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/epsilon for a small 0 < epsilon << 1, but also move to infinity for epsilon -> 0. Moreover, the model under consideration is as a random field at a fixed time 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 epsilon with a precisely given constant.