31,668 research outputs found
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 for both orientable and non-orientable rooted
connected maps of Euler characteristic , 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 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 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
Asymptotic enumeration and limit laws for graphs of fixed genus
It is shown that the number of labelled graphs with n vertices that can be
embedded in the orientable surface S_g of genus g grows asymptotically like
where , and is the exponential growth rate of planar graphs. This generalizes the
result for the planar case g=0, obtained by Gimenez and Noy.
An analogous result for non-orientable surfaces is obtained. In addition, it
is proved that several parameters of interest behave asymptotically as in the
planar case. It follows, in particular, that a random graph embeddable in S_g
has a unique 2-connected component of linear size with high probability
Enumeration of N-rooted maps using quantum field theory
A one-to-one correspondence is proved between the N-rooted ribbon graphs, or
maps, with e edges and the (e-N+1)-loop Feynman diagrams of a certain quantum
field theory. This result is used to obtain explicit expressions and relations
for the generating functions of N-rooted maps and for the numbers of N-rooted
maps with a given number of edges using the path integral approach applied to
the corresponding quantum field theory.Comment: 27 pages, 7 figure
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
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