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2013 The local semicircle law for a general class of random matrices
László Erdős, Antti Knowles, Horng-Tzer Yau, Jun Yin
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Electron. J. Probab. 18: 1-58 (2013). DOI: 10.1214/EJP.v18-2473


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.


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László Erdős. Antti Knowles. Horng-Tzer Yau. Jun Yin. "The local semicircle law for a general class of random matrices." Electron. J. Probab. 18 1 - 58, 2013.


Accepted: 29 May 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1373.15053
MathSciNet: MR3068390
Digital Object Identifier: 10.1214/EJP.v18-2473

Primary: 15B52
Secondary: 82B44

Keywords: eigenvalue rigidity , Local semicircle law , Random band matrix , Universality

Vol.18 • 2013
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