22 research outputs found
Unified bijections for maps with prescribed degrees and girth
This article presents unified bijective constructions for planar maps, with
control on the face degrees and on the girth. Recall that the girth is the
length of the smallest cycle, so that maps of girth at least are
respectively the general, loopless, and simple maps. For each positive integer
, we obtain a bijection for the class of plane maps (maps with one
distinguished root-face) of girth having a root-face of degree . We then
obtain more general bijective constructions for annular maps (maps with two
distinguished root-faces) of girth at least . Our bijections associate to
each map a decorated plane tree, and non-root faces of degree of the map
correspond to vertices of degree of the tree. As special cases we recover
several known bijections for bipartite maps, loopless triangulations, simple
triangulations, simple quadrangulations, etc. Our work unifies and greatly
extends these bijective constructions. In terms of counting, we obtain for each
integer an expression for the generating function
of plane maps of girth with root-face of
degree , where the variable counts the non-root faces of degree .
The expression for was already obtained bijectively by Bouttier, Di
Francesco and Guitter, but for the expression of is new. We
also obtain an expression for the generating function
\G_{p,q}^{(d,e)}(x_d,x_{d+1},...) of annular maps with root-faces of degrees
and , such that cycles separating the two root-faces have length at
least while other cycles have length at least . Our strategy is to
obtain all the bijections as specializations of a single "master bijection"
introduced by the authors in a previous article. In order to use this approach,
we exhibit certain "canonical orientations" characterizing maps with prescribed
girth constraints
A note on irreducible maps with several boundaries
We derive a formula for the generating function of d-irreducible bipartite
planar maps with several boundaries, i.e. having several marked faces of
controlled degrees. It extends a formula due to Collet and Fusy for the case of
arbitrary (non necessarily irreducible) bipartite planar maps, which we recover
by taking d=0. As an application, we obtain an expression for the number of
d-irreducible bipartite planar maps with a prescribed number of faces of each
allowed degree. Very explicit expressions are given in the case of maps without
multiple edges (d=2), 4-irreducible maps and maps of girth at least 6 (d=4).
Our derivation is based on a tree interpretation of the various encountered
generating functions.Comment: 18 pages, 8 figure
A simple formula for the series of constellations and quasi-constellations with boundaries
We obtain a very simple formula for the generating function of bipartite
(resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed
lengths, which generalizes certain expressions obtained by Eynard in a book to
appear. The formula is derived from a bijection due to Bouttier, Di Francesco
and Guitter combined with a process (reminiscent of a construction of Pitman)
of aggregating connected components of a forest into a single tree. The formula
naturally extends to -constellations and quasi--constellations with
boundaries (the case corresponding to bipartite maps).Comment: 23 pages, full paper version of v1, with results extended to
constellations and quasi constellation
A bijection for rooted maps on general surfaces (extended abstract)
International audienceWe 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, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent for both orientable and non-orientable maps of Euler characteristic and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor ¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Nous étendons la bijection de Marcus et Schaeffer entre quadrangulations biparties orientables (de manière équivalente: cartes enracinées) et cartes à une face étiquetées orientables à toutes les surfaces, orientables ou non. Cette construction générale requiert des idées nouvelles et est plus délicate que dans le cas particulier orientable, mais permet des utilisations similaires. Elle donne donc une interprétation combinatoire uniforme de l’exposant de comptage pour les cartes orientables et non-orientables de caractéristique d’Euler , et de l’algébricité des fonctions génératrices. Elle montre l’universalité du facteur de normalisation ¼ pour la métrique des cartes, sur toutes les surfaces: le profil et le rayon d’une quadrangulation enracinée pointée sur une surface fixée converge en distribution. Enfin, elle ouvre à la voie à l’étude des surfaces Browniennes pour toute 2-variété compacte
On the distance-profile of random rooted plane graphs
International audienceWe study the distance-profile of the random rooted plane graph Gn with n edges (by a plane graph we mean a planar map with no loops nor multiple edges). Our main result is that the profile and radius of Gn (with respect to the root-vertex), rescaled by (2n) 1/4 , converge to explicit distributions related to the Brownian snake. A crucial ingredient of our proof is a bijection we have recently introduced between rooted outer-triangular plane graphs and rooted eulerian triangulations, combined with ingredients from Chassaing and Schaeffer (2004), Bousquet-Mélou and Schaeffer (2000), and Addario-Berry and Albenque (2013). We also show that the result for plane graphs implies similar results for random rooted loopless maps and general maps
The number of corner polyhedra graphs
Corner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs that can be skeleton of a corner polyhedron: as planar maps, they are the duals of some particular bipartite triangulations, which we call hereafter corner triangulations.In this paper we count corner polyhedral graphs by determining the generating function of the corner triangulations with respect to the number of vertices: we obtain an explicit rational expression for it in terms of the Catalan gen- erating function. We first show that this result can be derived using Tutte's classical compositional approach. Then, in order to explain the occurrence of the Catalan series we give a direct algebraic decomposition of corner triangu- lations: in particular we exhibit a family of almond triangulations that admit a recursive decomposition structurally equivalent to the decomposition of binary trees. Finally we sketch a direct bijection between binary trees and almond triangulations. Our combinatorial analysis yields a simpler alternative to the algorithm of Eppstein and Mumford for endowing a corner polyhedral graph with the cycle cover structure needed to realize it as a polyhedral graph
A bijection for rooted maps on general surfaces (extended abstract)
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, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent for both orientable and non-orientable maps of Euler characteristic and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor ¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold
Tridiagonalized GUE matrices are a matrix model for labeled mobiles
It is well-known that the number of planar maps with prescribed vertex degree
distribution and suitable labeling can be represented as the leading
coefficient of the -expansion of a joint cumulant of traces of
powers of an -by- GUE matrix. Here we undertake the calculation of this
leading coefficient in a different way. Firstly, we tridiagonalize the GUE
matrix in the manner of Trotter and Dumitriu-Edelman and then alter it by
conjugation to make the subdiagonal identically equal to . Secondly, we
apply the cluster expansion technique (specifically, the
Brydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statistical
mechanics. Thirdly, by sorting through the terms of the expansion thus
generated we arrive at an alternate interpretation for the leading coefficient
related to factorizations of the long cycle . Finally, we
reconcile the group-theoretical objects emerging from our calculation with the
labeled mobiles of Bouttier-Di Francesco-Guitter.Comment: 42 pages, LaTeX, 17 figures. The present paper completely supercedes
arXiv1203.3185 in terms of methods but addresses a different proble
Irreducible Metric Maps and Weil-Petersson Volumes
Contains fulltext :
253439.pdf (Publisher’s version ) (Open Access)24 juni 202