# On the Correlation of Critical Points and Angular Trispectrum for Random Spherical Harmonics

## TL;DR

This paper proves a Central Limit Theorem for the critical points of random spherical harmonics at high energy, revealing their asymptotic full correlation with the sample trispectrum and nodal length.

## Contribution

It establishes a novel link between critical points and the sample trispectrum, advancing understanding of high-energy behavior of random spherical harmonics.

## Key findings

- Critical points are asymptotically fully correlated with the sample trispectrum.
- Total number of critical points and nodal length are fully correlated at high energy.
- The paper proves a Central Limit Theorem for critical points in this context.

## Abstract

We prove a Central Limit Theorem for the Critical Points of Random Spherical Harmonics, in the High-Energy Limit. The result is a consequence of a deeper characterizations of the total number of critical points, which are shown to be asymptotically fully correlated with the sample trispectrum, i.e., the integral of the fourth Hermite polynomial evaluated on the eigenfunctions themselves. As a consequence, the total number of critical points and the nodal length are fully correlated for random spherical harmonics, in the high-energy limit.