Fluctuations of Level Curves for Time-Dependent Spherical Random Fields
Annales Henri Lebesgue, Volume 7 (2024), pp. 583-620.

Metadata

Keywords Sphere-cross-time random fields, Level curves and nodal lines, Berry’s cancellation, Central and non-Central Limit Theorems

Abstract

The investigation of the behaviour for geometric functionals of random fields on manifolds has drawn recently considerable attention. In this paper, we extend this framework by considering fluctuations over time for the level curves of general isotropic Gaussian spherical random fields. We focus on both long and short memory assumptions; in the former case, we show that the fluctuations of u-level curves are dominated by a single component, corresponding to a second-order chaos evaluated on a subset of the multipole components for the random field. We prove the existence of cancellation points where the variance is asymptotically of smaller order; these points do not include the nodal case u=0, in marked contrast with recent results on the high-frequency behaviour of nodal lines for random eigenfunctions with no temporal dependence. In the short memory case, we show that all chaoses contribute in the limit, no cancellation occurs and a Central Limit Theorem can be established by Fourth-Moment Theorems and a Breuer–Major argument.


References

[AT07] Adler, Robert J.; Taylor, Jonathan E. Random Fields and Geometry, Springer Monographs in Mathematics, Springer, 2007 | Zbl

[AW09] Azaïs, Jean-Marc; Wschebor, Mario Level Sets and Extrema of Random Processes and Fields, John Wiley & Sons, 2009 | DOI | Zbl

[Ber02] Berry, M. V. Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature, J. Phys. A. Math. Gen., Volume 35 (2002) no. 13, pp. 3025-3038 | DOI | Zbl

[Ber17] Berg, C. From Schoenberg coefficients to Schoenberg functions, Constr. Approx., Volume 5 (2017) no. 2, pp. 217-241 | DOI | Zbl

[BM83] Breuer, Peter; Major, Peter Central limit theorems for non-linear functionals of Gaussian fields, J. Multivariate Anal., Volume 13 (1983), pp. 425-441 | DOI | Zbl

[Chr17] Christakos, G. Spatiotemporal Random Fields: Theory and Applications, Theory and applications, Elsevier, 2017

[CM18] Cammarota, Valentina; Marinucci, Domenico A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions, Ann. Probab., Volume 46 (2018), pp. 3188-3288 | DOI | Zbl

[CM20] Cammarota, Valentina; Marinucci, Domenico A reduction principle for the critical values of random spherical harmonics, Stochastic Processes Appl., Volume 130 (2020) no. 4, pp. 2433-2470 | DOI | Zbl

[DM79] Dobrushin, Roland L.; Major, Peter Non-Central limit theorems for non-linear functionals of Gaussian fields, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 50 (1979), pp. 27-52 | DOI | Zbl

[DNPR19] Dalmao, Federico; Nourdin, Ivan; Peccati, Giovanni; Rossi, Maurizia Phase Singularities in Complex Arithmetic Random Waves, Electron. J. Probab., Volume 24 (2019), 71, 45 pages | DOI | Zbl

[EL16] Estrade, Anne; León, José R. A central limit theorem for the Euler characteristic of a Gaussian excursion set, Ann. Probab., Volume 44 (2016) no. 6, pp. 3849-3878 | DOI | Zbl

[KL01] Kratz, Marie F.; León, José R. Central limit theorems for level functionals of stationary Gaussian processes and fields, J. Theor. Probab., Volume 14 (2001) no. 3, pp. 639-672 | DOI | Zbl

[Leo88] Leonenko, Nikolai N. On the accuracy of the normal approximation of functionals of strongly correlated Gaussian random fields, Mat. Zametki, Volume 43 (1988) no. 2, pp. 283-299 | DOI | Zbl

[Leo18] Leonenko, Nikolai N. Estimation of the covariance function of Gaussian isotropic random fields on spheres, related Rosenblatt-type distributions and the cosmic variance problem, Electron. J. Stat., Volume 12 (2018) no. 2, pp. 3114-3146 | DOI | Zbl

[LO13] Leonenko, Nikolai N.; Olenko, Andriy Tauberian and Abelian theorems for long-range dependent random fields, Methodol. Comput. Appl. Probab., Volume 15 (2013) no. 4, pp. 715-742 | DOI | Zbl

[LP11] Lavancier, Frédéric; Philippe, Anne Some convergence results on quadratic forms for random fields and application to empirical covariances, Probab. Theory Relat. Fields, Volume 149 (2011) no. 3-4, pp. 493-514 | DOI | Zbl

[LRMT17] Leonenko, Nikolai N.; Ruiz-Medina, María D.; Taqqu, Murad S. Rosenblatt distribution subordinated to Gaussian random fields with long-range dependence, Stochastic Anal. Appl., Volume 35 (2017) no. 1, pp. 144-177 | DOI | Zbl

[Mar23] Marinucci, Domenico Some Recent Developments on the Geometry of Random Spherical Eigenfunctions, European Congress of Mathematics, EMS Press, Berlin, 2023, pp. 337-365 | DOI | MR | Zbl

[MM20] Ma, Chunsheng; Malyarenko, Anatoliy Time-varying isotropic vector random fields on compact two-point homogeneous spaces, J. Theor. Probab., Volume 33 (2020) no. 1, pp. 319-339 | DOI | Zbl

[MP11] Marinucci, Domenico; Peccati, Giovanni Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications, London Mathematical Society Lecture Note Series, 389, Cambridge University Press, 2011 | DOI | Zbl

[MPRW16] Marinucci, Domenico; Peccati, Giovanni; Rossi, Maurizia; Wigman, Igor Non-universality of nodal length distribution for arithmetic random waves, Geom. Funct. Anal., Volume 26 (2016) no. 3, pp. 926-960 | DOI | Zbl

[MRV21] Marinucci, Domenico; Rossi, Maurizia; Vidotto, Anna Non-Universal Fluctuations of the Empirical Measure for Isotropic Stationary Fields on 𝕊 2 ×, Ann. Appl. Probab., Volume 31 (2021) no. 5, pp. 2311-2349 | DOI | Zbl

[MRW20] Marinucci, Domenico; Rossi, Maurizia; Wigman, Igor The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 56 (2020) no. 1, pp. 374-390 | DOI | Zbl

[NP12] Nourdin, Ivan; Peccati, Giovanni Normal Approximations Using Malliavin Calculus: from Stein’s Method to Universality, Cambridge Tracts in Mathematics, 192, Cambridge University Press, 2012 | DOI | Zbl

[NPR19] Nourdin, Ivan; Peccati, Giovanni; Rossi, Maurizia Nodal statistics of planar random waves, Commun. Math. Phys., Volume 369 (2019) no. 1, pp. 99-151 | DOI | Zbl

[PV20] Peccati, Giovanni; Vidotto, Anna Gaussian random measures generated by Berry’s nodal sets, J. Stat. Phys., Volume 178 (2020) no. 4, pp. 996-1027 | DOI | Zbl

[Ros19] Rossi, Maurizia Random nodal lengths and Wiener chaos, Probabilistic Methods in Geometry, Topology and Spectral Theory (Contemporary Mathematics), Volume 739, American Mathematical Society; Centre de Recherches Mathématiques (CRM), 2019, pp. 155-169 | DOI | Zbl

[RW08] Rudnick, Zeév; Wigman, Igor On the volume of nodal sets for eigenfunctions of the Laplacian on the torus, Ann. Henri Poincaré, Volume 9 (2008) no. 1, pp. 109-130 | DOI | Zbl

[Sze75] Szegö, Gabor Orthogonal polynomials, Colloquium Publications, 23, American Mathematical Society, 1975 | Zbl

[Taq75] Taqqu, Murad S. Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 31 (1975), pp. 287-302 | DOI | Zbl

[Taq79] Taqqu, Murad S. Convergence of Integrated Processes of Arbitrary Hermite Rank, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 50 (1979), pp. 53-83 | DOI | Zbl

[VT13] Veillette, Mark S.; Taqqu, Murad S. Properties and numerical evaluation of the Rosenblatt distribution, Bernoulli, Volume 19 (2013) no. 3, pp. 982-1005 | DOI | Zbl

[Wig10] Wigman, Igor Fluctuations of the nodal length of random spherical harmonics, Commun. Math. Phys., Volume 298 (2010) no. 3, pp. 787-831 | DOI | Zbl

[Wig23] Wigman, Igor On the nodal structures of random fields: a decade of results, J. Appl. Comput. Topol. (2023) | DOI