PUB - Publikationen an der Universität Bielefeld

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 vertical bar X-11 vertical bar(4+delta) =: mu(4+delta) < infinity 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.