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

Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...

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, tess ...

. Cellular automata are a discrete-time

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

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 (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...

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

random dynamical system In the mathematical field of dynamical systems, 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 ...

. 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 arises from local interactions between parts of an initially disordered system. The process can be spontaneous when suffic ...

. 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. Stochastic processes are widely used as mathematical models of systems and phenomena that a ...

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 countable-infinite graph G and a local state space, a compact metric space S . Mor ...

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-see ...

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 Toom's rule is a 2-dimensional cellular automaton model created by Andrei Toom Andrei Leonovich Toom (in Russian: Андрей Леонович Тоом), also known as André Toom, (1942 Tashkent, Soviet Union - 2022 Queens, New York City) was ...

Relation to lattice random fields

PCA may be used to simulate the

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

ferromagnetism Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials a ...

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