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

Abstract

We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ 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 $n^{-\frac{1}{2}} {\bf X}$ 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.