Bethe ansatz solution of discrete time stochastic processes with fully parallel update

Abstract

We present the Bethe ansatz solution for the discrete time zero range and asymmetric exclusion processes with fully parallel dynamics. The model depends on two parameters: $p$, the probability of single particle hopping, and $q$, the deformation parameter, which in the general case, $|q|<1$, is responsible for long range interaction between particles. The particular case $q=0$ corresponds to the Nagel-Schreckenberg traffic model with $v_{\max }=1$. As a result, we obtain the largest eigenvalue of the equation for the generating function of the distance travelled by particles. For the case $q=0$ the result is obtained for arbitrary size of the lattice and number of particles. In the general case we study the model in the scaling limit and obtain the universal form specific for the Kardar-Parisi-Zhang universality class. We describe the phase transition occurring in the limit $p\to 1$ when $q<0$.

43 pages, 3 figures, final version, published in Journal Stat. Phys. in Online First, avalable at http://dx.doi.org/10.1007/s10955-006-9023-0

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• Determinant solution for the Totally Asymmetric Exclusion Process with parallel update II. Ring geometry
• Emergence of jams in the generalized totally asymmetric simple exclusion process
• Current fluctuations and large deviations for periodic TASEP on the relaxation scale
• KPZ fluctuations in finite volume
• Current statistics in the q-boson zero range process
• Zero Range Process and Multi-Dimensional Random Walks
• Crossover scaling functions in the asymmetric avalanche process