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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References

  • Bai, Z. D.; Miao, Baiqi; Tsay, Jhishen. Convergence rates of the spectral distributions of large Wigner matrices. Int. Math. J. 1 (2002), no. 1, 65–90.
  • Cacciapuoti, C., Maltsev, A., Schlein, B.: Local Marchenko-Pastur Law at the Hard Edge of Sample Covariance Matrices. Preprint. arxiv:1206.1730
  • Chatterjee, Sourav. A generalization of the Lindeberg principle. Ann. Probab. 34 (2006), no. 6, 2061–2076.
  • Davies, E. B. The functional calculus. J. London Math. Soc. (2) 52 (1995), no. 1, 166–176.
  • Erdős, Łászló; Knowles, Antti. Quantum diffusion and delocalization for band matrices with general distribution. Ann. Henri Poincaré 12 (2011), no. 7, 1227–1319.
  • Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral Statistics of Erdős-Rényi Graphs I: Local Semicircle Law. To appear in Annals Prob. Preprint. Arxiv:1103.1919
  • Erdős, Łászló; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. Spectral statistics of ErdÅ‘s-Rényi Graphs II: Eigenvalue spacing and the extreme eigenvalues. Comm. Math. Phys. 314 (2012), no. 3, 587–640.
  • Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Delocalization and Diffusion Profile for Random Band Matrices. Preprint. Arxiv:1205.5669
  • Erdős, L., Knowles, A., Yau, H.-T.: Averaging Fluctuations in Resolvents of Random Band Matrices. Preprint. Arxiv:1205.5664
  • Erdős, Łászló; Péché, Sandrine; Ramírez, José A.; Schlein, Benjamin; Yau, Horng-Tzer. Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63 (2010), no. 7, 895–925.
  • Erdős, Łászló; Ramírez, José A.; Schlein, Benjamin; Yau, Horng-Tzer. Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15 (2010), no. 18, 526–603.
  • Erdős, Łászló; Schlein, Benjamin; Yau, Horng-Tzer. Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 (2009), no. 3, 815–852.
  • Erdős, Łászló; Schlein, Benjamin; Yau, Horng-Tzer. Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 (2009), no. 2, 641–655.
  • Erdős, Łászló; Schlein, Benjamin; Yau, Horng-Tzer. Universality of random matrices and local relaxation flow. Invent. Math. 185 (2011), no. 1, 75–119.
  • Erdős, Łászló; Schlein, Benjamin; Yau, Horng-Tzer; Yin, Jun. The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 1, 1–46.
  • Erdős, Łászló; Yau, Horng-Tzer. Universality of local spectral statistics of random matrices. Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 3, 377–414.
  • Erdős, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. To appear in Prob. Theor. Rel. Fields. Preprint arXiv:1001.3453.
  • Erdős, Łászló; Yau, Horng-Tzer; Yin, Jun. Universality for generalized Wigner matrices with Bernoulli distribution. J. Comb. 2 (2011), no. 1, 15–81.
  • Erdős, Łászló; Yau, Horng-Tzer; Yin, Jun. Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229 (2012), no. 3, 1435–1515.
  • Guionnet, A.; Zeitouni, O. Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5 (2000), 119–136 (electronic).
  • Helffer, B.; Sjöstrand, J. Équation de Schrödinger avec champ magnétique et équation de Harper. (French) [The Schrodinger equation with magnetic field, and the Harper equation] Schrödinger operators (Sønderborg, 1988), 118–197, Lecture Notes in Phys., 345, Springer, Berlin, 1989.
  • Feldheim, Ohad N.; Sodin, Sasha. A universality result for the smallest eigenvalues of certain sample covariance matrices. Geom. Funct. Anal. 20 (2010), no. 1, 88–123.
  • Fyodorov, Yan V.; Mirlin, Alexander D. Scaling properties of localization in random band matrices: a $\sigma$-model approach. Phys. Rev. Lett. 67 (1991), no. 18, 2405–2409.
  • V. A. Marcenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Sbornik: Mathematics 1 (1967), 457–483.
  • Mehta, Madan Lal. Random matrices. Second edition. Academic Press, Inc., Boston, MA, 1991. xviii+562 pp. ISBN: 0-12-488051-7
  • Pillai, N.S. and Yin, J.: Universality of covariance matrices. Preprint arXiv:1110.2501
  • Sodin, Sasha. The spectral edge of some random band matrices. Ann. of Math. (2) 172 (2010), no. 3, 2223–2251.
  • Spencer, Thomas. Random banded and sparse matrices. The Oxford handbook of random matrix theory, 471–488, Oxford Univ. Press, Oxford, 2011.
  • Tao, Terence; Vu, Van. Random matrices: universality of local eigenvalue statistics. Acta Math. 206 (2011), no. 1, 127–204.
  • Tao, T. and Vu, V.: Random matrices: Sharp concentration of eigenvalues. Preprint arXiv:1201.4789
  • Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 (1955), 548–564.