Approximate Normality of High-Energy Hyperspherical Eigenfunctions

TL;DR

This paper rigorously quantifies the Gaussian-like behavior of high-energy eigenfunctions on spheres, providing bounds on their deviation from Gaussianity and exploring implications for non-Gaussian models.

Contribution

It establishes quantitative bounds on the deviation from Gaussianity for hyperspherical eigenfunctions at high energies, advancing the understanding of Berry's heuristic.

Findings

Derived explicit bounds on the distance to Gaussianity for spherical harmonics.

Showed that high-energy eigenfunctions approximate Gaussian behavior under certain conditions.

Discussed applications to non-Gaussian models in related contexts.

Abstract

The Berry heuristic has been a long standing \emph{ansatz} about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see Berry 1977). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace-Beltrami operator on the normalized $d$-dimensional sphere - also known as spherical harmonics). Some applications to non-Gaussian models are also discussed.