Local Semicircle Law and Complete Delocalization for Wigner Random Matrices |
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Authors: | László?Erd?s Benjamin?Schlein Horng-Tzer?Yau |
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Institution: | (1) Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany;(2) Department of Mathematics, Harvard University, Cambridge, MA 02138, USA |
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Abstract: | We consider N × N Hermitian random matrices with independent identical distributed 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 . 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 ℓ
∞-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.
Supported by Sofja-Kovalevskaya Award of the Humboldt Foundation. On leave from Cambridge University, UK.
Partially supported by NSF grant DMS-0602038. |
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Keywords: | |
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