Random Morse functions and spectral geometry
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 $\varepsilon>0$. We first prove that as $\varepsilon\to 0$ the expected distribution of critical values of this random function approaches a universal measure on $\mathbb{R}$, independent of $g$, that can be explicitly described in terms the expected distribution of eigenvalues of the Gaussian Wigner ensemble of random $(m+1)\times (m+1)$ 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 $\varepsilon$.
 Publication:

arXiv eprints
 Pub Date:
 September 2012
 arXiv:
 arXiv:1209.0639
 Bibcode:
 2012arXiv1209.0639N
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 Mathematics  Probability;
 15B52;
 42C10;
 53C65;
 58K05;
 58J50;
 60D05;
 60G15;
 60G60
 EPrint:
 47 pages (changed the title, fixed typos, substantially revised the introduction, updated references). arXiv admin note: text overlap with arXiv:1201.4972