We investigate Stein-Malliavin approximations for nonlinear functionals of geometric interest for random eigenfunctions on the unit d-dimensional sphere Sd, d ≥ 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 ex- cursion 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 Theo- rems 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 S2; our results are new or improve the existing bounds even in this 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.

This talk is based on the paper Stein-Malliavin Approximations for Nonlinear Func- tionals of Random Eigenfunctions on Sd, joint work with Domenico Marinucci.