Stochastic cellular automaton
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Stochastic cellular automata or 'probabilistic cellular automata' (PCA) or 'random cellular automata' or locally interacting Markov chains are an important extension of cellular automaton. Cellular automata are a discrete-time dynamical system of interacting entities, whose state is discrete.
The state of the collection of entities is updated at each discrete time according to some simple homogeneous rule. All entities' states are updated in parallel or synchronously. Stochastic Cellular Automata are CA whose updating rule is a stochastic one, which means the new entities' states are chosen according to some probability distributions. It is a discrete-time random dynamical system. From the spatial interaction between the entities, despite the simplicity of the updating rules, complex behaviour may emerge like self-organization. As mathematical object, it may be considered in the framework of stochastic processes as an interacting particle system in discrete-time.
PCA as Markov stochastic processes
As discrete-time Markov process, PCA are defined on a product space (cartesian product) where is a finite or infinite graph, like and where is a finite space, like for instance or . The transition probability has a product form where and is a probability distribution on . In general some locality is required where with a finite neighbourhood of k. See  for a more detailed introduction following the probability theory's point of view.
Examples of stochastic cellular automaton
Majority cellular automaton
Relation to random fields
Cellular Potts model
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