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 the first author has investigated critical sets of random functions on a compact Riemann manifold.
Random Morse functions and spectral geometry
arXiv: Differential Geometry
Mathematics