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Nodal lengths in shrinking domains for random eigenfunctions on \(S^2\). (English) Zbl 1507.60069

Summary: We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: for example, the length of the zero set \(\mathcal{Z}_{\ell,r_{\ell}}:=\mathcal{Z}^{B_{r_{\ell}}}(T_{\ell})=\operatorname{len}(\{x\in S^2\cap B_{r_{\ell}}:T_{\ell}(x)=0\})\), where \(B_{r_{\ell}}\) is the spherical cap of radius \(r_{\ell} \). We show that the variance of the nodal length is logarithmic in the high energy limit; moreover, it is asymptotically fully equivalent, in the \(L^2\)-sense, to the “local sample trispectrum”, namely, the integral on the ball of the fourth-order Hermite polynomial. This result extends and generalizes some recent findings for the full spherical case. As a consequence a Central Limit Theorem is established.

MSC:

60G60 Random fields
60F05 Central limit and other weak theorems
62R30 Statistics on manifolds

References:

[1] Adler, R.J. and Taylor, J.E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. New York: Springer. · Zbl 1149.60003
[2] Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Hoboken, NJ: Wiley. · Zbl 1168.60002
[3] Benatar, J., Marinucci, D. and Wigman, I. (2020). Planck-scale distribution of nodal length of arithmetic random waves. J. Anal. Math. To appear. · Zbl 1458.81020
[4] Bérard, P. (1985). Volume des ensembles nodaux des fonctions propres du laplacien. In Bony-Sjöstrand-Meyer Seminar, 1984-1985 IV.1-IV.9. Palaiseau: École Polytech. · Zbl 1002.58517
[5] Berry, M.V. (1977). Regular and irregular semiclassical wavefunctions. J. Phys. A 10 2083-2091. · Zbl 0377.70014 · doi:10.1088/0305-4470/10/12/016
[6] Bleher, P., Shiffman, B. and Zelditch, S. (2000). Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142 351-395. · Zbl 0964.60096 · doi:10.1007/s002220000092
[7] Bleher, P., Shiffman, B. and Zelditch, S. (2001). Universality and scaling of zeros on symplectic manifolds. In Random Matrix Models and Their Applications. Math. Sci. Res. Inst. Publ. 40 31-69. Cambridge: Cambridge Univ. Press. · Zbl 1129.58303
[8] Bourgain, J. and Rudnick, Z. (2011). On the geometry of the nodal lines of eigenfunctions of the two-dimensional torus. Ann. Henri Poincaré 12 1027-1053. · Zbl 1227.58008 · doi:10.1007/s00023-011-0098-z
[9] Buckley, J. and Wigman, I. (2016). On the number of nodal domains of toral eigenfunctions. Ann. Henri Poincaré 17 3027-3062. · Zbl 1361.35051 · doi:10.1007/s00023-016-0476-7
[10] Cammarota, V. and Marinucci, D. (2018). A quantitative central limit theorem for the Euler-Poincaré characteristic of random spherical eigenfunctions. Ann. Probab. 46 3188-3228. · Zbl 1428.60067 · doi:10.1214/17-AOP1245
[11] Cammarota, V., Marinucci, D. and Wigman, I. (2016). Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics. Proc. Amer. Math. Soc. 144 4759-4775. · Zbl 1351.60061 · doi:10.1090/proc/13299
[12] Cammarota, V., Marinucci, D. and Wigman, I. (2016). On the distribution of the critical values of random spherical harmonics. J. Geom. Anal. 26 3252-3324. · Zbl 1353.60020 · doi:10.1007/s12220-015-9668-5
[13] Cheng, S.Y. (1976). Eigenfunctions and nodal sets. Comment. Math. Helv. 51 43-55. · Zbl 0334.35022 · doi:10.1007/BF02568142
[14] Donnelly, H. and Fefferman, C. (1988). Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93 161-183. · Zbl 0659.58047 · doi:10.1007/BF01393691
[15] Krishnapur, M., Kurlberg, P. and Wigman, I. (2013). Nodal length fluctuations for arithmetic random waves. Ann. of Math. (2) 177 699-737. · Zbl 1314.60101 · doi:10.4007/annals.2013.177.2.8
[16] Logunov, A. (2018). Nodal sets of Laplace eigenfunctions: Polynomial upper estimates of the Hausdorff measure. Ann. of Math. (2) 187 221-239. · Zbl 1384.58020 · doi:10.4007/annals.2018.187.1.4
[17] Logunov, A. (2018). Nodal sets of Laplace eigenfunctions: Proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture. Ann. of Math. (2) 187 241-262. · Zbl 1384.58021 · doi:10.4007/annals.2018.187.1.5
[18] Logunov, A. and Malinnikova, E. (2015). On ratios of harmonic functions. Adv. Math. 274 241-262. · Zbl 1369.31008 · doi:10.1016/j.aim.2015.01.009
[19] Marinucci, D., Peccati, G., Rossi, M. and Wigman, I. (2016). Non-universality of nodal length distribution for arithmetic random waves. Geom. Funct. Anal. 26 926-960. · Zbl 1347.60013 · doi:10.1007/s00039-016-0376-5
[20] Marinucci, D., Rossi, M. and Wigman, I. (2020). The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics. Ann. Inst. Henri Poincaré Probab. Stat. 56 374-390. · Zbl 1465.60044 · doi:10.1214/19-AIHP964
[21] Marinucci, D. and Wigman, I. (2014). On nonlinear functionals of random spherical eigenfunctions. Comm. Math. Phys. 327 849-872. · Zbl 1322.60030 · doi:10.1007/s00220-014-1939-7
[22] Neuheisel, J.D. (2000). The Asymptotic Distribution of Nodal Sets on Spheres. Ph.D. thesis, The Johns Hopkins Univ.
[23] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge: Cambridge Univ. Press. · Zbl 1266.60001
[24] Nourdin, I., Peccati, G. and Rossi, M. (2019). Nodal statistics of planar random waves. Comm. Math. Phys. 369 99-151. · Zbl 1431.60025 · doi:10.1007/s00220-019-03432-5
[25] Rossi, M. (2015). The geometry of spherical random fields. Ph.D. thesis. Available at arXiv:1603.07575v1.
[26] Rossi, M. (2019). Random nodal lengths and Wiener chaos. In Probabilistic Methods in Geometry, Topology and Spectral Theory. Contemp. Math. 739 155-169. Providence, RI: Amer. Math. Soc. · Zbl 1458.60058
[27] Rossi, M. and Wigman, I. (2018). Asymptotic distribution of nodal intersections for arithmetic random waves. Nonlinearity 31 4472-4516. · Zbl 1394.60053 · doi:10.1088/1361-6544/aaced4
[28] Rudnick, Z. and Wigman, I. (2008). On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Ann. Henri Poincaré 9 109-130. · Zbl 1142.60029 · doi:10.1007/s00023-007-0352-6
[29] Szego, G. (1975). Orthogonal Polynomials, 4th ed. Colloquium Publications 23. Providence, RI: Amer. Math. Soc. · Zbl 0305.42011
[30] Todino, A.P. (2020). Supplement to “Nodal lengths in shrinking domains for random eigenfunctions on \(S^2\).” https://doi.org/10.3150/20-BEJ1216SUPP
[31] Wigman, I. (2009). On the distribution of the nodal sets of random spherical harmonics. J. Math. Phys. 50 Art. ID 013521. · Zbl 1200.58021 · doi:10.1063/1.3056589
[32] Wigman, I. (2010). Fluctuations of the nodal length of random spherical harmonics. Comm. Math. Phys. 298 787-831. · Zbl 1213.33019 · doi:10.1007/s00220-010-1078-8
[33] Yau, S.-T. (1982). Survey on partial differential equations in differential geometry. In Seminar on Differential Geometry. Ann. of Math. Stud. 102 3-71. Princeton, NJ: Princeton Univ. Press. · Zbl 0478.53001
[34] Yau, S.
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