Random field

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.^[1]