Limiting Behavior of Eigenvectors of Large Wigner Matrices

Journal of Statistical Physics

Volumn 146, Issue 3, 2012, Pages 519-549

  Bai, Z D ^a   Pan, G M ^b  

^a NORTHEAST NORMAL UNIVERSITY   (China)

^b NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

Author keywords

Central limit theorem; Eigenvectors; Empirical spectral distribution function; Semicircular law; Wigner matrices

Indexed keywords

EID: 84855864044     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-011-0407-4     Document Type: Article

References (20)

1. Bai, Z. D.: Methodologies in spectral analysis of large dimensional random matrices, a review. Stat. Sin. 9, 611-677 (1999).
2. Bai, Z. D., Silverstein, J. W.: No eigenvalues outside the support of the limiting spectral distribution of large dimensional random matrices. Ann. Probab. 26, 316-345 (1998).
3. Bai, Z. D., Yao, J. F.: On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11, 1059-1092 (2005).
4. Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995).
5. Burkholder, D.: Distribution function inequalities for martingales. Ann. Probab. 1, 19-42 (1973).
6. Erdös, L., Schlein, B., Yau, H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37(3), 815-852 (2009).
7. Erdös, L., Yau, H.-T., Yin, J.: Rigidity of eigenvalues of generalized Wigner matrices. Preprint (2011).
8. Girko, V., Kirsch, W., Kutzelnigg, A.: A necessary and sufficiently conditions for the semicircle law. Random Oper. Stoch. Equ. 2, 195-202 (1994).
9. Khorunzhy, A. M., Khoruzhenko, B. A., Pastur, L. A.: Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37, 5033-5060 (1996).
10. Knowles, A., Yin, J.: Eigenvector distribution of Wigner matrices. Preprint (2011).
11. Lytova, A., Pastur, L.: Fluctuations of matrix elements of regular functions of Gaussian random matrices. J. Stat. Phys. 134, 1 (2009).
12. Pizzo, A., Renfrew, D., Soshnikov, A.: Fluctuations of matrix entries of regular functions of Wigner matrices. Preprint (2011).
13. Pizzo, A., Renfrew, D., Soshnikov, A.: On finite rank deformations of Wigner matrices. Ann. Inst. Henri Poincaré (2011).
14. Rourke, O., Renfrew, D., Soshnikov, A.: On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries.
15. Sinai, Y., Soshnikov, A.: Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bol. Soc. Bras. Mat. (N. S.) 29, 1-24 (1998).
16. Silverstein, J. W.: On the eigenvectors of large dimensional sample covariance matrices. J. Multivar. Anal. 30, 1-16 (1989).
17. Silverstein, J. W.: Weak convergence of random functions defined by the eigenvectors of sample covariance matrices. Ann. Probab. 18, 1174-1193 (1990).
18. Tao, T., Vu, V.: Random matrices: universality of eigenvectors. Preprint (2011).
19. Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548-564 (1955).
20. Wigner, E.: On the distributions of the roots for certain symmetric matrices. Ann. Math. 67, 325-327 (1958).