2,480 research outputs found
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
Asymptotic expansion of the multi-orientable random tensor model
Three-dimensional random tensor models are a natural generalization of the
celebrated matrix models. The associated tensor graphs, or 3D maps, can be
classified with respect to a particular integer or half-integer, the degree of
the respective graph. In this paper we analyze the general term of the
asymptotic expansion in N, the size of the tensor, of a particular random
tensor model, the multi-orientable tensor model. We perform their enumeration
and we establish which are the dominant configurations of a given degree.Comment: 27 pages, 24 figures, several minor modifications have been made, one
figure has been added; accepted for publication in "Electronic Journal of
Combinatorics
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-constellations.Comment: v5: minor revision after reviewers comments, 33 pages, added a
refinement by degree of the Harer-Zagier formula and more details in some
proof
Rectangular Matrix Models and Combinatorics of Colored Graphs
We present applications of rectangular matrix models to various combinatorial
problems, among which the enumeration of face-bicolored graphs with prescribed
vertex degrees, and vertex-tricolored triangulations. We also mention possible
applications to Interaction-Round-a-Face and hard-particle statistical models
defined on random lattices.Comment: 42 pages, 11 figures, tex, harvmac, eps
A generalization of the quadrangulation relation to constellations and hypermaps
Constellations and hypermaps generalize combinatorial maps, i.e. embedding of
graphs in a surface, in terms of factorization of permutations. In this paper,
we extend a result of Jackson and Visentin (1990) stating an enumerative
relation between quadrangulations and bipartite quadrangulations. We show a
similar relation between hypermaps and constellations by using a result of
Littlewood on factorization of characters. A combinatorial proof of
Littlewood's result is also given. Furthermore, we show that coefficients in
our relation are all positive integers, hinting possibility of a combinatorial
interpretation. Using this enumerative relation, we recover a result on the
asymptotic behavior of hypermaps in Chapuy (2009).Comment: 19 pages, extended abstract published in the proceedings of FPSAC
201
Analyticity of the Free Energy of a Closed 3-Manifold
The free energy of a closed 3-manifold is a 2-parameter formal power series
which encodes the perturbative Chern-Simons invariant (also known as the LMO
invariant) of a closed 3-manifold with gauge group U(N) for arbitrary . We
prove that the free energy of an arbitrary closed 3-manifold is uniformly
Gevrey-1. As a corollary, it follows that the genus part of the free energy
is convergent in a neighborhood of zero, independent of the genus. Our results
follow from an estimate of the LMO invariant, in a particular gauge, and from
recent results of Bender-Gao-Richmond on the asymptotics of the number of
rooted maps for arbitrary genus. We illustrate our results with an explicit
formula for the free energy of a Lens space. In addition, using the Painlev\'e
differential equation, we obtain an asymptotic expansion for the number of
cubic graphs to all orders, stengthening the results of Bender-Gao-Richmond.Comment: This is a contribution to the Special Issue on Deformation
Quantization, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Combinatorics of bicubic maps with hard particles
We present a purely combinatorial solution of the problem of enumerating
planar bicubic maps with hard particles. This is done by use of a bijection
with a particular class of blossom trees with particles, obtained by an
appropriate cutting of the maps. Although these trees have no simple local
characterization, we prove that their enumeration may be performed upon
introducing a larger class of "admissible" trees with possibly doubly-occupied
edges and summing them with appropriate signed weights. The proof relies on an
extension of the cutting procedure allowing for the presence on the maps of
special non-sectile edges. The admissible trees are characterized by simple
local rules, allowing eventually for an exact enumeration of planar bicubic
maps with hard particles. We also discuss generalizations for maps with
particles subject to more general exclusion rules and show how to re-derive the
enumeration of quartic maps with Ising spins in the present framework of
admissible trees. We finally comment on a possible interpretation in terms of
branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction
and discussion/conclusion extended, minor corrections, references adde
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