The defect of toral Laplace eigenfunctions and arithmetic random waves

Pär Kurlberg, Igor Wigman, Nadav Yesha

Research output: Contribution to journalArticlepeer-review


We study the defect (or ‘signed area’) distribution of standard toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, either for deterministic eigenfunctions averaged w.r.t. the spatial variable, or in a random Gaussian scenario (‘arithmetic random waves’). In either case we exploit the associated symmetry of the eigenfunctions to show that the expectation (spatial or Gaussian) vanishes. In the deterministic setting, we prove that the variance of the defect of flat eigenfunctions, restricted to balls shrinking above the Planck scale, vanishes for ‘most’ energies. Hence the defect of eigenfunctions restricted to most of the said balls is small. We also construct ‘esoteric’ eigenfunctions with large defect variance, by choosing our eigenfunctions so that to mimic the situation on the hexagonal torus, thus breaking the symmetries associated to the standard torus. In the random Gaussian setting, we establish various upper and lower bounds for the defect variance w.r.t. the Gaussian probability measure. A crucial ingredient in the proof of the lower bound is the use of Schmidt’s subspace theorem.

Original languageEnglish
Article number6651
Pages (from-to)6651-6684
Issue number9
StatePublished - Sep 2021

Bibliographical note

Publisher Copyright:
© 2021 IOP Publishing Ltd & London Mathematical Society.


  • Defect distribution
  • Laplace eigenfunctions
  • Signed measure
  • Standard torus

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics


Dive into the research topics of 'The defect of toral Laplace eigenfunctions and arithmetic random waves'. Together they form a unique fingerprint.

Cite this