Abstract
We rigorously prove a form of disorder-resistance for a class of one-dimensional
cellular automaton rules, including some that arise as boundary dynamics of two-dimensional
solidification rules. Specifically, when started from a random initial seed on an interval
of length $L$, with probability tending to one as $L\to\infty$, the evolution is a
replicator. That is, a region of space-time of density one is filled with a spatially and
temporally periodic pattern, punctuated by a finite set of other finite patterns repeated
at a fractal set of locations. On the other hand, the same rules exhibit provably more
complex evolution from some seeds, while from other seeds their behavior is apparently
chaotic. A principal tool is a new variant of percolation theory, in the context of
additive cellular automata from random initial states.
