Markov Chain Analysis for One-Dimensional Asynchronous Cellular Automata

Alexandru Agapie (), Robin Höns () and Heinz Mühlenbein ()
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Alexandru Agapie: Fraunhofer Institute AIS
Robin Höns: Fraunhofer Institute AIS
Heinz Mühlenbein: Fraunhofer Institute AIS

Methodology and Computing in Applied Probability, 2004, vol. 6, issue 2, 181-201

Abstract: Abstract We consider the finite homogeneous Markov chain induced by a class of one-dimensional asynchronous cellular automata—automata that are allowed to change only one cell per iteration. Furthermore, we confine to totalistic automata, where transitions depend only on the number of 1s in the neighborhood of the current cell. We consider three different cases: (i) size of neighborhood equals length of the automaton; (ii) size of neighborhood two, length of automaton arbitrary; and (iii) size of neighborhood three, length of automaton arbitrary. For each case, the associated Markov chain proves to be ergodic. We derive simple-form stationary distributions, in case (i) by lumping states with respect to the number of 1s in the automaton, and in cases (ii) and (iii) by considering the number of 0–1 borders within the automaton configuration. For the three-neighborhood automaton, we analyze also the Markov chain at the boundary of the parameter domain, and the symmetry of the entropy. Finally, we show that if the local transition rule is exponential, the stationary probability is the Boltzmann distribution of the Ising model.

Keywords: finite homogeneous Markov chains; stationary distribution; lumpability; Boltzmann distribution; Ising model; cellular automata (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1023/B:MCAP.0000017712.55431.96

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