Open Access
August 2018 On the local semicircular law for Wigner ensembles
Friedrich Götze, Alexey Naumov, Alexander Tikhomirov, Dmitry Timushev
Bernoulli 24(3): 2358-2400 (August 2018). DOI: 10.3150/17-BEJ931


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.


Download Citation

Friedrich Götze. Alexey Naumov. Alexander Tikhomirov. Dmitry Timushev. "On the local semicircular law for Wigner ensembles." Bernoulli 24 (3) 2358 - 2400, August 2018.


Received: 1 April 2016; Revised: 1 February 2017; Published: August 2018
First available in Project Euclid: 2 February 2018

zbMATH: 06839269
MathSciNet: MR3757532
Digital Object Identifier: 10.3150/17-BEJ931

Keywords: delocalization , Local semicircle law , mean spectral distribution , random matrices , rate of convergence , rigidity , Stieltjes transform

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 3 • August 2018
Back to Top