2,400 research outputs found
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
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
A combinatorial approach to jumping particles: the parallel TASEP
article en revisionIn this paper we continue the combinatorial study of models of particles jumping on a row of cells which we initiated with the standard totally asymmetric simple exclusion process or TASEP (Journal of Combinatorial Theory, Series A, 110(1):1–29, 2005). We consider here the parallel TASEP, in which particles can jump simultaneously. On the one hand, the interest in this process comes from highway traffic modeling: it is the only solvable special case of the Nagel-Schreckenberg automaton, the most popular model in that context. On the other hand, the parallel TASEP is of some theoretical interest because the derivation of its stationary distribution, as appearing in the physics literature, is harder than that of the standard TASEP. 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
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
A combinatorial approach to jumping particles II: general boundary conditions
International audienceWe consider a model of particles jumping on a row, the TASEP. From the point of view of combinatorics a remarkable feauture of this Markov chain is that Catalan numbers are involved in several entries of its stationary distribution. In a companion paper, we gave a combinatorial interpretaion and a simple proof of these observations in the simplest case where the particles enter, jump and exit at the same rate. In this paper we show how to deal with general rates
Discrete Particle Swarm Optimization for the minimum labelling Steiner tree problem
Particle Swarm Optimization is an evolutionary method inspired by the
social behaviour of individuals inside swarms in nature. Solutions of the problem are
modelled as members of the swarm which fly in the solution space. The evolution is
obtained from the continuous movement of the particles that constitute the swarm
submitted to the effect of the inertia and the attraction of the members who lead the
swarm. This work focuses on a recent Discrete Particle Swarm Optimization for combinatorial optimization, called Jumping Particle Swarm Optimization. Its effectiveness is
illustrated on the minimum labelling Steiner tree problem: given an undirected labelled
connected graph, the aim is to find a spanning tree covering a given subset of nodes,
whose edges have the smallest number of distinct labels
Tableaux combinatorics for the asymmetric exclusion process
The partially asymmetric exclusion process (PASEP) is an important model from
statistical mechanics which describes a system of interacting particles hopping
left and right on a one-dimensional lattice of sites. It is partially
asymmetric in the sense that the probability of hopping left is times the
probability of hopping right. Additionally, particles may enter from the left
with probability and exit from the right with probability .
In this paper we prove a close connection between the PASEP and the
combinatorics of permutation tableaux. (These tableaux come indirectly from the
totally nonnegative part of the Grassmannian, via work of Postnikov, and were
studied in a paper of Steingrimsson and the second author.) Namely, we prove
that in the long time limit, the probability that the PASEP is in a particular
configuration is essentially the generating function for permutation
tableaux of shape enumerated according to three statistics. The
proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on
the {\it matrix ansatz} for the PASEP model.
As an application, we prove some monotonicity results for the PASEP. We also
derive some enumerative consequences for permutations enumerated according to
various statistics such as weak excedence set, descent set, crossings, and
occurences of generalized patterns.Comment: Clarified exposition, more general result, new author (SC), 19 pages,
6 figure
An Inhomogeneous Multispecies TASEP on a Ring
We reinterpret and generalize conjectures of Lam and Williams as statements
about the stationary distribution of a multispecies exclusion process on the
ring. The central objects in our study are the multiline queues of Ferrari and
Martin. We make some progress on some of the conjectures in different
directions. First, we prove their conjectures in two special cases by
generalizing the rates of the Ferrari-Martin transitions. Secondly, we define a
new process on multiline queues, which have a certain minimality property. This
gives another proof for one of the special cases; namely arbitrary jump rates
for three species.Comment: 21 pages, 1 figure. major changes in exposition; definitions
clarified and terminology made more self-containe
Stationary currents in particle systems with constrained hopping rates
We study the effect on the stationary currents of constraints affecting the
hopping rates in stochastic particle systems. In the framework of Zero Range
Processes with drift within a finite volume, we discuss how the current is
reduced by the presence of the constraint and deduce exact formulae, fully
explicit in some cases. The model discussed here has been introduced in Ref.
[1] and is relevant for the description of pedestrian motion in elongated dark
corridors, where the constraint on the hopping rates can be related to
limitations on the interaction distance among pedestrians
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