Abstract
We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure ν ij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution ν ij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.
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Anderson G., Guionnet A., Zeitouni O.: An Introduction to Random Matrices Studies in advanced mathematics, 118. Cambridge University Press, Cambridge (2009)
Bleher P., Its A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185–266 (1999)
Deift, P.: Orthogonal polynomials and random matrices: a Riemann–Hilbert approach. Courant Lecture Notes in Mathematics, vol. 3. American Mathematical Society, Providence (1999)
Deift, P., Gioev, D.: Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Mathematics, vol. 18. American Mathematical Society, Providence (2009)
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. Commun. Pure Appl. Math. 52, 1335–1425 (1999)
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)
Dyson F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)
Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdős-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues. arXiv:1103.3869 (2011, preprint)
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)
Erdős L., Schlein B., Yau H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287, 641–655 (2009)
Erdős L., Schlein B., Yau H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Notices 2010(3), 436–479 (2010)
Erdős, L., Schlein, B., Yau, H.-T.: Universality of random matrices and local relaxation flow. Invent. Math. arXiv:0907.5605 (2011, to appear, preprint)
Erdős, L., Ramirez, J., Schlein, B., Yau, H.-T.: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electr. J. Prob. 15, 526–604 (2010)
Erdős, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. H. Poincaré Probab. Stat. arXiv:0911.3687 (2011, to appear, preprint)
Erdős, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. arXiv:1001.3453 (2011, preprint)
Erdős, L., Yau, H.-T., Yin, J.: Universality for generalized Wigner matrices with Bernoulli distribution. J. Combin. arXiv:1003.3813 (2011, to appear, preprint)
Erdős, L., Yau, H.-T., Yin, J.: Rigidity of eigenvalues of generalized Wigner matrices. arXiv:1007.4652 (2011, preprint)
Gustavsson J.: Gaussian Fluctuations of Eigenvalues in the GUE. Ann. Inst. H. Poincaré Probab. Stat. 41(2), 151–178 (2005)
Johansson K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001)
Johansson, K.: Universality for certain Hermitian Wigner matrices under weak moment conditions. arXiv:0910.4467 (2011, preprint)
O’Rourke S.: Gaussian fluctuations of eigenvalues in Wigner random matrices. J. Stat. Phys. 138(6), 1045–1066 (2009)
Pastur L., Shcherbina M.: Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130(2), 205–250 (2008)
Sinai Y., Soshnikov A.: A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge. Funct. Anal. Appl. 32(2), 114–131 (1998)
Sodin, S.: The spectral edge of some random band matrices. arXiv:0906.4047 (2011, preprint)
Soshnikov A.: Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207(3), 697–733 (1999)
Tao, T., Vu, V.: Random matrices: universality of the local eigenvalue statistics. Acta Math. arXiv:0906.0510 (2011, to appear, preprint)
Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics up to the edge. arXiv:0908.1982 (2011, preprint)
Tao, T., Vu, V.: Random matrices: universal properties of eigenvectors. arXiv:1103.2801 (2011, preprint)
Tracy C., Widom H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)
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A. Knowles was partially supported by NSF grant DMS-0757425 and J. Yin was partially supported by NSF grant DMS-1001655.
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Knowles, A., Yin, J. Eigenvector distribution of Wigner matrices. Probab. Theory Relat. Fields 155, 543–582 (2013). https://doi.org/10.1007/s00440-011-0407-y
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DOI: https://doi.org/10.1007/s00440-011-0407-y