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A reduction principle for the critical values of random spherical harmonics

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  • Cammarota, Valentina
  • Marinucci, Domenico

Abstract

We study here the random fluctuations in the number of critical points with values in an interval I⊂R for Gaussian spherical eigenfunctions fℓ, in the high energy regime where ℓ→∞. We show that these fluctuations are asymptotically equivalent to the centred L2-norm of fℓ times the integral of a (simple and fully explicit) function over the interval under consideration. We discuss also the relationships between these results and the asymptotic behaviour of other geometric functionals on the excursion sets of random spherical harmonics.

Suggested Citation

  • Cammarota, Valentina & Marinucci, Domenico, 2020. "A reduction principle for the critical values of random spherical harmonics," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2433-2470.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2433-2470
    DOI: 10.1016/j.spa.2019.07.006
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    References listed on IDEAS

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    1. Cammarota, V. & Wigman, I., 2017. "Fluctuations of the total number of critical points of random spherical harmonics," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3825-3869.
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    Cited by:

    1. Valentina Cammarota & Domenico Marinucci, 2022. "On the Correlation of Critical Points and Angular Trispectrum for Random Spherical Harmonics," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2269-2303, December.
    2. Elena Di Bernardino & Céline Duval, 2022. "Statistics for Gaussian random fields with unknown location and scale using Lipschitz‐Killing curvatures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 143-184, March.
    3. Vidotto, Anna, 2021. "A note on the reduction principle for the nodal length of planar random waves," Statistics & Probability Letters, Elsevier, vol. 174(C).

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    1. Valentina Cammarota & Domenico Marinucci, 2022. "On the Correlation of Critical Points and Angular Trispectrum for Random Spherical Harmonics," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2269-2303, December.
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