  We study the maximum norm behavior of $L^2$-normalized random Fourier cosine
series with a prescribed large wave number. Precise bounds of this type are an
important technical tool in estimates for spinodal decomposition, the
celebrated phase separation phenomenon in metal alloys. We derive rigorous
asymptotic results as the wave number converges to infinity, and shed light on
the behavior of the maximum norm for medium range wave numbers through
numerical simulations. Finally, we develop a simplified model for describing
the magnitude of extremal values of random Neumann Fourier series. The model
describes key features of the development of maxima and can be used to predict
them. This is achieved by decoupling magnitude and sign distribution, where the
latter plays an important role for the study of the size of the maximum norm.
Since we are considering series with Neumann boundary conditions, particular
care has to be placed on understanding the behavior of the random sums at the
boundary.
