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 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. 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 simple 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 . Given a configuration τ , let M(τ) be the set of walls on which the previous mapping θ can effectively do something: inner walls with a neighborhood of the form •|◦, or borders with a neighborhood of the form |◦ or •|. The definition of θ can be extended to any subset {i1, . . . , ik} of M(τ) by setting θ(τ, i1, . . . , ik) = θ(θ(τ, i1, . . . , ik−1), ik). Observe that this extended mapping can be interpreted as performing moves at walls i1, . . . , ik in parallel since the basic mapping acts only locally and M(τ) never contains two adjacent walls. A pair (τ, A) with A ⊂ M(τ) will be referred to as an active configuration, and from now on θ is considered as a mapping from the set of active configurations into the set of TASEP configurations. In the previous work [4], we dealt with several variants of sequential TASEP. In particular, the standard sequential TASEP with open boundaries is a Markov chain S on the set of TASEP configurations with n cells whose dynamic is defined as follows in terms of θ: • Let τ = S(t) be the current configuration. Date: April 10, 2007. 1