Open Access
July 2015 Percolation and disorder-resistance in cellular automata
Janko Gravner, Alexander E. Holroyd
Ann. Probab. 43(4): 1731-1776 (July 2015). DOI: 10.1214/14-AOP918


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. "Percolation and disorder-resistance in cellular automata." Ann. Probab. 43 (4) 1731 - 1776, July 2015.


Received: 1 May 2013; Revised: 1 January 2014; Published: July 2015
First available in Project Euclid: 3 June 2015

zbMATH: 1322.60214
MathSciNet: MR3353814
Digital Object Identifier: 10.1214/14-AOP918

Primary: 37B15 , 60K35

Keywords: Additivity , cellular automaton , ether , percolation , quasireplicator , replicator

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 4 • July 2015
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