Percolation and disorder-resistance in cellular automata

Janko Gravner, Alexander Holroyd

Research output: Contribution to journalArticle (Academic Journal)peer-review

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→∞, 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.
Original languageEnglish
Pages (from-to)1731-1776
JournalAnnals of Probability
Volume43
Issue number4
DOIs
Publication statusPublished - 3 Jun 2015

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