In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as ${\displaystyle \mathbb {R} ^{n}}$). That is, it is a function ${\displaystyle f(x)}$ that takes on a random value at each point ${\displaystyle x\in \mathbb {R} ^{n}}$(or some other domain). It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold.

Given a probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$, an X-valued random field is a collection of X-valued random variables indexed by elements in a topological space T. That is, a random field F is a collection

where each ${\displaystyle F_{t}}$ is an X-valued random variable.

In its discrete version, a random field is a list of random numbers whose indices are identified with a discrete set of points in a space (for example, n-dimensional Euclidean space). Suppose there are four random variables, ${\displaystyle X_{1}}$, ${\displaystyle X_{2}}$, ${\displaystyle X_{3}}$, and ${\displaystyle X_{4}}$, located in a 2D grid at (0,0), (0,2), (2,2), and (2,0), respectively. Suppose each random variable can take on the value of −1 or 1, and the probability of each random variable's value depends on its immediately adjacent neighbours. This is a simple example of a discrete random field.

More generally, the values each ${\displaystyle X_{i}}$ can take on might be defined over a continuous domain. In larger grids, it can also be useful to think of the random field as a "function valued" random variable as described above. In quantum field theory the notion is generalized to a random functional, one that takes on random values over a space of functions .

Several kinds of random fields exist, among them the Markov random field (MRF), Gibbs random field, conditional random field (CRF), and Gaussian random field. In 1974, Julian Besag proposed an approximation method relying on the relation between MRFs and Gibbs RFs.^{[citation needed]}

An MRF exhibits the Markov property

for each choice of values ${\displaystyle (x_{j})_{j}}$. Here each ${\displaystyle \partial _{i}}$ is the set of neighbors of ${\displaystyle i}$. In other words, the probability that a random variable assumes a value depends on its immediate neighboring random variables. The probability of a random variable in an MRF^{[clarification needed]} is given by