Year: 2014
Type: preprint
Abstract: We numerically investigate the distribution of extrema of 'chaotic' Laplacian eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a) we count extrema on grid graphs with a small number of randomly added edges and show the behavior to coincide with the 1957 prediction of Longuet-Higgins for the continuous case and (b) compute the regularity of their spatial distribution using \textit{discrepancy}, which is a classical measure from the theory of Monte Carlo integration. The first part suggests that grid graphs with randomly added edges should behave like two-dimensional surfaces with ergodic geodesic flow; in the second part we show that the extrema are more regularly distributed in space than the grid $\mathbb{Z}^2$.
Source: arXiv (Cornell University)
Authors: Pausinger, Florian, Steinerberger, Stefan
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2285740213