Probability Theory and Related Fields

, Volume 163, Issue 1–2, pp 1–59 | Cite as

Bounds for the Stieltjes transform and the density of states of Wigner matrices

  • Claudio Cacciapuoti
  • Anna Maltsev
  • Benjamin Schlein
Article

Abstract

We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from Erdős et al. (Adv Math 229(3):1435–1515, 2012; Electron J Probab 18(59):1–58, 2013), by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate \((\log N)/N\) and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of Erdős et al. (Adv Math 229(3):1435–1515, 2012; Electron J Probab 18(59):1–58, 2013), Götze and Tikhomirov (2013).

Keywords

Random matrices Wigner matrices Rigidity of the eigenvalues  Rate of convergence 

Mathematics Subject Classification

60B20 60B12 47B80 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Claudio Cacciapuoti
    • 1
    • 2
  • Anna Maltsev
    • 3
  • Benjamin Schlein
    • 1
  1. 1.Hausdorff Center for Mathematics, Institute for Applied MathematicsUniversity of BonnBonnGermany
  2. 2.Dipartimento di Scienza e Alta TecnologiaUniversità dell’InsubriaComoItaly
  3. 3.Department of MathematicsUniversity of BristolBristolUK

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