3,597 research outputs found
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
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
Large Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension
We review an approach which aims at studying discrete (pseudo-)manifolds in
dimension and called random tensor models. More specifically, we
insist on generalizing the two-dimensional notion of -angulations to higher
dimensions. To do so, we consider families of triangulations built out of
simplices with colored faces. Those simplices can be glued to form new building
blocks, called bubbles which are pseudo-manifolds with boundaries. Bubbles can
in turn be glued together to form triangulations. The main challenge is to
classify the triangulations built from a given set of bubbles with respect to
their numbers of bubbles and simplices of codimension two. While the colored
triangulations which maximize the number of simplices of codimension two at
fixed number of simplices are series-parallel objects called melonic
triangulations, this is not always true anymore when restricting attention to
colored triangulations built from specific bubbles. This opens up the
possibility of new universality classes of colored triangulations. We present
three existing strategies to find those universality classes. The first two
strategies consist in building new bubbles from old ones for which the problem
can be solved. The third strategy is a bijection between those colored
triangulations and stuffed, edge-colored maps, which are some sort of hypermaps
whose hyperedges are replaced with edge-colored maps. We then show that the
present approach can lead to enumeration results and identification of
universality classes, by working out the example of quartic tensor models. They
feature a tree-like phase, a planar phase similar to two-dimensional quantum
gravity and a phase transition between them which is interpreted as a
proliferation of baby universes
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
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