Localization and delocalization for heavy tailed band matrices

Benaych-Georges, Florent; Péché, Sandrine

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1385-1403

We consider some random band matrices with band-width $N^{\mu }$ whose entries are independent random variables with distribution tail in $x^{-\alpha }$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $\alpha <2(1+{\mu }^{-1})$, the largest eigenvalues have order $N^{(1+\mu )/\alpha }$, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices by Soshnikov in (Electron. Comm. Probab. 9 (2004) 82-91, In Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles (2006) 351-364) when $\alpha <2$ and by Auffinger et al. in (Ann. Inst. H. Poincarè Probab. Statist. 45 (2005) 589-610) when $\alpha <4$). On the other hand, when $\alpha >2(1+{\mu }^{-1})$, the largest eigenvalues have order $N^{\mu /2}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.

On considère des matrices aléatoires à structure bande dont la bande a pour largeur $N^{\mu }$ et dont les coefficients sont indépendants à queue de distribution en $x^{-\alpha }$. On s’intéresse aux plus grandes valeurs propres et aux vecteurs propres associés et prouve la transition de phase suivante. D’une part, quand $\alpha <2(1+{\mu }^{-1})$, les plus grandes valeurs propres ont pour ordre $N^{(1+\mu )/\alpha }$, sont asymptotiquement distribuées selon un processus de Poisson et les vecteurs propres associés sont essentiellement portés par deux coordonnées (ce phénomène a déjà été remarqué pour des matrices pleines par Soshnikov dans (Electron. Comm. Probab. 9 (2004) 82-91, In Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles (2006) 351-364) quand $\alpha <2$, et par Auffinger et al. dans (Ann. Inst. H. Poincarè Probab. Statist. 45 (2005) 589-610) quand $\alpha <4$). D’autre part, quand $\alpha >2(1+{\mu }^{-1})$, les plus grandes valeurs propres ont pour ordre $N^{\mu /2}$ et la plupart des vecteurs propres de la matrice sont délocalisés, i.e. approximativement uniformément distribués sur leurs $N$ coordonnées.

Classification : 15A52, 60F05
Keywords: random matrices, band matrices, heavy tailed random variables

[1] A. Auffinger, G. Ben Arous and S. Péché. Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Ann. Inst. Henri Poincaré Probab. Statist. 45 (3) (2009) 589-610. | Zbl | MR | Numdam

[2] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices, 2nd edition. Springer, New York, 2009. | Zbl | MR

[3] R. Bhatia. Matrix Analysis. Graduate Texts in Mathematics 169. Springer, New York, 1997. | Zbl | MR

[4] R. Bhatia. Perturbation Bounds for Matrix Eigenvalues. Classics in Applied Mathematics 53. Reprint of the 1987 original. SIAM, Philadelphia, PA, 2007. | Zbl | MR

[5] G. Ben Arous and A. Guionnet. The spectrum of heavy tailed random matrices. Comm. Math. Phys. 278 (3) (2007) 715-751. | Zbl | MR

[6] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1989. | Zbl | MR

[7] C. Bordenave and A. Guionnet. Localization and delocalization of eigenvectors for heavy-tailed random matrices. Preprint. Available at arXiv:1201.1862. | Zbl | MR

[8] P. Cizeau and J.-P. Bouchaud. Theory of Lévy matrices. Phys. Rev. E 50 (1994) 1810-1822.

[9] L. Devroye and G. Lugosi. Combinatorial Methods in Density Estimation. Springer, New York, 2001. | Zbl | MR

[10] L. Erdös. Universality of Wigner random matrices: A survey of recent results. Uspekhi Mat. Nauk 66 (2011) 67-198. | Zbl | MR

[11] L. Erdös and A. Knowles. Quantum diffusion and eigenfunction delocalization in a random band matrix model. Comm. Math. Phys. 303 (2) (2011) 509-554. | Zbl | MR

[12] L. Erdös and A. Knowles. Quantum diffusion and delocalization for band matrices with general distribution. Ann. Henri Poincaré 12 (7) (2011) 1227-1319. | Zbl | MR

[13] L. Erdös, A. Knowles, H.-T. Yau and J. Yin. Delocalization and diffusion profile for random band matrices. Comm. Math. Phys. 323 (2013) 367-416. | Zbl | MR

[14] L. Erdös, B. Schlein and H. T. Yau. Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 (2009), 641-655. | Zbl | MR

[15] L. Erdös, B. Schlein and H. T. Yau. Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Prob. 37 (2009) 815-852. | Zbl | MR

[16] L. Erdös, B. Schlein and H. T. Yau. Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. 2010 (2010) 436-479. | Zbl | MR

[17] W. Feller. An Introduction to Probability Theory and Its Applications, vol. II, 2nd edition. Wiley, New York, 1966. | Zbl | MR

[18] Y. V. Fyodorov and A. D. Mirlin. Scaling properties of localization in random band matrices: A $\sigma$-model approach. Phys. Rev. Lett. 67 (18) (1991) 2405-2409. | Zbl | MR

[19] A. Knowles and J. Yin. Eigenvector distribution of Wigner matrices. Probab. Theory Related Fields 155 (2013) 543-582. | Zbl | MR

[20] A. Knowles and J. Yin. The outliers of a deformed Wigner matrix. Preprint, 2012. Available at arXiv:1207.5619v1. | MR

[21] M. R. Leadbetter, G. Lindgren and H. Rootzén. Extremes and Related Properties of Random Sequences and Processes. Springer, New York, 1983. | Zbl | MR

[22] S. Péché and A. Soshnikov. Wigner random matrices with non-symmetrically distributed entries. J. Stat. Phys. 129 (5-6) (2007) 857-884. | Zbl | MR

[23] R. Resnick. Extreme Values, Regular Variation and Point Processes. Springer, New York, 1987. | Zbl | MR

[24] J. Schenker. Eigenvector localization for random band matrices with power law band width. Comm. Math. Phys. 290 (3) (2009) 1065-1097. | Zbl | MR

[25] Y. Sinai and A. Soshnikov. A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. (Russian). Funktsional. Anal. i Prilozhen. 32 (2) (1998) 56-79. 96; translation in Funct. Anal. Appl. 32 (1998), no. 2, 114-131. | Zbl | MR

[26] S. Sodin. The spectral edge of some random band matrices. Ann. Math. 172 (2010) 2223-2251. | Zbl | MR

[27] Y. Sinai and A. Soshnikov. Central limit theorem for traces of large random symmetric matrices. Bol. Soc. Brasil. Mat. 29 (1) (1998) 1-24. | Zbl | MR

[28] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (3) (1999) 697-733. | Zbl | MR

[29] A. Soshnikov. Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9 (2004) 82-91. | Zbl | MR | EuDML

[30] A. Soshnikov. Poisson statistics for the largest eigenvalues in random matrix ensembles. In Mathematical Physics of Quantum Mechanics 351-364. Lecture Notes in Phys. 690. Springer, Berlin, 2006. | Zbl | MR

[31] T. Spencer. Random banded and sparse matrices. In The Oxford Handbook of Random Matrix Theory 471-488. Oxford Univ. Press, Oxford, 2011. | Zbl | MR

[32] T. Tao and V. Vu. Random matrices: Universal properties of eigenvectors. Random Matrices Theory Appl. 1 (2012) 1150001. | Zbl | MR

[33] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 (2) (2010) 549-572. | Zbl | MR