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

TL;DR

This survey reviews recent advances in understanding the high energy asymptotics of nonlinear functionals of Gaussian eigenfunctions on spheres, highlighting CLT results for various cases.

Contribution

It provides a unified presentation of CLT results for nonlinear functionals of Gaussian eigenfunctions on spheres, including new quantitative bounds for Hermite rank two functionals.

Findings

Quantitative CLT for Hermite rank two functionals

CLT for the defect on $ ext{S}^2$ in the nodal case

Asymptotic moment analysis of Gegenbauer polynomials

Abstract

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