Local Universality for Zeros and Critical Points of Monochromatic Random Waves

This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves ϕ_λ of frequency λ on a compact, smooth, Riemannian manifold (M, g) as λ→ ∞. We prove global variance estimates for the measures of integration over the zeros and critical points of ϕ_λ. These global estimates hold for a wide class of manifolds—for example when (M, g) has no conjugate points—and rely on new local variance estimates on zeros and critical points of ϕ_λ in balls of radius ≈ λ^{- 1} around a fixed point. Our local results hold under conditions about the structure of geodesics that are generic in the space of all metrics on M.