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

Ferenc Oravecz*, Zeév Rudnick, Igor Wigman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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, equal to l/√2π. Our main result is that the variance of Leray measure is asymptotically 1/4pi;N, as N → ∞, at least in dimensions d = 2 and d ≥ 5.

Original languageEnglish
Pages (from-to)299-335+IX
JournalAnnales de l'Institut Fourier
Issue number1
StatePublished - 2008


  • Eigenfunctions of the laplacian
  • Leray measure
  • Nodal sets
  • Trigonometric polynomials


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