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Planck-scale distribution of nodal length of arithmetic random waves

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Abstract

We study the nodal length of random toral Laplace eigenfunctions (“arithmetic random waves”) restricted to decreasing domains (“shrinking balls”), all the way down to Planck scale. We find that, up to a natural scaling, for “generic” energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use of bounds for the so-called spectral quasi-correlations, i.e., unusually small sums of lattice points lying on the same circle.

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Author notes

  1. Jacques Benatar

    Present address: Current address School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

Authors and Affiliations

  1. Department of Mathematics, King’s College London, London, WC2R 2LS, UK

    Jacques Benatar

  2. Department of Mathematics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Roma, Italy

    Domenico Marinucci

  3. Department of Mathematics, King’s College London, London, WC2R 2LS, UK

    Igor Wigman

Authors
  1. Jacques Benatar
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  2. Domenico Marinucci
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  3. Igor Wigman
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Correspondence to Igor Wigman.

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Benatar, J., Marinucci, D. & Wigman, I. Planck-scale distribution of nodal length of arithmetic random waves. JAMA 141, 707–749 (2020). https://doi.org/10.1007/s11854-020-0114-7

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  • DOI: https://doi.org/10.1007/s11854-020-0114-7

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