Nodal Lengths in Shrinking Domains for Random Eigenfunctions on $\mathbb{S}^2$

TL;DR

This paper studies how the lengths of zero sets of random spherical harmonics behave in shrinking regions on the sphere, revealing a logarithmic variance growth and establishing a CLT for these nodal lengths.

Contribution

It extends previous results by analyzing nodal lengths in shrinking domains, showing their variance is logarithmic and deriving a CLT, thus advancing understanding of local eigenfunction behavior.

Findings

Variance of nodal length is logarithmic in high energy limit.

Nodal length asymptotically matches the local sample trispectrum.

A Central Limit Theorem for nodal lengths in shrinking domains is proved.

Abstract

We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: i.e., the length of the zero set $\mathcal{Z}_{\ell,r_\ell} := \mathcal{Z}^{B_{r_{\ell}}}(T_\ell)=\text{len}(\{x \in \mathbb{S}^2 \cap B_{r_\ell}: T_\ell(x)=0 \})$, 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.