Ballistic Annihilation and Deterministic Surface Growth

A model of deterministic surface growth studied by Krug and Spohn, a model of the annihilating reaction A + B ! inert studied by Elskens and Frisch, a one-dimensional tree-color cyclic cellular automaton studied by Fisch and a particular automaton that has the number 184 in the classiication of Wolfram can be studied via a certain cellular automaton with stochastic initial data. This automaton is deened by the following rules: At time t = 0, one particle is put at each integer point of IR. To each particle, a velocity is assigned in such a way that it may be either +1 or ?1 with probabilities 1=2, independently of the velocities of the other particles. As time goes, each particle moves along IR at the velocity assigned to it, and annihilates when collides with another particle. In the present paper we compute the distribution of this automaton for each time t 2 IN. We then use this result to obtain the hydrodynamic limit for the surface proole from the model of deterministic surface growth mentioned above. We also show the relation of this limit process to the process which we call moving local minimum of Brownian motion. The latter is the process B min x ; x 2 IR, deened by B min x := minfB y ; x ? 1 y x + 1g for every x 2 IR, where B x ; x 2 IR, is the standard Brownian motion with B 0 = 0.