On optimal bounds in the local semicircle law under four moment condition

We consider symmetric random matrices with independent mean zero and unit variance entries in the upper triangular part. Assuming that the distributions of matrix entries have finite moment of order four, we prove optimal bounds for the distance between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law. Application concerning the convergence rate in probability of the empirical spectral distribution to the semicircle law is discussed as well.