# Gaussian random field

In statistics, a **Gaussian random field** (GRF) is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process. 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]}

## Construction[edit]

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.

Suppose *f*(*x*) is the value of a GRF at a point *x* in some *D*-dimensional space. If we make a vector of the values of *f* at *N* points, *x*_{1}, ..., *x*_{N}, in the *D*-dimensional space, then the vector (*f*(*x*_{1}), ..., *f*(*x*_{N})) will always be distributed as a multivariate Gaussian.

## See also[edit]

## References[edit]

**^**Peacock, John (1999).*Cosmological Physics*. Cambridge University Press. p. 342. ISBN 0-521-41072-X – via Google Books.

## External links[edit]

- For details on the generation of Gaussian random fields using Matlab, see circulant embedding method for Gaussian random field.