On the correlation between nodal and boundary lengths for random spherical harmonics

TL;DR

This paper investigates the relationship between the nodal length and boundary measure of random spherical harmonics, revealing asymptotic independence that becomes complete after controlling for the eigenfunctions' L^2-norm.

Contribution

It demonstrates the asymptotic zero correlation between nodal length and boundary measure, and the asymptotic one after controlling for the eigenfunctions' L^2-norm.

Findings

Correlation between nodal length and boundary measure is asymptotically zero.

Controlling for the eigenfunctions' L^2-norm makes the correlation asymptotically one.

Provides insight into the dependence structure of random spherical harmonics.

Abstract

We study the correlation between the nodal length of random spherical harmonics and the measure of the boundary for excursion sets at any non-zero level. We show that the correlation is asymptotically zero, while the partial correlation after controlling for the random $L^2$-norm on the sphere of the eigenfunctions is asymptotically one.