Results 1  10
of
65
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 ..."
Abstract

Cited by 86 (19 self)
 Add to MetaCart
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.
Application of the τfunction theory of Painlevé equations to random matrices
 PV, PIII, the LUE, JUE and CUE
, 2002
"... Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidim ..."
Abstract

Cited by 42 (16 self)
 Add to MetaCart
Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a nonnegative
Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
, 1997
"... Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on RN. The definition and properties of these generalized Hermite systems extend naturally those of their classi ..."
Abstract

Cited by 37 (7 self)
 Add to MetaCart
Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on RN. The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In case of the symmetric group SN, our setting includes the polynomial eigenfunctions of certain CalogeroSutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl’s Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive oneparameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.
DUNKL OPERATORS: THEORY AND APPLICATIONS
, 2002
"... These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the D ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable particle systems of CalogeroMoserSutherland type, and discuss some systems of orthogonal polynomials associated with them. A major part is devoted to positivity results for the intertwining operator and the Dunkl kernel, the Dunkltype heat semigroup, and related probabilistic aspects. The notes conclude with recent results on the asymptotics of the Dunkl kernel.
Common algebraic structure for the CalogeroSutherland models
"... We investigate common algebraic structure for the rational and trigonometric CalogeroSutherland models by using the exchangeoperator formalism. We show that the set of the Jack polynomials whose arguments are Dunkltype operators provides an orthogonal basis for the rational case. One dimensional ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
We investigate common algebraic structure for the rational and trigonometric CalogeroSutherland models by using the exchangeoperator formalism. We show that the set of the Jack polynomials whose arguments are Dunkltype operators provides an orthogonal basis for the rational case. One dimensional quantum integrable models with longrange interaction have attracted much interest, because of not only their physical significance, but also their beautiful mathematical structure. One of such models is the Sutherland (trigonometric) model, which describes interacting particles on a circle[1]. The total momentum and Hamiltonian of the model are respectively given by N ∑ 1 ∂ Ps =, j=1 i ∂θj
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
Quantum CalogeroMoser Models: Integrability For All Root Systems
, 2000
"... The issues related to the integrability of quantum CalogeroMoser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the roo ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
The issues related to the integrability of quantum CalogeroMoser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix L, i.e. ∑ µ,ν∈R (Ln)µν, in which R is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based
The CalogeroSutherland Model And Polynomials With Prescribed Symmetry
 Nucl. Phys. B
, 1997
"... Introduction The Schrodinger operator + fi(fi=2 \Gamma 1) ; 0 x j L (1.1) describes quantum particles on a line of length L interacting through a 1=r pair potential with periodic boundary conditions, or equivalently quantum particles on a circle of circumference length L (h ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
Introduction The Schrodinger operator + fi(fi=2 \Gamma 1) ; 0 x j L (1.1) describes quantum particles on a line of length L interacting through a 1=r pair potential with periodic boundary conditions, or equivalently quantum particles on a circle of circumference length L (hence the superscript (C)) with the pair potential proportional to the inverse square of the chord length. It is one of a number of quantum many body systems in one dimension which are of the CalogeroSutherland type, meaning that the ground state (i.e. eigenstate with the smallest eigenvalue E 0 ) is of the form e \Gammafi W=2 W consisting of one and two body terms only. Explicitly, for (1.1) we have W = W log je 2ix k =L \Gamma e j: (1.2) email: tbaker@maths.mu.oz.au email: matpjf@maths.mu.oz.au In studying the integrability properties of (1.1) it is useful [1] to generalize the Schrodinger operator to include the exchange operator M jk for coordinates x j and x k :