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We consider $N\times N$ Hermitian random matrices with i.i.d. entries. Thematrix is normalized so that the average spacing between consecutiveeigenvalues is of order $1/N$. We study the connection between eigenvaluestatistics on microscopic energy scales $\eta\ll1$ and (de)localizationproperties of the eigenvectors. Under suitable assumptions on the distributionof the single matrix elements, we first give an upper bound on the density ofstates on short energy scales of order $\eta \sim\log N/N$. We then prove thatthe density of states concentrates around the Wigner semicircle law on energyscales $\eta\gg N^{-2/3}$. We show that most eigenvectors are fully delocalizedin the sense that their $\ell^p$-norms are comparable with$N^{{1}/{p}-{1}/{2}}$ for $p\ge2$, and we obtain the weaker bound$N^{{2}/{3}({1}/{p}-{1}/{2})}$ for all eigenvectors whose eigenvalues areseparated away from the spectral edges. We also prove that, with a probabilityvery close to one, no eigenvector can be localized. Finally, we give an optimalbound on the second moment of the Green function.

Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices