2010
DOI: 10.1088/1751-8113/43/45/455209
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Percolating level sets of the adjacency eigenvectors ofd-regular graphs

Abstract: One of the most surprising discoveries in quantum chaos was that nodal domains of eigenfunctions of quantum-chaotic billiards and maps in the semi-classical limit display critical percolation. Here we extend these studies to the level sets of the adjacency eigenvectors of d-regular graphs. Numerical computations show that the statistics of the largest level sets (the maximal connected components of the graph for which the eigenvector exceeds a prescribed value) depend critically on the level. The critical leve… Show more

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Cited by 4 publications
(8 citation statements)
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“…Spectral properties of large regular discrete graphs have been studied in [28,22,17,33,30] but eigenfunctions have attracted attention only recently. A statistical study of the auto-correlations and the level sets of eigenvectors appeared in the papers [9,10] that introduce a random wave model (see also [16] for a random wave model on metric graphs). The paper [5] has pioneered the study of quantum ergodicity on large regular graphs -that is to say, the study of the spatial 1991 Mathematics Subject Classification.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
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“…Spectral properties of large regular discrete graphs have been studied in [28,22,17,33,30] but eigenfunctions have attracted attention only recently. A statistical study of the auto-correlations and the level sets of eigenvectors appeared in the papers [9,10] that introduce a random wave model (see also [16] for a random wave model on metric graphs). The paper [5] has pioneered the study of quantum ergodicity on large regular graphs -that is to say, the study of the spatial 1991 Mathematics Subject Classification.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…. 10 Note that B has cardinality n(q + 1) if G has n vertices and is (q + 1)-regular. 11 Or, equivalently, if the graph is bi-partite.…”
Section: Proof Of Theorem 17mentioning
confidence: 99%
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“…Simulations suggest that the eigenvalue spacing distribution has the same limit as that of the Wigner matrices. A limiting Gaussian wave character of eigenvectors have also been conjectured [14][15][16]. Some fine properties of eigenvalues and eigenvectors can indeed be proved for a single permutation matrix; see [33] and [4].…”
mentioning
confidence: 99%
“…In the statement below [∞] will refer to N. There exists a polynomial basis {f i , i ∈ N} (depending on d) such that for any K ∈ N ∪ {∞}, the process (tr f k (G(s + t)), k ∈ [K], t ≥ 0) converges in law, as s tends to infinity, to the Markov process (N k (t), k ∈ [K], t ≥ 0) of Theorem 1. [The polynomials are given explicitly in (16).] Hence, for any polynomial f , the process (tr f (G(s + t))) converges to a linear combination of the coordinate processes of (N k (t), k ∈ N).…”
mentioning
confidence: 99%