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Doklady Mathematics

, Volume 93, Issue 3, pp 248–250 | Cite as

Local semicircle law under weak moment conditions

  • F. GötzeEmail author
  • A. A. Naumov
  • A. N. Tikhomirov
  • D. A. Timushev
Mathematics

Abstract

Symmetric random matrices are considered whose upper triangular entries are independent identically distributed random variables with zero mean, unit variance, and a finite moment of order 4 + δ, δ > 0. It is shown that the distances between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law are of order lnn/nv, where v is the distance to the real axis in the complex plane. Applications concerning the convergence rate in probability to the semicircle law, localization of eigenvalues, and delocalization of eigenvectors are discussed.

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • F. Götze
    • 1
    Email author
  • A. A. Naumov
    • 2
  • A. N. Tikhomirov
    • 3
  • D. A. Timushev
    • 3
  1. 1.University of BielefeldBielefeldGermany
  2. 2.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia
  3. 3.Komi Center of Science, Ural BranchRussian Academy of SciencesSyktyvkarRussia

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