Local Extrema in Quantum Chaos

TL;DR

This paper numerically studies the distribution of extrema in chaotic Laplacian eigenfunctions on 2D manifolds, showing their behavior aligns with classical predictions and their extrema are more evenly spaced than regular grids.

Contribution

It demonstrates that grid graphs with random edges mimic chaotic surface behavior and analyzes the spatial regularity of extrema using discrepancy measures.

Findings

Extrema count matches Longuet-Higgins' 1957 prediction.

Extrema are more regularly distributed than on regular grids.

Randomly augmented grid graphs behave like ergodic 2D surfaces.

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 $\mathbb{Z}^2$.