On the approximate normality of eigenfunctions of the Laplacian

TL;DR

This paper establishes bounds on how close eigenfunctions of the Laplacian on compact manifolds are to Gaussian distributions, with applications to spherical harmonics and eigenfunctions on flat tori.

Contribution

It provides a new bound on the total variation distance between eigenfunction distributions and Gaussian distributions, extending understanding of their approximate normality.

Findings

Eigenfunctions of the Laplacian can be close to Gaussian in distribution.

Explicit bounds depend on eigenvalues and gradient norms.

Applications include spherical harmonics and eigenfunctions on flat tori.

Abstract

The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and eigenvalue $-\mu$, then $$d_{TV}(f(X),Z)\le\frac{2}{\mu}\E\big|\|\nabla f(X)\|^2-\E\|\nabla f(X) \|^2\big|.$$ This result is applied to construct specific examples of spherical harmonics of arbitrary (odd) degree which are close to Gaussian in distribution. A second application is given to random linear combinations of eigenfunctions on flat tori.