On the correlation between critical points and critical values for random spherical harmonics
Authors:
V. Cammarota and A. P. Todino
Journal:
Theor. Probability and Math. Statist. 106 (2022), 41-62
MSC (2020):
Primary 60G60, 62M15, 42C10, 33C55, 60D05
DOI:
https://doi.org/10.1090/tpms/1164
Published electronically:
May 16, 2022
MathSciNet review:
4438443
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Additional Information
Abstract: We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval $I \subset \mathbb {R}$. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random $L^2$-norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics.
References
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- Valentina Cammarota, Nodal area distribution for arithmetic random waves, Trans. Amer. Math. Soc. 372 (2019), no. 5, 3539–3564. MR 3988618, DOI 10.1090/tran/7779
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- V. Cammarota, D. Marinucci and M. Rossi, Lipschitz–Killing Curvatures for Arithmetic Random Waves, arXiv:2010.14165
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- Fedor Nazarov and Mikhail Sodin, On the number of nodal domains of random spherical harmonics, Amer. J. Math. 131 (2009), no. 5, 1337–1357. MR 2555843, DOI 10.1353/ajm.0.0070
- Giovanni Peccati and Anna Vidotto, Gaussian random measures generated by Berry’s nodal sets, J. Stat. Phys. 178 (2020), no. 4, 996–1027. MR 4064212, DOI 10.1007/s10955-019-02477-z
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 2, 2nd ed., Gordon & Breach Science Publishers, New York, 1988. Special functions; Translated from the Russian by N. M. Queen. MR 950173
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- Zeév Rudnick, Igor Wigman, and Nadav Yesha, Nodal intersections for random waves on the 3-dimensional torus, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 6, 2455–2484 (English, with English and French summaries). MR 3580177, DOI 10.5802/aif.3068
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- Anna Vidotto, A note on the reduction principle for the nodal length of planar random waves, Statist. Probab. Lett. 174 (2021), Paper No. 109090, 5. MR 4237481, DOI 10.1016/j.spl.2021.109090
- Igor Wigman, Fluctuations of the nodal length of random spherical harmonics, Comm. Math. Phys. 298 (2010), no. 3, 787–831. MR 2670928, DOI 10.1007/s00220-010-1078-8
- Igor Wigman, On the nodal lines of random and deterministic Laplace eigenfunctions, Spectral geometry, Proc. Sympos. Pure Math., vol. 84, Amer. Math. Soc., Providence, RI, 2012, pp. 285–297. MR 2985322, DOI 10.1090/pspum/084/1362
References
- R. J. Adler and J. E. Taylor, Random Fields and Geometry, Springer Monographs in Mathematics, Springer, New York, 2007. MR 2319516
- D. Beliaev, V. Cammarota and I. Wigman, No repulsion between critical points for planar Gaussian random fields, Electron. Commun. Probab. 25 (2020), Paper No. 82, 13 pp. MR 4187723
- M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), no. 12, 2083–2091. MR 489542
- J. Benatar, D. Marinucci and I. Wigman, Planck-scale distribution of nodal length of arithmetic random waves, J. Anal. Math. 141 (2020), no. 2, 707–749. MR 4179775
- J. Buckley and I. Wigman, On the Number of Nodal Domains of Toral Eigenfunctions. Ann. Henri Poincaré 17 (2016), no. 11, 3027–3062. MR 3556515
- V. Cammarota, Nodal Area Distribution for Arithmetic Random Waves, Trans. Amer. Math. Soc. 372 (2019), no. 5, 3539–3564. MR 3988618
- V. Cammarota and D. Marinucci, A reduction principle for the critical values of random spherical harmonics, Stochastic Process. Appl. 130 (2020), no. 4, 2433–2470. MR4074706 MR 4074706
- V. Cammarota and D. Marinucci, On the correlation of critical points and angular trispectrum for random spherical harmonics, Journal of Theoretical Probability (2019), https://doi.org/10.1007/s10959-021-01136-y
- V. Cammarota and D. Marinucci, A Quantitative Central Limit Theorem for the Euler–Poincaré Characteristic of Random Spherical Eigenfunctions, Ann. Probab. 46 (2018), no. 6, 3188–3228. MR 3857854
- V. Cammarota, D. Marinucci and M. Rossi, Lipschitz–Killing Curvatures for Arithmetic Random Waves, arXiv:2010.14165
- V. Cammarota, D. Marinucci and I. Wigman, Fluctuations of the Euler-Poincaré Characteristic for Random Spherical Harmonics, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4759–4775. MR 3544528
- V. Cammarota, D. Marinucci and I. Wigman, On the distribution of the critical values of random spherical harmonics, J. Geom. Anal. 26 (2016), no. 4, 3252–3324. MR 3544960
- V. Cammarota and I. Wigman, Fluctuations of the total number of critical points of random spherical harmonics, Stochastic Process. Appl. 127 (2017), no. 12, 3825–3869. MR 3718098
- E. Estrade and J. R. Leon, A Central Limit Theorem for the Euler Characteristic of a Gaussian Excursion Set, Ann. Probab. 44 (2016), no. 6, 3849–3878. MR 3572325
- R. Feng and R. J. Adler, Critical Radius and Supremum of Random Spherical Harmonics, Ann. Probab. 47 (2019), no. 2, 1162–1184. MR 3916945
- A. Granville and I. Wigman, Planck-scale Mass Equidistribution of Toral Laplace Eigenfunctions. Comm. Math. Phys. 355 (2017), no. 2, 767–802. MR 3681390
- X. Han, Small Scale Equidistribution of Random Eigenbases, Comm. Math. Phys. 349 (2017), no. 1, 425–440. MR 3592754
- M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves, Ann. of Math. (2) 177 (2013), no. 2, 699–737. MR 3010810
- A. Logunov, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Ann. of Math. (2) 187 (2018), no. 1, 221–239. MR 3739231
- D. Marinucci, A Central Limit Theorem and Higher Order Results for the Angular Bispectrum, Probab. Theory Related Fields 141 (2008), no. 3-4, 389–409. MR 2391159
- D. Marinucci, G. Peccati, M. Rossi and I. Wigman, Non-universality of Nodal Length Distribution for Arithmetic Random Waves, Geom. Funct. Anal. 26 (2016), no. 3, 926–960. MR 3540457
- D. Marinucci and M. Rossi, On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics, Ann. Henri Poincaré 22 (2021), no. 1, 275–307. MR 4201595
- D. Marinucci and M. Rossi, Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on $S^{d}$, J. Funct. Anal. 268 (2015), no. 8, 2379–2420. MR 3318653
- D. Marinucci, M. Rossi and I. Wigman, The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics, Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020), no. 1, 374–390. MR 4058991
- D. Marinucci and I. Wigman, On the Area of Excursion Sets of Spherical Gaussian Eigenfunctions, J. Math. Phys. 52 (2011), no. 9, 093301, 21 pp. MR 2867816
- D. Marinucci and I. Wigman, The Defect Variance of Random Spherical Harmonics, Journal of Physics A-Mathematical and Theoretical 44, no. 35, (2011).
- D. Marinucci and I. Wigman, On Nonlinear Functionals of Random Spherical Eigenfunctions, Comm. Math. Phys. 327 (2014), no. 3, 849–872. MR 3192051
- F. Nazarov and M. Sodin, On the Number of Nodal Domains of Random Spherical Harmonics, Amer. J. Math. 131 (2009), no. 5, 1337–1357. MR 2555843
- G. Peccati and A. Vidotto, Gaussian random measures generated by Berry’s nodal sets, J. Stat. Phys. 178 (2020), no. 4, 996–1027. MR 4064212
- A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and series.Vol. 2. Special functions, Second edition, Gordon & Breach Science Publishers, New York, 1986. MR 950173
- Z. Rudnick and I. Wigman, Nodal Intersections for Random Eigenfunctions on the Torus, Amer. J. Math. 138 (2016), no. 6, 1605–1644. MR3595496 MR 3595496
- Z. Rudnick, I. Wigman and N. Yesha, Nodal Intersections for Random Waves on the 3-Dimensional Torus, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 6, 2455–2484. MR 3580177
- A. P. Todino, Nodal Lengths in Shrinking Domains for Random Eigenfunctions on $S^2$, Bernoulli 26 (2020), no. 4, 3081–3110. MR 4140538
- A. P. Todino, A Quantitative Central Limit Theorem for the Excursion Area of Random Spherical Harmonics over Subdomains of $S^2$, Math. Phys. 60 (2019), no. 2, 023505, 33 pp. MR 3916834
- A. Vidotto, A note on the reduction principle for the nodal length of planar random waves, Statist. Probab. Lett. 174 (2021), Paper No. 109090, 5 pp. MR 4237481
- I. Wigman, Fluctuations of the Nodal Length of Random Spherical Harmonics, Comm. Math. Phys. 298 (2010), no. 3, 787–831. MR 2670928
- I. Wigman, On the Nodal Lines of Random and Deterministic Laplace Eigenfunctions, Spectral geometry, 285–297, Proc. Sympos. Pure Math., 84, Amer. Math. Soc., Providence, RI. MR 2985322
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Additional Information
V. Cammarota
Affiliation:
Department of Statistics, Sapienza University of Rome, Piazzale Aldo Moro, 5, 00185 Rome, Italy
MR Author ID:
829478
Email:
valentina.cammarota@uniroma1.it
A. P. Todino
Affiliation:
Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
Email:
anna.todino@polito.it
Keywords:
Critical points,
spherical harmonics,
partial correlation,
Wiener–Chaos expansion
Received by editor(s):
July 31, 2021
Accepted for publication:
October 21, 2021
Published electronically:
May 16, 2022
Additional Notes:
The first author received funding from Sapienza University research project RM120172B80031BE, Geometry of Random Fields. The second author was partially supported by Progetto di Eccellenza, Dipartimento di Scienze Matematiche, Politecnico di Torino, CUP: E11G18000350001 and by GNAMPA-INdAM (project: Stime asintotiche: principi di invarianza e grandi deviazioni).
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv
