A combinatorial approach to jumping particles: The parallel TASEP

In 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 exclusion process or TASEP (Journal of Combinatorial Theory, Series A, to appear). 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. Résumé. L’objet de cet article est de poursuivre l’étude combinatoire d’une famille de modèles de particules sauteuses que nous avons commencé avec le cas du processus d’exclusion totalement asymétrique standard, ou TASEP (Journal of Combinatorial Theory, Series A, to appear). Nous traitons ici le TASEP parallèle, dans lequel les particules peuvent sauter simultanément. L’étude de ce processus est motivée par les nombreux travaux de modélisation du trafic automobile qui portent sur l’automate de Nagel-Schreckenberg: le TASEP parallèle est en effet la seule instance de cet automate stochastique dont la distribution stationnaire soit connue. De plus, le TASEP parallèle présente l’intérêt théorique que la détermination de sa distribution stationnaire par des méthodes de physique mathématique est plus délicate que pour le TASEP standard. Nous utilisons une approche combinatoire qui étend l’approche que nous avions développée pour le TASEP standard. En particulier nous montrons que cette distribution peut-être décrite en termes de raffinements des nombres de Catalan. 1. Jumping particles and the TASEP family The aim of this article is to continue the combinatorial study of a family of models of particles jumping on a row of cells that are known in the physics and probability literature as one dimensional totally asymmetric exclusion processes (TASEPs for short). In order to define TASEPs we first introduce a set of configurations and some rules. A TASEP configuration is a row of n cells, separated by n+ 1 walls (the leftmost and rightmost ones are borders). Each cell is occupied by one particle, and each particle has a type, black or white (see Figure 1). Figure 1. A TASEP configuration with n = 10 cells, 5 black particles, and 5 white particles. The transitions of the TASEP are based on a mapping θ that modifies a configuration τ near a wall i to produce a configuration θ(τ, i). Given a pair (τ, i) the following rules define its image θ(τ, i): a. Rule •|◦ → ◦|•: If the wall i separates a black particle (on its left) and a white particle (on its right), then two particles swap to give θ(τ, i). b. Rule |◦ → |•: If the wall is the left border (i = 0) and the leftmost cell contains a white particle, this white particle leaves the row and it is replaced by a black particle. c. Rule •| → ◦|: If the wall is the right border (i = n) and the rightmost cell contains a black particle, this black particle leaves the row and it is replaced by a white particle. d In the other cases, nothing happens, θ(τ, i) = τ . Date: November 22, 2004. ∗ Supported by a post-doctoral grant of the CNRS. † Supported in part by EC’s IHRP Programm, within the Research Training Network Algebraic Combinatorics in Europe, grant HPRN-CT-2001-00272. 1