×

zbMATH — the first resource for mathematics

Rigidity of eigenvalues of generalized Wigner matrices. (English) Zbl 1238.15017
Let \(H= (h_{ij})^N_{i,j=1}\) be an \(N \times N\) Hermitian or symmetric matrix where the matrix elements \(h_{ij} = {\bar h}_{ij}, i \leq j\), are independent random variables given by a probability measure \(\nu_{ij}\) with mean zero and variance \(\sigma^2_{ij} \geq 0\). The variances satisfy the normalization condition \(\sum_{i=1}^N \sigma^2_{ij}=1\) for any fixed \(j\) and there is a constant \(c > 0\) such that \(c \leq N \sigma^2_{ij} \leq c^{-1}\). It is also assumed that probability distributions \(\nu_{ij}\) have a uniform subexponential decay.
In this paper, it is proved that the Stieltjes transform of the empirical eigenvalue distribution of \(H\) is given by the Wigner semicircle law uniformly up to edges of the spectrum with an error of order \((N \eta)^{-1}\) where \(\eta\) is the imaginary part of the spectral parameter in the Stieltjes transform. From this strong local semicircle law three important consequences follow:
(1) Rigidity of eigenvalues: If \(\gamma_{j,N}\) denotes the classical location of the \(j\)-th eigenvalue under the semicircle law ordered in increasing order, then the \(j\)-th eigenvalue \(\lambda_i\) is close to \(\gamma_{j,N}\) in the sense that for some positive constants \(C, c\) is \[ \begin{split} {\mathbf P}(\exists j: |\lambda_i - \gamma_{j,N}| \geq (\text{log} N)^{C \text{log log}N} [\text{min}(j,N - j + 1)]^{-1/3} N^{-2/3})\\ \leq C\text{exp}[-(\text{log} N)^{c \text{log log}N}]\end{split} \] for \(N\) large enough.
(2) The proof of F. J. Dyson’s conjecture [J. Math. Phys. 3, 1191–1198 (1962; Zbl 0111.32703)], which states that the Dyson Brownian motion reaches local equilibrium at time \(t \sim N^{-1 + \delta}\) for arbitrary small \(\delta\).
(3) The edge universality in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are equal in the large \(N\) limit provided that the second moments of the two ensembles are identical.

MSC:
15B52 Random matrices (algebraic aspects)
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60B20 Random matrices (probabilistic aspects)
60F15 Strong limit theorems
60J65 Brownian motion
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alon, N.; Krivelich, M.; Vu, V., On the concentration of eigenvalues of random symmetric matrices, Israel J. math., 131, 259-267, (2002) · Zbl 1014.15016
[2] Anderson, G.; Guionnet, A.; Zeitouni, O., An introduction to random matrices, Stud. adv. math., vol. 118, (2009), Cambridge University Press
[3] Anderson, G.; Zeitouni, O., A CLT for a band matrix model, Probab. theory related fields, 134, 2, 283-338, (2006) · Zbl 1084.60014
[4] Auffinger, A.; Ben Arous, G.; Péché, S., Poisson convergence for the largest eigenvalues of heavy-tailed matrices, Ann. inst. H. Poincaré probab. statist., 45, 3, 589-610, (2009) · Zbl 1177.15037
[5] Bai, Z.D.; Miao, B.; Tsay, J., Convergence rates of the spectral distributions of large Wigner matrices, Int. math. J., 1, 1, 65-90, (2002) · Zbl 0987.60050
[6] Ben Arous, G.; Péché, S., Universality of local eigenvalue statistics for some sample covariance matrices, Comm. pure appl. math., LVIII, 1-42, (2005) · Zbl 1075.62014
[7] Biroli, G.; Bouchaud, J.-P.; Potters, M., On the top eigenvalue of heavy-tailed random matrices, Europhys. lett., 78, 10001, (2007) · Zbl 1244.82029
[8] Bleher, P.; Its, A., Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of math., 150, 185-266, (1999) · Zbl 0956.42014
[9] Brézin, E.; Hikami, S.; Brézin, E.; Hikami, S., Spectral form factor in a random matrix theory, Nuclear phys. B, Phys. rev. E, 55, 4067-4083, (1997)
[10] Deift, P., Orthogonal polynomials and random matrices: A Riemann-Hilbert approach, Courant lect. notes math., vol. 3, (1999), American Mathematical Society Providence, RI
[11] Deift, P.; Gioev, D., Random matrix theory: invariant ensembles and universality, Courant lect. notes math., vol. 18, (2009), American Mathematical Society Providence, RI · Zbl 1171.15023
[12] Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. pure appl. math., 52, 1335-1425, (1999) · Zbl 0944.42013
[13] Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. pure appl. math., 52, 1491-1552, (1999) · Zbl 1026.42024
[14] Disertori, M.; Pinson, H.; Spencer, T., Density of states for random band matrices, Comm. math. phys., 232, 83-124, (2002) · Zbl 1019.15014
[15] Dyson, F.J., A Brownian-motion model for the eigenvalues of a random matrix, J. math. phys., 3, 1191-1198, (1962) · Zbl 0111.32703
[16] Erdős, L.; Péché, G.; Ramírez, J.; Schlein, B.; Yau, H.-T., Bulk universality for Wigner matrices, Comm. pure appl. math., 63, 7, 895-925, (2010) · Zbl 1216.15025
[17] Erdős, L.; Schlein, B.; Yau, H.-T., Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann. probab., 37, 3, 815-852, (2009) · Zbl 1175.15028
[18] Erdős, L.; Schlein, B.; Yau, H.-T., Local semicircle law and complete delocalization for Wigner random matrices, Comm. math. phys., 287, 641-655, (2009) · Zbl 1186.60005
[19] Erdős, L.; Schlein, B.; Yau, H.-T., Wegner estimate and level repulsion for Wigner random matrices, Int. math. res. not., 2010, 3, 436-479, (2010) · Zbl 1204.15043
[20] Erdős, L.; Schlein, B.; Yau, H.-T., Universality of random matrices and local relaxation flow, Invent. math., 185, 1, 75-119, (2011) · Zbl 1225.15033
[21] L. Erdős, B. Schlein, H.-T. Yau, J. Yin, The local relaxation flow approach to universality of the local statistics for random matrices, Ann. Inst. H. Poincaré Probab. Statist., in press, arXiv:0911.3687. · Zbl 1285.82029
[22] Erdős, L.; Yau, H.-T.; Yin, J., Universality for generalized Wigner matrices with Bernoulli distribution, J. comb., 1, 2, 15-85, (2011) · Zbl 1235.15029
[23] L. Erdős, H.-T. Yau, J. Yin, Bulk universality for generalized Wigner matrices, Probab. Theory Related Fields, in press, arXiv:1001.3453.
[24] Guionnet, A., Large deviation upper bounds and central limit theorems for band matrices, Ann. inst. H. Poincaré probab. statist., 38, 341-384, (2002) · Zbl 0995.60028
[25] Gustavsson, J., Gaussian fluctuations of eigenvalues in the GUE, Ann. inst. H. Poincaré probab. statist., 41, 2, 151-178, (2005) · Zbl 1073.60020
[26] Johansson, K., Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, Comm. math. phys., 215, 3, 683-705, (2001) · Zbl 0978.15020
[27] Johansson, K., Universality for certain Hermitian Wigner matrices under weak moment conditions, preprint · Zbl 1279.60014
[28] Mehta, M.L., Random matrices, (1991), Academic Press New York · Zbl 0594.60067
[29] OʼRourke, S., Gaussian fluctuations of eigenvalues in Wigner random matrices, J. stat. phys., 138, 6, 1045-1066, (2009)
[30] Pastur, L.; Shcherbina, M., Bulk universality and related properties of Hermitian matrix models, J. stat. phys., 130, 2, 205-250, (2008) · Zbl 1136.15015
[31] Péché, S.; Soshnikov, A., Wigner random matrices with non-symmetrically distributed entries, J. stat. phys., 129, 5-6, 857-884, (2007) · Zbl 1139.82019
[32] Péché, S.; Soshnikov, A., On the lower bound of the spectral norm of symmetric random matrices with independent entries, Electron. commun. probab., 13, 280-290, (2008) · Zbl 1189.15046
[33] Ruzmaikina, A., Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries, Comm. math. phys., 261, 2, 277-296, (2006) · Zbl 1130.82313
[34] Sinai, Y.; Soshnikov, A., A refinement of wignerʼs semicircle law in a neighborhood of the spectrum edge, Funct. anal. appl., 32, 2, 114-131, (1998) · Zbl 0930.15025
[35] Sodin, S., The Tracy-Widom law for some sparse random matrices, J. stat. phys., 136, 5, 834-841, (2009) · Zbl 1177.82066
[36] Sodin, S., The spectral edge of some random band matrices, Ann. of math., 172, 3, 2223-2251, (2010) · Zbl 1210.15039
[37] Soshnikov, A., Universality at the edge of the spectrum in Wigner random matrices, Comm. math. phys., 207, 3, 697-733, (1999) · Zbl 1062.82502
[38] Soshnikov, A., Poisson statistics for the largest eigenvalues of Wigner matrices with heavy tails, Electron. commun. probab., 9, 82-91, (2004) · Zbl 1060.60013
[39] T. Spencer, Review article on random band matrices, in preparation.
[40] Tao, T.; Vu, V., Random matrices: universality of the local eigenvalue statistics, Acta math., 206, 1, 127-204, (2011) · Zbl 1217.15043
[41] Tao, T.; Vu, V., Random matrices: universality of local eigenvalue statistics up to the edge, preprint · Zbl 1202.15038
[42] Tao, T.; Vu, V., Random matrices: localization of the eigenvalues and the necessity of four moments, preprint · Zbl 1247.15035
[43] Tracy, C.; Widom, H., Level-spacing distributions and the Airy kernel, Comm. math. phys., 159, 151-174, (1994) · Zbl 0789.35152
[44] Tracy, C.; Widom, H., On orthogonal and symplectic matrix ensembles, Comm. math. phys., 177, 3, 727-754, (1996) · Zbl 0851.60101
[45] Wigner, E., Characteristic vectors of bordered matrices with infinite dimensions, Ann. of math., 62, 548-564, (1955) · Zbl 0067.08403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.