On the least singular value of random symmetric matrices

Electronic Journal of Probability

Volumn 17, Issue , 2012, Pages

  Nguyen, Hoi H ^a  

^a UNIVERSITY OF PENNSYLVANIA   (United States)

Author keywords

Least singular values; Random symmetric matrices

Indexed keywords

EID: 84864852114     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v17-2165     Document Type: Article

References (27)

1. A. Barvinok, Polynomial time algorithms to approximate permanents and mixed discriminantswithin a simply exponential factor, Random Structures Algorithms 14 (1999), 29-61. MR-1662270
2. K. Costello, T. Tao and V. Vu, Random symmetric matrices are almost surely non-singular, Duke Math. J. 135 (2006), 395-413. MR-2267289
3. K. P. Costello and V. Vu, Concentration of random determinants and permanent estimators, Siam J. Discrete Math 23 (2009), 1356-1371. MR-2556534
4. L. Erdo{double acute}s, Universality of Wigner random matrices: a survey of recent results, arxiv. org/abs/1004.0861. MR-2859190
5. L. Erdo{double acute}s, B. Schlein and H-T. Yau, Wegner estimate and level repulsion for Wigner randommatrices, Int. Math. Res. Notices, 3 (2010), 436-479. MR-2587574
6. P. Erdo{double acute}s, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. 51 (1945), 898-902. MR-0014608
7. S. Friedland, B. Rider, and O. Zeitouni, Concentration of permanent estimators for certainlarge matrices, Ann. Appl. Probab., 14 (2004) 1559-1576. MR-2071434
8. A. Guionnet and O. Zeitouni, Concentration of the spectral measure for large matrices, Electron. Comm. Probab.5 (2000) 119-136. MR-1781846
9. G. Halász, Estimates for the concentration function of combinatorial number theory andprobability, Period. Math. Hungar. 8 (1977), no. 3-4, 197-211. MR-0494478
10. D. Kleitman, On a lemma of Littlewood and Offord on the distributions of linear combinationsof vectors, Advances in Math. 5 (1970), 155-157. MR-0265923
11. J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. III. Rec. Math. Mat. Sbornik N. S. 12, (1943), 277-286. MR-0009656
12. H. Nguyen, Inverse Littlewood-Offord problems and the singularity of random symmetricmatrices, Duke Mathematics Journal Vol. 161, 4 (2012), 545-586.
13. H. Nguyen, A continuous variant of the inverse Littlewood-Offord problem for quadraticforms, to appear in Contribution to Discrete Mathematics.
14. H. Nguyen and V. Vu, Optimal Littlewood-Offord theorems, Advances in Math., Vol. 226 6(2011), 5298-5319. MR-2775902
15. A. Odlyzko, On subspaces spanned by random selections of ±1 vectors, J. Combin. TheorySer. A 47 (1988), no. 1, 124-133. MR-0924455
16. M. Rudelson and R. Vershynin, The Littlewood-Offord Problem and invertibility of randommatrices, Advances in Mathematics 218 (2008), 600-633. MR-2407948
17. S. Smale, On the efficiency of algorithms of analysis, Bullentin of the AMS (13) (1985),87-121. MR-0799791
18. D. A. Spielman and S. H. Teng, Smoothed analysis of algorithms: why the simplex algorithmusually takes polynomial time, J. ACM 51 (2004), no. 3, 385-463. MR-2145860
19. T. Tao and V. Vu, On ±1 matrices: singularity and determinant, Random Structures Algorithms28 (2006), 1-23. MR-2187480
20. T. Tao and V. Vu, On the singularity probability of random Bernoulli matrices, Journal of theA. M. S 20 (2007), 603-673. MR-2291914
21. T. Tao and V. Vu, Random matrices: The Circular Law, Communication in ContemporaryMathematics 10 (2008), 261-307. MR-2409368
22. T. Tao and V. Vu, From the Littlewood-Offord problem to the circular law: universality ofthe spectral distribution of random matrices, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 3,377-396. MR-2507275
23. T. Tao and V. Vu, Inverse Littlewood-Offord theorems and the condition number of randommatrices, Annals of Mathematics (2) 169 (2009), no 2, 595-632. MR-2480613
24. T. Tao and V. Vu, Smooth analysis of the condition number and the least singular value, Mathematics of Computation, 79 (2010), 2333-2352. MR-2684367
25. T. Tao and V. Vu, Random matrices: universality of local eigenvalue statistics, Acta Math.206 (2011), no. 1, 127-204. MR-2784665
26. T. Tao and V. Vu, Additive Combinatorics, Cambridge Univ. Press, 2006. MR-2289012
27. R. Vershynin, Invertibility of symmetric random matrices, to appear in Random Structureand Algorithms.