A Reduction Principle for the Critical Values of Random Spherical Harmonics

TL;DR

This paper establishes a reduction principle linking the fluctuations in the count of critical points of Gaussian spherical harmonics to the $L^2$-norm, revealing their asymptotic behavior in the high-energy limit.

Contribution

It introduces a novel reduction principle that simplifies the analysis of critical point fluctuations for Gaussian spherical eigenfunctions at high energies.

Findings

Fluctuations are asymptotically proportional to the $L^2$-norm of the eigenfunctions.

Derived explicit formulas relating critical point counts to eigenfunction norms.

Connected these results to the behavior of geometric functionals on excursion sets.

Abstract

We study here the random fluctuations in the number of critical points with values in an interval $I\subset \mathbb{R}$ for Gaussian spherical eigenfunctions $\left\{f_{\ell }\right\} $, in the high energy regime where $\ell \rightarrow \infty $. We show that these fluctuations are asymptotically equivalent to the centred $L^{2}$-norm of $\left\{ f_{\ell }\right\} $ times the integral of a (simple and fully explicit) function over the interval under consideration. We discuss also the relationships between these results and the asymptotic behaviour of other geometric functionals on the excursion sets of random spherical harmonics.