The spherical random waves model
The spherical random waves model is a random field defined on the sphere that depends on an integer λ, the frequency parameter. There are several possible definitions of this model, which are equivalent up to a (random) scaling factor. One can define it as a rotationally invariant distribution on the eigenspace of the Laplace operator on the sphere, associated with the eigenvalue λ(λ+1). Alternatively, it can be described as a random linear combination of spherical harmonics of degree λ with i.i.d. Gaussian coefficients.
As the frequency parameter λ goes to infinity, the model exhibit a local scaling limit (at scale 1/λ) given by the model of planar random waves. It is a Gaussian process on the plane whose spectral measure is the uniform probability distribution on the unit circle. Despite this local behavior, the striginess observed on the right picture (simulation of a spherical random wave of frequency λ= 500) suggests that the model of spherical random waves may exhibit long-range dependencies.
I wrote some MATLAB codes, inspired by the simulations of Alex Barnett, to simulate various quantities related to the spherical random waves model. The source codes can be found here (coming soon). The videos have been processed with Blender and encoded in WebM format with an alpha channel (for transparency). There may be some issues displaying them properly in older web browsers.
Frequency increment
As the frequency increases, a generic spherical random wave becomes increasingly oscillatory. The following videos show the heatmap of a spherical random wave along with its positive and negative values as the frequency parameter increases.
Band enlargement
The large band spherical random waves model is defined in a similar manner. Instead of taking a Gaussian combination of spherical harmonics of degree λ, this model is defined by a Gaussian combination of spherical harmonics of degree lower than λ. The resulting model is less oscillatory and has shorter range dependencies. The following videos show a smooth transition between the large band model and the true spherical random wave model.
Moving wave
Coming soon.
Level increment
Coming soon.