Results 1  10
of
78
Matrix models for circular ensembles
 Int. Math. Res. Not
"... Abstract. The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by E β n(f) = 1 · · · f(e ..."
Abstract

Cited by 47 (6 self)
 Add to MetaCart
Abstract. The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by E β n(f) = 1 · · · f(e
The efficient evaluation of the hypergeometric function of a matrix argument
 Math. Comp
"... Abstract. We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the ..."
Abstract

Cited by 28 (10 self)
 Add to MetaCart
Abstract. We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix. 1.
Beta ensembles, stochastic Airy spectrum, and a diffusion”, Preprint, 2006. Available at http://arxiv.org/abs/ math.PR/0607331
"... 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 ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
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. 1
CMV: The unitary analogue of Jacobi matrices
 Comm. Pure Appl. Math
"... Abstract. We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
Abstract. We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of wellknown properties of Jacobi matrices: foliation by coadjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (AblowitzLadik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite AblowitzLadik hierarchy and describe the longtime behaviour of this system. 1.
The importance of Selberg integral
 Bull. Amer. Math. Soc
"... Abstract. It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an ndimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a que ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
Abstract. It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an ndimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the Macdonald conjectures. It further initiated the study of qanalogues, which in turn enriched the Macdonald conjectures. We review these developments and proceed to exhibit the sustained prominence of the Selberg integral as evidenced by its central role in random matrix theory, Calogero–Sutherland quantum manybody systems, Knizhnik–Zamolodchikov equations, and multivariable orthogonal polynomial
CMV matrices in random matrix theory and integrable systems: A survey
 J. Phys. A: Math. Gen
, 2006
"... Abstract. We present a survey of recent results concerning a remarkable class of unitary matrices, the CMV matrices. We are particularly interested in the role they play in the theory of random matrices and integrable systems. Throughout the paper we also emphasize the analogies and connections to J ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
Abstract. We present a survey of recent results concerning a remarkable class of unitary matrices, the CMV matrices. We are particularly interested in the role they play in the theory of random matrices and integrable systems. Throughout the paper we also emphasize the analogies and connections to Jacobi matrices. 1.
Tails of condition number distributions
 SIAM J. Matrix Anal. Appl
"... Abstract. Let κ be the condition number of an mbyn matrix with independent standard Gaussian entries, either real (β = 1) or complex (β = 2). The major result is the existence of a constant C (depending on m, n, β) such that P [κ> x] < C x −β for all x. As x → ∞, the bound is asymptotically tight. ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
Abstract. Let κ be the condition number of an mbyn matrix with independent standard Gaussian entries, either real (β = 1) or complex (β = 2). The major result is the existence of a constant C (depending on m, n, β) such that P [κ> x] < C x −β for all x. As x → ∞, the bound is asymptotically tight. An analytic expression is given for the constant C, and simple estimates are given, one involving a TracyWidom largest eigenvalue distribution. All of the results extend beyond real and complex entries to general β.
The betaJacobi matrix model, the CS decomposition, and generalized singular value problems
 Foundations of Computational Mathematics
, 2007
"... Abstract. We provide a solution to the βJacobi matrix model problem posed by Dumitriu and the first author. The random matrix distribution introduced here, called a matrix model, is related to the model of Killip and Nenciu, but the development is quite different. We start by introducing a new matr ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
Abstract. We provide a solution to the βJacobi matrix model problem posed by Dumitriu and the first author. The random matrix distribution introduced here, called a matrix model, is related to the model of Killip and Nenciu, but the development is quite different. We start by introducing a new matrix decomposition and an algorithm for computing this decomposition. Then we run the algorithm on a Haardistributed random matrix to produce the βJacobi matrix model. The Jacobi ensemble on R n, parameterized by β> 0, a> −1, and b> −1, is the probability distribution whose density is proportional to Q β 2 i λ (a+1)−1
Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations
"... We derive Painlevé–type expressions for the distribution of the m th largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. The work of Johnstone and Soshnikov (see [7], [10]) implies the immediate relevance of our formulas for the m th largest eigenvalue o ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
We derive Painlevé–type expressions for the distribution of the m th largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. The work of Johnstone and Soshnikov (see [7], [10]) implies the immediate relevance of our formulas for the m th largest eigenvalue of the appropriate Wishart distribution. 1