2010
DOI: 10.1103/physreve.82.031135
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Localization transition in symmetric random matrices

Abstract: We study the behavior of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully connected Lévy matrices. We derive a critical line separating localized from extended states in the case of Lévy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.

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Cited by 65 publications
(126 citation statements)
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“…The quantity g represents the diagonal elements of the resolvent on the cavity graph [22,35]. Substituting Eq.…”
Section: B Random Regular Graphsmentioning
confidence: 99%
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“…The quantity g represents the diagonal elements of the resolvent on the cavity graph [22,35]. Substituting Eq.…”
Section: B Random Regular Graphsmentioning
confidence: 99%
“…For random regular graphs with uniform edges, in which all eigenvectors are delocalized [27][28][29], we show that σ 2 (λ) = 0 for any λ. On the other hand, for random graph models with localized eigenvectors [22][23][24]30,31], the prefactor σ 2 (λ) exhibits a maximum for a certain λ, while it vanishes for |λ| → 0. These results indicate that the linear scaling of the variance is a consequence of the uncorrelated nature of the eigenvalues in the localized regions of the spectrum.…”
Section: Introductionmentioning
confidence: 99%
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