Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law

Götze, Friedrich; Tikhomirov, Alexander A.

doi:10.48550/arxiv.1407.2780

Cited by 4 publications

(20 citation statements)

References 22 publications

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“…Under additional assumptions (1.7) was proved in [17], [28] and [18]. Comparing our result with [23]Theorem 3.6] note that we reduced the logarithmic factor and give explicit dependence on δ.…”

Section: Introduction and Main Resultssupporting

confidence: 63%

“…The first proof of a result of this type follows from a combination of arguments in a series of papers [10], [8], [23] (we sketched the underlying main ideas in the introduction of [15]). In [15] we gave a self-contained proof based on the method from [21], [18] while at the same time reducing the power of log n from β = c log log n to β = 2. Our work and some crucial bounds of our proof were motivated by the methods used in a recent paper of C. Cacciapuoti, A. Maltsev and B. Schlein, [5], where the authors improved the log-factor dependence in (1.2) in the sub-Gaussian case.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The proof is the direct consequence of the previous lemma and we omit it. For details the interested reader is referred to [18][Corollary 2.1].…”

Section: Rate Of Convergence In Probabilitymentioning

confidence: 99%

“…Proof of Theorem 1.1. We proceed as in the proof of Theorem 1.1 in [18]. We choose in Corollary 2.2 the following values for the parameters v 0 , ε and V .…”

Section: Rate Of Convergence In Probabilitymentioning

confidence: 99%

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Local semicircle law under moment conditions. Part II: Localization and delocalization

Götze,

Naumov,

Tikhomirov

2015

Preprint

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We consider a random symmetric matrix X = [X jk ] n j,k=1 with upper triangular entries being independent identically distributed random variables with mean zero and unit variance. We additionally suppose that E |X 11 | 4+δ =: µ 4+δ < C for some δ > 0 and some absolute constant C. Under these conditions we show that the typical Kolmogorov distance between the empirical spectral distribution function of eigenvalues of n −1/2 X and Wigner's semicircle law is of order 1/n up to some logarithmic correction factor. As a direct consequence of this result we establish that the semicircle law holds on a short scale. Furthermore, we show for this finite moment ensemble rigidity of eigenvalues and delocalization properties of the eigenvectors. Some numerical experiments are included illustrating the influence of the tail behavior of the matrix entries when only a small number of moments exist.

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Section: Introduction and Main Resultssupporting

confidence: 63%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The proof is the direct consequence of the previous lemma and we omit it. For details the interested reader is referred to [18][Corollary 2.1].…”

Section: Rate Of Convergence In Probabilitymentioning

confidence: 99%

“…Proof of Theorem 1.1. We proceed as in the proof of Theorem 1.1 in [18]. We choose in Corollary 2.2 the following values for the parameters v 0 , ε and V .…”

Section: Rate Of Convergence In Probabilitymentioning

confidence: 99%

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Local semicircle law under moment conditions. Part II: Localization and delocalization

Götze,

Naumov,

Tikhomirov

2015

Preprint

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“…The result (1.3) under the conditions (C0) was proved in a series of papers [11], [9], [31] with an n-dependent value β = c log log n. In [18] we gave a self-contained proof based on the methods developed in [28], [23] while at the same time reducing the power of log n from β = c log log n to β = 2. Our work and some crucial bounds of our proof were motivated by the methods used in a recent paper of C. Cacciapuoti, A. Maltsev and B. Schlein, [8], where the authors improved the log-factor dependence in (1.3) in the sub-Gaussian case.…”

Section: Introduction and Main Resultsmentioning

confidence: 88%

On the local semicircular law for Wigner ensembles

Götze¹,

Naumov²,

Tikhomirov³

et al. 2018

Bernoulli

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We consider a random symmetric matrix X = [X jk ] n j,k=1 with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that E |X 11 | 4+δ =: µ 4+δ < ∞ for some δ > 0. The aim of this paper is to significantly extend a recent result of the authors [18] and show that with high probability the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix n − 1 2 X and Wigner's semicircle law is of order (nv) −1 log n, where v denotes the distance to the real line in the complex plane. We apply this result to the rate of convergence of the ESD to the distribution function of the semicircle law as well as to rigidity of eigenvalues and eigenvector delocalization significantly extending a recent result by Götze, Naumov and Tikhomirov [19]. The result on delocalization is optimal by comparison with GOE ensembles. Furthermore the techniques of this paper provide a new shorter proof for the optimal O(n −1 ) rate of convergence of the expected ESD to the semicircle law.

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Local Semicircle Law under Moment Conditions: The Stieltjes Transform, Rigidity, and Delocalization

Götze

Naumov

Tikhomirov³

2018

Theory Probab. Appl.

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We consider a random symmetric matrix X = [X jk ] n j,k=1 in which the upper triangular entries are independent identically distributed random variables with mean zero and unit variance. We additionally suppose that E |X 11 | 4+δ =: µ 4+δ < ∞ for some δ > 0. Under these conditions we show that the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix n − 1 2 X and Wigner's semicircle law is of order (nv) −1 , where v is the distance in the complex plane to the real line. Furthermore we outline applications which are deferred to a subsequent paper, such as the rate of convergence in probability of the ESD to the distribution function of the semicircle law, rigidity of the eigenvalues and eigenvector delocalization.

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Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law

Cited by 4 publications

References 22 publications

Local semicircle law under moment conditions. Part II: Localization and delocalization

Local semicircle law under moment conditions. Part II: Localization and delocalization

On the local semicircular law for Wigner ensembles

Local Semicircle Law under Moment Conditions: The Stieltjes Transform, Rigidity, and Delocalization

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