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We consider $N\times N$ Hermitian random matrices with independent identicaldistributed entries. The matrix is normalized so that the average spacingbetween consecutive eigenvalues is of order 1/N. Under suitable assumptions onthe distribution of the single matrix element, we prove that, away from thespectral edges, the density of eigenvalues concentrates around the Wignersemicircle law on energy scales $\eta \gg N^{-1} (\log N)^8$. Up to thelogarithmic factor, this is the smallest energy scale for which the semicirclelaw may be valid. We also prove that for all eigenvalues away from the spectraledges, the $\ell^\infty$-norm of the corresponding eigenvectors is of order$O(N^{-1/2})$, modulo logarithmic corrections. The upper bound $O(N^{-1/2})$implies that every eigenvector is completely delocalized, i.e., the maximumsize of the components of the eigenvector is of the same order as their averagesize. In the Appendix, we include a lemma by J. Bourgain which removes one ofour assumptions on the distribution of the matrix elements.

Local semicircle law and complete delocalization for Wigner random matrices