This article is cited in 8 papers

Local semicircle law under moment conditions: Stieltjes transform, rigidity and delocalization

F. Götze^a, A. A. Naumov^bc, A. N. Tikhomirov<sup>d

a</sup> Bielefeld University, Department of Mathematics
^b Skolkovo Institute of Science and Technology
^c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
^d Komi Scientific Center of Ural Branch of RAS

Abstract: We consider a random symmetric matrix ${X} = [X_{jk}]_{j,k=1}^n$ where the upper triangular entries are independent identically distributed random variables with zero mean and unit variance. We additionally suppose that ${{E}} |X_{11}|^{4 + \delta} =: \mu_{4+\delta} < \infty$ for some $\delta > 0$. 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 such as the rate of convergence of the ESD to the distribution function of the semicircle law, rigidity of the eigenvalues, and eigenvector delocalization.

Keywords: random matrices, local semicircle law, Stieltjes transform.

Received: 28.11.2016
Accepted: 20.01.2017

DOI: 10.4213/tvp5092

 English version: Theory of Probability and its Applications, 2018, 62:1, 58–83

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