Quantitative gap universality for Wigner matrices

We obtain the explicit rate of convergence N−1 2+ε for the gaps between eigenvalues of generalized Wigner matrices in the bulk of the spectrum, for distributions of matrix entries possibly atomic and supported on enough points, quantifying the universality result by Erdős and Yau [17]. The proof proceeds by a Green function comparison, coupled with the relaxation estimate from [5]. In particular, we extend the 4 moment matching method [34] to arbitrary moments, allowing to compare resolvents down to the submicroscopic scale N−3 2+ε. This method also gives universality of the smallest gaps between eigenvalues for the Hermitian symmetry class, providing a universal, optimal separation of eigenvalues for discrete random matrices with entries supported on Ω(1) points.