On the distribution of the nodal sets of random spherical harmonics

We study the volume of the nodal set of eigenfunctions of the Laplacian on the m-dimensional sphere. It is well known that the eigenspaces corresponding to E(n) =n(n+m-1) are the spaces epsilon(n) of spherical harmonics of degree n of dimension N. We use the multiplicity of the eigenvalues to endow epsilon(n) with the Gaussian probability measure and study the distribution of the m-dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to root E(n). One of our main results is bounding the variance of the volume to be O(En/root N). In addition to the volume of the nodal set, we study its Leray measure. We find that its expected value is n independent. We are able to determine that the asymptotic form of the variance is (const)/N. (C) 2009 American Institute of Physics. [DOI: 10.1063/1.3056589]