  We study the two-dimensional snake-like pattern that arises in phase
separation of alloys described by spinodal decomposition in the Cahn-Hilliard
model. These are somewhat universal patterns due to an overlay of
eigenfunctions of the Laplacian with a similar wave-number. Similar structures
appear in other models like reaction-diffusion systems describing animal coats'
patterns or vegetation patterns in desertification.
  Our main result studies random functions given by cosine Fourier series with
independent Gaussian coefficients, that dominate the dynamics in the
Cahn-Hilliard model. This is not a cosine process, as the sum is taken over
domains in Fourier space that not only grow and scale with a parameter of order
$1/\varepsilon$, but also move to infinity. Moreover, the model under
consideration is neither stationary nor isotropic.
  To study the pattern size of nodal domains we consider the density of zeros
on any straight line through the spatial domain. Using a theorem by Edelman and
Kostlan and weighted ergodic theorems that ensure the convergence of the moving
sums, we show that the average distance of zeros is asymptotically of order
$\varepsilon$ with a precisely given constant.

Probability;Analysis of PDEs