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Local semicircle law under moment conditions. Part I: The Stieltjes transformстатья
- Авторы: Naumov A.A., Tikhomirov A.N., Goetze F.
- Журнал: ArXiv e-prints
- Номер: 1510.07350
- Год издания: 2015
- Первая страница: 1
- Последняя страница: 52
- Аннотация: We consider a random symmetric matrix $\X=[X_{jk}]_{j,k=1}^n$ in which the upper triangular entries are independent identically distributed random variables with mean zero and unit variance. We additionally suppose that $\E|X_{11}|^{4+\delta}=:\mu_4<\infty$ for some $\delta>0$. Under these conditions we show that the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-1/2}\X$ and Wigner's semicircle law is of order $(nv)^{−1}$, where $v$ is the distance in the complex plane to the real line. Furthermore we outline applications which are deferred to a subsequent paper, such as the rate of convergence in probability of the ESD to the distribution function of the semicircle law, rigidity of the eigenvalues and eigenvector delocalization.
- Добавил в систему: Наумов Алексей Александрович