Ballistic annihilation and deterministic surface growth

Abstract

A model of deterministic surface growth studied by Krug and Spohn, a model of the annihilating reactionA+B→inert studied by Elskens and Frisch, a one-dimensional three-color cyclic cellular automaton studied by Fisch, and a particular automaton that has the number 184 in the classification of Wolfram can be studied via a cellular automaton with stochastic initial data called ballistic annihilation. This automaton is defined by the following rules: At timet=0, one particle is put at each integer point of ℝ. To each particle, a velocity is assigned in such a way that it may be either +1 or −1 with probabilities 1/2, independent of the velocities of the other particles. As time goes on, each particle moves along ℝ at the velocity assigned to it and annihilates when it collides with another particle. In the present paper we compute the distribution of this automaton for each timet ∈ ℕ. We then use this result to obtain the hydrodynamic limit for the surface profile 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 processB ^min _x ,x ∈ ℝ, defined byB ^min _x ≔min{B _y ;x−1≤y≤x+1} for everyx ∈ ℝ, whereB _x ,x ∈ ℝ, is the standard Brownian motion withB ₀=0.