Results 1  10
of
21
Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence
 ANN. STATIST
, 2008
"... Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A+B) −1 B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppos ..."
Abstract

Cited by 30 (2 self)
 Add to MetaCart
Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A+B) −1 B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that m and n grow in proportion to p. We show that after centering and scaling, the distribution is approximated to secondorder, O(p −2/3), by the Tracy–Widom law. The results are obtained for both complex and then realvalued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.
Universality for orthogonal and symplectic Laguerretype ensembles
 J. Statist. Phys
, 2007
"... Abstract. We give a proof of the Universality Conjecture for orthogonal (β = 1) and symplectic (β = 4) random matrix ensembles of Laguerretype in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We give a proof of the Universality Conjecture for orthogonal (β = 1) and symplectic (β = 4) random matrix ensembles of Laguerretype in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels Kn,β, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (β = 2) Laguerretype ensembles have been proved by the fourth author in [23]. The varying weight case at the hard spectral edge
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 ..."
Abstract

Cited by 15 (11 self)
 Add to MetaCart
(Show Context)
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
Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic
, 2007
"... In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N × N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to s ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N × N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, a matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk. In addition, aspects of the asymptotic expansion of the smoothed density, which involves delta functions at the endpoints of the support, are interpreted microscopically. PACS numbers: 02.50.Cw,05.90.+m,02.30.Gp 1
CORRELATION KERNELS FOR DISCRETE SYMPLECTIC AND ORTHOGONAL ENSEMBLES
, 712
"... Abstract. In [41] H. Widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials. We obtain similar results for discrete ensembles with rational discrete logarithmic derivative, and compute explicitly correlatio ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
(Show Context)
Abstract. In [41] H. Widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials. We obtain similar results for discrete ensembles with rational discrete logarithmic derivative, and compute explicitly correlation kernels associated to the classical Meixner and Charlier weights. Contents
RiemannHilbert approach to gap probabilities for the Bessel process. arXiv:1306.5663
, 2013
"... We consider the gap probability for the Bessel process in the singletime and multitime case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels a ́ la ItsIzerginKorepinSlavnov and thus related to suitable R ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We consider the gap probability for the Bessel process in the singletime and multitime case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels a ́ la ItsIzerginKorepinSlavnov and thus related to suitable RiemannHilbert problems. In the singletime case, we construct a Lax pair formalism and we derive a Painleve ́ III equation related to the Fredholm determinant. 1
RANDOM MATRIX ENSEMBLES ASSOCIATED TO COMPACT SYMMETRIC SPACES
, 2001
"... Abstract. We introduce random matrix ensembles that correspond to the in finite families of irreducible Riemannian symmetric spaces of type I. In particular, we recover the Circular Orthogonal and Symplectic Ensembles of Dyson, and find other families of (unitary, orthogonal and symplectic) ensemble ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We introduce random matrix ensembles that correspond to the in finite families of irreducible Riemannian symmetric spaces of type I. In particular, we recover the Circular Orthogonal and Symplectic Ensembles of Dyson, and find other families of (unitary, orthogonal and symplectic) ensembles of Jacobi type. We discuss the universal and weakly universal features of the global and local correlations of the levels in the bulk and at the “hard” edge of the spectrum (i. e., at the “central points ” ±1 on the unit circle). Previously known results are extended, and we find new simple formulas for the Bessel Kernels that describe the local correlations at a hard edge. 1.