Year: 2012
Type: preprint
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 $\varepsilon$.
Source: arXiv (Cornell University)
Author: Nicolaescu, Liviu I.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W1945973881