EIGENVALUE VARIANCE BOUNDS FOR WIGNER AND COVARIANCE RANDOM MATRICES

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdos, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected $2$-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices.

[1]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[2]  Alexander Tikhomirov,et al.  On the rate of convergence to the semi-circular law , 2011, 1109.0611.

[3]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[4]  C. Villani.  Topics in Optimal Transportation , 2003 .

[5]  N. Pillai,et al.  Universality of covariance matrices , 2011, 1110.2501.

[6]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[7]  P. Forrester,et al.  Interrelationships between orthogonal, unitary and symplectic matrix ensembles , 1999, solv-int/9907008.

[8]  A. Guionnet,et al.  An Introduction to Random Matrices , 2009 .

[9]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[10]  T. Tao,et al.  Random matrices: Sharp concentration of eigenvalues , 2012, 1201.4789.

[11]  Sean O’Rourke,et al.  Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices , 2009, 0909.2677.

[12]  H. Yau,et al.  Rigidity of eigenvalues of generalized Wigner matrices , 2010, 1007.4652.

[13]  Z. Su.  Gaussian Fluctuations in Complex Sample Covariance Matrices , 2006 .

[14]  M. Ledoux,et al.  Small deviations for beta ensembles , 2009, 0912.5040.

[15]  A. Soshnikov.  Universality at the Edge of the Spectrum¶in Wigner Random Matrices , 1999, math-ph/9907013.

[16]  Y. Peres,et al.  Determinantal Processes and Independence , 2005, math/0503110.

[17]  T. Tao,et al.  Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge , 2009, 0908.1982.

[18]  Friedrich Götze,et al.  The rate of convergence for spectra of GUE and LUE matrix ensembles , 2005 .

[19]  A. Guionnet,et al.  CONCENTRATION OF THE SPECTRAL MEASURE FOR LARGE MATRICES , 2000 .

[20]  Jonas Gustavsson.  Gaussian fluctuations of eigenvalues in the GUE , 2004 .

[21]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[22]  T. Tao,et al.  Random Matrices: the Distribution of the Smallest Singular Values , 2009, 0903.0614.

[23]  L. Pastur,et al.  Eigenvalue Distribution of Large Random Matrices , 2011 .

[24]  Increasing subsequences and the hard-to-soft edge transition in matrix ensembles , 2002, math-ph/0205007.

[25]  Ke Wang.  RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES UP TO THE EDGE , 2011, 1104.4832.

[26]  Z. D. Bai,et al.  Necessary and Sufficient Conditions for Almost Sure Convergence of the Largest Eigenvalue of a Wigner Matrix , 1988 .

[27]  T. Tao,et al.  Random covariance matrices: Universality of local statistics of eigenvalues , 2009, 0912.0966.

[28]  F. Götze,et al.  Rate of convergence to the semicircular law for the Gaussian unitary ensemble , 2003 .

[29]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[30]  S. Bobkov,et al.  On Concentration of Empirical Measures and Convergence to the Semi-circle Law , 2010 .

[31]  Random matrices: Universality of eigenvectors , 2011 .

[32]  Friedrich Götze,et al.  Rate of convergence to the semi-circular law , 2003 .

[33]  M. Meckes,et al.  Concentration and convergence rates for spectral measures of random matrices , 2011, 1109.5997.