Fluctuations of the Nodal Length of Random Spherical Harmonics

Abstract

Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics.It is known that the expected length is of order n. It is natural to conjecture that the variance should be of order n, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order log n. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines.

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

“…Nn ; this situation (customarily called arithmetic Berry's cancellation, see [KKW]) is similar to the cancellation phenomenon observed by Berry in a different setting, see [Be,W1].…”

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

“…To be clear, as given in Marinucci et al (2017), as → ∞, and, setting L: = + 1 2 , we have that More explicitly, it was shown in Marinucci and Wigman (2014) that, as → ∞, to find a better approximation, we evaluate numerically the constant see the Appendix for some analytic results. Hence, we conclude that, up to smaller order terms, Then, the variance of the scaled sample trispectrum M is asymptotically given by Finally, let us recall that these results, as in Marinucci et al (2017), Rossi (2015), Todino (2018b) and Wigman (2010), refer to the boundary length, not to the first Lipschitz-Killing curvature; there is hence a difference of a factor 2 in the expected value, and a factor 4 in the variance. The values in the Table 2 refer to the Lipschitz-Killing curvature; hence, they have been normalized accordingly.…”

“…For u = 0, the leading term in the previous expression disappears (the so‐called Berry's cancellation phenomenon, see Berry (), Wigman ()) and the variance is of smaller order; more precisely, we have that (Wigman, )(the same happens for shrinking subdomains of the sphere, see Todino ()). It is important to notice that the difference between the leading and remainder terms is here only of logarithmic order, and we hence expect a less precise approximation (in relative terms) in the simulations.…”

“…To compute the expected value, it is enough to exploit the Gaussian Kinematic Formula; as before, note that we shall normalize by 4π in the simulations (see Table 2) so that we obtain Likewise, using results in Marinucci et al (2017), Rossi (2015) and Wigman (2010), we have for the leading stochastic term and using again (18), the variance can easily be seen to be Again, in the simulations below (see Table 2), normalizing the boundary length by 4π divides by a factor 16π 2 , leading (up to negligible terms) to a variance of order 128 u 4 e −u 2 . For u = 0, the leading term in the previous expression disappears (the so-called Berry's cancellation phenomenon, see Berry (2002), Wigman (2010)) and the variance is of smaller order; more precisely, we have that (Wigman, 2010) (the same happens for shrinking subdomains of the sphere, see Todino (2018b)). It is important to notice that the difference between the leading and remainder terms is here only of logarithmic order, and we hence expect a less precise approximation (in relative terms) in the simulations.…”

Random nodal lengths and Wiener chaos

“…Note that the mean (5.36) has the same form as in the toral or spherical case (see Remark 2.3), and the asymptotic variance (5.37) is of lower order than expectedas predicted by Berry in [Ber02]. Thanks to the symmetry of nodal lines on the two dimensional sphere [Wig10, §1.6.2], the result on the asymptotic variance for the nodal length on S 2 (2.13) obtained by Wigman in [Wig10] is of course consistent to Berry's prediction ((5.37) or [Ber02]).…”