On the norm and eigenvalue distribution of large random matrices

Annals of Probability

Volumn 27, Issue 2, 1999, Pages 913-944

  Boutet De Monvel, Anne ^a   Khorunzhy, Alexei ^b  

^a INSTITUT DE MATHÉMATIQUES DE JUSSIEU   (France)

^b B VERKIN INSTITUTE FOR LOW TEMPERATURE PHYSICS AND ENGINEERING   (Ukraine)

Author keywords

Eigenvalue distribution; Limiting theorems; Random matrices; Spectral norm

Indexed keywords

EID: 0040118450     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1022677390     Document Type: Article

References (33)

1. ANDERSON, T. W. (1984). Introduction to Multivariate Statistical Analysis. Wiley, New York.
2. BAI, Z.D. and YIN, Y. Q. (1988). Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16 1729-1741.
3. BEREZIN, F. (1973). Several notes on the Wigner distribution. Theoret. and Math. Phys. 17 1163-1176.
4. BOUGEROL, P. and LACROIX, J. (1985). Products of Random Matrices with Applications to Schroedinger Operator. Birkhäuser, Boston.
5. BOUTET DE MONVEL, A. and SHCHERBINA, M. (1995). On the norm of random matrices. Math. Notes 57 475-484.
6. BOUTET DE MONVEL, A., PASTUR, L. and SHCHERBINA, M. (1995). On the statistical mechanics approach in the random matrix theory: integrated density of states. J. Statist. Phys. 79 585-613.
7. BOVIER, A., GAYRARD, V. and PICCO, P. (1995). Large deviation principles for the Hopfleld model and the Kac-Hopfield model. Probab. Theory Related Fields 101 511-546.
8. BRONK, B. V. (1964). Accuracy of the semicircle approximation for the density of eigenvalues of random matrices. J. Math. Phys. 5 215-220.
9. COHEN, J. E., KESTEN, H. and NEWMAN C. M., eds. (1986). Random. Matrices and Their Applications. Amer. Math. Soc., Providence, RI.
10. CRISANTI, A., PALADIN, G. and VULPIANI, A. (1993). Products of Random Matrices in Statistical Physics. Springer, Berlin.
11. DI FRANCESCO, P., GINSPARG, P. and ZINN-JUSTIN, J. (1995). 2D gravity and random matrices. Phys. Rep. 254 1-133.
12. DONOGHUE, W. (1974). Monotone Matrix Functions and Analytic Continuation. Springer, Berlin.
13. GEMAN, S. (1980). A limit theorem for the norm of random matrices. Ann. Probab. 8 1318-1328.
14. GIRKO, V. L. (1975). Random Matrices. Naukova Dumka, Kiev (in Russian).
15. GIRKO, V. L. (1988). Spectral Theory of Random Matrices. Nauka, Moscow (in Russian).
16. GRENANDER, U. and SZEGÖ, G. (1958). Toeplitz Forms and Their Applications. Univ. California Press, Berkeley.
17. ISOPI, M. and NEWMAN, C. (1992). The triangle law for Lyapunov exponents of large random matrices. Comm. Math. Phys. 143 591-598.
18. KHORUNZHY, A. (1996). Eigenvalue distribution of large random matrices with correlated entries. Matematich. Fizika, Analiz i Geometria 3 80-101.
19. KHORUNZHY, A. and PASTUR, L. (1993). Limits of infinite interaction radius, dimensionality and number of components for random operators with off-diagonal randomness. Comm. Math. Phys. 153 605-646.
20. KHORUNZHY, A. and PASTUR, L. (1994). On the eigenvalue distribution of the deformed Wigner ensemble of random matrices. Adv. Soviet Math. 19 97-127.
21. KHORUNZHY, A., KHORUZHENKO, B. and PASTUR, L. (1996). Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 5033-5060.
22. LIFSHITZ, I., GREDESKUL, S. and PASTUR, L. (1988). Introduction in the Theory of Disordered Systems. Wiley, New York.
23. LOÈVE, M. (1978). Probability Theory 2. Springer, Berlin.
24. MARCHENKO, V. A. and PASTUR, L. A. (1967). Eigenvalue distribution of certain ensembles of random matrices. Mat. Sbornik 72 507-536 (English translation in Math. USSR Sb. 1 457-483).
25. MARCHENKO, V. A. and PASTUR, L. A. (1967). Eigenvalue distribution of certain ensembles of random matrices. Mat. Sbornik 72 507-536 (English translation in Math. USSR Sb. 1 457-483).
26. MEHTA, M. L. (1991). Random Matrices. Academic Press, New York.
27. PASTUR, L. and FIGOTIN, A. (1992). Spectra of Random and Almost Periodic Operators. Springer, New York.
28. PORTER, C. E., ed. (1965). Statistical Theory of Spectra: Fluctuations. Academic Press, New York.
29. SINAI, YA. and SOSHNIKOV, A. (1998). Central limit theorem for traces of large symmetric matrices with independent matrix elements. Bol. Soc. Bras. Mat. 29 1-24.
30. TRACY, C. A. and WIDOM, H. (1994). Level spacing distribution and the Airy kernel. Comm. Math. Phys. 161 289-309.
31. VOICULESCU, D., DYKEMA, K. J. and NICA, A. (1992). Free Random Variables. Amer. Math. Soc., Providence, RI.
32. WEGNER, F. (1979). Disordered system with n orbitals per site: n = ∞ limit. Phys. Rev. B 19 783-792.
33. WIGNER, E. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. 62 548-564.