Skip to main content
SpringerLink
Log in

Fluctuations of the Nodal Length of Random Spherical Harmonics

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

An Erratum to this article was published on 10 November 2011

Abstract

Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics.

It is known that the expected length is of order n. It is natural to conjecture that the variance should be of order n, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order log n. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for “generic” linear statistics of the nodal lines.

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. Andrews, G.E., Askey, R., Roy, R.: Special functions Encyclopedia of Mathematics and its Applications 71. Cambridge: Cambridge University Press, 1999

  2. Bérard, P.: Volume des ensembles nodaux des fonctions propres du laplacien. Bony-Sjostrand-Meyer seminar, 1984–1985, Exp. No. 14, 10 pp., École Polytech., Palaiseau, 1985

  3. Berry M.V.: Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A 35, 3025–3038 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Bleher P., Shiffman B., Zelditch S.: Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142(2), 351–395 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of zeros on symplectic manifolds. In: Random Matrix Models and their Applications, Math. Sci. Res. Inst. Publ. 40, Cambridge: Cambridge Univ. Press, 2001, pp. 31–69

  6. Brüning J.: Über Knoten Eigenfunktionen des Laplace-Beltrami Operators. Math. Z. 158, 15–21 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brüning J., Gromes D.: Über die Länge der Knotenlinien schwingender Membranen. Math. Z. 124, 79–82 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheng S.Y.: Eigenfunctions and nodal sets. Comm. Math. Helv. 51, 43–55 (1976)

    Article  MATH  Google Scholar 

  9. Cramér, H., Leadbetter, M.R.: Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications. Reprint of the 1967 original. Mineola, NY: Dover Publications, Inc., 2004

  10. Donnelly H., Fefferman C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93, 161–183 (1988)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Federer H.: Curvature measures. Trans. Amer. Math. Soc. 93, 418–491 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  12. Forrester P.J., Honner G.: Exact statistical properties of the zeros of complex random polynomials. J. Phys. A 32(16), 2961–2981 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, 80. Basel: Birkhäuser Verlag, 1984

  14. Krishnapur, M., Wigman, I.: Fluctuations of the Nodal Length of Random Eigenfunctions of the Laplacian on the Torus. In preparation

  15. Neuheisel, J.: The Asymptotic Distribution of Nodal Sets on Spheres. Johns Hopkins Ph.D. thesis, 2000

  16. Rudnick Z., Wigman I.: On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Ann. Henri Poincaré 9(1), 109–130 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Szego, G.: Orthogonal Polynomials. Fourth edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. Providence, RI: Amer. Math. Soc., 1975

  18. Shiffman B., Zelditch S.: Number variance of random zeros on complex manifolds. Geom. Funct. Anal. 18(4), 1422–1475 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Shiffman, B., Zelditch, S.: Number variance of random zeros on complex manifolds, II: smooth statistics. available online http://arxiv.org/abs/0711.1840v1[math.CV], 2007

  20. Sodin M., Tsirelson B.: Random complex zeroes. I. Asymptotic normality. Israel J. Math. 144, 125–149 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Toth J.A., Wigman I.: Counting open nodal lines of random waves on planar domains. IMRN 2009, 3337–3365 (2009)

    MATH  MathSciNet  Google Scholar 

  22. Toth, J.A., Wigman, I.: Universality of length distribution of nodal lines of random waves on generic surfaces. In progress (2009)

  23. Wigman I.: On the distribution of the nodal sets of random spherical harmonics. J. Math. Phys 50, 013521 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  24. Wigman, I.: Volume fluctuations of the nodal sets of random Gaussian subordinated spherical harmonics. In preparation

  25. Yau, S.T.: Survey on partial differential equations in differential geometry. In: Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton, NJ: Princeton Univ. Press, 1982, pp. 3–71

  26. Yau, S.T.: Open problems in geometry. In: Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math. 54, Part 1, Providence, RI: Amer. Math. Soc., 1993, pp. 1–28

  27. Zelditch, S.: Real and Complex zeros of Riemannian Random Waves. To appear in the Proceedings of the Conference, “Spectral Analysis in Geometry and Number Theory on the occasion of Toshikazu Sunada’s 60th birthday”, to appear in the Contemp. Math. Series, available online http://arxiv.org/abs/0803.4334v1[math.Sp], 2008

Download references

Author information

Author notes

  1. Igor Wigman

    Present address: Institutionen för Matematik, Kungliga Tekniska Högskolan (KTH), Lindstedtsvägen 25, 10044, Stockholm, Sweden

Authors and Affiliations

  1. Centre de Recherches Mathématiques (CRM), Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, H3C 3J7, Canada

    Igor Wigman

Authors
  1. Igor Wigman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor Wigman.

Additional information

Communicated by S. Zelditch

The author is supported by a CRM ISM fellowship, Montréal and the Knut and Alice Wallenberg Foundation, grant KAW.2005.0098.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00220-011-1367-x

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wigman, I. Fluctuations of the Nodal Length of Random Spherical Harmonics. Commun. Math. Phys. 298, 787–831 (2010). https://doi.org/10.1007/s00220-010-1078-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-010-1078-8

Keywords

Search

Navigation