Stein–Malliavin approximations for nonlinear functionals of random eigenfunctions on Sd

We investigate Stein–Malliavin approximations for nonlinear functionals of geometric interest for random eigenfunctions on the unit d-dimensional sphere $S^d$, $d\ge 2$. All our results are established in the high energy limit, i.e. as the corresponding eigenvalues diverge. In particular, we prove a quantitative Central Limit Theorem for the excursion volume of Gaussian eigenfunctions; this goal is achieved by means of several results of independent interest, concerning the asymptotic analysis for the variance of moments of Gaussian eigenfunctions, the rates of convergence in various probability metrics for Hermite subordinated processes, and quantitative Central Limit Theorems for arbitrary polynomials of finite order or general, square-integrable, nonlinear transforms. Some related issues were already considered in the literature for the 2-dimensional case $S^2$; our results are new or improve the existing bounds even in these special circumstances. Proofs are based on the asymptotic analysis of moments of all order for Gegenbauer polynomials, and make extensive use of the recent literature on so-called fourth-moment theorems by Nourdin and Peccati.