Local Universality for Zeros and Critical Points of Monochromatic Random Waves

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).