Skip to main content
SpringerLink
Log in

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

“Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincaré 9(1):109–130, 2008; Krishnapur et al. in Annals of Mathematics (2) 177(2):699–737, 2013). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener–Itô chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series. Washington, DC (1964).

  2. Berry M.V.: Statistics of nodal lines and points in chaotic quantum billiards: Perimeter corrections, fluctuations, curvature. Journal of Physics A, 35, 3025–3038 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. V. Cammarota, D. Marinucci, and I. Wigman. Fluctuations of the Euler-Poncaré characteristic for random spherical harmonics. In: Proceedings of the American Mathematical Society (2015) (in press). arXiv:1504.01868.

  4. Cilleruelo J.: The distribution of the lattice points on circles. Journal of Number Theory, 43(2), 198–202 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Donnelly H., Fefferman C.: Nodal sets of eigenfunctions on Riemannian manifolds. Inventiones Mathematicae, 93, 161–183 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dudley R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  7. P. Erdos and R.R. Hall. On the angular distribution of Gaussian integers with fixed norm. Discrete Mathematics, (1–3)200 (1999), 87–94 (Paul Erdös memorial collection).

  8. Kratz M.F., León J.R.: Central limit theorems for level functionals of stationary Gaussian processes and fields. Journal of Theoretical Probability, 14(3), 639–672 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Krishnapur M., Kurlberg P., Wigman I.: Nodal length fluctuations for arithmetic random waves. Annals of Mathematics (2), 177(2), 699–737 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Kurlberg and I. Wigman. On probability measures arising from lattice points on circles. Mathematische Annalen (2015) (in press). arXiv:1501.01995.

  11. E. Landau. Uber die Einteilung der positiven Zahlen nach vier Klassen nach der Mindestzahl der zu ihrer addition Zusammensetzung erforderlichen Quadrate. In: Archiv der Mathematik und Physics, III (1908).

  12. Marinucci D., Rossi M.: Stein–Malliavin approximations for nonlinear functionals of random eigenfunctions on S d. Journal of Functional Analysis, 268(8), 2379–2420 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Marinucci and I. Wigman. On the excursion sets of spherical Gaussian eigenfunctions. Journal of Mathematical Physics, 52 (2011), 093301. arXiv:1009.4367.

  14. I. Nourdin and G. Peccati. Normal Approximations with Malliavin Calculus. From Stein’s Method to Universality. Cambridge University Press, Cambridge (2012).

  15. Nourdin I., Rosinski J.: Asymptotic independence of multiple Wiener–Itô integrals and the resulting limit laws. Annals of Probability, 42(2), 497–526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Peccati G., Taqqu M.S.: Wiener Chaos: Moments, Cumulants and Diagrams. Springer, Berlin (2010)

    MATH  Google Scholar 

  17. M. Rossi. The Geometry of Spherical Random Fields. Ph.D. thesis, University of Rome Tor Vergata (2015). arXiv:1603.07575.

  18. Rudnick Z., Wigman I.: On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Annales de l’Insitute Henri Poincaré 9(1), 109–130 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wigman I.: Fluctuations of the nodal length of random spherical harmonics. Communications in Mathematical Physics 298(3), 787–831 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wigman I.: On the nodal lines of random and deterministic Laplace eigenfunctions. Proceedings of the International Conference on Spectral Geometry, Dartmouth College, 84, 285–298 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. S.T. Yau. Survey on partial differential equations in differential geometry. In: Seminar on Differential Geometry. Annals of Mathematics Studies, Vol. 102. Princeton University Press, Princeton (1982), pp. 3–71.

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tor Vergata, Via delle Ricerca Scientifica, 1, 00133, Rome, Italy

    Domenico Marinucci

  2. Unité de Recherche en Mathématiques, Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359, Luxembourg City, Luxembourg

    Giovanni Peccati & Maurizia Rossi

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

    Igor Wigman

Authors
  1. Domenico Marinucci
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giovanni Peccati
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maurizia Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Igor Wigman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Domenico Marinucci.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marinucci, D., Peccati, G., Rossi, M. et al. Non-Universality of Nodal Length Distribution for Arithmetic Random Waves. Geom. Funct. Anal. 26, 926–960 (2016). https://doi.org/10.1007/s00039-016-0376-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-016-0376-5

Keywords and phrases

Mathematics Subject Classification

Search

Navigation