Stochastic cellular automata or probabilistic cellular automata (PCA) or random cellular automata or locally interacting

Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...

s are an important extension of

cellular automaton A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...

. Cellular automata are a discrete-time

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

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 Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...

one, which means the new entities' states are chosen according to some probability distributions. It is a discrete-time

random dynamical system In mathematics, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are characterized by a state space ''S'', a set of maps \Gamma from ''S'' into itself t ...

. From the spatial interaction between the entities, despite the simplicity of the updating rules,

complex behaviour Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

may emerge like

self-organization Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order and disorder, order arises from local interactions between parts of an initially disordered system. The process can be spont ...

. As mathematical object, it may be considered in the framework of

stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...

as an

interacting particle system In probability theory, an interacting particle system (IPS) is a stochastic process (X(t))_ on some configuration space \Omega= S^G given by a site space, a countably-infinite-order graph G and a local state space, a compact metric space S ...

in discrete-time. See for a more detailed introduction.

PCA as Markov stochastic processes

As discrete-time Markov process, PCA are defined on a

product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...

E=\prod_ S_k

(cartesian product) where

G

is a finite or infinite graph, like

\mathbb Z

and where

S_k

is a finite space, like for instance

S_k=\

S_k=\

. The transition probability has a product form

P(d\sigma ,  \eta) = \otimes_ p_k(d\sigma_k ,  \eta)

where

\eta \in E

and

p_k(d\sigma_k ,  \eta)

is a probability distribution on

S_k

. In general some locality is required

p_k(d\sigma_k ,  \eta)=p_k(d\sigma_k ,  \eta_)

where

\eta_=(\eta_j)_

with

a finite neighbourhood of k. See P.-Y. Louis PhD
/ref> for a more detailed introduction following the probability theory's point of view.

Examples of stochastic cellular automaton

Majority cellular automaton

There is a version of the majority cellular automaton with probabilistic updating rules. See the Toom's rule.

Relation to lattice random fields

PCA may be used to simulate the

Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...

ferromagnetism Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

.. Some categories of models were studied from a statistical mechanics point of view.

Cellular Potts model

There is a strong connection between probabilistic cellular automata and the

cellular Potts model In computational biology, a Cellular Potts model (CPM, also known as the Glazier-Graner-Hogeweg model) is a computational model of cells and tissues. It is used to simulate individual and collective cell behavior, tissue morphogenesis and cancer de ...

in particular when it is implemented in parallel.

Non Markovian generalization

The

Galves–Löcherbach model The Galves–Löcherbach model (or GL model) is a mathematical model for a biological neural network, network of neurons with intrinsic stochasticity. In the most general definition, a GL network consists of a countable number of elements (ide ...

is an example of a generalized PCA with a non Markovian aspect.