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21
Beta ensembles, stochastic Airy spectrum, and a diffusion
, 2008
"... We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the lowlying eigenvalues of the random Schrödinger operator − d2 dx 2 + x + 2 √ β b ′ x restricted to the positive halfline, where b ′ x is white noise. In doing so we extend the definit ..."
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Cited by 67 (9 self)
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We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the lowlying eigenvalues of the random Schrödinger operator − d2 dx 2 + x + 2 √ β b ′ x restricted to the positive halfline, where b ′ x is white noise. In doing so we extend the definition of the TracyWidom(β) distributions to all β> 0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a onedimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how TracyWidom laws arise from a functional central limit theorem.
Global spectrum fluctuations for the βHermite and βLaguerre ensembles via matrix models
 J. Math. Phys
, 2006
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Spectral density asymptotics for Gaussian and Laguerre βensembles in the exponentially small region
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EIGENVALUE SEPARATION IN SOME RANDOM MATRIX MODELS
, 2008
"... The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount c/(2N) 1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support ..."
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Cited by 5 (2 self)
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The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount c/(2N) 1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided c> 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the size of the matrices are fixed and c → ∞, and higher rank analogues of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions, and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogues an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.
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.
Scaling Limits of Correlations of Characteristic Polynomials for the Gaussian βEnsemble with
 External Source, Int. Math. Res. Notices
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UNIVERSALITY AT ZERO AND THE SPECTRUM EDGE FOR FIXED TRACE GAUSSIAN βENSEMBLES OF RANDOM MATRICES
, 2009
"... Consider fixed trace Gaussian βensembles (GβEs), closely related to Gaussian βensembles. For all β, we prove that universal limits of correlation functions for fixed trace GβEs and GβEs are equivalent at zero and the edge of the spectrum. As corollaries, we prove universality at zero and the spe ..."
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Cited by 2 (1 self)
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Consider fixed trace Gaussian βensembles (GβEs), closely related to Gaussian βensembles. For all β, we prove that universal limits of correlation functions for fixed trace GβEs and GβEs are equivalent at zero and the edge of the spectrum. As corollaries, we prove universality at zero and the spectrum edge for fixed trace GOE, GUE and GSE.
Moments of the Gaussian β ensembles and the largeN expansion of the densities
 J. Math. Phys
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