I have spent the past ten days in Saint Flour, an inaccessible but picturesque town in the rolling hills of Cantal, in the middle of France, and venue for perhaps the most notable summer school in probability. My highlight has been the course ‘Aspects of Random Maps’ delivered by Gregory Miermont, and I thought I should write a few posts about points of interest encountered during the lectures and private study.
A map is an embedding of a connected graph onto a surface. We typically do not care about the nature of this embedding up to homeomorphisms of the surface which preserve orientations of the map. One advantage for doing this is that the set of maps now might be countable, and the set of maps with n edges might be finite. This can be proved by considering a map to be obtained by glueing together polygonal faces. Some potential glueings are impossible, and some are equivalent, but for a fixed number of edges, the number of such sets of polygons and a possible glueing is finite. In fact we can be much more precise than this about how to describe precisely the legal glueing through a triple of permutations, but I won’t discuss this here.
I haven’t yet given a complete definition of a map. We want a typical large map, that is a map with a large number of vertices and edges, to be topologically roughly the same as the surface it is embedded into. In particular, the map needs to encode the geometric features of the surface. So a small triangle on the surface of a torus should not be considered a map. To rigorise this, we demand that any face of a map should be a topological disc, in particular, it should be simply connected. Since the torus itself is not simply connected, this excludes our triangle example. Note a single vertex on a torus is also excluded.
Although it goes against the usual order of definitions, it might be helpful to think of a map as an embedding which satisfies Euler’s formula: V – E + F = 2 -2g, where g is the genus of the surface. For a connected planar graph, induction is on the number of edges and vertices is the typical way to prove this result. The inductive step works the same on a more general surface, but it is less clear what the base case should be. Another consequence of the definition is that we should work on the sphere rather than the plane. From now on, this is our surface of interest.
We begin by considering to be the family of rooted plane maps with n edges. The root is a distinguished oriented edge. Our aim is to count the size of this set.
Before doing this, we digress onto the topic of rooted plane trees. Note that any (rooted) tree in the classical sense is planar, but in a rooted plane tree, we also specify the geometric ordering of the offspring. For example, if the root has two offspring, of which one has precisely one offspring and the other has none, we consider these as two separate cases.
So now, if we denote by the number of plane trees with k edges, we can define a generating function via . If the root vertex v has no offspring, this gives one possibility corresponding to k=0. Otherwise, there is a well-defined left-most offspring of the root, called u. Then u and its descendents form a plane tree, and v and its descendents apart from those through u also form a plane tree. So after accounting for the edge between u and v, we obtain
whenever A(z) is defined. We now can apply whichever is our favourite method of showing that this is the generating function of the Catalan numbers, .
There is a more complicated version of this generating function argument due to Tutte that allows us to enumerate . It is convenient to work with a second variable in the generating function that encodes the degree of the root face. The resulting equation of generating functions is less well-known but using the Lagrange inversion formula gives the explicit expression
Although there are extra terms, this motivates seeking a bijection between maps, and some version of rooted plane trees, perhaps decorated with some extra information. As in many cases, this will turn out to be possible. The bijection we end up with will not just help us enumerate the maps, but will also allow us to control a lot more information about distances in the map, which will be particularly useful when we try to take limits.
The first observation is that given a map, we can construct a dual map, by placing fresh vertices somewhere in the middle of each face, and joining a pair of these if the corresponding faces in the original graph share an edge.
Alternatively, we can place the same fresh vertices in the middle of each face, then join each new vertex to an original vertex, if that original vertex lies on the face corresponding to the new vertex. If you focus in on an original edge, it is clear that it is now surrounded by a ‘diamond’ (if you’ve drawn the diagram in a natural way) of new edges. Removing the original edges thus leaves us with a quadrangulation. This procedure is called the ‘trivial bijection’ between and , the family of rooted quadrangulations with n faces. Note that the root in such a quadrangulation is an identified directed edge, rather than a vertex. We haven’t yet specified how to describe the root of the resulting quadrangulation. It suffices to take the first new edge which lies clockwise of the root edge in the original graph, seen from the ‘tail’ of the root, which is of course oriented.
In this, and the bijections which follow, the natural questions to ask are: a) is the inverse obvious? and b) what happens to self-loops and isthmuses? Here, the inverse really is obvious. Any quadrangulation is bipartite, hence two-colourable, so we need to fix one colour and join the two vertices of that colour within each face to recover the original graph. The root tells us which colour we need to take. As for the second question, first we should say that an isthmus is an edge which has the same face on both sides. This causes no problems in this particular bijection. For the self-loops, we get a sort of Pacman-like quadrangle, with two ‘outer-edges’ between the same two vertices, and an edge between one of the outer vertices and some internal vertex. This edge contributes twice to the degree of the face.
The upshot of this is that for a simple enumeration, it suffices to prove that . This may not look like we have achieved much, but we can now apply Euler’s formula to any quadrangulation in this set to deduce that the number of vertices present is n+2. If we consider the set , where now we identify a particular vertex in the quadrangulation, it suffices to prove that , where is the nth Catalan number as before. Now we have the most efficient setup to look for a bijection with some type of decorated plane tree as discussed before.