Electronic Journal of Probability

The local semicircle law for a general class of random matrices

László Erdős, Antti Knowles, Horng-Tzer Yau, and Jun Yin

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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.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 59, 58 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15B52: Random matrices
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Random band matrix local semicircle law universality eigenvalue rigidity

This work is licensed under a Creative Commons Attribution 3.0 License.


Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. The local semicircle law for a general class of random matrices. Electron. J. Probab. 18 (2013), paper no. 59, 58 pp. doi:10.1214/EJP.v18-2473. https://projecteuclid.org/euclid.ejp/1465064284

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