Gaussian random field

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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[]

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, x₁, ..., x_N, in the D-dimensional space, then the vector (f(x₁), ..., f(x_N)) will always be distributed as a multivariate Gaussian.

References[]

^ Peacock, John. Cosmological Physics, Cambridge University Press, 1999. ISBN 0-521-41072-X^{[page needed]}

External links[]

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

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