On the distribution of the nodal sets of random spherical harmonics

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 $\frac{const}{\eigspcdim}$.

47 pages, accepted for publication in the Journal of Mathematical Physics. Lemmas 2.5, 2.11 were proved for any dimension, some other, suggested by the referee, modifications and corrections, were made

References in corpus (1)

• On the volume of nodal sets for eigenfunctions of the Laplacian on the torus