Simple form of the stationary distribution for 3D cellular automata in a special case

3D cellular automata can be analyzed by means of finite homogeneous Markov chains. If the automaton is allowed to change only one cell per iteration, and the transition probability depends linearly on the number of ones in the neighborhood, the Markov chain has two attractors at all zeroes and all o...

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Bibliographic Details
Published in:Physica A 2010-07, Vol.389 (13), p.2495-2499
Main Author: Agapie, Alexandru
Format: Article
Language:English
Subjects:
Borders
Cellular automata
Chains
Exact solutions
Finite homogeneous Markov chain
Interacting particle system
Ising model
Markov chains
Mathematical analysis
Statistical mechanics
Three dimensional
Voter model
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Summary:3D cellular automata can be analyzed by means of finite homogeneous Markov chains. If the automaton is allowed to change only one cell per iteration, and the transition probability depends linearly on the number of ones in the neighborhood, the Markov chain has two attractors at all zeroes and all ones. Otherwise–and this is the case we tackle–the chain is ergodic, thus allowing for the search of stationary distributions. This proves cumbersome in the general case, still, under detailed balance equation, the stationary distribution can be derived in closed form. The probability of a particular state is found to be exponential in the number of zero–one borders within the configuration.
ISSN:0378-4371
1873-2119
DOI:10.1016/j.physa.2010.03.011