Local Universality for Zeros and Critical Points of Monochromatic Random Waves

TL;DR

This paper demonstrates that the local behavior of zeros and critical points of monochromatic random waves on a manifold converges to that of Euclidean random waves as frequency increases, leading to global variance estimates.

Contribution

It establishes local universality results for zeros and critical points of monochromatic random waves on manifolds, connecting local behavior to Euclidean models and deriving global variance bounds.

Findings

Local zero set measures converge to Euclidean random wave measures

Critical point counts' moments converge to Euclidean models

Global variance estimates for zero sets and critical points

Abstract

This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves $\phi_\lambda$ of frequency $\lambda$ on a compact, smooth, Riemannian manifold $(M,g)$ as $\lambda \rightarrow \infty$. We prove that the measure of integration over the zero set of $\phi_\lambda$ restricted to balls of radius $\approx \lambda^{-1}$ converges in distribution to the measure of integration over the zero set of a frequency $1$ random wave on $\mathbb R^n$, where $n$ is the dimension of $M$. We also prove convergence of finite moments for the counting measure of the critical points of {\phi}{\lambda}, again restricted to balls of radius $\approx \lambda^{-1}$, to the corresponding moments for frequency $1$ random waves. We then patch together these local results to obtain new global variance estimates on the volume of the zero set and numbers of critical points of $\phi_\lambda$ on all of $M.$ Our local results hold under conditions about the structure of geodesics on $M$ that are generic in the space of all metrics on $M$, while our global results hold whenever $(M,g)$ has no conjugate points (e.g is negatively curved).