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On the distribution of the nodal sets of random spherical harmonics

Wigman, Igor 2009. On the distribution of the nodal sets of random spherical harmonics. Journal of Mathematical Physics 50 (1) , 013521. 10.1063/1.3056589

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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 En = n(n+m−1) are the spaces En of spherical harmonics of degree n of dimension N. We use the multiplicity of the eigenvalues to endow En 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 math. One of our main results is bounding the variance of the volume to be O(En/math). 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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Additional Information: 44 page article.
Publisher: American Institute of Physics
ISSN: 0022-2488
Last Modified: 26 Jun 2019 01:57

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