Consider N × N hermitian or symmetric random matrices H with independent entries, where the distribution of the (i, j) matrix element is given by the probability measure νij with zero expectation and with variance σ 2 ij . We assume that the variances satisfy the normalization condition i σ 2 ij = 1 for all j and that there is a positive constant c such that c ≤ N σ 2 ij ≤ c −1 . We further assume that the probability distributions νij have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of H is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order (N η) −1 where η is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If γj = γj,N denotes the classical location of the j-th eigenvalue under the semicircle law ordered in increasing order, then the j-th eigenvalue λj is close to γj in the sense that for some positive constants C, c(2) The proof of Dyson's conjecture [15] which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order N −1 up to logarithmic corrections. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large N limit provided that the second moments of the two ensembles are identical.

We consider N N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density .x/ D e U.x/ . We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U 2 C 6 .R/ with at most polynomially growing derivatives and .x/ Ä C e C jxj for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N ). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN → ∞ (with a speed at least logarithmic in N ), the density of eigenvalues of the Erdős-Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N −1 (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the ℓ ∞ -norms of the ℓ 2 -normalized eigenvectors are at most of order N −1/2 with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pN ≫ N 2/3 .

We consider a general class of N ×N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, maxi,j E|hij| 2 . As a consequence, we prove the universality of the local n-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width W N 1−εn with some εn > 0 and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [6,17,19].

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N ). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN ≫ N 2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.