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The Leray measure of nodal sets for random eigenfunctions on the torus

Ferenc OraveczZeév RudnickIgor Wigman — 2008

Annales de l’institut Fourier

We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in d 2 dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity 𝒩 . The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant,...

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