%0 Journal Article
%T The Local Semicircle Law for a General Class of Random Matrices
%A Laszlo Erdos
%A Antti Knowles
%A Horng-Tzer Yau
%A Jun Yin
%J Mathematics
%D 2012
%I arXiv
%X 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 [14] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \E \abs{h_{ij}}^2$. 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 $W\gg N^{1-\e_n}$ with some $\e_n>0$ 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 from [3,4,16].
%U http://arxiv.org/abs/1212.0164v4