Eigenvectors of random matrices of symmetric entry distributions
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- by Sean Meehan and Hoi Nguyen PDF
- Proc. Amer. Math. Soc. 147 (2019), 835-847 Request permission
Abstract:
In this short note we study a non-degeneration property of eigenvectors of symmetric random matrices with entries of symmetric sub-Gaussian distributions. Our result is asymptotically optimal under the sub-exponential regime.References
- P. Bourgade and H.-T. Yau, The eigenvector moment flow and local quantum unique ergodicity, Comm. Math. Phys. 350 (2017), no. 1, 231–278. MR 3606475, DOI 10.1007/s00220-016-2627-6
- Yael Dekel, James R. Lee, and Nathan Linial, Eigenvectors of random graphs: nodal domains, Random Structures Algorithms 39 (2011), no. 1, 39–58. MR 2839984, DOI 10.1002/rsa.20330
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Local semicircle law and complete delocalization for Wigner random matrices, Comm. Math. Phys. 287 (2009), no. 2, 641–655. MR 2481753, DOI 10.1007/s00220-008-0636-9
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann. Probab. 37 (2009), no. 3, 815–852. MR 2537522, DOI 10.1214/08-AOP421
- Hoi Nguyen, Terence Tao, and Van Vu, Random matrices: tail bounds for gaps between eigenvalues, Probab. Theory Related Fields 167 (2017), no. 3-4, 777–816. MR 3627428, DOI 10.1007/s00440-016-0693-5
- Sean O’Rourke and Behrouz Touri, On a conjecture of Godsil concerning controllable random graphs, SIAM J. Control Optim. 54 (2016), no. 6, 3347–3378. MR 3585024, DOI 10.1137/15M1049622
- S. O’Rourke and B. Touri, Controllability of random systems: universality and minimal controllability, arxiv.org/abs/1506.03125.
- Sean O’Rourke, Van Vu, and Ke Wang, Eigenvectors of random matrices: a survey, J. Combin. Theory Ser. A 144 (2016), 361–442. MR 3534074, DOI 10.1016/j.jcta.2016.06.008
- Mark Rudelson and Roman Vershynin, The Littlewood-Offord problem and invertibility of random matrices, Adv. Math. 218 (2008), no. 2, 600–633. MR 2407948, DOI 10.1016/j.aim.2008.01.010
- Mark Rudelson and Roman Vershynin, Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62 (2009), no. 12, 1707–1739. MR 2569075, DOI 10.1002/cpa.20294
- Terence Tao and Van Vu, Random matrices: universal properties of eigenvectors, Random Matrices Theory Appl. 1 (2012), no. 1, 1150001, 27. MR 2930379, DOI 10.1142/S2010326311500018
- Terence Tao and Van Vu, Random matrices have simple spectrum, Combinatorica 37 (2017), no. 3, 539–553. MR 3666791, DOI 10.1007/s00493-016-3363-4
- Roman Vershynin, Invertibility of symmetric random matrices, Random Structures Algorithms 44 (2014), no. 2, 135–182. MR 3158627, DOI 10.1002/rsa.20429
- Van Vu and Ke Wang, Random weighted projections, random quadratic forms and random eigenvectors, Random Structures Algorithms 47 (2015), no. 4, 792–821. MR 3418916, DOI 10.1002/rsa.20561
Additional Information
- Sean Meehan
- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio
- MR Author ID: 1076920
- Email: meehan.73@osu.edu
- Hoi Nguyen
- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio
- MR Author ID: 833497
- Email: nguyen.1261@math.osu.edu
- Received by editor(s): May 26, 2017
- Received by editor(s) in revised form: March 26, 2018
- Published electronically: October 31, 2018
- Additional Notes: The authors were supported by NSF grant DMS 1600782
- Communicated by: Zhen-Qing Chen
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 835-847
- MSC (2010): Primary 60B20, 97K50; Secondary 05C50
- DOI: https://doi.org/10.1090/proc/14284
- MathSciNet review: 3894921