Generation of random fields defined on a cluster of points when the random field has a given site-to-site correlation (Q1077148)

It is well known that a stationary random field \(M\) on the lattice of points \(\underline 1\) in \(Z^n\) can be represented as \(M(\underline 1) = \int \exp \{i\underline1 \cdot \underline q\}\,d\zeta (\underline q),\) where \(\zeta\) is a process on \(T^n = ((0,2\pi])^n\) with orthogonal increments. The covariance function is given by \[ \mathrm{Cov}(M(\underline1),M({\underline1}')) = \int \exp \{i(\underline1- {\underline1}') \cdot \underline q\} F (d\underline q). \tag{1} \] The author applies these basic results to simulate on a finite lattice a normal random field (possibly transformed) with given correlation function only depending on the separation of the lattice points. The torus \(T^n\) is replaced by a finite set and the method is approximated in the sense that (1) in practice is inverted by a trial and error method. Physical applications are simulation of exchange fields and AB-alloys.