CYCLIC CELLULAR AUTOMATA AND RELATED PROCESSES

A cyclic cellular automaton is a discrete-time process defined on the state space {0, 1, ..., N -1}Z. Starting from a random initial condition, at each time step, each site looks at its nearest neighbors. If the state at site x is i, then site x will change state to i + 1 (mod N) if it locates state i+1 (mod N) among its neighbors; otherwise, site x will remain in state i. Cyclic cellular automata have properties akin to cyclic particle systems, introduced by Bramson and Griffeath, which are continuous-time Markov processes on {0, 1, ..., N -1}Z with a stochastic evolution rather than a deterministic one. In one dimension, some clustering results about cyclic cellular automata are presented; the analogous problems in the context of cyclic particle systems are still unresolved. Also, we discuss the consequences of these clustering results in the context of one-dimensional systems of particles with various interaction mechanisms. Finally, we touch upon the work currently being done on two-dimensional cyclic particle systems and cellular automata. © 1990.