## BALLISTIC ANNIHILATION AND DETERMINISTIC SURFACE GROWTH (PS file)

### Vladimir Belitsky and Pablo A. Ferrari

\noindent {\bf Abstract.}\ \ A model of deterministic surface growth studied by Krug and Spohn, a model of the annihilating reaction $A+B\rightarrow inert$ 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 classification of Wolfram can be studied via a certain cellular automaton with stochastic initial data. This automaton is defined by the following rules: At time $t=0$, one particle is put at each integer point of $\real$. 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 $\real$ 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 $t\in \natural$. 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 process $B^{\min}_x, x\in \real$, defined by $B^{\min}_x:=\min\{B_y; x-1\leq y\leq x+1\}$ for every $x\in \real$, where $B_x, x\in \real$, is the standard Brownian motion with $B_0=0$. \vskip4mm
\noindent {\sl Keywords.} Cellular automaton, deterministic model of surface growth, ballistic annihilation, three-color cyclic cellular automaton, annihilating two-species reaction, hydrodynamic limit, moving local minimum of Brownian motion. \vskip 1truemm \noindent {\sl AMS 1991 classification numbers.} 60K35, 82C22, 60J65. \vskip 3truemm