Nodal lengths in shrinking domains for random eigenfunctions on S²

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 Zℓ,rℓ:= ZBrℓ (Tℓ) = len({x ∈ S2 ∩ Brℓ: Tℓ(x) = 0}), where Brℓ is the spherical cap of radius rℓ. We show that the variance of the nodal length is logarithmic in the high energy limit; moreover, it is asymptotically fully equivalent, in the L2-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.