The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics

Abstract

We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell }$ of high degree $\ell \to \infty$, i.e. the length of their zero set $f_{\ell }^{-1}(0)$. It is found that the nodal lengths are asymptotically equivalent, in the $L^2$-sense, to the "sample trispectrum", i.e., the integral of $H_4(f_{\ell }(x))$, the fourth-order Hermite polynomial of the values of $f_{\ell }$. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.