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.

We consider the time evolution of a system of N identical bosons whose interaction potential is rescaled by N −1 . We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the meanfield limit N → ∞ the quantum N -body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum N -body dynamics to the Hartree dynamics.

We develop a new method for deriving local laws for a large class of random matrices. It is applicable to many matrix models built from sums and products of deterministic or independent random matrices. In particular, it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity. For definiteness, we present the method for sample covariance matrices of the form Q . .= T XX * T * , where T is deterministic and X is random with independent entries. We prove that with high probability the resolvent of Q is close to a deterministic matrix, with an optimal error bound and down to optimal spectral scales.As an application, we prove the edge universality of Q by establishing the Tracy-Widom-Airy statistics of the eigenvalues of Q near the soft edges. This result applies in the single-cut and multi-cut cases. Further applications include the distribution of the eigenvectors and an analysis of the outliers and BBP-type phase transitions in finite-rank deformations; they will appear elsewhere.We also apply our method to Wigner matrices whose entries have arbitrary expectation, i.e. we consider W + A where W is a Wigner matrix and A a Hermitian deterministic matrix. We prove the anisotropic local law for W + A and use it to establish edge universality.