domains, one of which has diameter ≤ n 1/4 ε.
Proof.We argue by contradiction, assuming there exist simple loops γ n made of edges of S n , with diameters o(n 1/4 ) as n → ∞, such that the two Jordan domains bounded by γ n are of diameters ≥ n 1/4 ε, where ε > 0 is some fixed constant.Let l n be the minimal label on γ n , i.e. the distance from the root vertex x * to γ n .Then all the labels of vertices that are in a connected component D n of S n \ γ n not containing x * are all larger than l n , since a geodesic from x * to any such vertex must pass through γ n .
The intuitive idea of the proof is the following.Starting from the root of the tree T n , follow a maximal simple path in T n that enters in D n at some stage.If all such paths remained in D n after entering, then all the descendents of the entrance vertices would have labels larger than that of the entrance vertex, a property of zero probability under the limiting Brownian snake measure, see [10,Lemma 2.2] and Lemma 2 below.Thus, some of these paths must go out of D n after entering, but they can do it only by passing through γ n , which entails that strict ancestors in T n will be at distance o(n 1/4 ), and this is prohibited by Proposition 2. This is summed up in Figure 1, which gathers some of the notations to come.
We proceed to the rigorous proof.Take a vertex y n in D n .As a vertex of T n , it can be written in the form x(j n ) for some j n .Let j ′ n be the first integer j ≥ j n such that x(j) is at d n -distance ≤ 1 from γ n .Such a j exists because of the way edges of T n are drawn (entailing that the ancestral path in T n from x n to the root of T n must itself pass at distance ≤ 1 from γ n , since the root of T n is at distance 1 from x * and x n lies in D n ) and the label l(x(j ′ n )) is at most max z∈γn l(z) + 1 = l n + o(n 1/4 ).Moreover, for k ∈ [j n , j ′ n ], the vertex x(k) is in D n , and in particular, its label is ≥ l n .Applying the bound (1) to the times j n , j ′ n , we get that d n (y n , x(j ).Since by hypothesis the diameter of D n is at least n 1/4 ε, it is thus possible to choose y n with label l(y n ) ≥ l n + n 1/4 ε/2.
We let x n be the first ancestor of y n in T n lying at d n -distance ≤ 1 from γ n , so that l(x n ) = l n + o(n 1/4 ).Take i n < j n such that i n is a time encoding x n , so that 0 1 Figure 1: Illustration of the proof.The surface S n is depicted as a sphere with a bottleneck circled by γ n (thick line).The root edge of the quadrangulation is drawn at the bottom, and the tree T n originates from its tip.In dashed lines are represented the two branches of T n that are useful in the proof: one enters the component D n , and the other goes out after entering, identifying strict ancestors in the limit Up to further extraction, we may and will assume that Then s ≤ t and s ≤ u for u ∈ [s, t].More precisely, we have Z s = l and Z t ≥ l + (9/8) 1/4 ε/2, which implies s < t, and s < t .In terms of the continuum tree encoded by , this amounts to the fact that s, t encode two vertices such that the first is an ancestor of the second, and that are not the same because the snake Z takes distinct values at these points.We will need the following technical statement: Lemma 2 Assume that s > 0. With probability 1, there exist η > 0 and integers i ′ n , k n , r n with i n ≤ i ′ n < k n < r n < j n such that i ′ n /2n → s ′ ∈ [s, t), that satisfy for n large enough: We claim that this lemma is enough to obtain 1-regularity of the convergence, and hence to conclude by Theorem 2 that the limit (S, d) is a sphere.First choose ε < diam S/3 to avoid trivialities.Let γ n be a loop in S n with diameter ≤ n 1/4 δ.The boundary of the union of the closures of faces of q n that are hit by γ n is made of pairwise disjoint simple loops of edges of S n .If x, y are elements of this union of faces, and since a face of S n has diameter less than 3, there exist points x ′ , y ′ of γ n at distance at most 3 from x, y respectively, so that the diameters of these loops all are ≤ n 1/4 δ + 6.By the Jordan Curve Theorem, each of these loops splits S n into two simply connected components, one of which has diameter ≤ n 1/4 ε, and one of which contains γ n entirely.It suffices to justify that these two properties (being of diameter ≤ n 1/4 ε and containing γ n ) hold simultaneously for some loop in the family to conclude that γ n is homotopic to 0 in its ε-neighborhood.So assume the contrary: the component not containing γ n of every loop is of diameter ≤ n 1/4 ε.By definition, any point in the complement of the union of these components is at distance at most 3 from some point of γ n Take x, y such that d n (x, y) = diam (S n ).Then there exist points x ′ , y ′ in γ n at distance at most n 1/4 ε + 3 respectively from x, y, and we conclude that d n (x ′ , y ′ ) ≥ diam (S n ) − 6 − 2n 1/4 ε > n 1/4 δ for n large enough by our choice of ε, a contradiction.