The local semicircle law for a general class of random matrices

We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \mathbb{E} \left|h_{ij}\right|^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width  $W\gg N^{1-\varepsilon_n}$ with some $\varepsilon_n>0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments.