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29
Fluctuations of eigenvalues and second order Poincaré inequalities
, 2007
"... Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified t ..."
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Cited by 49 (5 self)
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Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out; some of them are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.
Sample eigenvalue based detection of highdimensional signals in white noise using relatively few samples
, 2007
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Sparse regular random graphs: Spectral density and eigenvectors
"... Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to ..."
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Cited by 26 (2 self)
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Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized. 1.
GAUSSIAN FLUCTUATIONS FOR β ENSEMBLES.
, 2007
"... Abstract. We study the Circular and Jacobi βEnsembles and prove Gaussian fluctuations for the number of points in one or more intervals in the macroscopic scaling limit. 1. ..."
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Cited by 12 (0 self)
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Abstract. We study the Circular and Jacobi βEnsembles and prove Gaussian fluctuations for the number of points in one or more intervals in the macroscopic scaling limit. 1.
Duality of real and quaternionic random matrices, Electron
 J. Probab
"... Abstract. We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner and Wishart families of random matrices the result g ..."
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Cited by 8 (0 self)
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Abstract. We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner and Wishart families of random matrices the result gives the duality between moments of these families and the corresponding real Wigner and Wishart families. 1.
Spectral density asymptotics for Gaussian and Laguerre βensembles in the exponentially small region
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Limit Theorems for BetaJacobi Ensembles
"... Abstract For a large betaJacobi ensemble determined by several parameters, under certain restrictions among them, we obtain both the bulk and the edge scaling limits. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the Central Limit Theorems of th ..."
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Cited by 7 (3 self)
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Abstract For a large betaJacobi ensemble determined by several parameters, under certain restrictions among them, we obtain both the bulk and the edge scaling limits. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the Central Limit Theorems of the eigenvalues, and the limiting distributions of the empirical distributions of the eigenvalues. 1
Global Fluctuations for Linear Statistics of βJacobi Ensembles, arXiv: 1203.6103
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STURM SEQUENCES AND RANDOM EIGENVALUE DISTRIBUTIONS
"... Abstract. This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation of eigenvalue distributions of random matrix ensembles. We first explore the use of Sturm sequences to efficiently compute histograms of eigenvalues for symmetric tri ..."
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Cited by 4 (2 self)
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Abstract. This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation of eigenvalue distributions of random matrix ensembles. We first explore the use of Sturm sequences to efficiently compute histograms of eigenvalues for symmetric tridiagonal matrices and apply these ideas to random matrix ensembles such as the βHermite ensemble. Using our techniques, we reduce the time to compute a histogram of the eigenvalues of such a matrix from O(n 2 + m) to O(mn) time where n is the dimension of the matrix and m is the number of bins (with arbitrary bin centers and widths) desired in the histogram (m is usually much smaller than n). Second, we derive analytic formulas in terms of iterated multivariate integrals for the eigenvalue distribution and the largest eigenvalue distribution for arbitrary symmetric tridiagonal random matrix models. As an example of the utility of this approach, we give a derivation of both distributions for the βHermite random matrix ensemble (for general β). Third, we explore the relationship between the Sturm sequence of a random matrix and its shooting eigenvectors. We show using Sturm sequences that, assuming the eigenvector contains no zeros, the number of sign changes in a shooting eigenvector of parameter λ is equal to the number of eigenvalues greater than λ. Finally, we use the techniques presented in the first section to experimentally demonstrate a O(log n) growth relationship between the variance of histogram bin values and the order of the βHermite matrix ensemble. This paper is dedicated to the fond memory of James T. Albrecht 1.
MOMENTS OF THE TRANSMISSION EIGENVALUES, PROPER DELAY TIMES AND RANDOM MATRIX THEORY II
"... Abstract. We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assumption that the LandauerBüttiker scatt ..."
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Cited by 4 (1 self)
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Abstract. We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assumption that the LandauerBüttiker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of Random Matrix Theory (RMT). The starting points are the finiten formulae that we recently discovered.53 Our analysis includes all the symmetry classes β ∈ {1, 2, 4}; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer74 and Altland and Zirnbauer.3 Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed. by Berkolaiko et al.9 and Berkolaiko and Kuipers.10;11 Our approach also applies to the Selberglike integrals. We calculate