This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves ϕλ 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 ϕλ. 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 ϕλ 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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics