Spanning forests in regular planar maps
We address the enumeration of -valent planar maps equipped with a spanning forest, with a weight per face and a weight per connected component of the forest. Equivalently, we count -valent maps equipped with a spanning tree, with a weight per face and a weight per internally active edge, in the sense of Tutte; or the (dual) -angulations equipped with a recurrent sandpile configuration, with a weight per vertex and a variable that keeps track of the level of the configuration. This enumeration problem also corresponds to the limit of the -state Potts model on -angulations.
Our approach is purely combinatorial. The associated generating function, denoted , is expressed in terms of a pair of series defined implicitly by a system involving doubly hypergeometric series. We derive from this system that is differentially algebraic in , that is, satisfies a differential equation in with polynomial coefficients in and . This has recently been proved to hold for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof.
For , we study the singularities of and the corresponding asymptotic behaviour of its th coefficient. For , we find the standard asymptotic behaviour of planar maps, with a subexponential term in . At we witness a phase transition with a term . When , we obtain an extremely unusual behaviour in . To our knowledge, this is a new “universality class” for planar maps.
A planar map is a proper embedding of a connected graph in the sphere. The enumeration of planar maps has received a continuous attention in the past 60 years, first in combinatorics with the pionneering work of Tutte , then in theoretical physics , where maps are considered as random surfaces modelling the effect of quantum gravity, and more recently in probability theory [36, 38]. General planar maps have been studied, as well as sub-families obtained by imposing constraints of higher connectivity, or prescribing the degrees of vertices or faces (e.g., triangulations). Precise definitions are given below.
Several robust enumeration methods have been designed, from Tutte’s recursive approach (e.g. ), which leads to functional equations for the generating functions of maps, to the beautiful bijections initiated by Schaeffer , and further developed by physicists and combinatorics alike [11, 19], via the powerful approach based on matrix integrals . See for instance  for a more complete (though non-exhaustive) bibliography.
Beyond the enumerative and asymptotic properties of planar maps, which are now well understood, the attention has also focussed on two more general questions: maps on higher genus surfaces [6, 24], and maps equipped with an additional structure. The latter question is particularly relevant in physics, where a surface on which nothing happens (“pure gravity”) is of little interest. For instance, one has studied maps equipped with a polymer , with an Ising model [19, 34, 16, 18] or more generally a Potts model, with a proper colouring [46, 47], with loops models [14, 13], with a spanning tree , or percolation on planar maps [2, 10].
In particular, several papers have been devoted in the past 20 years to the study of the Potts model on families of planar maps [4, 15, 26, 30, 33, 49]. In combinatorial terms, this means counting maps equipped with a colouring in colours, according to the size (e.g., the number of edges) and the number of monochromatic edges (edges whose endpoints have the same colour). Up to a change of variables, this also means counting maps weighted by their Tutte polynomial, a bivariate combinatorial invariant which has numerous interesting specializations. By generalizing Tutte’s formidable solution of properly coloured triangulations (1973-1982), it has recently been proved that the Potts generating function is differentially algebraic, that is, satisfies a (non-linear) differential equation111with respect to the size variable with polynomial coefficients [9, 8, 17]. This holds at least for general planar maps and for triangulations (or dualy, for cubic maps).
The method that yields these differential equations is extremely involved, and does not shed much light on the structure of -coloured maps. Moreover, one has not been able, so far, to derive from these equations the asymptotic behaviour of the number of coloured maps, nor the location of phase transitions.
The aim of this paper is to remedy these problems — so far for a one-variable specialization of the Tutte polynomial. This specialization is obtained by setting to one of the variables, or by taking (in an adequate way) the limit in the Potts model. Combinatorially, we are simply counting maps (in this paper, -valent maps) equipped with a spanning forest. We call them forested maps. This problem has already been studied in  via a random matrix approach, but with no explicit solution. The generating function that we obtain keeps track of the size of the map (the number of faces; variable ) and of the number of trees in the forest (minus one; variable ). The specialization thus counts maps equipped with a spanning tree and was determined a long time ago by Mullin .
Here is an outline of the paper. We begin in Section 2 with general definitions on maps, and on the Tutte polynomial. We recall some of its combinatorial descriptions, and underline in particular that the series , once expanded in powers of and , has non-negative coefficients and admits several combinatorial interpretations. This important observation implies that the natural domain of the parameter is rather than . In Section 3, we obtain in a purely combinatorial manner an expression of in terms of the solution of a system of two functional equations. In Section 4 we derive from this system that is differentially algebraic in , and give explicit differential equations for cubic () and 4-valent () maps. Section 5 is a combinatorial interlude explaining why all series occurring in our equations, like itself, still have non-negative coefficients when .
The rest of the paper is devoted to asymptotic results, still for and : when , forested maps follow the standard asymptotic behaviour of planar maps () but then there is a phase transition at (where one counts maps equipped with a spanning tree), and a very unusual asymptotic behaviour in holds when . To our knowledge, this is the first time a class of planar maps exhibits this asymptotic behaviour. This proves in particular that is not D-finite, that is, does not satisfy any linear differential equation in for these values of (nor for a generic value of ). This is in contrast with the case , for which the generating function of maps equipped with a spanning forest is known to be D-finite.
Our key tool is the singularity analysis of : its basic principle is to derive the asymptotic behaviour of the coefficients of a series from the singular behaviour of near its dominant singularities (i.e., singularities of minimal modulus). The first case we study (4-valent maps with ) is simple: first, one of the two series involved in our system vanishes; the remaining one, denoted , satisfies an inversion equation for which the (unique) dominant singularity of is such that lies in the domain of analyticity of . One obtains for a “standard” square root singularity. This is well understood and almost routine. Two ingredients make the other cases significantly harder:
when , is a singularity of ,
These difficulties, which culminate when and , are addressed in Sections 6 and 7. Section 6 establishes general results on implicitly defined series. Section 7 focusses on the inversion equation in the case where (up to translation) has a singularity at . One then applies these results to the asymptotic analysis of forested maps in Sections 8 (4-valent maps) and 10 (cubic maps). Section 9 exploits the results of Section 8 to study some properties of large random maps equipped with a spanning forest or a spanning tree.
We conclude in Section 11 with a few comments.
2.1. Planar maps
A planar map is a proper embedding of a connected graph (possibly with loops and multiple edges) in the oriented sphere, considered up to continuous deformation. All maps in this paper are planar, and we often omit the term “planar”. A face is a (topological) connected component of the complement of the embedded graph. Each edge consists of two half-edges, each incident to an endpoint of the edge. A corner is an ordered pair of half-edges incident to the same vertex, such that immediately follows in counterclockwise order. The degree of a vertex or a face is the number of corners incident to it. A vertex of degree is called -valent. One-valent vertices are also called leaves. A map is -valent if all vertices are -valent. A rooted map is a map with a marked corner , called the root and indicated by an arrow in our figures. The root vertex is the vertex incident to the root. The root half-edge is and the root edge is the edge supporting . This way of rooting maps is equivalent to the more standard way where one marks the root edge and orients it from to its other half-edge. All maps of the paper are rooted, and we often omit the term “rooted”. The dual of a map , denoted , is the map obtained by placing a vertex of in each face of and an edge of across each edge of ; see Figure 1(a). The dual of a -valent map is a map with all faces of degree , also called -angulation.
A (plane) tree is a planar map with a unique face. A tree is -valent if all non-leaf vertices have degree . We consider the edges leading to the leaves as half-edges, as suggested by Figure 1(b). A leaf-rooted tree (resp. corner-rooted) is a tree with a marked leaf (resp. corner). The number of -valent leaf-rooted (resp. corner-rooted) trees with leaves is denoted by (resp. ) (the notation should be and , but we consider as a fixed integer, ). These numbers are well-known [42, Thm. 5.3.10]: they are unless with , and in this case,
Let be a rooted planar map with vertex set . A spanning forest of is a graph where is a subset of edges of forming no cycle. Each connected component of is a tree, and the root component is the tree containing the root vertex. We say that the pair is a forested map. We denote by the generating function of -valent forested maps, counted by faces (variable ) and non-root components (variable ):
where denotes the number of faces and the number of components. When ,
The coefficient means that there are 10 trivalent (or cubic) forested maps with 3 faces: 6 in which the forest is a tree, and 4 in which it has two components (Figure 2).
2.2. Forest counting, the Tutte polynomial, and related models
Let be a graph with vertex set and edge set . The Tutte polynomial of is the following polynomial in two indeterminates (see e.g. ):
where the sum is over all spanning subgraphs of (equivalently, over all subsets of edges) and , and denote respectively the number of vertices, edges and connected components. The quantity is the cyclomatic number of , that is, the minimal number of edges one has to delete from to obtain a forest.
Note that we write although the value of the Tutte polynomial only depends on the underlying graph of , not on the embedding.
Even though this is not clear from (4), the polynomial has non-negative coefficients in and . This was proved combinatorially by Tutte , who showed that counts spanning trees of according to two parameters, called internal and external activities. Other combinatorial descriptions of , in terms of other notions of activity, were given later. Let us present the one due to Bernardi, which is nicely related to maps . Following Mullin , we call tree-rooted map a map equipped with a spanning tree . Walking around in counter-clockwise order, starting from the root, defines a total order on the edges: the first edge that is met is the smallest one, and so on (Figure 3). An edge is internally active if it belongs to and is minimal in its cocycle; that is, all the edges such that is a tree are larger than . It is externally active if it does not belong to and is minimal in the cycle created by adding to . Denoting by and the numbers of internally and externally edges, one has:
A non-obvious property of this description is that it only depends on the underlying graph of .
Returning to (5), we thus obtain a second description of :
In particular, it makes sense combinatorially to write and take .
We now give four more descriptions of in terms of the dual -angulations. For any planar map , it is known that
we first derive from (5) that
counts -angulations equipped with a connected (spanning) subgraph , by the vertex number of and the cyclomatic number of . Also, the “dual” expression of (6) reads
where the sum runs over all recurrent configurations , and is the level of . Hence
also counts -angulations equipped with a recurrent configuration of the sandpile model, by the vertex number of and the level of .
Our final interpretation is in terms of the Potts model. Take . A -colouring of the vertices of is a map . An edge of is monochromatic if its endpoints share the same colour. Every loop is thus monochromatic. The number of monochromatic edges is denoted by . The partition function of the Potts model on counts colourings by the number of monochromatic edges:
The Potts model is a classical magnetism model in statistical physics, which includes (for ) the famous Ising model (with no magnetic field) . Of course, is the chromatic polynomial of . More generally, it is not hard to see that is always a polynomial in and , and a multiple of . Let us define the reduced Potts polynomial by
Fortuin and Kasteleyn established the equivalence of with the Tutte polynomial :
for . Setting , we obtain, for a connected graph
Returning to (2.2) finally gives
2.3. Formal power series
Let be a power series in one variable with coefficients in a field . We say that is D-finite if it satisfies a (non-trivial) linear differential equation with coefficients in (the ring of polynomials in ). More generally, it is D-algebraic if there exist a positive integer and a non-trivial polynomial such that .
A -variate power series with coefficients in is D-finite if its partial derivatives (of all orders) span a finite dimensional vector space over .
3. Generating functions for forested maps
Fix . We give here a system of equations that defines the generating function of -valent forested maps, or, more precisely, the series that counts forested maps with a marked face. We also give simpler systems for two variants of , involving no derivative.
3.1. -Valent maps
Let , and be the following doubly hypergeometric series:
where and are given by (1) and denotes the trinomial coefficient .
There exists a unique pair of power series in with constant term and coefficients in that satisfy
The generating function of -valent forested maps is characterized by and
1. These equations allow us to compute the first terms in the expansion of , for any fixed . This is how we obtained (3).
2. When is even, then for all . In particular, all terms occurring in the definition (11) of are multiples of , so that . The simplified system reads:
where we require to be even if is odd.
This series counts -valent maps equipped with a spanning tree, and
this expression was already proved by Mullin .
4. The series and are explicited when and in Sections 4.2 and 4.3, respectively.
In order to prove Theorem 3.1, we first relate to the generating function of planar maps counted by the distribution of their vertex degrees. More precisely, let be the generating function of rooted planar maps, with a weight per face, per non-root vertex of degree and if the root vertex has degree .
The series is related to through:
The idea is to contract each tree of a spanning forest, incident to half-edges, into a -valent vertex. It is adapted from [19, Appendix A], where the authors study 4-valent forested maps for which the root edge is not in the forest. It can also be seen as an extension of Mullin’s construction for maps equipped with a spanning tree . Finally, it also appears in .
Let us now get into the details. First, let us recall that rooted maps have no symmetries: all vertices, edges and half-edges are distinguishable. In particular, one can fix, for every rooted planar map (with arbitrary valences) a total order on its half-edges. This order may have a combinatorial significance — a good choice is the order in which half-edges are visited when applying the construction of  — but can also be arbitrary.
We now describe a bijection , illustrated in Figure 4, between forested -valent maps and pairs formed of a map and a collection of -valent trees associated with the vertices of , such that the tree associated with the root vertex of is corner-rooted, the others are leaf-rooted, and the number of leaves of is the degree of in .
The map is obtained by contracting all edges of the forest (Figure 4(b)). The arrow that marks the root corner remains at the same place. Now split into two half-edges each edge of that is not in : this gives a collection of -valent trees, each of them being naturally associated with a vertex of . The half-edges of these trees form together the edges of (Figure 4(c)). If is the root vertex of , then inherits the corner-rooting of . Otherwise, we root at the smallest of its half-edges, for the total order on half-edges of .
The following properties are readily checked:
has leaves if has degree in ,
and have the same number of faces,
the number of vertices of is the number of components of .
Let us now prove that is bijective. To recover the forested map from the contracted map and the associated collection of trees, we inflate each vertex of into the corresponding tree . If is the root vertex of , the root corner of must coincide with the root corner of . Otherwise, the root half-edge of is put on the smallest of the half-edges incident to in . This proves the injectivity of . Since this reverse construction can be applied to any map with a corresponding collection of trees, is also surjective. ∎
Proof of Theorem 3.1.
In a recent paper, Bouttier and Guitter  have expressed the series via a system of equations, established bijectively222Strictly speaking, they do not take the vertex or face number into account, but both are prescribed by the distribution of vertex degrees.. Their expression is actually fairly complicated [21, Eq. (1.4)], but the series , which counts maps with a marked face, has a much simpler expression [21, Eq. (2.6)]:
where and, by [21, Eq. (2.5)],
3.2. Quasi--valent maps ( odd)
A map is said to be quasi--valent if all its vertices have degree , apart from one vertex which is a leaf. Such maps exist only when is odd. They are naturally rooted at their leaf: the root corner is the unique corner incident to the leaf and the root edge is the unique edge incident to the leaf. Let denote the generating function of quasi--valent forested maps counted by faces () and non-root components () (see Figure 5).
As in the previous subsection, the key of this result is to relate to a well-understood generating function of maps — here, the generating function that counts planar maps rooted at leaf, with a weight per face and per -valent non-root vertex.
The following analogue of Lemma 3.2 holds for quasi--valent forested maps:
The bijection used in the proof of Lemma 3.2 shows that the series counts quasi--valent forested maps such that the root edge is not in the forest. (With the notation used in that proof, the root vertex of , of degree 1, remains a trivial tree during the inflation step). To each such forested map, we can add the root edge to the forest. The resulting forested map has one less component, hence the factor . ∎
Proof of Proposition 3.3.
This gives the first part of Proposition 3.3. For the second part, we observe that is by definition the coefficient of in the series defined above Lemma 3.2. Hence it follows from (17) that (this can also be derived combinatorially from ). ∎
3.3. When the root edge is outside the forest
We now focus on forested maps such that the root edge is outside the forest. Let denote the associated generating function.
The generating function of -valent forested maps where the root edge is outside the forest is
When is even, then and the first double sum disappears. In this case, we also have a very simple expression of :
Again, the key of this result is to relate to a well-understood generating function of maps — here, the generating function that counts rooted planar maps with a weight per face and per vertex of degree .
The following analogue of Lemma 3.2 holds:
Let us consider again the bijection used in the proof of Lemma 3.2: the fact that the root edge of is not in the forest means that, in the corner-rooted tree associated with the root vertex of , the root half-edge is a leaf. It is then equivalent to root this tree at this leaf. ∎
Proof of Proposition 3.5.
The first part of the proposition follows from the known characterization of (see [20, Eq. (2.1)]):
where is given by (20) and
with and satisfying (18). This gives the first part of the proposition.
4. Differential equations
The equations established in the previous section imply that series counting regular forested maps are D-algebraic. We compute explicitly a few differential equations.
4.1. The general case
We start from the expression (14) of (as we always differentiate with respect to , we simply denote for ). We first observe that the doubly hypergeometric series , , are D-finite (this follows from the closure properties of D-finite power series ).
Then, by differentiating (12) and (13) with respect to , we obtain rational expressions of and in terms of and the partial derivatives and , evaluated at , for . (Indeed, differentiating (12) and (13) gives a linear system in and . Its determinant is a power series in with coefficients in . It is non-zero, since it equals at .)
Let be the field . Using (14) and the previous point, it is now easy to prove by induction that for all , there exists a rational expression of in terms of
with coefficients in . But since , and are D-finite, the above set of series spans a vector space of finite dimension over . Therefore there exist elements in this space, and rational functions , such that for all .
Since the transcendance degree [35, p. 254] of over is (at most) , the series , for , are algebraically dependent, so that (and thus ) is D-algebraic.
4.2. The 4-valent case
The series , and their derivatives lie in a 3-dimensional vector space over spanned (for instance) by , and . This follows from the following equations, which are easily checked:
By the argument described above, we can now express first , and then and all its derivatives as rational functions of , , and . But since , this means a rational function of , , and . We compute the explicit expressions of , and , eliminate and from these three equations, and this gives a differential equation of order 2 and degree 7 satisfied by , the details of which are not particularly illuminating:
As discussed in Section 11, we conjecture that does not satisfy a differential equation of order 2.
4.3. The cubic case
We start from the expression of given in Theorem 3.1. We now have to deal with series , and in two variables:
Let us first observe that