The genus distribution of cubic graphs and asymptotic number of rooted cubic maps with high genus

Let $C_{n,g}$ be the number of rooted cubic maps with $2n$ vertices on the orientable surface of genus $g$. We show that the sequence $(C_{n,g}:g\ge 0)$ is asymptotically normal with mean and variance asymptotic to $(1/2)(n-\ln n)$ and $(1/4)\ln n$, respectively. We derive an asymptotic expression for $C_{n,g}$ when $(n-2g)/\ln n$ lies in any closed subinterval of $(0,2)$. Using rotation systems and Bender's theorem about generating functions with fast-growing coefficients, we derive simple asymptotic expressions for the numbers of rooted regular maps, disregarding the genus. In particular, we show that the number of rooted cubic maps with $2n$ vertices, disregarding the genus, is asymptotic to $\frac{3}{\pi}\,n!6^n$

The combinatorics of the Jack parameter and the genus series for topological maps

Informally, a rooted map is a topologically pointed embedding of a graph in a surface. This thesis examines two problems in the enumerative theory of rooted maps. The b-Conjecture, due to Goulden and Jackson, predicts that structural similarities between the generating series for rooted orientable maps with respect to vertex-degree sequence, face-degree sequence, and number of edges, and the corresponding generating series for rooted locally orientable maps, can be explained by a unified enumerative theory. Both series specialize M(x,y,z;b), a series defined algebraically in terms of Jack symmetric functions, and the unified theory should be based on the existence of an appropriate integer valued invariant of rooted maps with respect to which M(x,y,z;b) is the generating series for locally orientable maps. The conjectured invariant should take the value zero when evaluated on orientable maps, and should take positive values when evaluated on non-orientable maps, but since it must also depend on rooting, it cannot be directly related to genus. A new family of candidate invariants, η, is described recursively in terms of root-edge deletion. Both the generating series for rooted maps with respect to η and an appropriate specialization of M satisfy the same differential equation with a unique solution. This shows that η gives the appropriate enumerative theory when vertex degrees are ignored, which is precisely the setting required by Goulden, Harer, and Jackson for an application to algebraic geometry. A functional equation satisfied by M and the existence of a bijection between rooted maps on the torus and a restricted set of rooted maps on the Klein bottle show that η has additional structural properties that are required of the conjectured invariant. The q-Conjecture, due to Jackson and Visentin, posits a natural combinatorial explanation, for a functional relationship between a generating series for rooted orientable maps and the corresponding generating series for 4-regular rooted orientable maps. The explanation should take the form of a bijection, ϕ, between appropriately decorated rooted orientable maps and 4-regular rooted orientable maps, and its restriction to undecorated maps is expected to be related to the medial construction. Previous attempts to identify ϕ have suffered from the fact that the existing derivations of the functional relationship involve inherently non-combinatorial steps, but the techniques used to analyze η suggest the possibility of a new derivation of the relationship that may be more suitable to combinatorial analysis. An examination of automorphisms that must be induced by ϕ gives evidence for a refinement of the functional relationship, and this leads to a more combinatorially refined conjecture. The refined conjecture is then reformulated algebraically so that its predictions can be tested numerically

Generating functions of bipartite maps on orientable surfaces

We compute, for each genus $g\geq 0$, the generating function $L_g\equiv L_g(t;p_1,p_2,\dots)$ of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables such that for each $g$, $L_g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $F_g$ of rooted bipartite maps. The form of the result is strikingly similar to the Goulden/Jackson/Vakil and Goulden/Guay-Paquet/Novak formulas for the generating functions of classical and monotone Hurwitz numbers respectively, which suggests stronger links between these models. Our result complements recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of dessins d'enfants. Our proofs borrow some ideas from Eynard's "topological recursion" that he applied in particular to even-faced maps (unconventionally called "bipartite maps" in his work). However, the present paper requires no previous knowledge of this topic and comes with elementary (complex-analysis-free) proofs written in the perspective of formal power series.Comment: 31 pages, 2 figure

Enumerative Applications of Integrable Hierarchies

Countably infinite families of partial differential equations such as the Kadomtzev - Petviashvili (KP) hierarchy and the B-type KP (BKP) hierarchy have received much interest in the mathematical and theoretical physics community for over forty years. Recently there has been much interest in the application of these families of partial differential equations to a variety of problems in enumerative combinatorics. For example, the generating series for monotone Hurwitz numbers, studied by Goulden, Guay-Paquet and Novak, is known to be a solution to the KP hierarchy. Using this fact along with some additional constraints we may find a second order, quadratic differential equation for the generating series for simple monotone Hurwitz numbers (a specialization of the problem considered in corresponding to factorizations of the identity). In addition, asymptotic analysis can be performed and it may be shown that the asymptotic behaviour of the simple monotone Hurwitz numbers is governed by the map asymptotics constants studied by Bender, Canfield and Gao. In their enumerative study of various families of maps, Bender, Canfield and Gao proved that for maps embedded in an orientable surface the asymptotic behaviour could be completely determined up to some constant depending only on genus and that similarly for maps embedded on a non-orientable surface the asymptotic behaviour could be determined up to a constant depending on the Euler characteristic of the surface. However, the only known way of computing these constants was via a highly non-linear recursion, making the determination of these constants very difficult. Using the integrable hierarchy approach to enumerative problems, Goulden and Jackson derived a quadratic recurrence for the number of rooted triangulations on an orientable surface of fixed genus. This result was then used by Bender, Richmond and Gao to show that the generating series for the orientable map asymptotics constants was given by a solution to a nonlinear differential equation called the Painlev\'e I equation. This gave a method for computing the orientable map asymptotics constants which was significantly simpler than any previously known method. A remaining open problem was whether a suitable integrable hierarchy could be found which could be applied to the corresponding problems in the non-orientable case. Using the BKP hierarchy of partial differential equations applied to the enumeration of rooted triangulations on all surfaces (orientable or non-orientable) we find a cubic recursion for the number of such triangulations and, as a result, we find a nonlinear differential equation which determines the non-orientable map asymptotics constants. In this thesis we provide a detailed development of both the KP and BKP hierarchies. We also discuss three different applications of these hierarchies, the two mentioned above (monotone Hurwitz numbers and rooted triangulations on locally orientable surfaces) as well as orientable bipartite quadrangulations

A bijection for rooted maps on general surfaces

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This general construction requires new ideas and is more delicate than the special orientable case, but it carries the same information. In particular, it leads to a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rooted connected maps of Euler characteristic $2-2h$, and of the algebraicity of their generating functions, similar to the one previously obtained in the orientable case via the Marcus-Schaeffer bijection. It also shows that the renormalization factor $n^{1/4}$ for distances between vertices is universal for maps on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation on any fixed surface converge in distribution when the size $n$ tends to infinity. Finally, we extend the Miermont and Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our construction opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Comment: v2: 55 pages, 22 figure

Feynman Diagrams and Rooted Maps

The Rooted Maps Theory, a branch of the Theory of Homology, is shown to be a powerful tool for investigating the topological properties of Feynman diagrams, related to the single particle propagator in the quantum many-body systems. The numerical correspondence between the number of this class of Feynman diagrams as a function of perturbative order and the number of rooted maps as a function of the number of edges is studied. A graphical procedure to associate Feynman diagrams and rooted maps is then stated. Finally, starting from rooted maps principles, an original definition of the genus of a Feynman diagram, which totally differs from the usual one, is given.Comment: 20 pages, 30 figures, 3 table