On the correlation between critical points and critical values for random spherical harmonics

TL;DR

This paper investigates the relationship between the total critical points and those with specific values in random spherical harmonics, revealing asymptotic independence and dependence under different conditions.

Contribution

It demonstrates that the correlation between total and interval-restricted critical points vanishes asymptotically, but becomes perfect after controlling for the eigenfunctions' $L^2$-norm.

Findings

Correlation between total and interval-restricted critical points is asymptotically zero.

Partial correlation, controlling for $L^2$-norm, tends to one.

Results extend understanding of geometric properties of random spherical harmonics.

Abstract

We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval $I \subset \mathbb{R}$. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random $L^2$-norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics.