 0852 Laszlo Erdos, Benjamin Schlein, HorngTzer Yau
 Local semicircle law and complete
delocalization for Wigner random matrices
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Mar 19, 08

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Abstract. We consider $N\times N$ Hermitian random matrices with i.i.d.
entries. The matrix is 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 prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales $\eta \gg N^{1} (\log N)^8$. Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, 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 maximum size of the components of the eigenvector is of the
same order as their average size.
In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements.
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