Stochastic cellular automata or probabilistic cellular automata (PCA) or random cellular automata or locally interacting Markov chains[1][2] 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.
As discrete-time Markov process, PCA are defined on a product space
is a finite or infinite graph, like
is a finite space, like for instance
The transition probability has a product form
In general some locality is required
a finite neighbourhood of k. See [4] for a more detailed introduction following the probability theory's point of view.
There is a version of the majority cellular automaton with probabilistic updating rules.
PCA may be used to simulate the Ising model of ferromagnetism in statistical mechanics.
[5] Some categories of models were studied from a statistical mechanics point of view.
There is a strong connection[6] between probabilistic cellular automata and the cellular Potts model in particular when it is implemented in parallel.
The Galves–Löcherbach model is an example of a generalized PCA with a non Markovian aspect.