Local Semicircle Law under Weak Moment Conditions

F. Götze^a, A. A. Naumov^b, A. N. Tikhomirov^c, and D. A. Timushev^c

Presented by Academician of the RAS I.A. Ibragimov December 15, 2015

Received December 29, 2015

Abstract—Symmetric random matrices are considered whose upper triangular entries are independent identi-
cally distributed random variables with zero mean, unit variance, and a finite moment of order 4 + , > 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/n, where 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 delocal-
ization of eigenvectors are discussed.

DOI: 10.1134/S1064562416030029

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