Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices

In this paper, we study the complex Wigner matrices $M_{n}=\frac{1}{\sqrt{n}}W_{n}$ whose eigenvalues are typically in the interval [−2,2]. Let λ1≤λ2⋯≤λn be the ordered eigenvalues of Mn. Under the assumption of four matching moments with the Gaussian Unitary Ensemble (GUE), for test function f 4-times continuously differentiable on an open interval including [−2,2], we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold u in the bulk of the Wigner semicircle law as $\mathcal{A}_{n}[f; u]=\sum_{l=1}^{n}f(\lambda_{l})\mathbf{1}_{\{\lambda_{l}\leq u\}}$. And the second one is $\mathcal{B}_{n}[f; k]=\sum_{l=1}^{k}f(\lambda_{l})$ with positive integer k=kn such that k/n→y∈(0,1) as n tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from $\mathcal{B}_{n}[f; \lfloor nt\rfloor]$. The main difficulty is to deal with the linear eigenvalue statistics for the test functions with several non-differentiable points. And our main strategy is to combine the Helffer-Sjöstrand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on $\mathcal{A}_{n}[f;u]$ for the real Wigner matrices will also be briefly discussed.

[1]  Jianfeng Yao,et al.  On the convergence of the spectral empirical process of Wigner matrices , 2005 .

[2]  Lebowitz,et al.  Gaussian fluctuation in random matrices. , 1994, Physical review letters.

[3]  T. Tao,et al.  Random matrices: Universality of local eigenvalue statistics , 2009, 0906.0510.

[4]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[5]  K. Mclaughlin,et al.  Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration , 2002, math-ph/0211022.

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

[7]  R. Bass.  Convergence of probability measures , 2011 .

[8]  Van Vu,et al.  A note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices , 2011 .

[9]  Limiting laws of linear eigenvalue statistics for Hermitian matrix models , 2006, math/0608719.

[10]  L. Pastur,et al.  CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES , 2008, 0809.4698.

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

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

[13]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

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

[15]  Sourav Chatterjee,et al.  Fluctuations of eigenvalues and second order Poincaré inequalities , 2007, 0705.1224.

[16]  T. Tao,et al.  A central limit theorem for the determinant of a Wigner matrix , 2011, 1111.6300.

[17]  Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry , 2005, math-ph/0508031.

[18]  On Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices with Non-identically Distributed Entries , 2011, 1104.1663.

[19]  H. Yau,et al.  Bulk universality for generalized Wigner matrices , 2010, 1001.3453.

[20]  Alexander Soshnikov.  Gaussian limit for determinantal random point fields , 2000 .

[21]  H. Yau,et al.  Universality of Sine-Kernel for Wigner Matrices with a Small Gaussian Perturbation , 2009, 0905.2089.

[22]  E. Davies,et al.  Spectral Theory and Differential Operators: Index , 1995 .

[23]  Horng-Tzer Yau,et al.  Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices , 2007, 0711.1730.

[24]  Rate of Convergence to the Semicircle Law for the Gaussian Orthogonal Ensemble , 2008 .

[25]  L. Pastur.  Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models , 2006 .

[26]  Babak Hassibi,et al.  New Null Space Results and Recovery Thresholds for Matrix Rank Minimization , 2010, ArXiv.

[27]  Sandrine Dallaporta,et al.  EIGENVALUE VARIANCE BOUNDS FOR WIGNER AND COVARIANCE RANDOM MATRICES , 2012, 1203.1597.

[28]  S. Dallaporta.  Eigenvalue variance bounds for Wigner random matrices , 2012 .

[29]  Mariya Shcherbina,et al.  Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices , 2011, 1101.3249.

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

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

[32]  Alexander Soshnikov,et al.  Central limit theorem for traces of large random symmetric matrices with independent matrix elements , 1998 .

[33]  M. M. Menon,et al.  Computing Partial Eigenvalue Sum in Electronic Structure Calculations , 1998 .

[34]  T. W. Anderson.  ASYMPTOTIC THEORY FOR PRINCIPAL COMPONENT ANALYSIS , 1963 .

[35]  Babak Hassibi,et al.  Improved thresholds for rank minimization , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[36]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

[38]  Zhidong Bai,et al.  CLT for Linear Spectral Statistics of Wigner matrices , 2009 .