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

DOI10.3150/20-BEJ1216arXiv1807.11787MaRDI QIDQ6304861

Anna Paola Todino

Publication date: 31 July 2018

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 $mathcal{Z}_{ell,r_{e}ll}:=mathcal{Z}^{B_{r_{ell}}}(T_{e}ll)=extlen(xinmathbb{S}^{2}cap{B}_{r_{e}ll}:T_{e}ll(x)=0)$, where $B_{r_{ell}}$ is the spherical cap of radius $r_{e}ll$. 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.

Mathematics Subject Classification ID

Random fields (60G60) Statistics on manifolds (62R30) Central limit and other weak theorems (60F05)

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