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SUMMARY:Local semicircle law and complete delocalization for Wigner random
  matrices - Yau\, HT (Harvard)
DTSTART:20080818T143000Z
DTEND:20080818T153000Z
UID:TALK13147@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We consider $N	imes N$ Hermitian random matrices with independ
 ent identical distributed entries. The matrix is normalized so that the av
 erage spacing between consecutive eigenvalues is of order 1/N. Under suita
 ble assumptions on the distribution of the single matrix element\, we prov
 e that\, away from the spectral edges\,\nthe density of eigenvalues concen
 trates around the Wigner semicircle law on energy scales $ta gg N^{-1} (l
 og N)^8$. 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 eig
 envalues away from the spectral edges\, the $ll^infty$-norm of the\ncorre
 sponding eigenvectors is of order $O(N^{-1/2})$\, modulo logarithmic corre
 ctions. The upper bound $O(N^{-1/2})$ implies that every eigenvector is co
 mpletely delocalized\, i.e.\, the maximum size of the components of the ei
 genvector is of the same order as their average size.
LOCATION:Seminar Room 1\, Newton Institute
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