Asymptotic distribution of entry times in a cellular automaton with annihilating particles

Cellular automata are a simple formal model for complex systems. They consist of an infinite number of identical finite automata arranged over a regular lattice (here Z). Each automaton updates its state according to its own state and the one of a fixed set of neighboring automata according to a local rule. All updates are synchronous. The simplicity of the model contrasts with the great variety of different dynamical behaviors. Indeed, exactly this rich variety of behaviors and the ease of being simulated on computers made CA fortune. Actually, they are used in almost all scientific disciplines ranging from Mathematics to Computer Science and Natural Sciences. In particular, in Biology, Physics and Economics, they can be used as a discrete counterpart (in the sense of time) of interacting particle systems (IPS). The advantage of modeling IPS by CA is that one can have information not only about limit distributions and particle densities but also on their spatial distribution. On the other hand, as we have already mentioned, the dynamical behavior of CA is complex and not fully understood and IPS can help to understand the dynamics of some CA whenever it can be described in terms of particles or signals that move in a neutral background and interact on encounters. The general concept of a signal or particle (in the context of CA) has been elaborated in Formenti and Kůrka (2007).