Mathematical Physics and Harmonic Analysis Seminar

Date: September 27, 2019

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: Blake Keeler, UNC

Title: Random Waves and the Spectral Function on Manifolds without Conjugate Points

Abstract: In this talk, we discuss off-diagonal Weyl asymptotics on a compact manifold M, with the goal of understanding the statistical properties of monochromatic random waves. These waves can be thought of as randomized "approximate eigenfunctions," and their statistics are completely determined by an associated covariance kernel which coincides exactly with a rescaled version of the spectral function of the Laplace-Beltrami operator. We will prove that in the geometric setting of manifolds without conjugate points, one can obtain a logarithmic improvement in the two-point Weyl law for this spectral function, provided one restricts to a shrinking neighborhood of the diagonal in M x M. This then implies that the covariance kernel of a monochromatic random wave locally converges to a universal limit at a logarithmic rate as we take the frequency parameter to infinity. This result generalizes the work of Berard, who obtained the logarithmic improvement in the on-diagonal case for manifolds with nonpositive curvature.