2009

Random covariance matrices: Universality of local statistics of eigenvalues

Tao, Terence, Vu, Van

Understand

We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$ whose entries are i.i.d.

random variables of mean zero, variance one, and having finite $C_0$th moment for some sufficiently large constant $C_0$.
The main result of this paper is a Four Moment theorem for i.i.d.
covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math.

Built on

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Then

Erdős, LászlóL., Péché, SandrineS., Ramírez, José A.J. A., Schlein, BenjaminB. andYau, Horng-TzerH.-T. (2010). Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63 895–925
2010
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Erdős, LászlóL., Ramírez, JoséJ., Schlein, BenjaminB., Tao, TerenceT., Vu, VanV. andYau, Horng-TzerH.-T. (2010). Bulk universality for Wigner Hermitian matrices with subexponential decay. Math. Res. Lett. 17 667–674
2010
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Erdős, LászlóL., Ramírez, José A.J. A., Schlein, BenjaminB. andYau, Horng-TzerH.-T. (2010). Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15 526–603
2010
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Erdős, LászlóL., Schlein, BenjaminB. andYau, Horng-TzerH.-T. (2010). Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. IMRN 3 436–479
2010
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Erdős, L.L., Yau, H. T.H. T. andYin, J.J. (2010). Bulk universality for generalized Wigner matrices. Preprint. Available at arXiv:1001.3453
Original
2010
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Erdős, L.L., Yau, H. T.H. T. andYin, J.J. (2010). Rigidity of eigenvalues of generalized Wigner matrices. Preprint. Available at arXiv:1007.4652
Original
2010
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Feldheim, Ohad N.O. N. andSodin, SashaS. (2010). A universality result for the smallest eigenvalues of certain sample covariance matrices. Geom. Funct. Anal. 20 88–123
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Tao, TerenceT. andVu, VanV. (2010). Random matrices: The distribution of the smallest singular values. Geom. Funct. Anal. 20 260–297
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Tao, TerenceT. andVu, VanV. (2010). Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 549–572
2010
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Tao, T.T. andVu, V.V. (2011). Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 127–204
2011
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