Local semicircle law under weak moment conditions
- 39 Downloads
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.
Unable to display preview. Download preview PDF.
- 11.F. Götze, A. Naumov, and A. Tikhomirov, “Local semicircle law under moment conditions. Part I: The Stieltjes transform,” arXiv:1510.07350, 1–54 (2015).Google Scholar
- 13.F. Götze and A. Tikhomirov, Probab. Theory Relat. Fields, 1–71 (2015). http://link.springer.com/article/10.1007%2Fs00440-015-0629-5Google Scholar
- 15.F. Götze, A. Naumov, and A. Tikhomirov, “Local semicircle law under moment conditions. Part II: Localization and delocalization,” arXiv:1511.00862, 1–33 (2015).Google Scholar