Eigenvector statistics of Lévy matrices

Lopatto, Patrick; Marcinek, Jake

doi:10.1214/20-aop1493

Cited by 19 publications

(33 citation statements)

References 68 publications

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“…This phenomenon is identified in [25], where Gaussian fluctuations are proved for N 2/9 d N 1/3 . In [23], Gaussian fluctuations are established in the full polynomial regime N o (1) d N 1/3 . The goal of this paper is to analyse the regime d log N .…”

Section: Introductionmentioning

confidence: 99%

“…In [2], full delocalization is proved in a compact interval around the origin, and the authors even establish GOE local eigenvalue statistics in the same spectral region. In [1], the law of the eigenvector components of Lévy matrices is computed. Moreover, the fluctuations of the extreme eigenvalues are determined in [8,33], where they are shown to form asymptotically a Poisson process with power law intensity measure.…”

Section: Introductionmentioning

confidence: 99%

“…It is proved in [5] that for √ log N d < b * log N the spectrum of H splits into two phases: a delocalized phase in the bulk of the spectrum and a semilocalized phase in its complement. The delocalized phase is characterized by completely delocalized eigenvectors w, in the sense that w 2 ∞ / w 2 2 N −1+o (1) . The semilocalized phase is characterized by eigenvectors satisfying w 2 ∞ / w 2 2 N −γ(w)+o (1) for some (explicit) γ(w) < 1.…”

Section: Introductionmentioning

confidence: 99%

“…The delocalized phase is characterized by completely delocalized eigenvectors w, in the sense that w 2 ∞ / w 2 2 N −1+o (1) . The semilocalized phase is characterized by eigenvectors satisfying w 2 ∞ / w 2 2 N −γ(w)+o (1) for some (explicit) γ(w) < 1.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs

Alt¹,

Ducatez²,

Knowles³

2021

Preprint

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We consider the adjacency matrix A of the Erdős-Rényi graph on N vertices with edge probability d/N . For (log log N ) 4 d log N , we prove that the eigenvalues near the spectral edge form asymptotically a Poisson point process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale d log N , the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known distribution. Together with [5], our result establishes the coexistence of a fully delocalized phase and a fully localized phase in the spectrum of A. The proof relies on a three-scale rigidity argument, which characterizes the fluctuations of the eigenvalues in terms of the fluctuations of sizes of spheres of radius 1 and 2 around vertices of large degree.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs

Alt¹,

Ducatez²,

Knowles³

2021

Preprint

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“…When µ ≤ 0, then P(H) cannot be normalized, while for µ ≥ 2, the distribution has a finite variance and corresponds to the Gaussian orthogonal ensemble (GOE) case. Such matrices have been introduced and called "Lévy matrices" in Reference [4], where an Anderson delocalization-localization transition from the GOE to the Poisson distribution was proposed and observed for µ < 1 as a function of energies as well; see discussion in Reference [6][7][8]. It should be noted that such a situation takes place also in dynamical systems such as quantum chaos, where the quantum spectrum follows either chaotic or regular dynamics of corresponding classical counterparts, e.g., [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Quantum Walks in Hilbert Space of Lévy Matrices: Recurrences and Revivals

Iomin

2021

Fractal Fract

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The quantum evolution of wave functions controlled by the spectrum of Lévy random matrices is considered. An analytical treatment of quantum recurrences and revivals in the Hilbert space is performed in the framework of a theory of almost periodic functions. It is shown that the statistics of quantum recurrences in the Hilbert space of quantum systems is sensitive to the statistics of the corresponding quantum spectrum. In particular, it is shown that both the Poisson energy level statistics and the Brody distribution correspond to the power law of the quantum recurrences, while the Wigner–Dyson and Lévy–Smirnov statistics of the energy spectra are responsible for the exponential statistics of the quantum returns of the wave function.

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Delocalization Transition for Critical Erdős–Rényi Graphs

Alt

Ducatez

Knowles

2021

Commun. Math. Phys.

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We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph $${\mathbb {G}}(N,d/N)$$ G ( N , d / N ) , where d is of order $$\log N$$ log N . We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent $$\gamma (\varvec{\mathrm {w}})$$ γ ( w ) of an eigenvector $$\varvec{\mathrm {w}}$$ w , defined through $$\Vert \varvec{\mathrm {w}} \Vert _\infty / \Vert \varvec{\mathrm {w}} \Vert _2 = N^{-\gamma (\varvec{\mathrm {w}})}$$ ‖ w ‖ ∞ / ‖ w ‖ 2 = N - γ ( w ) . Our results remain valid throughout the optimal regime $$\sqrt{\log N} \ll d \leqslant O(\log N)$$ log N ≪ d ⩽ O ( log N ) .

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Eigenvector statistics of Lévy matrices

Cited by 19 publications

References 68 publications

Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs

Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs

Quantum Walks in Hilbert Space of Lévy Matrices: Recurrences and Revivals

Delocalization Transition for Critical Erdős–Rényi Graphs

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