Eigenvectors of the discrete Laplacian on regular graphs - a statistical approach

TL;DR

This paper investigates the statistical properties of eigenvectors of random regular graphs, suggesting they follow a Gaussian distribution and exploring their nodal structures beyond percolation models.

Contribution

It introduces a statistical approach to characterize eigenvector structures, including correlations and Gaussianity, for eigenvectors of random regular graphs.

Findings

Eigenvectors exhibit Gaussian distribution characteristics.

Correlations between eigenvector components are analyzed.

Nodal patterns cannot be fully explained by percolation models.

Abstract

In an attempt to characterize the structure of eigenvectors of random regular graphs, we investigate the correlations between the components of the eigenvectors associated to different vertices. In addition, we provide numerical observations, suggesting that the eigenvectors follow a Gaussian distribution. Following this assumption, we reconstruct some properties of the nodal structure which were observed in numerical simulations, but were not explained so far. We also show that some statistical properties of the nodal pattern cannot be described in terms of a percolation model, as opposed to the suggested correspondence for eigenvectors of 2 dimensional manifolds.