Local semicircle law under moment conditions. Part I: The Stieltjes transform

Abstract

We consider a random symmetric matrix $\mathbf{X}=[X_{jk}{]}_{j,k=1}^n$ in which the upper triangular entries are independent identically distributed random variables with mean zero and unit variance. We additionally suppose that $\mathbb{E}|X_{11}{|}^{4+\delta }=:{\mu }_4<\infty$ 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 $n^{-\frac{1}{2}}\mathbf{X}$ 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.

54 pages. Some misprints were corrected and a detailed comparison with results for ensembles with more than four moments resulting from the combination of earlier papers has been added

References in corpus (2)

• Local semicircle law and complete delocalization for Wigner random matrices
• Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law

Cited by in corpus (3)

• Local semicircle law under moment conditions. Part II: Localization and delocalization
• Bulk Universality for Generalized Wigner Matrices With Few Moments
• Local semicircle law under fourth moment condition