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Bulk universality for generalized Wigner matrices
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  • Published: 06 October 2011

Bulk universality for generalized Wigner matrices

  • László Erdős1,
  • Horng-Tzer Yau2 &
  • Jun Yin2 

Probability Theory and Related Fields volume 154, pages 341–407 (2012)Cite this article

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Abstract

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure ν ij with a subexponential decay. Let \({\sigma_{ij}^2}\) be the variance for the probability measure ν ij with the normalization property that \({\sum_{i} \sigma^2_{ij} = 1}\) for all j. Under essentially the only condition that \({c\le N \sigma_{ij}^2 \le c^{-1}}\) for some constant c > 0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M −1.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Munich, Theresienstr. 39, 80333, Munich, Germany

    László Erdős

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Horng-Tzer Yau & Jun Yin

Authors
  1. László Erdős
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  2. Horng-Tzer Yau
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  3. Jun Yin
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Corresponding author

Correspondence to Horng-Tzer Yau.

Additional information

L. Erdős was partially supported by SFB-TR 12 Grant of the German Research Council; H-. T. Yau was partially supported by NSF Grants DMS-0602038, 0757425, 0804279; J. Yin was partially supported by NSF Grants DMS-100165.

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Erdős, L., Yau, HT. & Yin, J. Bulk universality for generalized Wigner matrices. Probab. Theory Relat. Fields 154, 341–407 (2012). https://doi.org/10.1007/s00440-011-0390-3

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  • Received: 12 February 2010

  • Revised: 22 July 2011

  • Published: 06 October 2011

  • Issue Date: October 2012

  • DOI: https://doi.org/10.1007/s00440-011-0390-3

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Keywords

  • Random band matrix
  • Local semicircle law
  • Sine kernel

Mathematics Subject Classification (2010)

  • 15B52
  • 82B44
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