Tangent nodal sets for random spherical harmonics

TL;DR

This paper investigates the distribution of points where a fixed vector field is tangent to the nodal set of random spherical harmonics on the sphere, revealing a universal asymptotic behavior independent of the vector field.

Contribution

It establishes the asymptotic expected count of tangent points for any fixed vector field, demonstrating a form of universality in their distribution.

Findings

Expected tangent point count asymptotic to eigenvalue

Leading coefficient independent of vector field

Universality of distribution up to lower order terms

Abstract

In this note, we consider a fixed vector field $V$ on $S^2$ and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where $V$ is also tangent. We show that the expected value of the corresponding counting function is asymptotic to the eigenvalue with a leading coefficient that is independent of the vector field $V$. This demonstrates, in some form, a universality for vector fields up to lower order terms.