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

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Zeev Rudnick, Igor Wigman

Original languageEnglish
Pages (from-to)109-130
Number of pages22
JournalAnnales Henri Poincare
Volume9
Issue number1
DOIs
PublishedFeb 2008

King's Authors

Abstract

We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4 pi(2) E with growing multiplicity N -> infinity, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is const root E. Our main result is that the variance of the volume normalized by root E is bounded by O(1/root N), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.

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