Random Morse functions and spectral geometry

2012/09/04 by Liviu I. Nicolaescu, Nicolaescu, Liviu I. · 2 citations
Computer Science · Mathematics · #15B52 #42C10 #53C65 #58J50 #58K05 #60D05 #60G15 #60G60 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Random Matrices and Applications #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1209.0639

openalex publication_date 2012/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 explicitly 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.

Citations

• The spectral function of an elliptic operator

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