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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064284

Digital Object Identifier
doi:10.1214/EJP.v18-2473

Mathematical Reviews number (MathSciNet)
MR3068390

Zentralblatt MATH identifier
1373.15053

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

Keywords
Random band matrix local semicircle law universality eigenvalue rigidity

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

Citation

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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