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res:
bibo_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
n ≫ 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 -tempℓ∞-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.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: László
foaf_name: László Erdös
foaf_surname: Erdös
foaf_workInfoHomepage: http://www.librecat.org/personId=4DBD5372-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0001-5366-9603
- foaf_Person:
foaf_givenName: Benjamin
foaf_name: Schlein, Benjamin
foaf_surname: Schlein
- foaf_Person:
foaf_givenName: Horng
foaf_name: Yau, Horng-Tzer
foaf_surname: Yau
bibo_doi: 10.1007/s00220-008-0636-9
bibo_issue: '2'
bibo_volume: 287
dct_date: 2009^xs_gYear
dct_publisher: Springer@
dct_title: Local semicircle law and complete delocalization for Wigner random matrices@
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