Gaussian random field
Encyclopedia
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
.
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 densisty through a Fourier transformation.
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.
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, ..., xN, in the D-dimensional space, then the vector (f(x1), ..., f(xN)) will always be distributed as a multivariate Gaussian.
An important special case of a GRF is the Gaussian free field
.
Random field
A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....
involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...
.
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
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
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 densisty through a Fourier transformation.
With regard to applications of GRFs, the initial conditions of physical cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
generated by quantum mechanical fluctuations
Quantum fluctuation
In quantum physics, a quantum fluctuation is the temporary change in the amount of energy in a point in space, arising from Werner Heisenberg's uncertainty principle.According to one formulation of the principle,energy and time can be related by the relation...
during cosmic inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
are thought to be a GRF with a nearly scale invariant spectrum.
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, ..., xN, in the D-dimensional space, then the vector (f(x1), ..., f(xN)) will always be distributed as a multivariate Gaussian.
An important special case of a GRF is the Gaussian free field
Gaussian free field
In probability theory and statistical mechanics, the Gaussian free field is a Gaussian random field, a central model of random surfaces . A nice survey is ....
.