{"id":339764,"date":"2016-12-20T15:14:47","date_gmt":"2016-12-20T23:14:47","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=339764"},"modified":"2018-10-16T21:16:46","modified_gmt":"2018-10-17T04:16:46","slug":"percolation-disorder-resistance-cellular-automata","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/percolation-disorder-resistance-cellular-automata\/","title":{"rendered":"Percolation and Disorder-Resistance in Cellular Automata"},"content":{"rendered":"<p>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 <em>L<\/em>, with probability tending to one as <em>L<\/em> \u2192 \u221e, the evolution is a <em>replicator<\/em>. 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.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 \u2192 \u221e, the evolution is a replicator. 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