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In this short survey we recollect some of the recent results on the highenergy behavior (i.e., for diverging sequences of eigenvalues) of nonlinearfunctionals of Gaussian eigenfunctions on the $d$-dimensional sphere $\mathbbS^d$, $d\ge 2$. We present a quantitative Central Limit Theorem for a class offunctionals whose Hermite rank is two, which includes in particular theempirical measure of excursion sets in the non-nodal case. Concerning the nodalcase, we recall a CLT result for the defect on $\mathbb S^2$. The key tools areboth, the asymptotic analysis of moments of all order for Gegenbauerpolynomials, and so-called Fourth-Moment theorems.

On the High Energy Behavior of Nonlinear Functionals of Random Eigenfunctions on $\mathbb S^d$