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Communications in Mathematical Physics

, Volume 359, Issue 3, pp 869–913 | Cite as

Discretisation Schemes for Level Sets of Planar Gaussian Fields

  • D. Beliaev
  • S. Muirhead
Open Access
Article

Abstract

Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic compact manifolds. Our work is motivated by questions about the geometry of such random functions, in particular relating to the structure of their nodal and level sets. We study four discretisation schemes that extract information about level sets of planar Gaussian fields. Each scheme recovers information up to a different level of precision, and each requires a maximum mesh-size in order to be valid with high probability. The first two schemes are generalisations and enhancements of similar schemes that have appeared in the literature (Beffara and Gayet in Publ Math IHES, 2017.  https://doi.org/10.1007/s10240-017-0093-0; Mischaikow and Wanner in Ann Appl Probab 17:980–1018, 2007); these give complete topological information about the level sets on either a local or global scale. As an application, we improve the results in Beffara and Gayet (2017) on Russo–Seymour–Welsh estimates for the nodal set of positively-correlated planar Gaussian fields. The third and fourth schemes are, to the best of our knowledge, completely new. The third scheme is specific to the nodal set of the random plane wave, and provides global topological information about the nodal set up to ‘visible ambiguities’. The fourth scheme gives a way to approximate the mean number of excursion domains of planar Gaussian fields.

References

  1. 1.
    Adler, R.J.: The Geometry of Random Fields. Classics in Applied Mathematics. SIAM, Philadelphia (2010)Google Scholar
  2. 2.
    Alexander K.S.: Boundedness of level lines for two-dimensional random fields. Ann. Probab. 24, 1653–1674 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Azaïs J., Wschebor M.: Level Sets and Extrema of Random Processes and Fields. Wiley, (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Beffara, V., Gayet, D.: Percolation of random nodal lines. Publ. Math. IHES (2017)  https://doi.org/10.1007/s10240-017-0093-0
  5. 5.
    Beliaev D., Kereta Z.: On the Bogomolny–Schmit conjecture. J. Phys. A Math. Theor. 46(45), 455003 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berry M.V.: Regular and irregular semiclassical wavefunctions. J. Phys. A Math. Gen. 10(12), 2083 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bogomolny E., Schmit C.: Percolation model for nodal domains of chaotic wave functions. Phys. Rev. Lett. 88, 114102 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    Cheng, D., Schwartzman, A.: Expected number and height distribution of critical points of smooth isotropic Gaussian random fields. arXiv:1511.06835 (to appear in Bernoulli) (2015)
  9. 9.
    Fyodorov Y.V.: Complexity of random energy landscapes, glass transitions and absolute value of spectral determinant of random matrices. Phys. Rev. Lett. 92, 240601 (2004)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Konrad, K.: Asymptotic statistics of nodal domains of quantum chaotic billiards in the semiclassical limit. Senior thesis, Dartmouth College (2012)Google Scholar
  11. 11.
    Mischaikow K., Wanner T.: Probabilistic validation of homology computations for nodal domains. Ann. Appl. Probab. 17, 980–1018 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nazarov F., Sodin M.: On the number of nodal domains of random spherical harmonics. Am. J. Math. 131(5), 1337–1357 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nazarov F., Sodin M.: Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions. J. Math. Phys. Anal. Geom. 12(3), 205–278 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Sun X.P.: Conditionally positive definite functions and their application to multivariate interpolations. J. Approx. Theory 74(2), 159–180 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tassion V.: Crossing probabilities for Voronoi percolation. Ann. Probab. 44(5), 3385–3398 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wendland H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005)Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.King’s College LondonLondonUK

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