Gaussian Random Measures Generated by Berry's Nodal Sets

TL;DR

This paper proves that, as energy increases, the normalized nodal set lengths of Berry's random wave models converge to a Gaussian distribution, revealing their deep connection to random measures and Wiener sheets.

Contribution

It establishes the Gaussian limit for nodal set lengths of Berry's random waves and characterizes their covariance structure, extending to complex-valued waves.

Findings

Normalized nodal lengths converge to Gaussian vectors

Dominant chaotic projections converge to Wiener sheets

Results apply to both real and complex-valued random waves

Abstract

We consider vectors of random variables, obtained by restricting the length of the nodal set of Berry's random wave model to a finite collection of (possibly overlapping) smooth compact subsets of $\mathbb{R}^2$. Our main result shows that, as the energy diverges to infinity and after an adequate normalisation, these random elements converge in distribution to a Gaussian vector, whose covariance structure reproduces that of a homogeneous independently scattered random measure. A by-product of our analysis is that, when restricted to rectangles, the dominant chaotic projection of the nodal length field weakly converges to a standard Wiener sheet, in the Banach space of real-valued continuous mappings over a fixed compact set. An analogous study is performed for complex-valued random waves, in which case the nodal set is a locally finite collection of random points.