On 3-dimensional Berry's model

TL;DR

This paper investigates the geometric properties of 3D Berry's random wave models, focusing on the expected length of dislocation lines, their variance, and distribution, including special cases like monochromatic waves and models with non-standard asymptotics.

Contribution

It provides the first detailed analysis of the dislocation lines in 3D Berry's models, deriving asymptotic variance, distribution, and establishing a central limit theorem under certain conditions.

Findings

Expected length of dislocation lines computed for isotropic and anisotropic cases

Asymptotic variance and distribution derived for the isotropic case

Non-central limit theorem established for a power law model with unusual behavior

Abstract

This work aims to study the dislocation or nodal lines of 3D Berry's random wave model. Their expected length is computed both in the isotropic and anisotropic cases, being them compared. Afterwards, in the isotropic case the asymptotic variance and distribution of the length are obtained as the domain grows to the whole space. Under some integrability condition on the covariance function, a central limit theorem is established. The study includes the Berry's monochromatic random waves, the Bargmann-Fock model and the Black-Body radiation as well as a power law model that exhibits an unusual asymptotic behaviour and yields a non-central limit theorem.