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

• 成果类型: Article
• 署名作者: Cacciapuoti, Claudio; Maltsev, Anna; Schlein, Benjamin
• 署名单位: University of Bonn; University of Bristol
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0586-4
• 发表日期: 2015
• 页码: 1-59
• 关键词: bulk universality; spectral statistics; local statistics; semicircle law; delocalization; edge; fluctuations; eigenvalues; diffusion

摘要

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 ErdAs 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 and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of ErdAs et al. (Adv Math 229(3):1435-1515, 2012; Electron J Probab 18(59):1-58, 2013), Gotze and Tikhomirov (2013).