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Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices
 Electron. J. Prob
, 2011
"... Abstract. Consider a deterministic selfadjoint matrix Xn with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalised ..."
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Abstract. Consider a deterministic selfadjoint matrix Xn with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalised eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix Xn so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the nonperturbed model and fluctuate in the same scale. We generalize these results to the case when Xn is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the socalled matrix models.
Central limit theorem for linear eigenvalue statistics of random matrices with . . .
, 2009
"... We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X ∗ X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X ..."
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Cited by 11 (0 self)
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We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X ∗ X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, ∞) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5). This is done by using a simple “interpolation trick ” from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially C 5 test function. Here the variance of statistics contains an additional term proportional to κ4. The proofs of all limit theorems follow essentially the same scheme.
ON FLUCTUATIONS OF EIGENVALUES OF RANDOM PERMUTATION MATRICES
, 2013
"... Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting nonuniversality phenomenon. Though they have bounded variance, their fluctuations are asymptotically nonGaussian but infinitely divisible. The fluctuations are asymptoticall ..."
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Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting nonuniversality phenomenon. Though they have bounded variance, their fluctuations are asymptotically nonGaussian but infinitely divisible. The fluctuations are asymptotically Gaussian for less smooth linear statistics for which the variance diverges. The degree of smoothness is measured in terms of the quality of the trapezoidal approximations of the integral of the observable.
On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices
"... We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U and V(von Mises) statistics of eigenvalues of random matrices as their size tends to in nity. We show rst that for a certain class of test functions (kernels), determining the statistics, the validity of these limitin ..."
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We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U and V(von Mises) statistics of eigenvalues of random matrices as their size tends to in nity. We show rst that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices, The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent. 1
From random matrices to quasiperiodic Jacobi matrices via orthogonal polynomials
 MR2220042 (2006m:47057) Licensed to Penn St Univ, University Park. Prepared on Fri Jul 5 09:15:32 EDT 2013 for download from IP 130.203.136.75. License
"... We present an informal review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with respect to the Szegö weight and polynomials orthonormal on R with ..."
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We present an informal review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with respect to the Szegö weight and polynomials orthonormal on R with respect to varying weights and having the same union of intervals as the set of oscillations of asymptotics. In both cases we construct double infinite Jacobi matrices with generically quasiperiodic coefficients and show that each of them is an isospectral deformation of another. Related results on asymptotic eigenvalue distribution of a class of random matrices of large size are also shortly discussed. 1
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, 711
"... Central limit theorem for linear eigenvalue statistics of orthogonally invariant matrix models ..."
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Central limit theorem for linear eigenvalue statistics of orthogonally invariant matrix models