Bernoulli

  • Bernoulli
  • Volume 24, Number 3 (2018), 2358-2400.

On the local semicircular law for Wigner ensembles

Friedrich Götze, Alexey Naumov, Alexander Tikhomirov, and Dmitry Timushev

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Abstract

We consider a random symmetric matrix $\mathbf{X}=[X_{jk}]_{j,k=1}^{n}$ with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that $\mathbb{E}|X_{11}|^{4+\delta}=:\mu_{4+\delta}<\infty$ for some $\delta>0$. The aim of this paper is to significantly extend a recent result of the authors Götze, Naumov and Tikhomirov (2015) and show that with high probability the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-\frac{1}{2}}\mathbf{X}$ 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 (2015). 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.

Article information

Source
Bernoulli, Volume 24, Number 3 (2018), 2358-2400.

Dates
Received: April 2016
Revised: February 2017
First available in Project Euclid: 2 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1517540477

Digital Object Identifier
doi:10.3150/17-BEJ931

Mathematical Reviews number (MathSciNet)
MR3757532

Zentralblatt MATH identifier
06839269

Keywords
delocalization local semicircle law mean spectral distribution random matrices rate of convergence rigidity Stieltjes transform

Citation

Götze, Friedrich; Naumov, Alexey; Tikhomirov, Alexander; Timushev, Dmitry. On the local semicircular law for Wigner ensembles. Bernoulli 24 (2018), no. 3, 2358--2400. doi:10.3150/17-BEJ931. https://projecteuclid.org/euclid.bj/1517540477


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