Optimal bounds for convergence of expected spectral distributions to the semi-circular law

Götze, Friedrich; Tikhomirov, Alexander

doi:10.1007/s00440-015-0629-5

Cited by 24 publications

(46 citation statements)

References 26 publications

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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: 97%

“…Recently Götze and Tikhomirov [28] proved the bound ∆ n = O(n −1 ) assuming that µ 8 < ∞ or µ 4 < ∞ combined with the assumption |X jk | ≤ Cn 1 4 a.s. Finally in [22] the following theorem was proved Theorem 1.6. Assume that the conditions (C0) hold.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…As before we denote Λ n (u + iv) := m n (u + iv) − s(u + iv). The bound for the first integral follows from [28][Inequality 3.11]…”

Section: −2αmentioning

confidence: 99%

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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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Section: Introduction and Main Resultsmentioning

confidence: 97%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On the local semicircular law for Wigner ensembles

Götze¹,

Naumov²,

Tikhomirov³

et al. 2018

Bernoulli

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show abstract

“…for some δ > 0, see e.g. [7], [5], [20], [13], [14], [15], [18] and [17]. In particular, the result of [17] implies that (1.4) holds with α(n) ≡ 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

Local Semicircle Law Under Fourth Moment Condition

2019

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We consider a random symmetric matrix X = [X jk ] n j,k=1 with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that max jk E |X jk | 4+δ < ∞, δ > 0, it was proved in [17] that with high probability the typical distance between the Stieltjes transforms mn(z), z = u + iv, of the empirical spectral distribution (ESD) and the Stieltjes transforms msc(z) of the semicircle law is of order (nv) −1 log n. The aim of this paper is to remove δ > 0 and show that this result still holds if we assume that max jk E |X jk | 4 < ∞. We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors.

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“…The bound for E |T n (z)| p is given in Theorem 2.1. The crucial step in [21], [18] was to show that finiteness of eight moments suffices to show that max 1≤j≤n E |R jj (z)| p ≤ C p 0 for all 1 ≤ p ≤ C(nv) 1 4 and v ≥ v 0 . The proof of this fact was based on the descent method developed in [5][Lemma 3.4] (see Lemma 4.1 below), but used in proving bounds for moments of the diagonal entries R jj (z) only.…”

Section: Main Resultmentioning

confidence: 99%

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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Optimal bounds for convergence of expected spectral distributions to the semi-circular law

Cited by 24 publications

References 26 publications

On the local semicircular law for Wigner ensembles

On the local semicircular law for Wigner ensembles

Local Semicircle Law Under Fourth Moment Condition

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

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