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14
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 27 (3 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
"... ..."
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 4 (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.
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 3 (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.
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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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.
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 2 (2 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
Published for SISSA by
, 2011
"... Loop equations and topological recursion for the arbitraryβ twomatrix model ..."
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Loop equations and topological recursion for the arbitraryβ twomatrix model
FREE PROBABILITY OF TYPE B: ANALYTIC INTERPRETATION AND APPLICATIONS
"... Abstract. In this paper we give an analytic interpretation of free convolution of type B, and provide a new formula for its computation. This formula allows us to show that free additive convolution of type B is essentially a recasting of conditionally free convolution. We put in evidence several a ..."
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Abstract. In this paper we give an analytic interpretation of free convolution of type B, and provide a new formula for its computation. This formula allows us to show that free additive convolution of type B is essentially a recasting of conditionally free convolution. We put in evidence several aspects of this operation, the most significant being its apparition as an “intertwiner ” between derivation and free convolution of type A. We also show connections between several limit theorems in type A and type B free probability. Moreover, we show that the analytical picture fits very well with the idea of considering type B random variables as infinitesimal deformations to ordinary noncommutative random variables. Free probability theory was introduced by D. Voiculescu in the eighties (see e.g. [23]). Already in his early work [22] Voiculescu has found analytical ways for computation of a number of natural operations in his theory, such as free convolution. Later on, Speicher [21] found that the combinatorics of free probability theory has to do with (“type A”) noncrossing partitions. For example, he found a description of the relation between moments and cumulants in terms of the lattice NC (A)(n) of noncrossing partitions of {1, 2,..., n}; free independence could then be phrased in terms of vanishing of mixed cumulants. We refer the reader to [16] for a detailed description.
SAMPLE SIZE COGNIZANT DETECTION OF SIGNALS IN WHITE NOISE
, 704
"... The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a computationally simple, sample eigenvalue based procedure for estimating the number of highdimensional signals in white noise when there are relativel ..."
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The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a computationally simple, sample eigenvalue based procedure for estimating the number of highdimensional signals in white noise when there are relatively few samples. We highlight a fundamental asymptotic limit of sample eigenvalue based detection of weak highdimensional signals from a limited sample size and discuss its implication for the detection of two closely spaced signals. This motivates our heuristic definition of the effective number of identifiable signals. Numerical simulations are used to demonstrate the consistency of the algorithm with respect to the effective number of signals and the superior performance of the algorithm with respect to Wax and Kailath’s “asymptotically consistent ” MDL based estimator. Index Terms — Signal detection, eigeninference, random matrices
hepth/0512102 PUPT2184 Random Matrices and the Spectrum of Nflation
, 2005
"... Nflation is a promising embedding of inflation in string theory in which many string axions combine to drive inflation [1]. We characterize the dynamics of a general Nflation model with nondegenerate axion masses. Although the precise mass of a single axion depends on compactification details in ..."
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Nflation is a promising embedding of inflation in string theory in which many string axions combine to drive inflation [1]. We characterize the dynamics of a general Nflation model with nondegenerate axion masses. Although the precise mass of a single axion depends on compactification details in a complicated way, the distribution of masses can be computed with very limited knowledge of microscopics: the shape of the mass distribution is an emergent property. We use random matrix theory to show that a typical Nflation model has a spectrum of masses distributed according to the MarčenkoPastur law. This distribution depends on a single parameter, the number of axions divided by the dimension of the moduli space. We use this result to describe the inflationary dynamics and phenomenology of a general Nflation model. We produce an ensemble of models and use numerical integration to track the axions ’ evolution and the resulting scalar power spectrum. For realistic initial conditions, the power spectrum is considerably more red than in singlefield m2φ2 inflation. We conclude that random matrix models of Nflation are surprisingly tractable and have a rich phenomenology that differs in testable ways from