2016/10/28 by Yaiza Canzani, Boris Hanin, Canzani, Yaiza +1 · 1 citation
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1610.09438
This paper concerns the asymptotic behavior of zeros and critical points for\nmonochromatic random waves \φ_\λ of frequency \λ on a compact,\nsmooth, Riemannian manifold (M,g) as \λ \→ \∞. We prove\nthat the measure of integration over the zero set of \φ_\λ restricted\nto balls of radius \≈ \λ-1 converges in distribution to the\nmeasure of integration over the zero set of a frequency 1 random wave on\n mathbb Rn, where n is the dimension of M. We also prove convergence of\nfinite moments for the counting measure of the critical points of\n\φ\λ, again restricted to balls of radius \≈ \λ-1, to\nthe corresponding moments for frequency 1 random waves. We then patch\ntogether these local results to obtain new global variance estimates on the\nvolume of the zero set and numbers of critical points of \φ_\λ on all\nof M. Our local results hold under conditions about the structure of\ngeodesics on M that are generic in the space of all metrics on M, while our\nglobal results hold whenever (M,g) has no conjugate points (e.g is negatively\ncurved).\n