On the distribution of the nodal sets of random spherical harmonics

TL;DR

This paper investigates the distribution of the volume of nodal sets of random spherical harmonics on spheres, providing bounds on variance and analyzing the Leray measure's behavior as the eigenvalue grows.

Contribution

It introduces bounds on the variance of the nodal set volume and characterizes the asymptotic variance of the Leray measure for random spherical harmonics.

Findings

Expected volume proportional to 

Variance of volume bounded by O(/)

Asymptotic variance of Leray measure is proportional to 1/

Abstract

We study the length of the nodal set of eigenfunctions of the Laplacian on the $\spheredim$-dimensional sphere. It is well known that the eigenspaces corresponding to $\eigval=n(n+\spheredim-1)$ are the spaces $\eigspc$ of spherical harmonics of degree $n$, of dimension $\eigspcdim$. We use the multiplicity of the eigenvalues to endow $\eigspc$ with the Gaussian probability measure and study the distribution of the $\spheredim$-dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to $\sqrt{\eigval}$. One of our main results is bounding the variance of the volume to be $O(\frac{\eigval}{\sqrt{\eigspcdim}})$. In addition to the volume of the nodal set, we study its Leray measure. For every $n$, the expected value of the Leray measure is $\frac{1}{\sqrt{2\pi}}$. We are able to determine that the asymptotic form of the variance is $\frac{const}{\eigspcdim}$.