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On the Correlation of Critical Points and Angular Trispectrum for Random Spherical Harmonics

Author

Listed:
  • Valentina Cammarota

    (Sapienza University of Rome)

  • Domenico Marinucci

    (University of Rome Tor Vergata)

Abstract

We prove a Central Limit Theorem for the critical points of random spherical harmonics, in the high-energy limit. The result is a consequence of a deeper characterization of the total number of critical points, which are shown to be asymptotically fully correlated with the sample trispectrum, i.e. the integral of the fourth Hermite polynomial evaluated on the eigenfunctions themselves. As a consequence, the total number of critical points and the nodal length are fully correlated for random spherical harmonics, in the high-energy limit.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01136-y
    DOI: 10.1007/s10959-021-01136-y
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    References listed on IDEAS

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    1. Maurizia Rossi, 2019. "The Defect of Random Hyperspherical Harmonics," Journal of Theoretical Probability, Springer, vol. 32(4), pages 2135-2165, December.
    2. 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.
    3. 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.
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