# The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics

## TL;DR

This paper proves that the nodal length of high-degree random spherical harmonics behaves asymptotically like the sample trispectrum, leading to a quantitative CLT for the nodal length.

## Contribution

It establishes the asymptotic equivalence between the nodal length and the sample trispectrum for high-degree spherical harmonics, providing a new approach to analyze their distribution.

## Key findings

- Asymptotic equivalence in L^2 sense between nodal length and sample trispectrum.
- Quantitative CLT (Wasserstein distance) for the nodal length at high energy.
- Insight into the geometric properties of random spherical harmonics.

## Abstract

We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell}$ of high degree $\ell \rightarrow\infty$, i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It is found that the nodal lengths are asymptotically equivalent, in the $L^{2}$-sense, to the "sample trispectrum", i.e., the integral of $H_{4}(f_{\ell}(x))$, the fourth-order Hermite polynomial of the values of $f_{\ell}$. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.