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

TL;DR

This paper establishes functional limit theorems for Berry's random wave model's nodal lengths in high-energy limits, revealing Gaussian total disorder behavior and advancing understanding of nodal set fluctuations.

Contribution

It provides the first detailed analysis of the high-energy asymptotics of nodal lengths, including their convergence to a Gaussian total disorder field, using Wiener chaos projections and Gaussian process techniques.

Findings

Nodal lengths converge to a Gaussian total disorder field

High-energy fluctuations are characterized by second Wiener chaos projections

Established tightness and functional convergence for discretized nodal lengths

Abstract

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