Nodal lengths in shrinking domains for random eigenfunctions on \(S^2\). (English) Zbl 1507.60069
Summary: 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.
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