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

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.