Nodal lengths in shrinking domains for random eigenfunctions on $S^{2}$

Abstract

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