Delocalization for a class of random block band matrices Journal Article

Author(s): Bao, Zhigang; Erdős, László
Article Title: Delocalization for a class of random block band matrices
Affiliation IST Austria
Abstract: We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.
Keywords: supersymmetry; Local semicircle law; Random band matrix; delocalization; Green’s function comparison
Journal Title: Probability Theory and Related Fields
Volume: 167
Issue 3
ISSN: 1432-2064
Publisher: Springer  
Date Published: 2017-04-01
Start Page: 673
End Page: 776
Copyright Statement: CC BY 4.0
Sponsor: Open access funding provided by Institute of Science and Technology (IST Austria). Z. Bao was supported by ERC Advanced Grant RANMAT No. 338804; L. Erd ̋ os was partially supported by ERC Advanced Grant RANMAT No. 338804.
DOI: 10.1007/s00440-015-0692-y
Open access: yes (OA journal)
IST Austria Authors
  1. László Erdős
    102 Erdős
  2. Zhigang Bao
    7 Bao
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