Asymptotic expansion of the multi-orientable random tensor model

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expansion in N, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.


Introduction and motivation
Random tensor models (see [11] for a recent review) generalize in dimension three (and higher) the celebrated matrix models (see, for example, [3] for a review). Indeed, in the same way matrix models are related to combinatorial maps [14], tensor models in dimension three are related to tensor graphs or 3D maps. A certain case of random tensor models, the so-called colored tensor models, have been intensively studied in the recent years (see [7] for a review). The graphs associated to these models are regular edge-colored graphs. An important result is the asymptotic expansion in the limit N → ∞ (N being the size of the tensor), expansion which was obtained in [5]. The role played by the genus in the 2D case is played here by a distinct integer called the degree. The dominant graphs of this asymptotic expansion are the so-called melonic graphs, which correspond to particular triangulations of the three-dimensional sphere S 3 . Let us also mention here that a universality result generalizing matrix universality was obtained in the tensor case in [6].
A particularly interesting combinatorial approach for the study of these colored graphs was proposed recently by Gurȃu and Schaeffer in [9], where they analyze in detail the structure of colored graphs of fixed degree and perform exact and asymptotic enumeration. This analysis relies on the reduction of colored graphs to some terminal forms, called schemes. An important result proven in [9] is that the number of schemes of a given degree is finite (while the number of graphs of a given degree is infinite).
Nevertheless, a certain drawback of colored tensor models is that an important number of tensor graphs is discarded by the very definition of the model. Thus, a different type of model was initially proposed in [13], the 3D multi-orientable (MO) tensor model. This model is related to tensor graphs which correspond to 3D maps with a particular Eulerian orientation. The set of MO tensor graphs contains as a strict subset the set of colored tensor graphs. The asymptotic expansion in the limit N → ∞ for the MO tensor model was studied in [2], where it was shown that the same class of tensor graphs, the melonic ones, are the dominant graphs in this limit. The sub-dominant term of this expansion was then studied in detail in [10].
In this paper we implement a Gurȃu-Schaeffer analysis for the MO random tensor model. We investigate in detail the general term of the asymptotic expansion in the limit N → ∞. As in the colored case, this is done by defining appropriate terminal forms, the schemes. Nevertheless, our analysis is somehow more involved from a combinatorial point of view, since, as already mentioned above, a larger class of 3D maps has to be taken into consideration. Also an important difference with respect to the colored case is that the MO model allows for half-integer degrees, and not only integer degrees, as does the colored model. This leads to the the fact that the dominant schemes are different from the ones identified in [9] for the colored model (interestingly, in both cases, dominant schemes are naturally associated to rooted binary trees).
Let us also mention that the analysis of this paper may further allow for the implementation of the the so-called double scaling limit for the MO tensor model. This is a particularly important mechanism for matrix models (see again [3]), and which makes it possible to take, in a correlated way, the double limit N → ∞ and z → z c where z is a variable counting the number of vertices of the graph and z c is some critical point of the generating function of schemes of a given degree.

Preliminaries
In this section we recall the main definitions related to MO tensor graphs.
2.1. Multi-orientable tensor graphs. A map (also called a fat-graph) is a graph (possibly with loops and multiple edges, possibly disconnected) such that at each vertex v the cyclic order of the d incident half-edges (d being the degree of v) in clockwise (cw) order around v is specified. A corner of a map is defined as a sector between two consecutive half-edges around a vertex, so that a vertex of degree d has d incident corners. A 4-regular map is a map where all vertices have degree 4. A multi-orientable tensor graph, shortly called MO-graph thereafter, is a 4-regular map where each half-edge carries a sign, + or −, such that each edge has its two half-edges of opposite signs, and the two half-edges at each corner also have opposite signs. In addition, for convenience, the half-edges at each vertex are turned into 3 parallel strands, see Figure 1 for an example.
The strand in the middle is called internal, the two other ones are called external. An external strand is called left if it is on the left side of a positive half-edge or on the right side of a negative half-edge; an external strand is called right if it is on the right side of a positive half-edge or on the left side of a negative half-edge. A face of an MO-graph is a closed walk formed by a closed (cyclic) sequence of strands. External faces (faces formed by external strands) are the classical faces of the 4regular map, while internal faces (faces formed by internal strands), also called straight faces thereafter, are not faces of the 4-regular map. Note also that external faces are either made completely of left external strands, or are made completely of right external strands; accordingly external faces are called either left or right. We finally define a rooted MO-graph as a connected MO-graph with a marked edge, which is convenient (as in the combinatorial study of maps) to avoid symmetry issues (indeed, as for maps, MO graphs are unlabelled ). + + -- Figure 1. An MO-graph, with the 3 types of strands (left strands in blue, straight strands in green, right strands in red).

2.2.
The degree of an MO-graph. The degree of an MO-graph G is the quantity where c, V, F are respectively the numbers of connected components, vertices, and faces (including internal faces) of G. For G 1 and G 2 two MO-graphs, denote by G 1 + G 2 the MO-graph made of disconnected copies of G 1 and G 2 . Note that the quantities c, V, F for G result from adding the respective quantities from G 1 and G 2 . Hence we have: For two MO-graphs G 1 and G 2 , the degree of G 1 + G 2 is the sum of the degrees of G 1 and G 2 . In particular, the degree of an MO-graph is the sum of the degrees of its connected components.
That δ is nonnegative follows from the following observation [13]: G gives rise to three 4-regular maps G ,r (resp. G ,s , G r,s ), called the jackets of G, which are obtained from G by deleting straight faces (resp. deleting right faces, deleting left faces), see Figure 2 for an example. Note that G ,r is an orientable map, while G ,s and G r,s are typically only locally orientable (in a usual ribbon representation, edges have twists). Let g ,r , g ,s and g r,s be the respective genera of G ,r , G ,s , G r,s (since G ,r is orientable, g ,r ∈ Z + , while g ,s and g r,s are in 1 2 Z + ). Let F , F r , F s be the numbers of left faces, right faces, and straight faces of G. Denoting by E the number of edges of G, the Euler relation gives r + F ,s + F r,s and E = 2V (because the map is 4-regular) we conclude that δ = g ,r + g ,s + g r,s . We call g := g ,r ∈ Z + the (canonical) genus of G. An MO-graph G is called planar if g = 0.
2.3. MO-graphs as oriented 4-regular maps. Seeing edges as directed from the positive to the negative half-edge, MO-graphs may be seen as 4-regular maps where the edges are directed such that each vertex has two ingoing and two outgoing half-edges, and the two ingoing (resp. two outgoing) half-edges are opposite, see Figure 3 for an example. We call admissible orientations such orientations of 4regular maps. This point of view of MO-graphs as oriented 4-regular maps is quite convenient and will be adopted most of the time in the rest of the article (except when we need to locally follow the strand structure). Note that the left faces  are directed counterclockwise (ccw), and the right faces are directed cw; and the straight faces are closed walks that alternate in direction at each vertex they pass by.
Remark 2.1. Since straight faces alternate at each vertex they pass by, straight faces have even length.
Note also that admissible orientations are completely characterized by the property that the faces of the 4-regular map (not including the straight faces) are either cw or ccw; this is also equivalent to coloring the faces of the map in two possible colors, say red or blue, such that at each edge there is a blue face on one side and a red face on the other side (this is also directly visible given the definition with strands, where right faces are red and left faces are blue, see Figure 1). In other words, a 4-regular map can be endowed with an admissible orientation iff the dual of every of its connected components is a bipartite quadrangulation. If this is the case, and if there is a marked directed edge in each connected component, then there is a unique admissible orientation that fits with the prescribed edge directions (because we have already established the existence, and there is clearly a unique way to propagate the directions starting from the marked directed edges).
Recall that, in the context of maps, a rooted map means a connected map with a marked directed edge. The above paragraph yields the following statement, which we think is worth mentioning even if it will not be needed to perform the combinatorial study (scheme extraction and associated analysis) of MO-graphs: Remark 2.2. Rooted MO-graphs having genus g and m vertices identify to rooted 4-regular maps having genus g and m vertices, and with the property that the dual rooted quadrangulation is bipartite (these are themselves well-known to be in bijection with rooted connected maps having genus g and m edges). In genus 0 (planar case), all rooted 4-regular maps have this property.
We conclude this subsection with a simple lemma relating the degree and the genus to the number of straight faces: where δ and g are respectively the degree and the genus of G. Hence Λ ≤ 2δ.
Proof. Since the number E of edges is twice the number of vertices, we have p≥1 2pF Let F , F r be the numbers of left faces and right faces of G. We have seen in Section 2.2 that δ = g ,r + g ,s + g r,s , where g = g ,r and Substracting the sum of the two last equalities from the first one, we obtain: Remark 2.3. For g = 0, according to Remark 2.2, rooted MO-graphs correspond to rooted 4-regular planar maps. Seeing such a map as the planar projection of an entangled link that lives in the 3D space (vertices of the map correspond to crossings, where the under/over information is omitted), F s is classically interpreted as the number of knot-components. Lemma 2.1 gives F s = V /2+1−δ. Since F s ≥ 1 and δ ≥ 0, the extremal cases are: (1) δ = 0, in which case F s = V /2+1; (2) F s = 1, in which case δ = V /2 (the MO-graph of Figure 3, having V = 5, F s = 2, and thus δ = 3/2, is intermediate). While the case δ = 0 is combinatorially well-understood (as will be recalled in Section 3.1), the case F s = 1 is notoriously difficult: in that case, 4-regular planar maps are projected diagrams of one-component links (i.e., knot diagrams), see [12] and references therein.
We will go back to this remark in Section 7 (on generating functions), where we will establish, for each fixed δ ∈ 1 2 Z + , the asymptotic enumeration of rooted 4-regular planar maps with 2n vertices and with n + 1 − δ knot-components.
2.4. Regular colored graphs for D = 3 as a subfamily of MO-graphs. As recalled here from [13], MO-graphs form a superfamily of the well studied regular colored graphs of dimension 3. For D ≥ 2 a regular colored graph of dimension D is defined as a (D + 1)-regular bipartite graph G (vertices are either black or white) where the edges have a color in {0, . . . , D}, such that at each vertex the D + 1 incident edges have different colors. A rooted colored graph is a connected colored graph with a marked edge of color 0. For 0 ≤ i < j ≤ D, a face of type (i, j) in a colored graph of dimension D is a cycle made of edges that alternate colors in {i, j}; let F i,j be the number of faces of type (i, j) in G. Let c be the number of connected components, V = 2k the number of vertices and F the total number of   faces of G, i.e., F = 0≤i<j≤D F i,j . The degree of G is the integer δ given by Note that a colored graph has a canonical realization as a map, where the edge colors in cw (resp. ccw) order around black (resp. white) vertices are (0, 1, . . . , D). Let G be a rooted colored graph of dimension 3 that is canonically embedded. Orienting the edges of even color from the black to the white extremity and the edges of odd color from the white to the black extremity, we obtain a rooted MO-graphG, see Figure 4 for an example. Clearly this gives an injective mapping, since there is a unique way (when possible) to propagate the edge colors starting from the root-edge. Hence, rooted colored graphs of dimension 3 form a subfamily of rooted MO-graphs. In addition so thatG and G have the same total number of faces. Hence the degree formula for colored graphs is consistent with the degree formula for MO-graphs. 3. From MO-graphs to schemes 3.1. Extracting the melon-free core of a rooted MO-graph. From now on, it is convenient to consider that a "fake" vertex of degree 2, called the root-vertex, is inserted in the middle of the root-edge of any rooted MO-graph (so that the rootedge is turned into two edges). By convention also, it is convenient to introduce the following MO-graph: the cycle-graph is defined as an oriented self-loop carrying no vertex. The cycle-graph is connected, has V = 0, F = 3 (one face in each type), hence has degree 0. In its rooted version, the (rooted) cycle-graph is made of an oriented loop incident to the root-vertex. In a (possibly) rooted) MO-graph G, a melon is a triple edge (e 1 , e 2 , e 3 ) such that none of the 3 edges is the root-edge (if The reverse operation (where e is allowed to be a loop, and is allowed to be incident to the root-vertex if the MO-graph is rooted) is called the insertion of a melon at an edge. A rooted MO-graph G is called melonic if it can be reduced to the rooted cyclegraph by successive removals of melons, see Figure 6 for an example. An important remark is that, for a rooted melonic MO-graph, any greedy sequence of melon removals terminates at the cycle-graph. It is known [2] that rooted MO-graphs of degree 0 are exactly the rooted melonic graphs. For G a (possibly rooted) MOgraph, e an edge of G (possibly a loop, possibly incident to the root-vertex), and G a rooted MO-graph, define the operation of substituting e by G in G as in the following generic drawing: We have the following bijective statement, which is the counterpart for MOgraphs of [9, Theo. 4]: Proposition 3.1. Each rooted MO-graph G is uniquely obtained as a rooted melonfree MO-graph H -called the melon-free core of Gwhere each edge e is substituted by a rooted melonic MO-graph (possibly the cycle-graph, in which case e is unchanged). In addition, the degree of G equals the degree of H.
Proof. Let H be the rooted melon-free graph obtained from G after performing a greedy sequence of melon removals. Conversely, G is obtained from H where a sequence of melon insertions is performed. Hence G is equal to H where each edge e is substituted by a rooted melonic graph G e . This gives the existence of a melon-free core. Uniqueness of the melon-free core is given by the observation that any greedy sequence of melon removals starting from G has to progressively shell the melonic components G e , hence terminates at H. Finally, it is clear that G and L R S not 2-dipoles  H have the same degree, since a melon insertion preserves the degree (it preserves the number of connected components, increases the number of vertices by 2, and increases the number of faces by 3).

3.2.
Extracting the scheme of a rooted melon-free MO-graph. In this section we go a step further and define the process of extracting a scheme from a rooted melon-free MO-graph. Define a 2-dipole (shortly a dipole) in a (possibly rooted) MO-graph G as a face of length 2 incident to two distinct vertices, and not passing by the root if G is rooted. Accordingly, one has three distinct types of dipoles: L, R, or S, see Figure 7 (left part). As the figure shows, each dipole has two exterior half-edges on one side and two exterior half-edges on the other side. Notice also that a double edge does not necessarily delimit a dipole, as shown in Figure 7 (right part).
Remark 3.1. A face of length 2 is actually always incident to two distinct vertices, except in the MO-graphs that are made of one vertex and two loops. These two graphs, shown in Figure 8, are called the clockwise and the counterclockwise infinity graph, respectively. They have no dipole but have two faces of length 2.
In a (possibly rooted) MO-graph, define a chain as a sequence of dipoles d 1 . . . , d p such that for each 1 ≤ i < p, d i and d i+1 are connected by two edges involving two half-edges on the same side of d i and two half-edges on the same side of d i+1 , see Figure 9. A chain is called unbroken if all the p dipoles are of the same type. A broken chain is a chain which is not unbroken. A proper chain is a chain of at least two dipoles. A proper chain is called maximal if it cannot be extended into a larger proper chain. By very similar arguments as in Lemma 8 of [9] one obtains the following result:   Let G be a rooted melon-free MO-graph. The scheme of G is the graph obtained by simultaneously replacing any maximal proper chain of G by a so-called chainvertex, as shown in Figure 12, see also Figure 10 for an example.
Since a scheme is not an MO-graph (due to the presence of chain-vertices), we need to extend the definition and main properties seen so far. In the following, an MO-graph with chain-vertices is an oriented 4-regular map with two types of vertices: standard vertices and chain-vertices (each chain-vertex being labelled by either {L, R, S e , S o , B}), so as to satisfy the local rules of Figure 11. Such a graph is possibly rooted, i.e., has a fake-vertex of degree 2 in the middle of some edge. In order to compute the number of faces of an MO-graphG with chain-vertices, we need to specify locally the strand structure at every type of chain-vertex; as for MO-graphs, we imagine here that each edge is turned into 3 parallel strands and we have to specify at each chain-vertex how the incident strands go through the vertex. The specification is given by Figure 13; in the case of an unbroken chain-vertex the two strands that go through the chain-vertex are called the crossing strands at that chain-vertex. Then, as for classical MO-graphs, a face is a closed walk formed from strands. The degree δ ofG is defined as where c, V , F are as usual the numbers of connected components, standard vertices, and faces, and where U and B stand respectively for the numbers of unbroken chainvertices and broken chain-vertices.
An edge e ofG is said to be adjacent to a chain-vertex, if the two half-edges of e are the two half-edges on the same side of a chain-vertex ofG. ThenG is said to be melon-free if it has no melon nor an edge adjacent to a chain-vertex. It is easy to see that an MO-graph is melon-free iff its scheme is melon-free.
In a melon-free (possibly rooted) MO-graphG with chain-vertices, define a chain as a sequence d 1 . . . , d p of elements that are either dipoles (not passing by the root ifG is rooted) or chain-vertices, such that for each 1 ≤ i < p, d i and d i+1 are connected by two edges involving two half-edges on the same side of d i and two half-edges on the same side of d i+1 , see Figure 14. A proper chain is a chain of at least two elements.
Now define a reduced scheme as a rooted melon-free MO-graph with chainvertices and with no proper chain. By construction, the scheme of a rooted melonfree MO-graph (with no chain-vertices) is a reduced scheme. Claim 3.1 then easily yields the following bijective statement: Proposition 3.2. Every rooted melon-free MO-graph is uniquely obtained as a reduced scheme where each chain-vertex is consistently substituted by a chain of at least two dipoles (consistent means that if the chain-vertex is of type L, then the substituted chain is an unbroken chain of L-dipoles, etc).
The following result ensures that the degree definition for MO-graphs with chainvertices is consistent with the replacement of chains by chain-vertices: Corollary 3.1. A rooted melon-free MO-graph has the same degree as its scheme.

Finiteness of the set of reduced schemes at fixed degree
The main result in this section is the following: Similarly as in [9], this result will be obtained from two successive lemmas, proved respectively in Section 4.2 and Section 4.3: Lemma 4.2. For k ≥ 1 and δ ∈ 1 2 Z + , there is a constant n k,δ (depending only on k and δ) such that any connected unrooted MO-graph (without chain-vertices) of degree δ with at most k dipoles has at most n k,δ vertices.
Let us show how Lemmas 4.1 and 4.2 imply Proposition 4.1. Let S be a reduced scheme of degree δ, and let G(S) be the rooted MO-graph with no chain-vertices obtained by substituting each chain-vertex of S by a chain of dipoles of length 2 or 3 as shown in Figure 15. As already seen in Lemma 3.1, such substitutions preserve the degree, and the mapping S → G(S) is injective. In addition, the number of dipoles of G(S) is at most three times the total number of dipoles and chain-vertices of S, hence G(S) has at most 3·(7δ −1) dipoles according to Lemma 4.1. Unrooting G(S) might increase the number of dipoles by up to 3 (recall that dipoles are not counted in rooted MO-graphs if they pass by the root, and at most 3 dipoles can pass by the root, the case of 3 additional dipoles happening when the root-edge belongs to a melon). Hence, writing k := 3 · (7δ − 1) + 3, Lemma 4.2 ensures that G(S) has at most n k,δ vertices. Since there is an injective mapping from reduced schemes of degree δ to rooted MO-graphs of size bounded by the fixed quantity n k,δ , we conclude that the number of reduced schemes of degree δ is finite. We finally state the following "loop-removal" lemma, which will be useful at some points in the following: Lemma 4.3. Let G be an MO-graph of degree δ ∈ 1 2 Z + , let be a loop of G, and let G be the MO-graph obtained from G by erasing the loop and its incident vertex, as shown in Figure 16. Then G has degree δ − 1/2. Hence G has at most 2δ loops.
Proof. Clearly G has the same number of connected components as G, has one vertex less, and has one face less (which has length 1).

4.1.
Analysis of the removal of dipoles and of chain-vertices. We analyze here how the degree of an MO-graph with chain-vertices (not necessarily a reduced scheme, possibly with melons) evolves when removing a chain-vertex. We then have a similar analysis for the removal of a dipole.
Let G be an MO-graph with chain-vertices, and let m be a chain-vertex of G. The removal of m consists of the following operations (see the left part of Figure 17): (i) delete m from G; (ii) on each side of m, connect together the two detached legs (without creating a new vertex). Denote by G the resulting graph. The chain-vertex m is said to be non-separating if G has the same number of connected components as G, and separating otherwise (in which case G has one more connected component). In the separating case, it might be that one of the two resulting connected components is the cycle-graph (this happens if two half-edges on one side of m belong to the same edge). Let δ, c, V, F, U, B be the parameters for G (degree, numbers of connected components, standard vertices, faces, unbroken chain-vertices, broken chain-vertices), and let δ , V , F , U , B be the parameters for G .
Assume first that m is a broken chain-vertex. We clearly have V = V , F = F (the strand structure is the same before and after removal), U = U , B = B − 1 and c = c (resp. c = c + 1) if m is non-separating (resp. separating). Hence, if m is separating, then 2δ = 2δ + 6 − 6 = 2δ, so that δ = δ; and if m is non-separating, then 2δ = 2δ − 6, so that δ = δ − 3.
Assume now that m is an unbroken chain-vertex. We clearly have V = V , U = U − 1, B = B, and c = c (resp. c = c + 1) if m is non-separating (resp.  Figure 18. In the worst case, removing a dipole might decrease by 3 the total number of dipoles. separating). If the two crossing strands at m belong to a same face, then that face splits into two faces when removing m, hence F = F + 1; as opposed to that, if the two crossing strands belong to different faces, then these two faces are merged when removing m, hence F = F − 1. Moreover, the two crossing strands are easily seen to always be in the same face if m is separating. We conclude that, if m is separating, then 2δ = 2δ + 6 − 2 − 4 = 2δ, so that δ = δ; and if m is non-separating We now consider the operation of removing a dipole d from G, removal which consists in the following operations (the 3 cases are shown in the right-part of Figure 17): (i) delete the two vertices and the two edges of d from G; (ii) on each side of d, connect together the two detached legs (without creating a new vertex).
Let G be the resulting graph. Again, d is called non-separating if G has as many connected components as G, and separating otherwise. Observe that the operation of removing d is the same as substituting d by an unbroken chain-vertex m (of the same type as d, so that the substitution does not change the degree according to Lemma 3.1) and then removing m. From the discussion on the degree variation when removing m, we conclude that, if d is separating, then the degree is the same in G as in G, and if d is non-separating then the degree of G equals the degree of G minus a quantity that is either 1 or 3. This analysis thus leads to the following statement: Lemma 4.4. The degree is unchanged when removing a separating chain-vertex or a separating dipole (hence, by Claim 2.1, the degree is distributed among the resulting components). The degree decreases by 3 when removing a non-separating broken chain-vertex. The degree decreases by 1 or 3 when removing a non-separating unbroken chain-vertex or a non-separating dipole.

4.2.
Proof of Lemma 4.1. We show here Lemma 4.1, i.e., that for each reduced scheme G of degree δ, the sum N (G) of the numbers of dipoles and chain-vertices in G satisfies N (G) ≤ 7δ − 1.
As a first step, starting from G, as long as there is at least one non-separating dipole or one non-separating chain-vertex, delete it (as in Figure 17) and color the two resulting edges. Let G be the resulting graph with chain-vertices, and let q be the number of removals from G to G . According to Lemma 4.4, each non-separating removal decreases the degree by at least 1, hence q ≤ δ, G has degree at most δ − q, and G has at most 2q blue edges. During the transition from G to G , denote by N the sum of the numbers of chain-vertices and dipoles. As shown in Figure 18 Figure 19. The first drawing shows a reduced scheme G. Then, iteratively, one removes at each step a non-separating dipole or a non-separating chain-vertex (at each step the non-separating dipole or chain-vertex to be removed next is surrounded). Let G be the MO-graph with chain-vertices thus obtained (where colored edges are drawn bolder). As the last two drawings show, the removal of uncolored dipoles and chain-vertices (which are all separating) of G yields a tree of components (a tree edge is labelled c x if it comes from a chain-vertex of type x and is labelled d x if it comes from an uncolored dipole of type x).
removal might decrease N by 3, and this is actually the worst case. Hence, if we denote by N (G ) the value of N for G , we have Since all dipoles and chain-vertices of G are separating, G can be seen as a tree T of "components" (see the last row of Figure 19 for an example): the edges of T correspond to the N (G ) chain-vertices and uncolored dipoles (dipoles with no colored edge) of G , and the nodes of T correspond to the connected components of G left after these removals (note that some of these connected components might be cycle-graphs, including the component carrying the root). For a component C of T , the adjacency order of C is the number of adjacent components in T . An edge e of C resulting from a (removed) connection is called a marked edge. Note that a marked edge can be involved in several connections (for instance in Figure 19 there 16ÉRIC FUSY AND ADRIAN TANASA is an edge of a cycle-graph involved in 3 connections, and there is an edge of an infinity graph involved in 2 connections), hence the adjacency order of a component is at least as large as its number of marked edges. A marked edge e results from the merger of edges of G incident to separating chain-vertices or uncolored dipoles of G . Then e is called colored if at least one of these edges is a colored edge of G . A component is called uncolored if it bears neither a colored edge of G left untouched by removals, nor a colored marked edge. Since G has at most 2q colored edges, T has at most 2q colored components.
Similarly as in [9], an important remark is the following: Proof. The first unrooted MO-graphs of degree 0 are the cycle-graph (no vertex) and the so-called "quadruple-edge graph" (two vertices connected by 4 edges). And any other MO-graph of degree 0 can be obtained from the quadruple-edge graph by successive melon insertions. This easily implies that, if C is not the cycle-graph and has adjacency order smaller than 3 (hence has less than 3 marked edges), then C has a dipole not passing by any marked edge, a contradiction. If C is the cycle-graph and has adjacency order 1, then it clearly yields a melon in G, a contradiction.
Finally, if C is the cycle-graph and has adjacency order 2, then the two edges of T at C -each of which is either a separating dipole or a chain-vertex of G-together form a proper chain (of two elements) of G, a contradiction.
Denote by n 0 the number of uncolored components in G of degree 0 and not containing the root, n 0 the number of other components of degree 0 (colored or containing the root), and n + the number of components of positive degree. The degrees of the components add up to the degree of G (recall that, for separating removals, the degree is distributed among the components), and each component of positive degree has degree at least 1/2 , hence n + /2 ≤ δ − q. The components counted by n 0 have at least 3 neighbours in T , hence 3n 0 +n 0 +n + ≤ 2N (G ). Since T has at most 2q colored components, we have n 0 ≤ 2q + 1 (the +1 accounting for the component containing the root). And since T is a tree, we have N (G ) = n 0 + n 0 + n + − 1, hence, if we eliminate n 0 using 3n 0 + n 0 + n + ≤ 2N (G ), we obtain

Proof of Lemma 4.2.
We prove here Lemma 4.2, i.e., that for any δ ∈ 1 2 Z + and k ≥ 1, there exists some constant n k,δ such that any connected unrooted MOgraph (with no chain-vertices) of degree δ with at most k dipoles has at most n k,δ vertices, so that there are finitely many MO-graphs of degree δ with k dipoles. The bound n k,δ we obtain is linear in k and δ, with quite large multiplicative constants (we have not pushed to improve the bound, also to avoid making the proof too complicated).
The proof is a bit more technical than the analogous statement for colored graphs in [9], because in the MO model, only straight faces have even length, while left and right faces are allowed to have odd lengths. Precisely, our proof strategy is to show in a series of claims that some properties (e.g. being incident to a dipole) are satisfied only by a limited (bounded by a quantity depending only on k and δ) number of vertices. Hence if G could be arbitrarily large, it would have vertices not safisfying these properties. As we will see, this would yield the contradiction that G has to be reduced to a certain MO-graph with 6 vertices.
In all the proof, G denotes a connected unrooted MO graph with no chainvertices, of degree δ and with at most k dipoles; V, E, F are respectively the numbers of vertices, edges, and faces of G. Moreover, for p ≥ 1, F (p) s is the number of straight faces of length 2p. According to Remark 3.1, the only MO-graphs where the set of dipoles is not equal to the set of faces of length 2 are the two infinity graphs (both having one vertex). Hence we can assume that G is not equal to these two graphs. Thus F (1) s ≤ k, which gives: There are at most 2k vertices of G that are incident to a dipole.
Also, Lemma 2.1 directly implies Since p − 2 ≥ p/3 for p ≥ 3, the total length L = p≥3 2pF (p) s of straight faces of length larger than 4 is bounded by 12δ − 12 + 6k, so that we obtain:  Proof. It is easy to see that the only possibility for a straight face of length 4 to be self-intersecting is the configuration of Figure 20, in which case, there is a loop included in the straight face. The statement then just follows from the fact that G has at most 2δ loops, according to Lemma 4.3.
Given a non-self-intersecting cycle C of G, a chordal path for C is a path that starts from C, ends at C (possibly at the same vertex), and is outside of C inbetween. In the context of maps, we call a chordal path faulty if it starts on one side of C and ends on the other side of C. A cycle C is called faulty if it has a faulty chordal path, and, for k ≥ 1, is called k-faulty if it has a chordal path of length at most k. For C a faulty cycle, we denote by C the union of C and of a (canonically chosen) shortest possible faulty chordal path of C; C is called the faulty extension of C. Proof. For C a faulty cycle, C is clearly a topological minor of G of genus 1.
Hence, if we have s ≥ 1 vertex-disjoint faulty extensions of faulty cycles, these yield a topological minor of genus s. Since the genus of G is bounded by 2δ, we conclude that there can not be more than 2δ vertex-disjoint faulty extensions of faulty cycles.
In the case of a 4-faulty cycle C of length 4, C has at most 7 vertices. An easy calculation ensures that any given vertex v of G belongs to at most d := 96768 different faulty extensions of 4-faulty cycles (this can probably be improved, but we just aim at a certain fixed bound). Hence C can intersect at most 7d other faulty extensions of 4-cycles. We easily conclude that there are at most 14dδ 4faulty cycles of length 4 (otherwise one could construct incrementally a set of 2δ+1 4-faulty cycles whose faulty extensions are vertex-disjoint).
Let V exc be the set of vertices of G that are either incident to a dipole, or incident to a straight face of length larger than 4, or incident to a self-intersecting straight face of length 4, or incident to a 4-faulty cycle of length 4. The four claims above give (4) |V exc | ≤ (18 + 56d)δ + 8k − 12 =: h.
And let V 0 be the set of vertices in V exc or adjacent to a vertex in V exc . Since G is 4-regular, |V 0 | ≤ 5h. We are now going to show that if G has a vertex v not in V 0 , then G is actually reduced to having only 6 vertices. Since v / ∈ V exc , it is incident to two non-intersecting straight faces C 1 and C 2 , both of length 4. Since C 1 is not faulty, it has to meet C 2 at a vertex different from v (otherwise C 2 would be a faulty chordal path for C 1 ). Letv be an intersection of C 1 and C 2 different from v. For i ∈ {1, 2}, let d i be the distance ofv from v on C i (note that d i ∈ {1, 2}). If d 1 = d 2 = 1 then either the two corresponding edges e 1 , e 2 of C 1 and C 2 form an external dipole, or otherwise e 2 is a faulty chordal edge for C 1 . Both cases are excluded since v / ∈ V exc , hence d 1 = d 2 = 2. We write the cycle C 1 as (v, w 1 ,v, z 1 ) and the cycle C 2 as (v, w 2 ,v, z 2 ), see Figure 21. Let C(w 1 ) be the straight face (non-intersecting and of length 4, since w 1 / ∈ V exc ) passing by w 1 and different from C 1 . Since w 1 / ∈ V exc , C(w 1 ) is not faulty, hence it intersects the 4-cycle (v, w 1 ,v, w 2 ) at a vertex different from w 1 ; this other intersection must be at w 2 because v and v have their 4 incident edges already in C 1 ∪ C 2 . Similarly C(w 1 ) intersects the 4-cycle (v, w 1 ,v, z 2 ) at a vertex different from w 1 , and this other intersection must be z 2 ; and C(w 1 ) intersects the 4-cycle C 1 at a vertex different from w 1 , and this other intersection must be z 1 , see Figure 21. Hence G = C 1 ∪ C 2 ∪ C(w 1 ) forms a connected induced subgraph that saturates all its vertices (each of the 6 vertices of G is incident to 4 edges of G ). Hence G = G. In other words, if G has a vertex outside of V 0 , then G has 6 vertices.
We can now easily conclude the proof. Define n k,δ := max(6, 5h), and assume G has more than n k,δ vertices. Since |V 0 | ≤ 5h, G has a vertex not in V 0 , which implies that G is reduced to a graph with 6 vertices, giving a contradiction.

MO-graphs of degree 1/2
In this section we recover the results of [10] about the structure of MO-graphs of degree 1/2, using ingredients from the preceding sections.
Proposition 5.1. The only unrooted connected melon-free MO-graphs of degree 1/2 are the cw and the ccw infinity graph.
Proof. Assume there is a connected melon-free MO-graph G of degree 1/2 different from the cw or the ccw infinity graph. According to Remark 3.1 the set of dipoles of G equals the set of its faces of length 2. Note that in degree 1/2, the genus g has to be 0 (since g is a nonnegative integer bounded by δ). Hence Lemma 2.1 gives −F i.e., G has a dipole d (of type S). Since δ < 1, d has to be separating according to Corollary 4.1. Let G 1 and G 2 be the connected components resulting from the removal of d. Since d is separating, the degrees of G 1 and G 2 add up to 1/2, hence one has degree 0 and the other has degree 1/2. By convention let G 1 be the one of degree 0, considered as rooted at the edge resulting from the deletion of d. There are two cases: (i) if G 1 is the cycle-graph, then d is part of a melon of G, giving a contradiction; (ii) if G 1 is not the cycle-graph, then it has a melon (since rooted MO-graphs of degree 0 are melonic), giving again a contradiction.
This yields the following bijective result, where we recover [10, Theo. 3.1]: Corollary 5.1. Every unrooted connected MO-graph of degree 1/2 is uniquely obtained from the cw or the ccw infinity graph where each of the two edges is substituted by a rooted melonic graph.
Proof. All the arguments in the proof of Proposition 3.1 (extraction of a unique melon-free core) can be directly recycled to give the following statement: "for δ ∈ 1 2 Z * + , every unrooted connected MO-graph of degree δ is uniquely obtained as an unrooted connected melon-free MO-graph of degree δ where each edge is substituted by a rooted melonic graph", which -together with Proposition 5.1yields the result.
Another direct consequence of Proposition 5.1 is the following statement that will prove useful in the next section: Corollary 5.2. Every rooted connected MO-graph of degree 1/2 different from the (rooted) cw or ccw infinity graph has a dipole.
Proof. According to Proposition 5.1, the underlying unrooted MO-graph (obtained by erasing the fake-vertex of degree 2) has a melon, hence there is a dipole not including the marked edge; by definition this dipole is also a dipole of the rooted MO-graph (recall that dipoles of rooted MO-graphs are required not to pass by the root).

Dominant schemes
A reduced scheme S of degree δ ∈ 1 2 Z + is called dominant if it maximizes (over reduced schemes of degree δ) the number b of broken chain-vertices (as we will see in Section 7, b determines the singularity order of the generating function of rooted MO-graphs of degree δ and reduced scheme S, the larger b the larger the singularity order). In this section we show that, in each degree δ, the maximal number of broken chain-vertices is 4δ − 1, and we precisely determine what are the dominant schemes.
Let S be a reduced scheme of degree δ, and let b be the number of broken chainvertices of S. We now determine a bound for b in terms of δ. The discussion is very similar to the one of Section 4.2, with the difference that only the broken chain-vertices are removed. First, as long as there is a non-separating broken chainvertex, we remove it (as shown in Figure 17) and color the two resulting edges. Let q be the number of such removals until all broken chain-vertices are separating; at this stage, denote by S the resulting MO-graph with chain-vertices, which has degree δ = δ − 3q according to Lemma 4.4, and has at most 2q colored edges. Then S can be seen as a tree T of components which are obtained by removing all separating broken chain-vertices. It is also convenient to consider the treeT of components which are obtained by removing all separating chain-vertices and all separating uncolored dipoles -dipoles with no colored edge-of S (in T andT some components of degree 0 might be cycle-graphs). Note thatT is a refinement of T , i.e., each component C of T "occupies" a subtree T C ofT . For a component C of T (resp. ofT ), the adjacency order of C is defined as the number of adjacent components in T (resp. inT ). An edge e of C resulting from a separating removal for T (resp.T ) is called a marked edge. Such an edge e results from the merger of edges of S . Then e is called colored if at least one of these edges is a colored edge of S . And a component C of T (resp. ofT ) is called uncolored if it bears no colored edge (marked or unmarked). Since S has at most 2q colored edges, T (resp.T ) has at most 2q colored components.
Consider an uncolored component C of T of degree 0 and not containing the root. Note that C has no non-separating dipoles nor non-separating chain-vertices, by Corollary 4.1. In addition, by the arguments of Claim 4.1, any component of T C must have adjacency order at least 3, otherwise it would yield a melon or a proper chain in S. This clearly implies that C can not have adjacency order smaller than 3, and if it has adjacency order 3, then T C consists of a unique component.
Let n 0 be the number of components of T of degree 0, uncolored, and not containing the root. Let n 0 be the number of other components of degree 0. Let n + be the number of components of positive degree. Moreover, let p be the number of edges of T , which is also the number of broken chain-vertices of S , so that b = q +p.
The degrees of the components add up to the degree δ = δ − 3q of G (recall that, for separating removals, the degree is distributed among the components), and each component of positive degree has degree at least 1/2 , hence The components counted by n 0 have at least 3 neighbours in T , hence 3n 0 + n 0 + n + ≤ 2p.
The graph S has at most 2q blue edges, hence where the +1 accounts for the component containing the root. Since T is a tree, we have p = n 0 + n 0 + n + − 1, hence, eliminating n 0 using 3n 0 + n 0 + n + ≤ 2p, we obtain If a scheme S is such that b reaches the upper bound 4δ − 1, then it implies that all the above inequalities are tight, hence q = 0 (all broken chain-vertices are separating), n 0 = 1 (the component containing the root has degree 0), 3n 0 + n 0 + n + = 2p (all the components of positive degree and the component containing the root are leaves of T , and the other components of degree 0 have 3 neighbours in T ), and n + = 2δ (all positive degree components have degree 1/2).
We take a closer look at the components of degree 0. Let C be such a component not containing the root; note that C has no non-separating dipole nor non-separating chain-vertex according to Corollary 4.1. Since C has adjacency order 3 and is uncolored (because q = 0), the discussion above ensures that T C has a single component, hence C has no chain-vertex and any dipole of C must pass by a marked edge of C. Recall that an unrooted MO-graph of degree 0 is either the cycle-graph, or can be obtained from the "quadruple-edge" MO-graph (two vertices connected by 4 edges) by successive insertions of melons. Note that, as soon as at least one melon is inserted in the quadruple-edge MO-graph, there are two vertexdisjoint melons. Hence, an MO-graph of degree 0 with at most 3 marked edges and more than 2 vertices has a dipole that avoids the marked edges. It follows that C has to be either the cycle-graph or the quadruple-edge graph. Now, if C is the component containing the root, since it has only one adjacent component C , it has to be the (rooted) cycle-graph. Indeed, the case of a rooted quadruple-edge MOgraph is excluded; in that case, among the 4 edges of C, one carries the root, one is marked, and the two other ones form a dipole that, together with the chain-vertex connecting C to C , would form a proper chain of S, a contradiction.
We now take a closer look at components of positive degrees (of degree 1/2). Let C be such a component (which has a unique marked edge since it is at a non-root leaf of T ). Since C has degree 1/2 it can not have non-separating chain-vertices or non-separating unbroken chain-vertices, according to Corollary 4.1. And by the arguments of Claim 4.1, components of degree 0 of T C must have adjacency order at least 3. Since the degrees of components in T C add up to 1/2, this easily implies that the unique possibility is T C having a single node. Hence C has no dipole avoiding the marked edge nor chain-vertices, so that C must be the cw or the ccw infinity graph, according to Corollary 5.2.
Proposition 6.1. For δ ∈ 1 2 Z * + , the dominant schemes arise from rooted binary trees (see Figure 22 for an example) with 2δ+1 leaves, 2δ−1 inner nodes, and 4δ−1 edges, where the root-leaf is occupied by the rooted cycle-graph, the 2δ other leaves are occupied by (cw or ccw) infinity graphs, the 2δ − 1 inner nodes are occupied either by the cycle-graph or by the quadruple-edge graph, and the 4δ − 1 edges are occupied by separating broken chain-vertices.
Proof. In the analysis above we have seen that b ≤ 4δ − 1 and that any scheme with b = 4δ − 1 must be of the stated form. Conversely any scheme of the stated form is a valid scheme (no melon nor proper chain) of degree δ with b = 4δ − 1 chain-vertices. Hence these schemes are exactly the dominant schemes at degree δ. Each rooted binary tree with 2δ + 1 leaves gives rise to 4 2δ−1 2 2δ = 2 6δ−2 dominant schemes. Indeed, at each inner node, one has to decide whether it is occupied by the cycle-graph or by the quadruple-edge graph, and in the second case, one has to choose among the three possible configurations for the free edge (the unique edge not involved in any connection) since the straight face passing by the free edge can go either toward the left child, or the right child, or the parent. Finally, one has to decide at each non-root leaf if it is occupied by the cw or the ccw infinity graph.
Remark 6.1. As we have recalled in Section 2.4, regular colored graphs in dimension 3 naturally form a subfamily of MO-graphs. However, since half-integer degrees are not possible in the model of regular colored graphs, the dominant schemes differ; as shown in [9], in degree δ ∈ Z + , the dominant schemes are associated to rooted binary trees with δ +1 leaves (and δ −1 inner nodes), where the root-leaf is occupied by a root-melon, while the δ non-root leaves are occupied by the unique scheme of degree 1.

Generating functions and asymptotic enumeration
Let δ ∈ 1 2 Z + , and let S δ be the (finite) set of reduced schemes of degree δ. For each S ∈ S δ , let G (δ) S (u) be the generating function of rooted melon-free MOgraphs of reduced scheme S, where u marks half the number of non-root vertices (i.e., for p ∈ 1 2 Z + , a weight u p is given to rooted MO graphs with 2p non-root vertices). Let p be half the number of non-root standard vertices of S, b the number of broken chain-vertices, a the number of unbroken chain-vertices of type L or R, s e the number of even straight chain-vertices, and s o the number of odd straight chain-vertices. The generating functions for unbroken chains of type L (resp. R) is clearly u 2 /(1 − u), the one for even straight chains is u 2 /(1 − u 2 ), the one for odd straight chains is u 3 /(1 − u 2 ), and the one for broken chains is (3u Denoting by c the total number of chain-vertices and by s = s e + s o the total number of straight chain-vertices, this simplifies as It is well-known that the generating function of rooted melonic graphs is given by In addition, a rooted melon-free MO-graph with 2p non-root vertices has 4p + 1 edges (recall that the root-edge is split into two edges) where one can insert a rooted melonic graph. Therefore, defining U (z) := zT (z) 4 = T (z) − 1, the generating function F (δ) S (z) of rooted MO-graphs of reduced scheme S is given by and the generating function F (δ) (z) of rooted MO-graphs of degree δ is simply given by As discussed in [9], T (z) has its main singularity at z 0 := 3 3 /2 8 , T (z 0 ) = 4/3, and 1−3U (z) ∼ z→z0 2 3/2 3 −1/2 (1−z/z 0 ) 1/2 . Therefore, the dominant terms are those for which b is maximized. As shown in Section 6, these schemes are naturally associated to rooted binary trees with 2δ − 1 inner nodes, as stated in Proposition 6.1. For a fixed rooted binary tree γ with 2δ − 1 inner nodes, the total contribution of schemes arising from γ to the generating function of rooted melon-free MO-graphs of degree Indeed, such schemes have b = 4δ − 1, c = b, s = s o = 0, each of the 2δ non-root leaves is occupied either by the cw or ccw infinity graph, and each of the 2δ − 1 inner nodes is either occupied by the cycle-graph or the quadruple-edge graph that has 3 possible configurations.
Remark 7.1. According to Proposition 7.1, for δ ∈ Z and n ∈ Z, a (δ) n is asymptotically Θ(n 2δ−3/2 (2 8 /3 3 ) n ), where the multiplicative constant involves Cat 2δ−1 . As a comparison, as shown in [9], for δ ∈ Z the number c (δ) n of rooted colored graphs (in dimension 3) of degree δ with 2n vertices is asymptotically Θ(n δ−3/2 (2 8 /3 3 ) n ), where the multiplicative constant involves Cat δ−1 . We also deduce from these estimates that for fixed δ ∈ Z + , the probability that a random rooted MO-graph of degree δ with 2n vertices (where n ∈ Z + ) is a regular colored graph is Θ(n −δ ). Now, going back to Remark 2.3, we can also easily obtain the asymptotic enumeration under the planarity constraint, based on the following: Lemma 7.1. Each rooted melon-free MO-graph whose reduced scheme is dominant is planar.
Proof. As shown in Section 6, a rooted melon-free MO-graph G of degree δ ∈ 1 2 Z has a total of 2δ loops (one at each non-root leaf of the associated binary tree, see Figure 22). According to Lemma 4.3, erasing all these 2δ loops yields a (rooted) MO-graph G of degree 0, hence a planar MO-graph. Clearly the 2δ erased loops can be inserted back without breaking planarity, from which we conclude that G is planar.
Remark 7.2. Given any dominant scheme such as shown in Figure 22 (naturally drawn following the shape of a rooted binary tree), one can easily figure out how to retract/extend some parts of the drawing so as to obtain a planar redrawing, see Figure 23.
Corollary 7.1. For each fixed δ ∈ 1 2 Z + and for n ∈ δ + Z, the probability that a rooted MO-graph of degree δ with 2n vertices is planar tends to 1 as n → ∞. Proof. It just follows from the observation that the edge-substitution by melonic components clearly preserves planarity, so that all rooted MO-graphs arising from a dominating reduced scheme are planar. This concludes the proof since these rooted MO-graphs are the ones that dominate the asymptotic expansion.
According to Remark 2.3, in the planar case, rooted MO-graphs correspond (bijectively) to rooted 4-regular maps, the straight faces of the MO-graph identify to the knot-components of the map, and the number of straight faces is equal to V /2 + 1 − δ, with V the number of vertices and δ the degree. Under this rephrasing, Proposition 7.1 and Corollary 7.1 yield the following: Proposition 7.2. For n ∈ 1 2 Z + and k ∈ Z + , let b (k) n be the number of rooted 4-regular planar maps with 2n vertices and k knot-components. Then b (k) n = 0 for k > n + 1. In addition, for each fixed δ ∈ 1 2 Z + , and for n ∈ δ + Z, b (n+1−δ) n has the same asymptotic estimate as a (δ) n , given by (8).

Concluding remarks and perspectives
In this article we have shown that, similarly as in the colored model [9], the combinatorial study of MO-graphs can be done by extraction of so-called schemes, such that there are finitely many schemes in each fixed degree δ ∈ 1 2 Z + . We have also identified the dominant schemes in each degree, whose shapes are naturally associated to rooted binary trees. Having determined the dominant schemes we have obtained an explicit asymptotic estimate for the number of (rooted) MOgraphs of fixed degree as the number of vertices tends to infinity.
As already mentioned in the introduction, a first perspective for future work is the implementation of the double scale limit for the MO tensor model. For the sake of completeness, let us mention that the double scaling mechanism was implemented, for a different type of colored model, in [1], using quantum field theoretical-inspired methods (the so-called intermediate field method). From a probabilistic point a view, a different perspective for future work is to investigate whether the new dominant schemes we have exhibited in this paper correspond or not to phases different of the one of branched polymers (it was recently proved that the melon graphs correspond, from this point of view, to branched polymers [8]). Another appealing perspective is to extend the MO model (currently only developed in dimension D = 3) and results of this paper to higher dimensions. Finally, one can address the issue of proving counting theorems for 3D maps using tensor integral techniques, generalizing the fact that counting theorems for maps can be obtained using matrix integral techniques.