Probabilistic Estimates of the Maximum Norm of Random Neumann Fourier Series

TL;DR

This paper investigates the maximum norm behavior of random Fourier cosine series with Neumann boundary conditions, providing asymptotic bounds, numerical insights, and a simplified predictive model relevant for phase separation phenomena.

Contribution

It introduces rigorous asymptotic bounds, numerical analysis, and a simplified model for extremal values of random Neumann Fourier series, advancing understanding of their maximum norms.

Findings

Asymptotic bounds for maximum norm as wave number increases

Numerical simulations revealing medium-range wave number behavior

A simplified model predicting extremal value magnitudes

Abstract

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.