## The local semicircle law for a general class of random matrices.(English)Zbl 1373.15053

Summary: 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.

### MSC:

 15B52 Random matrices (algebraic aspects) 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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