New approach to evaluate the asymptotic distribution of particle systems expressed by probabilistic cellular automata

We propose some conjectures on the asymptotic distribution of the probabilistic Burgers cellular automaton (PBCA), which is defined by a simple rule of particle motion with a probabilistic parameter. Asymptotic distribution of configurations converges to a unique steady state for PBCA. We propose a new and widely-applicable approach to analyze probabilistic particle systems and apply it concretely to PBCA and its extensions. We introduce a conjecture on the distribution and derive the asymptotic probability expressed by the GKZ hypergeometric function. If the space size goes into infinity, we can evaluate the relationship between the density and flux of particles for infinite space. Moreover, we propose two extended systems of PBCA and analyze their asymptotic behavior.