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

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 $r_{\mathrm{\ell }}$. 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.