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

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

2008-04-17

arxiv.org/abs/0804.2771v1

Physics

Mathematical Physics

28 pages, 11 figures

Scientific paper

10.1088/1751-8113/41/43/435203

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.

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