An important special case of a GRF is the Gaussian free field.
With regard to applications of GRFs, the initial conditions of physical cosmology generated by quantum mechanical fluctuations during cosmic inflation are thought to be a GRF with a nearly scale invariant spectrum.
[1] One way of constructing a GRF is by assuming that the field is the sum of a large number of plane, cylindrical or spherical waves with uniformly distributed random phase.
Where applicable, the central limit theorem dictates that at any point, the sum of these individual plane-wave contributions will exhibit a Gaussian distribution.
This type of GRF is completely described by its power spectral density, and hence, through the Wiener–Khinchin theorem, by its two-point autocorrelation function, which is related to the power spectral density through a Fourier transformation.