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Delocalization of eigenvectors of random matrices with independent entries. (English) Zbl 1352.60007
Summary: We prove that an \(n\times n\) random matrix \(G\) with independent entries is completely delocalized. Suppose that the entries of \(G\) have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of \(G\) have all coordinates of magnitude \(O(n-1/2)\), modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices.

MSC:
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
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[1] F. Benaych-Georges and S. Péché, Localization and delocalization for heavy tailed band matrices , Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), 1385-1403. · Zbl 1307.15054
[2] A. Bloemendal, L. Erdős, A. Knowles, H.-T. Yau, and J. Yin, Isotropic local laws for sample covariance and generalized Wigner matrices , Electron. J. Probab. 19 (2014), no. 33. · Zbl 1288.15044
[3] C. Bordenave and A. Guionnet, Localization and delocalization of eigenvectors for heavy-tailed random matrices , Probab. Theory Related Fields 157 (2013), 885-953. · Zbl 1296.15019
[4] C. Cacciapuoti, A. Maltsev, and B. Schlein, Local Marchenko-Pastur law at the hard edge of sample covariance matrices , J. Math. Phys. 54 (2013), no. 043302. · Zbl 1282.15031
[5] L. Erdős, “Universality for random matrices and log-gases” in Current Developments in Mathematics 2012 , International Press, Somerville, Mass., 2013, 59-132. · Zbl 1291.15086
[6] L. Erdős and A. Knowles, Quantum diffusion and delocalization for band matrices with general distribution , Ann. Henri Poincaré 12 (2011), 1227-1319. · Zbl 1247.15033
[7] L. Erdős and A. Knowles, Quantum diffusion and eigenfunction delocalization in a random band matrix model , Comm. Math. Phys. 303 (2011), 509-554. · Zbl 1226.15024
[8] L. Erdős, A. Knowles, H.-T. Yau, and J. Yin, Spectral statistics of Erdős-Rényi graphs, II: Eigenvalue spacing and the extreme eigenvalues , Comm. Math. Phys. 314 (2012), 587-640. · Zbl 1251.05162
[9] L. Erdős, A. Knowles, H.-T. Yau, and J. Yin, Delocalization and diffusion profile for random band matrices , Comm. Math. Phys. 323 (2013), 367-416. · Zbl 1279.15027
[10] L. Erdős, A. Knowles, H.-T. Yau, and J. Yin, Spectral statistics of Erdős-Rényi graphs, I: Local semicircle law , Ann. Probab. 41 (2013), 2279-2375. · Zbl 1272.05111
[11] L. Erdős, B. Schlein, and H.-T. Yau, Local semicircle law and complete delocalization for Wigner random matrices , Comm. Math. Phys. 287 (2009), 641-655. · Zbl 1186.60005
[12] L. Erdős, B. Schlein, and H.-T. Yau, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices , Ann. Probab. 37 (2009), 815-852. · Zbl 1175.15028
[13] L. Erdős, B. Schlein, and H.-T. Yau, Wegner estimate and level repulsion for Wigner random matrices , Int. Math. Res. Not. IMRN 2010 , no. 3, 436-479. · Zbl 1204.15043
[14] L. Erdős and H.-T. Yau, Universality of local spectral statistics of random matrices , Bull. Amer. Math. Soc. (N.S.) 49 (2012), 377-414. · Zbl 1263.15032
[15] L. Erdős, H.-T. Yau, and J. Yin, Bulk universality for generalized Wigner matrices , Probab. Theory Related Fields 154 (2012), 341-407. · Zbl 1277.15026
[16] L. Erdős, H.-T. Yau, and J. Yin, Rigidity of eigenvalues of generalized Wigner matrices , Adv. Math. 229 (2012), 1435-1515. · Zbl 1238.15017
[17] J. Fischmann, W. Bruzda, B. A. Khoruzhenko, H.-J. Sommers, and K. Życzkowski, Induced Ginibre ensemble of random matrices and quantum operations , J. Phys. A 45 , no. 7 (2012), art. ID 075203. · Zbl 1241.81011
[18] P. J. Forrester and T. Nagao, Eigenvalue statistics of the real Ginibre ensemble , Phys. Rev. Lett. 99 (2007), no. 050603.
[19] Y. V. Fyodorov and H.-J. Sommers. Random matrices close to Hermitian or unitary: Overview of methods and results , J. Phys. A 36 , no. 12 (2003), 3303-3347. · Zbl 1069.82006
[20] G. H. Golub and C. F. Van Loan, Matrix Computations , 3rd ed., Johns Hopkins Univ. Press, Baltimore, 1996. · Zbl 0865.65009
[21] R. A. Janik, W. Norenberg, M. A. Nowak, G. Papp, and I. Zahed, Correlations of eigenvectors for non-Hermitian random-matrix models , Phys. Rev. E 60 (1999), 2699-2705.
[22] E. Kanzieper and G. Akemann, Statistics of real eigenvalues in Ginibre’s ensemble of random real matrices , Phys. Rev. Lett. 95 , no. 23 (2005), 230201. · Zbl 1136.82019
[23] A. Knowles and J. Yin, The isotropic semicircle law and deformation of Wigner matrices , Comm. Pure Appl. Math. 66 (2013), 1663-1750. · Zbl 1290.60004
[24] R. Latala, P. Mankiewicz, K. Oleszkiewicz, and N. Tomczak-Jaegermann, Banach-Mazur distances and projections on random subgaussian polytopes , Discrete Comput. Geom. 38 (2007), 29-50. · Zbl 1134.52016
[25] B. Mehlig and J. T. Chalker, Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles , J. Math. Phys. 41 (2000), 3233-3256. · Zbl 0977.82023
[26] M. Rudelson and R. Vershynin, The Littlewood-Offord problem and invertibility of random matrices , Adv. Math. 218 (2008), 600-633. · Zbl 1139.15015
[27] M. Rudelson and R. Vershynin, Smallest singular value of a random rectangular matrix , Comm. Pure Appl. Math. 62 (2009), 1707-1739. · Zbl 1183.15031
[28] M. Rudelson and R. Vershynin, “Non-asymptotic theory of random matrices: Extreme singular values” in Proceedings of the International Congress of Mathematicians, III , Hindustan Book Agency, New Delhi, 2010, 1576-1602. · Zbl 1227.60011
[29] M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-Gaussian concentration , Electron. Commun. Probab. 18 (2013), no. 82. · Zbl 1329.60056
[30] T. Tao and V. Vu, On random \(\pm1\) matrices: Singularity and determinant , Random Structures Algorithms 28 (2006), 1-23. · Zbl 1086.60008
[31] T. Tao and V. Vu, Random matrices: Universality of ESDs and the circular law , with an appendix by M. Krishnapur, Ann. Probab. 38 (2010), 2023-2065. · Zbl 1203.15025
[32] T. Tao and V. Vu, Random matrices: Universal properties of eigenvectors , Random Matrices Theory Appl. 1 (2012), no. 1150001. · Zbl 1248.15031
[33] M. Timme, F. Wolf, and T. Geisel, Topological speed limits to network synchronization , Phys. Rev. Lett. 92 , no. 7 (2004), art. ID 074101.
[34] L. V. Tran, V. H. Vu, and K. Wang, Sparse random graphs: Eigenvalues and eigenvectors , Random Structures Algorithms 42 (2013), 110-134. · Zbl 1257.05089
[35] R. Vershynin, “Introduction to the non-asymptotic analysis of random matrices” in Compressed Sensing , Cambridge Univ. Press, Cambridge, 2012, 210-268.
[36] V. Vu and K. Wang, Random weighted projections, random quadratic forms and random eigenvectors , Random Structures Algorithms, published electronically 2 July 2014. · Zbl 1384.60029
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