  We consider $N\times N$ Hermitian random matrices with independent
identically distributed entries (Wigner matrices). The matrices are normalized
so that the average spacing between consecutive eigenvalues is of order $1/N$.
Under suitable assumptions on the distribution of the single matrix element, we
first prove that, away from the spectral edges, the empirical density of
eigenvalues concentrates around the Wigner semicircle law on energy scales
$\eta \gg N^{-1}$. This result establishes the semicircle law on the optimal
scale and it removes a logarithmic factor from our previous result \cite{ESY2}.
We then show a
  Wegner estimate, i.e. that the averaged density of states is bounded.
Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in
agreement with the universality conjecture.
