Cyclic cellular automata and related processes
Abstract
A cyclic cellular automaton is a discretetime process defined on the state space {0, 1, …, N 1} ^{Z d}. 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 continuoustime Markov processes on {0, 1, …, N 1} ^{Z d} 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 onedimensional systems of particles with various interaction mechanisms. Finally, we touch upon the work currently being done on twodimensional cyclic particle systems and cellular automata.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 September 1990
 DOI:
 10.1016/01672789(90)90170T
 Bibcode:
 1990PhyD...45...19F