Optimal Bounds on the Stieltjes Transform of Wigner Matrices

Claudio Cacciapuoti, Anna Maltsev, Benjamin Schlein

Published 2013-11-02, updated 2014-08-21Version 3

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 [22,10], by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate $(\log N)/N$ and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of [22,10,23].