On Optimal Bounds in the Local Semicircle Law under Four Moment Condition

Abstract

We consider symmetric random matrices \({{{\mathbf{X}}}_{n}} = [{{X}_{{jk}}}]_{{j,k = 1}}^{n},n \geqslant 1\), whose upper triangular entries are independent random variables with zero mean and unit variance. Under the assumption \(\mathbb{E}{\text{|}}{{X}_{{jk}}}{{{\text{|}}}^{4}} < C\), j, k = 1, 2, ..., n, it is shown that the fluctuations of the Stieltjes transform m_n(z), \(z = u + i{v},{v} > 0,\) of the empirical spectral distribution function of the matrix \({{{\mathbf{X}}}_{n}}{\text{/}}\sqrt n \) about the Stieltjes transform \({{m}_{{{\text{sc}}}}}(z)\) of Wigner’s semicircle law are of order (n\({v}\))\(^{{ - 1}}\text{ln}n\). An application of the result obtained to the convergence rate in probability of the empirical spectral distribution function of \({{{\mathbf{X}}}_{n}}{\text{/}}\sqrt n \) to Wigner’s semicircle law in the uniform metric is discussed.