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

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.

28 pages, 11 figures

References in corpus (4)

• Nodal domains statistics - a criterion for quantum chaos
• Non-backtracking random walks mix faster
• Semiclassical Construction of Random Wave Functions for Confined Systems
• Geometric characterization of nodal domains: the area-to-perimeter ratio

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• Cycles and eigenvalues of sequentially growing random regular graphs
• Moments of the inverse participation ratio for the Laplacian on finite regular graphs
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• Many nodal domains in random regular graphs