Local semicircle law under moment conditions: Stieltjes transform, rigidity and delocalization

We consider a random symmetric matrix ${X} = [X_{jk}]_{j,k=1}^n$ where the upper triangular entries are independent identically distributed random variables with zero mean and unit variance. We additionally suppose that ${{E}} |X_{11}|^{4 + \delta} =: \mu_{4+\delta} < \infty$ 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 such as the rate of convergence of the ESD to the distribution function of the semicircle law, rigidity of the eigenvalues, and eigenvector delocalization.