Pattern Size in Gaussian Fields from Spinodal Decomposition

We study the two-dimensional snake-like pattern that arises at the onset of phase separation of alloys described by spinodal decomposition in the Cahn--Hilliard model. These are somewhat universal patterns due to an overlay of the most unstable pattern, which are eigenfunctions of the Laplacian all with a similar wave-number. Similar structures appear in other models like reaction-diffusion systems describing animal coats' patterns, hill formation in surface growth, or vegetation patterns in desertification. In order to study the early stages of spinodal decomposition we focus on the linearized equation. Our main result studies random functions given by cosine Fourier series with independent Gaussian coefficients, which 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 $1/\varepsilon$ for a small $0<\varepsilon\ll1$, but also move to infinity for $\varepsilon\to0$. Moreov...