Chaos and Monte Carlo Approximations of the Flip-Annihilation process

Abstract

The flip-annihilation process is a random particle process with one-dimensional local interaction in discrete time, initially presented by one of us, namely Toom in 2004. Its components are enumerated by integer numbers and every component has two states, “minus” and “plus”. At every time step two transformations occur. The first one, called “flip”, independently turns every minus into plus with probability β. The second one, called “annihilation”, acts thus: whenever a plus is a left neighbor of a minus, both disappear with probability α independently from other components. What is interesting about this process is that it is ergodic for β>α/2 and non-ergodic for β<α ²/250. It is natural to conjecture that there is some transition curve, which we call the true curve and denote by $\beta =\mathsf{true}(\alpha)$ , which separates the areas of ergodicity and non-ergodicity of this process from each other. The estimates, mentioned above, albeit rigorous, leave a large gap between them and the present article’s purpose is to obtain some closer, albeit non-rigorous, approximations of the true curve. We do it in two ways, one of which is a chaos approximation and the other is a Monte Carlo simulation. Thus we obtain two curves, which are much closer to each other than the rigorous estimations. Also we fill in, albeit only numerically, another shortcoming of the rigorous estimation β<α ²/250, namely that it leaves us uncertain whether the true curve has a zero or positive slope at the point α=β=0. Both approximate curves have a positive slope at α=0, as we hoped.