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We study the asymptotic behaviour of the nodal length of random$2d$-spherical harmonics $f_{\ell}$ of high degree $\ell \rightarrow\infty$,i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It is found that thenodal lengths are asymptotically equivalent, in the $L^{2}$-sense, to the"sample trispectrum", i.e., the integral of $H_{4}(f_{\ell}(x))$, thefourth-order Hermite polynomial of the values of $f_{\ell}$. A particularby-product of this is a Quantitative Central Limit Theorem (in Wassersteindistance) for the nodal length, in the high energy limit.

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