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

TL;DR

This paper provides an exact Bethe ansatz solution for discrete time stochastic processes with parallel updates, revealing phase transitions and universal behavior in certain limits, and connecting to traffic models and KPZ universality.

Contribution

It introduces a Bethe ansatz solution for discrete time zero range and exclusion processes with parallel dynamics, including new results for the eigenvalues and phase transitions.

Findings

Exact eigenvalues for the generating function of particle displacement.

Universal KPZ scaling behavior in the scaling limit.

Identification of phase transition at p→1 for q<0.

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 $v_{\mathrm{max}}=1$. 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 $q<0$.