1,237 research outputs found
Regular colored graphs of positive degree
Regular colored graphs are dual representations of pure colored D-dimensional
complexes. These graphs can be classified with respect to an integer, their
degree, much like maps are characterized by the genus. We analyse the structure
of regular colored graphs of fixed positive degree and perform their exact and
asymptotic enumeration. In particular we show that the generating function of
the family of graphs of fixed degree is an algebraic series with a positive
radius of convergence, independant of the degree. We describe the singular
behavior of this series near its dominant singularity, and use the results to
establish the double scaling limit of colored tensor models.Comment: Final version. Significant improvements made, main results unchange
A combinatorial approach to jumping particles
In this paper we consider a model of particles jumping on a row of cells,
called in physics the one dimensional totally asymmetric exclusion process
(TASEP). More precisely we deal with the TASEP with open or periodic boundary
conditions and with two or three types of particles. From the point of view of
combinatorics a remarkable feature of this Markov chain is that it involves
Catalan numbers in several entries of its stationary distribution. We give a
combinatorial interpretation and a simple proof of these observations. In doing
this we reveal a second row of cells, which is used by particles to travel
backward. As a byproduct we also obtain an interpretation of the occurrence of
the Brownian excursion in the description of the density of particles on a long
row of cells.Comment: 24 figure
Dissections, orientations, and trees, with applications to optimal mesh encoding and to random sampling
We present a bijection between some quadrangular dissections of an hexagon
and unrooted binary trees, with interesting consequences for enumeration, mesh
compression and graph sampling. Our bijection yields an efficient uniform
random sampler for 3-connected planar graphs, which turns out to be determinant
for the quadratic complexity of the current best known uniform random sampler
for labelled planar graphs [{\bf Fusy, Analysis of Algorithms 2005}]. It also
provides an encoding for the set of -edge 3-connected
planar graphs that matches the entropy bound
bits per edge (bpe). This solves a
theoretical problem recently raised in mesh compression, as these graphs
abstract the combinatorial part of meshes with spherical topology. We also
achieve the {optimal parametric rate} bpe
for graphs of with vertices and faces, matching in
particular the optimal rate for triangulations. Our encoding relies on a linear
time algorithm to compute an orientation associated to the minimal Schnyder
wood of a 3-connected planar map. This algorithm is of independent interest,
and it is for instance a key ingredient in a recent straight line drawing
algorithm for 3-connected planar graphs [\bf Bonichon et al., Graph Drawing
2005]
On universal singular exponents in equations with one catalytic parameter of order one
Equations with one catalytic variable and one univariate unkown, also known as discrete difference equations of order one, form a familly of combinatorially relevant functional equations first discussed in full generality by Bousquet-Mélou and Jehanne (2006) who proved that their power serie solutions are algebraic. Drmota, Noy and Yu (2022) recently showed that in the non linear case the singular expansions of these series have a universal dominant term of order 3/2, as opposed to the dominant square root term of generic -algebraic series. Their direct analysis of the cancellation underlying this behavior is a tour de force of singular analysis. We show that the result can instead be given a straightforward explanation by showing that the derivative of the solution series conforms to the standard square root singular behavior. Consequences also include an atypical, but generic in this situation, asymptotic behavior for the cumulated values of the underlying catalytic parameter
A combinatorial approach to jumping particles: the parallel TASEP
International audienceIn this paper we continue the combinatorial study of the TASEP. We consider here the parallel TASEP, in which particles jump simultaneously. We offer here an elementary derivation that extends the combinatorial approach we developed for the standard TASEP. In particular we show that this stationary distribution can be expressed in terms of refinements of Catalan numbers
The distribution of the number of small cuts in a random planar triangulation
International audienceWe enumerate rooted 3-connected (2-connected) planar triangulations with respect to the vertices and 3-cuts (2-cuts). Consequently, we show that the distribution of the number of 3-cuts in a random rooted 3-connected planar triangulation with vertices is asymptotically normal with mean and variance , and the distribution of the number of 2-cuts in a random 2-connected planar triangulation with vertices is asymptotically normal with mean and variance . We also show that the distribution of the number of 3-connected components in a random 2-connected triangulation with vertices is asymptotically normal with mean and variance
A combinatorial approach to jumping particles I: maximal flow regime
International audienceIn this paper we consider a model of particles jumping on a row of cells, called in physics the one dimensional totally asymmetric exclusion process (TASEP). More precisely we deal with the TASEP with two or three types of particles, with or without boundaries, in the maximal flow regime. From the point of view of combinatorics a remarkable feauture of these Markov chains is that they involve Catalan numbers in several entries of their stationary distribution. We give a combinatorial interpretation and a simple proof of these observations. In doing this we reveal a second row of cells, which is used by particles to travel backward. As a byproduct we also obtain an interpretation of the occurrence of the Brownian excursion in the description of the density of particles on a long row of cells
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