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273
Shape fluctuations and random matrices
, 1999
"... We study a certain random growth model in two dimensions closely related to the onedimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the TracyWidom largest eigenvalue distribution for the Gaussian Uni ..."
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Cited by 235 (10 self)
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We study a certain random growth model in two dimensions closely related to the onedimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the TracyWidom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE).
On the distribution of the largest eigenvalue in principal components analysis
 Ann. Statist
, 2001
"... Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribu ..."
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Cited by 196 (1 self)
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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The
Matrix models for betaensembles
 J. Math. Phys
, 2002
"... This paper constructs tridiagonal random matrix models for general (β> 0) βHermite (Gaussian) and βLaguerre (Wishart) ensembles. These generalize the wellknown Gaussian and Wishart models for β = 1,2,4. Furthermore, in the cases of the βLaguerre ensembles, we eliminate the exponent quantization ..."
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Cited by 86 (19 self)
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This paper constructs tridiagonal random matrix models for general (β> 0) βHermite (Gaussian) and βLaguerre (Wishart) ensembles. These generalize the wellknown Gaussian and Wishart models for β = 1,2,4. Furthermore, in the cases of the βLaguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.
General Orthogonal Polynomials
 in “Encyclopedia of Mathematics and its Applications,” 43
, 1992
"... Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed. ..."
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Cited by 60 (6 self)
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Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
Limiting Distributions for a Polynuclear Growth Model With External Sources
, 2000
"... The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources, which was considered by Pr ahofer and Spohn in [13]. Depending on the strength of the sources, the limiting distribution functions are either the TracyWidom func ..."
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Cited by 56 (9 self)
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The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources, which was considered by Pr ahofer and Spohn in [13]. Depending on the strength of the sources, the limiting distribution functions are either the TracyWidom functions of random matrix theory, or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.
The Asymptotics of Monotone Subsequences of Involutions
, 2001
"... We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the ..."
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Cited by 48 (5 self)
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We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the number of fixed points, (1) the TracyWidom distributions for the laxgest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of the authors in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the RiemannHilbert analysis for the orthogonal polynomials by Delft, Johansson and the first author in [3].
On the Distribution of the Largest Principal Component
 ANN. STATIST
, 2000
"... Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degr ..."
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Cited by 47 (0 self)
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Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n=p = 1: When centered by p = ( p n 1+ p p) 2 and scaled by p = ( p n 1+ p p)(1= p n 1+1= p p) 1=3 the distribution of x (1) approaches the TracyWidom law of order 1, which is dened in terms of the Painleve II dierential equation, and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to ...
Large n limit of Gaussian random matrices with external source, part III: double scaling limit in the critical case, in preparation
"... Abstract. We continue the study of the Hermitian random matrix ensemble with external source ..."
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Cited by 46 (14 self)
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Abstract. We continue the study of the Hermitian random matrix ensemble with external source
The RiemannHilbert approach to strong asymptotics for orthogonal polynomials on [1, 1]
"... We consider polynomials that are orthogonal on [1, 1] with respect to a modified Jacobi weight (1  x) # (1 + x) # h(x), with #, # > 1 and h real analytic and stricly positive on [1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [ ..."
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Cited by 45 (23 self)
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We consider polynomials that are orthogonal on [1, 1] with respect to a modified Jacobi weight (1  x) # (1 + x) # h(x), with #, # > 1 and h real analytic and stricly positive on [1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [1, 1], for the recurrence coe#cients and for the leading coe#cients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants. For the asymptotic analysis we use the steepest descent technique for RiemannHilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szego function associated with the weight and for the local analysis around the endpoints 1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions. 1 Supported by FWO research project G.0176.02 and by INTAS project 00272 2 Supported by NSF grant #DMS9970328 3 Supported by FWO research project G.0184.01 and by INTAS project 00272 4 Research Assistant of the Fund for Scientific Research  Flanders (Belgium) 1 1