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arXiv · 1304.7301

Percolation and disorder-resistance in cellular automata

Janko Gravner·Alexander E. Holroyd
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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.

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Janko Gravner, Alexander E. Holroyd. 2015-09-29. Percolation and disorder-resistance in cellular automata. https://doi.org/10.1214/14-aop918

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