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

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

Generating functions of bipartite maps on orientable surfaces (extended abstract)

International audienceWe compute, for each genus $g$ ≥ 0, the generating function $L$$g$ ≡ $L$$g$($t$;$p$1,$p$2,...) 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$$g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $L$$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 strengthens 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.Nous calculons, pour chaque genre $g$ ≥ 0, la série génératrice $L$$g$ ≡ $L$$g$($t$;$p$1,$p$2,...) des cartes bipartites (étiquetées) sur la surface orientable de genre $g$, avec contrôle des degrés des faces. On exhibe un changement de variable explicite tel que pour tout $g$, $L$$g$ est une fonction rationnelle des nouvelles variables, calculable par une récurrence explicite sur le genre. La même chose est vraie de la série génératrice $L$$g$ des cartes biparties enracinées. La forme du résultat est similaire aux formules de Goulden/Jackson/Vakil et Goulden/Guay-Paquet/Novak pour les séries génératrices de nombres de Hurwitz classiques et monotones, respectivement, ce qui suggère des liens plus forts entre ces modèles. Notre résultat renforce des résultats récents de Kazarian et Zograf, qui étudient le cas où le nombre de faces est borné, dans le formalisme équivalent des dessins d’enfants. Nos démonstrations utilisent deux idées de la “récurrence topologique” d’Eynard, qu’il a appliquée notamment aux cartes paires (appelées de manière non-standard “cartes biparties” dans son travail). Cela dit, ce papier ne requiert pas de connaissance préliminaire sur ce sujet, et nos démonstrations (sans analyse complexe) sont écrites dans le language des séries formelles

Formal multidimensional integrals, stuffed maps, and topological recursion

We show that the large N expansion in the multi-trace 1 formal hermitian matrix model is governed by the topological recursion of [Eynard and Orantin, 2007] with initial conditions. In terms of a 1d gas of eigenvalues, this model includes - on top of the squared Vandermonde - multilinear interactions of any order between the eigenvalues. In this problem, the initial data (W10,W20) of the topological recursion is characterized: for W10, by a non-linear, non-local Riemann-Hilbert problem on a discontinuity locus to determine ; for W20, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - W10 being the generating series of disks and W20 that of cylinders. In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name "stuffed maps". In a sense, our results complete the program of the "moment method" initiated in the 90s to compute the formal 1/N in the one hermitian matrix model.Comment: 33 pages, 6 figures ; v2, a correction and simplification in the final argument (Section 5

Asymptotic enumeration of constellations and related families of maps on orientable surfaces

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees. Our proofs rely on the generalisation to orientable surfaces of the Bouttier-Di Francesco-Guitter bijection, and on generating series methods. We show that each of the 2g fondamental cycles of the surface contributes a factor m between the numbers of m-hypermaps and m-constellations -- for example, large maps of genus g with even face degrees are bipartite with probability tending to 1/2^{2g}. A special case of our results implies former conjectures of Gao.Comment: 39 pages, 9 figure

Neural Surface Maps

Maps are arguably one of the most fundamental concepts used to define and operate on manifold surfaces in differentiable geometry. Accordingly, in geometry processing, maps are ubiquitous and are used in many core applications, such as paramterization, shape analysis, remeshing, and deformation. Unfortunately, most computational representations of surface maps do not lend themselves to manipulation and optimization, usually entailing hard, discrete problems. While algorithms exist to solve these problems, they are problem-specific, and a general framework for surface maps is still in need. In this paper, we advocate considering neural networks as encoding surface maps. Since neural networks can be composed on one another and are differentiable, we show it is easy to use them to define surfaces via atlases, compose them for surface-to-surface mappings, and optimize differentiable objectives relating to them, such as any notion of distortion, in a trivial manner. In our experiments, we represent surfaces by generating a neural map that approximates a UV parameterization of a 3D model. Then, we compose this map with other neural maps which we optimize with respect to distortion measures. We show that our formulation enables trivial optimization of rather elusive mapping tasks, such as maps between a collection of surfaces