Gaussian Random Measures Generated by Berry's Nodal Sets

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.

arXiv comment33 pages

References in corpus (7)

• Fluctuations of the nodal length of random spherical harmonics, erratum
• On the volume of nodal sets for eigenfunctions of the Laplacian on the torus
• On the distribution of the critical values of random spherical harmonics
• Phase Singularities in Complex Arithmetic Random Waves
• Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics
• Asymptotic distribution of nodal intersections for arithmetic random waves
• A Quantitative Central Limit Theorem for the Excursion Area of Random Spherical Harmonics over Subdomains of ${\mathbb{S}}^2$

Cited by in corpus (2)

• Nodal Statistics of Planar Random Waves
• Functional Convergence of Berry's Nodal Lengths: Approximate Tightness and Total Disorder