Bounds for the Stieltjes transform and the density of states of Wigner matrices

We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from Erdős et al. (Adv Math 229(3):1435–1515, 2012; Electron J Probab 18(59):1–58, 2013), by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate (Formula presented.) and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of Erdős et al. (Adv Math 229(3):1435–1515, 2012; Electron J Probab 18(59):1–58, 2013), Götze and Tikhomirov