On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics

Abstract

We study the correlation between the nodal length of random spherical harmonics and the length of a nonzero level set. We show that the correlation is asymptotically zero, while the partial correlation after removing the effect of the random $$L^2$$ L 2 -norm of the eigenfunctions is asymptotically one.

On 3-dimensional Berry’s model

“…Canzani and Hanin (2020) studied the universality phenomenon in general Riemannian manifolds. The reader can find results on arithmetic random waves defined on the flat torus (Cammarota, 2019;Dalmao et al, 2019) and on random spherical harmonics in Cammarota and Marinucci (2019); Fantaye et al (2019); Marinucci and Rossi (2021) and references therein, see also Rossi (2019) for a survey on both subjects. The nodal sets of Berry's planar random waves, i.e.…”

On the nodal structures of random fields -- a decade of results

“…While it is easy to check that fact via the cross-correlation route, it is intuitively clear from the corresponding result for the excursion sets [42]. Because of that it was also later observed [41] that the projection of the non-nodal length at level t = 0 to the 2nd Wiener chaos, dominating the fluctuations of the level curve length, is, up to the factor √ ce −t 2 /2 t 2 , equal to the squared norm of T ℓ . This is in contrast to the nodal case, that is invariant to products by a constant, hence the projection onto the 2nd Wiener chaos vanishes precisely, and the fluctuations of the nodal length are dominated by its projection onto the 4th Wiener chaos.…”

Fluctuations of nodal sets on the 3-torus and general cancellation phenomena

“…Part (ii) contains a general version of the chaos cancellation phenomenon observed e.g. in Wigman (2010); Marinucci and Rossi (2021); Krishnapur et al (2013); Dalmao et al (2019); Marinucci et al (2016); Nourdin et al (2019); Cammarota (2019). Its proof is deferred to Appendix A.…”

“…where g 2 L 2 (Z) := Z g(z) 2 µ(dz), (see also Rossi, 2021 andCammarota et al, 2020). In particular, the second order chaotic projection of J is a linear combination of the centred square (random) norms of the fields X…”