From a theoretical probabilistic point of view, a random field is a family of random variables indexed by a manifold.
Explain:
A stochastic process is a family of random variables {X(t)}t∈T{X(t)}t∈T, where for each tt, X(t)X(t) is a random variable, and tt varies in the set TT called the index set. Theoretically, the definition does not put any restriction on the index set TT, it can be any set. However, when we say stochastic process, 99% of time we are actually thinking tt as the time, hence, TT must be the real line or the set of integers or a part of them.
When this is not the case, most commonly, when TT is actually a higher dimensional Euclidean space or a part of it, or something like that (a “manifold”), then {X(t)}t∈T{X(t)}t∈Tis called a random field. The idea is that since the index is no longer one-dimensional, we can not think it as time, so we think it as space. As a result, we don’t get a “process”, we get a “field”. Thus what we get is a random surface, or a random multivariate function.