Direction distribution for nodal components of random band-limited functions on surfaces

TL;DR

This paper investigates the distribution of tangencies between nodal components of random band-limited functions and a vector field on a surface, revealing a universal law in the high-energy limit that is independent of specific surface or vector field details.

Contribution

It establishes a universal deterministic law governing tangency counts of nodal components for random band-limited functions on surfaces, independent of surface geometry and vector field.

Findings

Distribution supported on even integers

Universal law in high-energy limit

Independent of surface and vector field

Abstract

Let $(M,g)$ be a smooth compact Riemannian surface with no boundary. Given a smooth vector field $V$ with finitely many zeroes on $M$, we study the distribution of the number of tangencies to $V$ of the nodal components of random band-limited functions. It is determined that in the high-energy limit, these obey a universal deterministic law, independent of the surface $M$ and the vector field $V$, that is supported precisely on the even integers $2 \mathbb{Z}_{> 0}$.