Unicellular Maps Research Articles

Combinatorial proof for the rationality of the bivariate generating series of maps in positive genus

In this paper, we give the first combinatorial proof of a rationality scheme for the generating series of maps in positive genus enumerated by both vertices and faces. This scheme was first obtained by Bender, Canfield, and Richmond in 1993 by purely computational techniques. To do so, we rely on a bijection obtained by the second author in a previous work between those maps and a family of decorated unicellular maps. Our main contribution consists in a fine analysis of the family of unicellular maps. As a byproduct, we also obtain a new and simpler combinatorial proof of the rationality scheme for the generating series of maps enumerated by their number of edges, originally obtained computationally by Bender and Canfield in 1991 and combinatorially by the second author in 2019.

Unicellular maps vs. hyperbolic surfaces in large genus: Simple closed curves

Unicellular maps vs. hyperbolic surfaces in large genus: Simple closed curves

Random generation and scaling limits of fixed genus factorizations into transpositions

We study the asymptotic behaviour of random factorizations of the n-cycle into transpositions of fixed genus g>0. They have a geometric interpretation as branched covers of the sphere and their enumeration as Hurwitz numbers was extensively studied in algebraic combinatorics and enumerative geometry. On the probabilistic side, several models and properties of permutation factorizations were studied in previous works, in particular minimal factorizations of cycles into transpositions (which corresponds to the case g=0 of this work). Using the representation of factorizations via unicellular maps, we first exhibit an algorithm which samples an asymptotically uniform factorization of genus g in linear time. In a second time, we code a factorization as a process of chords appearing one by one in the unit disk, and we prove the convergence (as nrightarrow infty ) of the process associated with a uniform genus g factorization of the n-cycle. The limit process can be explicitly constructed from a Brownian excursion. Finally, we establish the convergence of a natural genus process, coding the appearance of the successive genera in the factorization.

Large expanders in high genus unicellular maps

We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been discovered, such as their local limit or their logarithmic diameter. In this work, we show that with high probability such a map contains a very large induced subgraph that is an expander.Mathematics Subject Classifications: 05C10, 05C48, 60C05, 60D05Keywords: Combinatorial maps, high genus, expander graphs

Short cycles in high genus unicellular maps

Nous étudions les grandes cartes aléatoires uniformes à une face dont le genre croît linéairement avec le nombre d’arêtes, qui sont un modèle de géométrie hyperbolique discrète. Dans des travaux précédents, plusieurs propriétés géométriques hyperboliques ont été étudiées. Dans le présent travail, nous étudions le nombre de petits cycles dans une carte unicellulaire uniforme de grand genre, et nous montrons qu’il converge vers une distribution de Poisson. En corollaire, nous obtenons la loi de la systole des cartes unicellulaires uniformes de grand genre. Nous obtenons également la distribution asymptotique des degrés des sommets dans une telle carte.

On breadth‐first constructions of scaling limits of random graphs and random unicellular maps

AbstractWe give alternate constructions of (i) the scaling limit of the uniform connected graphs with given fixed surplus, and (ii) the continuum random unicellular map of a given genus that start with a suitably tilted Brownian continuum random tree and make “horizontal” point identifications, at random heights, using the local time measures. Consequently, this can be seen as a continuum analogue of the breadth‐first construction of a finite connected graph. In particular, this yields a breadth‐first construction of the scaling limit of the critical Erdős–Rényi random graph which answers a question posed by Addario‐Berry, Broutin, and Goldschmidt. As a consequence of this breadth‐first construction, we obtain descriptions of the radii, the distance profiles, and the two point functions of these spaces in terms of functionals of tilted Brownian excursions.

A bijection for essentially 3-connected toroidal maps

We present a bijection for toroidal maps that are essentially 3-connected (3-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite toroidal maps with all faces of degree 4 except for a hexagonal root-face. We show that these maps are in bijection with certain well-characterized bipartite unicellular maps. Our bijection, closely related to the recent one by Bonichon and Lévêque for essentially 4-connected toroidal triangulations, can be seen as the toroidal counterpart of the one developed in the planar case by Fusy, Poulalhon and Schaeffer, and it extends the one recently proposed by Fusy and Lévêque for essentially simple toroidal triangulations. Moreover, we show that rooted essentially 3-connected toroidal maps can be decomposed into two pieces, a toroidal part that is treated by our bijection, and a planar part that is treated by the above-mentioned planar case bijection. This yields a combinatorial derivation for the bivariate generating function of rooted essentially 3-connected toroidal maps, counted by vertices and faces.

Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth

We present unified bijections for maps on the torus with control on the face-degrees and essential girth (girth of the periodic planar representation). A first step is to show that for d≥3 every toroidal d-angulation of essential girth d can be endowed with a certain ‘canonical’ orientation (formulated as a weight-assignment on the half-edges). Using an adaptation of a construction by Bernardi and Chapuy, we can then derive a bijection between face-rooted toroidal d-angulations of essential girth d (with the condition that, apart from the root-face contour, no other closed walk of length d encloses the root-face) and a family of decorated unicellular maps. The orientations and bijections can then be generalized, for any d≥1, to toroidal face-rooted maps of essential girth d with a root-face of degree d (and with the same root-face contour condition as for d-angulations), and they take a simpler form in the bipartite case, as a parity specialization. On the enumerative side we obtain explicit algebraic expressions for the generating functions of rooted essentially simple triangulations and bipartite quadrangulations on the torus. Our bijective constructions can be considered as toroidal counterparts of those obtained by Bernardi and the first author in the planar case, and they also build on ideas introduced by Despré, Gonçalves and the second author for essentially simple triangulations, of imposing a balancedness condition on the orientations in genus 1.

Combinatorial study of graphs arising from the Sachdev–Ye–Kitaev model

We consider the graphs involved in the theoretical physics model known as the colored Sachdev–Ye–Kitaev (SYK) model. We study in detail their combinatorial properties at any order in the so-called 1∕N expansion, and we enumerate these graphs asymptotically.Because of the duality between colored graphs involving q+1 colors and colored triangulations in dimension q, our results apply to the asymptotic enumeration of spaces that generalize unicellular maps – in the sense that they are obtained from a single building block – for which a higher-dimensional generalization of the genus is kept fixed.

Blossoming bijection for higher-genus maps

In 1997, Schaeffer described a bijection between Eulerian planar maps and some trees. In this work we generalize his work to a bijection between maps on an orientable surface of any fixed genus and some unicellular maps with the same genus. An important step of this construction is to exhibit a canonical orientation for maps, that allows to apply the same local opening algorithm as Schaeffer did.As an important byproduct, we obtain the first bijective proof of a result of Bender and Canfield from 1991, when they proved that the generating series of maps in higher genus is a rational function of the generating series of planar maps.

Encoding Toroidal Triangulations

Poulalhon and Schaeffer introduced an elegant method to linearly encode a planar triangulation optimally. The method is based on performing a special depth-first search algorithm on a particular orientation of the triangulation: the minimal Schnyder wood. Recent progress toward generalizing Schnyder woods to higher genus enables us to generalize this method to the toroidal case. In the plane, the method leads to a bijection between planar triangulations and some particular trees. For the torus we obtain a similar bijection but with particular unicellular maps (maps with only one face).

Large unicellular maps in high genus

Nous etudions la geometrie d’une carte aleatoire unicellulaire qui est distribuee uniformement sur l’ensemble de toutes les cartes unicellulaires dont le genre est proportionnel au nombre des arretes. Nous prouvons que la distance entre deux sommets choisis uniformement d’une telle carte est de l’ordre $\log n$ et le diametre est aussi de l’ordre $\log n$ avec une forte probabilite. Nous prouvons aussi une version quantitative du resultat que la carte est localement planaire avec une forte probabilite. L’ingredient principal de la preuve est une procedure d’exploration qui utilise une bijection due au Chapuy, Feray et Fusy (J. Combin. Theory Ser. A 120 (2013) 2064–2092).

A Bijection for Tricellular Maps

We give a bijective proof for a relation between unicellular, bicellular, and tricellular maps. These maps represent cell complexes of orientable surfaces having one, two, or three boundary components. The relation can formally be obtained using matrix theory (Dyson, 1949) employing the Schwinger-Dyson equation (Schwinger, 1951). In this paper we present a bijective proof of the corresponding coefficient equation. Our result is a bijection that transforms a unicellular map of genus g into unicellular, bicellular or tricellular maps of strictly lower genera. The bijection employs edge cutting, edge contraction, and edge deletion.

A simple model of trees for unicellular maps

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the “recursive part” of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure.All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer–Zagier recurrence formula, the Lehman–Walsh formula and the Goupil–Schaeffer formula. We also discuss several applications of our construction: we obtain a new proof of an identity related to covered maps due to Bernardi and the first author, and thanks to previous work of the second author, we give a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Finally, we show that our techniques apply partially to unicellular 3-constellations and to related objects that we call quasi-3-constellations.

A combinatorial way of counting unicellular maps and constellations

Our work is devoted to the bijective enumeration of the set of factorizations of a permutation into m factors with a given number of cycles. Previously, this major problem in combinatorics and its various specializations were considered mainly from the character theoretic or algebraic geometry point of view. Let us specially mention here the works of Harer and Zagier or Kontsevich. In 1988, Jackson reported a very general formula solving this problem. However, to the author’s own admission this result left little room for combinatorial interpretation and no bijective proof of it was known yet. In 2001, Lass found a combinatorial proof of the celebrated special case of Jackson’s formula known as the Harer–Zagier formula. This work was followed by Goulden and Nica, who presented in 2004 another combinatorial proof involving a direct bijection. In the past two years, we have introduced new sets of objects called partitioned maps and partitioned cacti, the enumeration of which allowed us to construct bijective proofs for more general cases of Jackson’s formula.

The local limit of unicellular maps in high genus

We show that the local limit of unicellular maps whose genus is proportional to the number of edges is a supercritical geometric Galton-Watson tree conditioned to survive. The proof relies on enumeration results obtained via the recent bijection given by the second author together with Feray and Fusy.

A simple model of trees for unicellular maps

We consider unicellular maps, or polygon gluings, of fixed genus. In FPSAC '09 the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the ``recursive part'' of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, or the Lehman-Walsh/Goupil-Schaeffer formulas. Thanks to previous work of the second author this also leads us to a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Nous considèrons des cartes orientèes à une face de genre fixé. à SFCA'09 le premier auteur a introduit une bijection rècursive envoyant une carte unicellulaire vers un arbre, ce qui permet d'obtenir des formules ènumèratives pour les cartes à une face (et en particulier la prèsence des nombres de Catalan). Dans l'article ici prèsent, et en nous appuyant sur la bijection ci-dessus, nous obtenons une incarnation très simple des cartes à une face comme des paires formèes d'un arbre plan et d'une permutation d'un certain type. Toutes les formules prècèdemment connues dècoulent aisèment de cette nouvelle incarnation, donnant des preuves bijectives dans un cadre unifié. Pour certaines de ces formules, telles que la rècurrence de Harer-Zagier ou les formules de Lehman-Walsh/Goupil-Schaeffer, nous obtenons la première preuve bijective connue. Par ailleurs, en combinant notre approche avec des travaux du second auteur, nous obtenons une nouvelle expression pour les polynômes de Stanley qui donnent certaines èvaluations des caractères du groupe symètrique.

An analogue of the Harer–Zagier formula for unicellular maps on general surfaces

A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is simply connected. In a famous article, Harer and Zagier established a formula for the generating function of unicellular maps counted according to the number of vertices and edges. The keystone of their approach is a counting formula for unicellular maps on orientable surfaces with n edges, and with vertices colored using every color in [ q ] (adjacent vertices are authorized to have the same color). We give an analogue of this formula for general (locally orientable) surfaces. Our approach is bijective and is inspired by Lassʼs proof of the Harer–Zagier formula. We first revisit Lassʼs proof and twist it into a bijection between unicellular maps on orientable surfaces with vertices colored using every color in [ q ] , and maps with vertex set [ q ] on orientable surfaces with a marked spanning tree. The bijection immediately implies Harer–Zagierʼs formula and a formula by Jackson concerning bipartite unicellular maps. It also shed a new light on constructions by Goulden and Nica, Schaeffer and Vassilieva, and Morales and Vassilieva. We then extend the bijection to general surfaces and obtain a correspondence between unicellular maps on general surfaces with vertices colored using every color in [ q ] , and maps on orientable surfaces with vertex set [ q ] with a marked planar submap. This correspondence gives an analogue of the Harer–Zagier formula for general surfaces. We also show that this formula implies a recursion formula due to Ledoux for the numbers of unicellular maps with given numbers of vertices and edges.

A new combinatorial identity for unicellular maps, via a direct bijective approach

A unicellular map, or one-face map, is a graph embedded in an orientable surface such that its complement is a topological disk. In this paper, we give a new viewpoint to the structure of these objects, by describing a decomposition of any unicellular map into a unicellular map of smaller genus. This gives a new combinatorial identity for the number ϵg(n) of unicellular maps with n edges and genus g. Unlike the Harer–Zagier recurrence formula, this identity is recursive in only one parameter (the genus).Iterating the construction gives an explicit bijection between unicellular maps and plane trees with distinguished vertices, which gives a combinatorial explanation (and proof) of the fact that ϵg(n) is the product of the n-th Catalan number by a polynomial in n. The combinatorial interpretation also gives a new and simple formula for this polynomial. Variants of the problem are considered, like bipartite unicellular maps, or unicellular maps with only cubic vertices.

A bijection for covered maps, or a shortcut between Harer–Zagierʼs and Jacksonʼs formulas

We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus g) with a marked unicellular spanning submap (which can have any genus in { 0 , 1 , … , g } ). Our main result is a bijection between covered maps with n edges and genus g and pairs made of a plane tree with n edges and a unicellular bipartite map of genus g with n + 1 edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in Bernardi (2007) [4]. Covered maps can also be seen as shuffles of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps. We also show that the bijection of Bouttier, Di Francesco and Guitter (2004) [8] (which generalizes a previous bijection by Schaeffer, 1998 [33]) between bipartite maps and so-called well-labeled mobiles can be obtained as a special case of our bijection.