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Random matrix theory
, 2005
"... Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We includ ..."
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Cited by 82 (4 self)
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Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.
A universality result for the smallest eigenvalues of certain sample covariance matrices
, 2009
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On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
, 2010
"... In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and their var ..."
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Cited by 36 (4 self)
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In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.
Developments in random matrix theory
 J. Phys. A: Math. Gen
, 2000
"... In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given. 1 1 ..."
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Cited by 24 (0 self)
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In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given. 1 1
Random covariance matrices: Universality of local statistics of eigenvalues up to the edge
 Random Matrices Theory Appl
"... Abstract. We study the universality of the eigenvalue statistics of the covariance matrices 1 ..."
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Cited by 15 (2 self)
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Abstract. We study the universality of the eigenvalue statistics of the covariance matrices 1
Universality in unitary random matrix ensembles when the soft edge meets the hard edge. arXiv:mathph/0701003
"... Dedicated to Percy Deift on the occasion of his sixtieth birthday Abstract. Unitary random matrix ensembles Z −1 n,N (detM)α exp(−N TrV (M))dM defined on positive definite matrices M, where α> −1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically va ..."
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Cited by 15 (3 self)
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Dedicated to Percy Deift on the occasion of his sixtieth birthday Abstract. Unitary random matrix ensembles Z −1 n,N (detM)α exp(−N TrV (M))dM defined on positive definite matrices M, where α> −1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we determine the scaling limits of the eigenvalue correlation kernel near 0 in the limit when n, N → ∞ such that n/N − 1 = O(n −2/3). For each value of α> −1 we find a oneparameter family of limiting kernels that we describe in terms of the HastingsMcLeod solution of the Painlevé II equation with parameter α + 1/2.
Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter
 In preparation
, 2002
"... A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is ..."
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Cited by 15 (11 self)
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A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously. 1
Nonintersecting squared Bessel paths: critical time and double scaling limit
 Comm. Math. Phys
"... We consider the double scaling limit for a model of n nonintersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a reg ..."
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Cited by 9 (2 self)
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We consider the double scaling limit for a model of n nonintersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n→ ∞ that intersects the hard edge at x = 0 at a critical time t = t∗. In a previous paper, the scaling limits for the positions of the paths at time t 6 = t ∗ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as n→ ∞ of the correlation kernel at critical time t ∗ and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a 3 × 3 matrix valued RiemannHilbert problem by the DeiftZhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular thirdorder linear differential equation,
TracyWidom law for the extreme eigenvalues of sample correlation matrices
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Correlation functions for random involutions
 INTERNAT. MATH. RES. NOTICES (2006), ARTICLEID
, 2005
"... Our interest is in the scaled joint distribution associated with kincreasing subsequences for random involutions with a prescribed number of fixed points. We proceed by specifying in terms of correlation functions the same distribution for a Poissonized model in which both the number of symbols in ..."
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Our interest is in the scaled joint distribution associated with kincreasing subsequences for random involutions with a prescribed number of fixed points. We proceed by specifying in terms of correlation functions the same distribution for a Poissonized model in which both the number of symbols in the involution, and the number of fixed points, are random variables. From this, a dePoissonization argument yields the scaled correlations and distribution function for the random involutions. These are found to coincide with the same quantities known in random matrix theory from the study of ensembles interpolating between the orthogonal and symplectic universality classes at the soft edge, the interpolation being due to a rank 1 perturbation.