Largest eigenvalues and eigenvectors of band or sparse random matrices

Florent Benaych-Georges (Université Paris Descartes)
Sandrine Péché (Université Paris Diderot)


In this text, we consider an $N$ by $N$ random matrix $X$ such that all but $o(N)$ rows of $X$ have $W$ non identically zero entries, the other rows having less than $W$ entries  (such as, for example, standard or cyclic band matrices). We always suppose that  $1 \ll W \ll N$. We first prove that if the entries are independent, centered,  have variance  one, satisfy a certain tail upper-bound condition and $W \gg (\log N)^{6(1+\alpha)}$, where $\alpha$ is a positive parameter depending on the distribution of the entries, then the  largest eigenvalue of $X/\sqrt{W}$ converges to the upper bound of its limit spectral distribution, that is $2$, as for Wigner matrices. This extends some previous results by Khorunzhiy and Sodin where less hypotheses were made on $W$, but more hypotheses were made about the law of the entries and the structure of the matrix. Then, under the same hypotheses, we prove a delocalization result for the eigenvectors of $X$, precisely  that most of them cannot be essentially localized on less than $W/\log(N)$ entries. This lower bound on the localization length has to be compared to the recent result by Steinerberger, which states that the localization length in the edge is $\ll W^{7/5}$ or there is strong interaction between two eigenvectors in an interval oflength $W^{7/5}$.

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Pages: 1-9

Publication Date: January 30, 2014

DOI: 10.1214/ECP.v19-3027


  • Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5 MR2760897
  • Bai, Zhidong; Silverstein, Jack W. Spectral analysis of large dimensional random matrices. Second edition. Springer Series in Statistics. Springer, New York, 2010. xvi+551 pp. ISBN: 978-1-4419-0660-1 MR2567175
  • F. Benaych-Georges, S. Péché Localization and delocalization for heavy tailed band matrices, to appear in Ann. Inst. Henri Poincaré Probab. Stat. arXiv:1210.7677
  • Bogachev, L. V.; Molchanov, S. A.; Pastur, L. A. On the density of states of random band matrices. (Russian) Mat. Zametki 50 (1991), no. 6, 31--42, 157; translation in Math. Notes 50 (1991), no. 5-6, 1232--1242 (1992) MR1150631
  • Erdős, László; Knowles, Antti. Quantum diffusion and eigenfunction delocalization in a random band matrix model. Comm. Math. Phys. 303 (2011), no. 2, 509--554. MR2782623
  • Erdős, László; Knowles, Antti. Quantum diffusion and delocalization for band matrices with general distribution. Ann. Henri Poincaré 12 (2011), no. 7, 1227--1319. MR2846669
  • Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. Delocalization and diffusion profile for random band matrices. Comm. Math. Phys. 323 (2013), no. 1, 367--416. MR3085669
  • Erdős, László; 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. MR2537522
  • 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. MR1130103
  • Füredi, Z.; Komlös, J. The eigenvalues of random symmetric matrices. Combinatorica 1 (1981), no. 3, 233--241. MR0637828
  • Khorunzhiy, Oleksiy. Estimates for moments of random matrices with Gaussian elements. Séminaire de probabilités XLI, 51--92, Lecture Notes in Math., 1934, Springer, Berlin, 2008. MR2483726
  • Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces. Isoperimetry and processes. Reprint of the 1991 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2011. xii+480 pp. ISBN: 978-3-642-20211-7 MR2814399
  • Schenker, Jeffrey. Eigenvector localization for random band matrices with power law band width. Comm. Math. Phys. 290 (2009), no. 3, 1065--1097. MR2525652
  • Sodin, Sasha. The spectral edge of some random band matrices. Ann. of Math. (2) 172 (2010), no. 3, 2223--2251. MR2726110
  • Spencer, Thomas. Random banded and sparse matrices. The Oxford handbook of random matrix theory, 471--488, Oxford Univ. Press, Oxford, 2011. MR2932643
  • S. Steinerberger. On Eigenvectors of Random Band Matrices with Large Band, arXiv:1307.5753

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