ABSTRACT. We study random Morse functions on a Riemann manifold (M m , g) defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric g. The randomness is determined by a fixed Schwartz function w and a small parameter ε > 0. We first prove that as ε → 0 the expected distribution of critical values of this random function approaches a universal measure on R, independent of g, that can be explictly described in terms the expected distribution of eigenvalues of the Gaussian Wigner ensemble of random (m + 1) × (m + 1) symmetric matrices. In contrast, we prove that the metric g and its curvature are determined by the statistics of the Hessians of the random function for small ε.
In REF Thm.1.7] it was shown that the geometry of a Riemann manifold is determined by the statistics of the differentials of random functions on it.
Random Morse functions and spectral geometry
arXiv: Differential Geometry
Mathematics