On the approximate normality of eigenfunctions of the Laplacian

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

$$d_{TV}(f(X),Z)\le\frac{2}{\mu}\mathbb{E}\big\vert\Vert\nabla f(X)\Vert^2-\mathbb{E}\Vert\nabla f(X) \Vert^2\big\vert.$$

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.