On the concentration of random multilinear forms and the universality of random block matrices

Probability Theory and Related Fields

Volumn 162, Issue 1-2, 2015, Pages 97-154

  Nguyen, Hoi H ^a   O’Rourke, Sean ^b  

^a OHIO STATE UNIVERSITY   (United States)

^b YALE UNIVERSITY   (United States)

Author keywords

11B25; 15A52; 15A63

Indexed keywords

EID: 84929961266     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-014-0567-7     Document Type: Article

References (37)

1. Bai, Z.D., Silverstein, J., Yin, Y.Q.: A note on the largst eigenvalue of large-dimensional sample covariance matrix. J. Multivar. Anal. 26, 166–168 (1988)
2. Bai, Z.D., Silverstein, J.: Spectral analysis of large dimensional random matrices. In: Mathematics Monograph Series, vol. 2. Science Press, Beijing (2006)
3. Bai, Z.D.: Circular law. Ann. Probab. 25, 494–529 (1997)
4. Benaych-Georges, F., Chapon, F.: Random right eigenvalues of Gaussian quaternionic matrices. Random Matrices Theory Appl. 1 (2012)
5. Bordenave, C., Chafai, D.: Around the circular law. Probab. Surv. 9, 1–89 (2012)
6. Costello, K., Tao, T., Vu, V.: Random symmetric matrices are almost surely non-singular. Duke Math. J. 135, 395–413 (2006)
7. Edelman, A.: The probability that a random real Gaussian matrix has $$k$$k real eigenvalues, related distributions, and the circular law. J. Multivar. Anal. 60, 203–232 (1997)
8. Erdős, P.: On a lemma of Littlewood and Offord. Bull. Am. Math. Soc. 51, 898–902 (1945)
9. Ginibre, J.: Statistical ensembles of complex, quaternion and real matrices. J. Math. Phys. 6, 440–449 (1965)
10. Girko, V.L.: Circular law. Theory Probab. Appl. 29, 694–706 (1984)
11. Girko, V.L.: The strong circular law, twenty years later, II. Random Oper. Stoch. Equ. 12(3), 255–312 (2004)
12. Goldsheid, I., Khoruzhenko, B.A.: The Thouless formula for random non-Hermitian Jacobi matrices. Isr. J. Math. 148, 331–346 (2005)
13. Götze, F., Tikhomirov, T.: The circular law for random matrices. Ann. Probab. 38(4), 1444–1491 (2010)
14. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
15. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1991)
16. Littlewood, J.E., Offord, A.C.: On the number of real roots of a random algebraic equation. III. Rec. Math. Mat. Sbornik N. S. 12, 277–286 (1943)
17. Mehta, M.L.: Random Matrices and the Statistical Theory of Energy Levels. Academic Press, Dublin (1967)
18. Mehta, M.L.: Random Matrices, 3rd edn. Elsevier/Academic Press, Amsterdam (2004)
19. Nguyen, H., O’Rourke, S.: The elliptic law. arXiv:1208.5883 [math.PR]
20. Nguyen, H.: Inverse Littlewood–Offord problems and the singularity of random symmetric matrices. Duke Math. J. 161(4), 545–586 (2012)
21. Nguyen, H., Vu, V.: Optimal Littlewood–Offord theorems. Adv. Math. 226(6), 5298–5319 (2011)
22. Nguyen, H., Vu, V.: Random matrices: law of the determinant. Ann. Probab. 42(1), 146–167 (2014)
23. Nguyen, H., Vu, V.: Small probability, inverse theorems, and applications. Bolyai Soc. Math. Stud. 25, 409–463 (2013)
24. O’Rourke, S., Renfrew, D.: Low rank perturbations of large elliptic random matrices. arXiv:1309.5326 [math.PR]
25. Pan, G., Zhou, W.: Circular law, extreme singular values and potential theory. J. Multivar. Anal. 101, 645–656 (2010)
26. Rudelson, M., Vershynin, R.: The Littlewood–Offord problem and invertibility of random matrices. Adv. Math. 218, 600–633 (2008)
27. Soshnikov, A.: A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Stat. Phys. 108, 1033–1056 (2002)
28. Tao, T., Vu, V.: From the Littlewood–Offord problem to the circular law: universality of the spectral distribution of random matrices. Bull. Am. Math. Soc. (N.S.) 46(3), 377–396 (2009)
29. Tao, T., Vu, V.: Inverse Littlewood–Offord theorems and the condition number of random matrices. Ann. Math. (2) 169(2), 595–632 (2009)
30. Tao, T., Vu, V.: On the singularity probability of random Bernoulli matrices. J. Am. Math. Soc. 20, 603–628 (2007)
31. Tao, T., Vu, V.: Random matrices: the circular law. Commun. Contemp. Math. 10, 261–307 (2008)
32. Tao, T., Vu, V.: Smooth analysis of the condition number and the least singular value. Math. Comput. 79, 2333–2352 (2010)
33. Tao, T., Vu, V.: Random matrices: universality of ESDs and the circular law. Ann. Probab. 38(5), 2023–2065 (2010)
34. Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics. Acta Math. 206, 127–204 (2011)
35. Vershynin, R.: Invertibility of symmetric random matrices. Random Struct. Algorithm 44, 135–182 (2014)
36. Wigner, E.P.: On the distributions of the roots of certain symmetric matrices. Ann. Math. 67, 325–327 (1958)
37. Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)