Functional Convergence of Berry’s Nodal Lengths: Approximate Tightness and Total Disorder

We consider Berry's random planar wave model (1977), and prove spatial functional limit theorems - in the high-energy for discretized truncated versions of field obtained by restricting its nodal length to rectangular domains. Our analysis is crucially based on a detailed study projection lengths onto so-called second Wiener chaos, whose fluctuations are given Gaussian total disorder indexed polygonal curves. Such an exact characterization then combined with moment estimates suprema stationary fields, tightness criterion Davydov Zikitis (2005).