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arXiv · 1506.02048

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

Timothy B. P. Clark·Adrian Del Maestro
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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.

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BibTeXRIS

Timothy B. P. Clark, Adrian Del Maestro. 2015-06-05. Moments of the inverse participation ratio for the Laplacian on finite regular graphs. https://doi.org/10.1088/1751-8121/aaebb2

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