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arXiv · 0805.2768

On the distribution of the nodal sets of random spherical harmonics

Igor Wigman
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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 $\frac{1}{\sqrt{2π}}$. We are able to determine that the asymptotic form of the variance is $\frac{const}{\eigspcdim}$.

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Igor Wigman. 2008-11-13. On the distribution of the nodal sets of random spherical harmonics. https://doi.org/10.1063/1.3056589

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