Wegner Estimate and Level Repulsion for Wigner Random Matrices

Erdős, László; Schlein, Benjamin; Yau, Horng‐Tzer

doi:10.1093/imrn/rnp136

Cited by 160 publications

(397 citation statements)

References 13 publications

(70 reference statements)

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“…304-305]. More recently, in agreement with the universality conjectures, the level repulsion was proved for the eigenvalues of a Wigner matrix [4].…”

Section: Introductionmentioning

confidence: 52%

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Eigenvalue Attraction

Movassagh¹

2015

J Stat Phys

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We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.

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“…304-305]. More recently, in agreement with the universality conjectures, the level repulsion was proved for the eigenvalues of a Wigner matrix [4].…”

Section: Introductionmentioning

confidence: 52%

“…It is easy to check that P (t) is differentiable everywhere 4 . More specifically, for > 0, it is smooth (C ∞ ) everywhere but on a set of measure zero (i.e., all t i ), where it is only C 1 .…”

Section: Smoothened Discrete Stochastic Processmentioning

confidence: 99%

Eigenvalue Attraction

Movassagh¹

2015

J Stat Phys

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“…In this subsection we prove the second part of Theorem 2.11. To prove (2.22), we follow the traditional path of [20,21,22]. Presumably, the same result can be obtained by a more detailed analysis of G ∆ jj than the one carried out in the previous subsection.…”

Section: Preliminariesmentioning

confidence: 57%

Extremal eigenvalues and eigenvectors of deformed Wigner matrices

Lee

Schnelli

2015

Probab. Theory Relat. Fields

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We consider random matrices of the form H = W + λV , λ ∈ R + , where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d. entries that are independent of W . We assume subexponential decay of the distribution of the matrix entries of W and we choose λ ∼ 1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+ ∈ R + such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ > λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N → ∞ if λ > λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ > λ+, while they are completely delocalized for λ < λ+. Similar results hold for the lowest eigenvalues.

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“…This includes both limits of random diagonal matrices without level repulsion, and those of random matrix ensembles within the GXE domains of attraction. The latter case includes a class of random Wigner matrices for which the result is established through a combination of the general criteria derived here with previous analytical results derived in [16,29,17] on the convergence of the local law to the scaling limit of the GUE ensemble.…”

Section: Introductionmentioning

confidence: 86%

“…It is easy to see that (1.3) suffices for the association with µ ω of the function 16) which is holomorphic over C + and which inherits the stationarity of µ ω . The above question can therefore be rephrased as asking under what conditions would K ω (z) be the derivative of a stationary random HP function.…”

Section: A Cocycle Criterionmentioning

confidence: 99%

On the ubiquity of the Cauchy distribution in spectral problems

Aizenman

Warzel

2014

Probab. Theory Relat. Fields

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We consider the distribution of the values at real points of random functions which belong to the Herglotz-Pick (HP) class of analytic mappings of the upper half plane into itself. It is shown that under mild stationarity assumptions the individual values of HP functions with singular spectra have a Cauchy type distribution. The statement applies to the diagonal matrix elements of random operators, and holds regardless of the presence or not of level repulsion, i.e. applies to both random matrix and Poisson-type spectra.

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Wegner Estimate and Level Repulsion for Wigner Random Matrices

Cited by 160 publications

References 13 publications

Eigenvalue Attraction

Eigenvalue Attraction

Extremal eigenvalues and eigenvectors of deformed Wigner matrices

On the ubiquity of the Cauchy distribution in spectral problems

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