Play with dGFF

Change the number of modes \(m\) and \(n\), the higher \(m\) and \(n\) the more the dGFF has a flavour of a continuous GFF.

Mathematica Code:

The following Mathematica code simulate a discrete Gaussian Free Field.
It is the code that appears in Scott Sheffield's Gaussian Free Field for Mathematicians (see https://arxiv.org/abs/math/0312099) and it is based on sampling the discrete Fourier Modes of the dGFF.

Manipulate[ Show[ListPlot3D[ Re[Fourier[ Table[(InverseErf[2 Random[] - 1] + I InverseErf[2 Random[] - 1])* If[j + k == 2, 0, 1/Sqrt[(Sin[(j - 1)*Pi/m]^2 + Sin[(k - 1)*Pi/n]^2)]], {j, m}, {k, n}]]], ColorFunction -> "ValentineTones", LabelStyle -> {FontFamily -> "CMU Serif", 11, GrayLevel[0]}], Boxed -> False, ImageSize -> Large], {{m, 100}, 2, 500}, {{n, 100}, 2, 500}]

An instance of dGFF