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The defect of toral Laplace eigenfunctions and arithmetic random waves
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-4734-5092
Kings Coll London, Dept Math, London WC2R 2LS, England..
Univ Haifa, Dept Math, IL-3498838 Haifa, Israel..
2021 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 34, no 9, p. 6651-6684Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
IOP PUBLISHING LTD , 2021. Vol. 34, no 9, p. 6651-6684
Keywords [en]
Laplace eigenfunctions, standard torus, signed measure, defect distribution
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-300824DOI: 10.1088/1361-6544/ac17c8ISI: 000686332100001Scopus ID: 2-s2.0-85114411845OAI: oai:DiVA.org:kth-300824DiVA, id: diva2:1598691
Note

QC 20210929

Available from: 2021-09-29 Created: 2021-09-29 Last updated: 2022-06-25Bibliographically approved

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Kurlberg, Pär

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CiteExportLink to record
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Citation style
  • apa
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  • de-DE
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