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.


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