Moments of the inverse participation ratio for the Laplacian on finite regular graphs

TL;DR

This paper studies the statistical behavior of the inverse participation ratio (IPR) for eigenvectors of the Laplacian on finite regular graphs, revealing universal limits and fluctuations due to localized modes.

Contribution

It provides an exact analysis of the first and second moments of the IPR for all eigenvectors on finite regular graphs, uncovering universal behavior as graph size grows.

Findings

Mean IPR approaches 3 as graph size increases.

Large fluctuations are caused by localized eigenmodes.

Universal behavior emerges in the large-$n$ limit.

Abstract

We investigate the first and second moments of the inverse participation ratio (IPR) for all eigenvectors of the Laplacian on finite random regular graphs with $n$ vertices and degree $z$. By exactly diagonalizing a large set of $z$-regular graphs, we find that as $n$ becomes large, the mean of the inverse participation ratio on each graph, when averaged over a large ensemble of graphs, approaches the numerical value $3$. This universal number is understood as the large-$n$ limit of the average of the quartic polynomial corresponding to the IPR over an appropriate $(n-2)$-dimensional hypersphere of $\mathbb{R}^n$. For a large, but not exhaustive ensemble of graphs, the mean variance of the inverse participation ratio for all graph Laplacian eigenvectors deviates from its continuous hypersphere average due to large graph-to-graph fluctuations that arise from the existence of highly localized modes.