arXiv subjects

T. H. Baker

Publications and source records attributed to T. H. Baker.

12 recordsLinked to original sources

arXiv2001

Random walks and random fixed-point free involutions

A bijection is given between fixed point free involutions of $\{1,2,...,2N\}$ with maximum decreasing subsequence size $2p$ and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points $l \ge 1$. In one class of walker configurations the maximum displacement of the right most walker is $p$. Because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the right most walker.

math.CO↗

arXiv1998

Transformation formulae for multivariable basic hypergeometric series

We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2ϕ_1$ transformation, the $q$-Pfaff-Kummer and Euler transformations, the $q$-Saalschütz summation formula and Sear's transformation for terminating, balanced ${}_4ϕ_3$ series. For bilateral series, we rederive Kaneko's analogue of the ${}_1ψ_1$ summation formula and give multivariable extensions of Bailey's ${}_2ψ_2$ transformations.

math.QA↗

arXiv1998

Random matrix ensembles with an effective extensive external charge

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite matrices. Here we consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter $β$, while for $β= 2$ (random matrices with unitary symmetry), the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class.

cond-mat.stat-mech↗

arXiv1997

Isomorphisms of type $A$ affine Hecke algebras and multivariable orthogonal polynomials

We examine two isomorphisms between affine Hecke algebras of type $A$ associated with parameters $q^{-1}$, $t^{-1}$ and $q$, $t$. One of them maps the non-symmetric Macdonald polynomials $E_η(x;q^{-1},t^{-1})$ onto $E_η(x;q,t)$, while the other maps them onto non-symmetric analogues of the multivariable Al-Salam & Carlitz polynomials. Using the properties of $E_η(x;q^{-1},t^{-1})$, the corresponding properties of these latter polynomials can then be elucidated.

q-alg↗

arXiv1997

Symmetric Jack polynomials from non-symmetric theory

The theory of non-symmetric Jack polynomials is developed independently of the theory of symmetric Jack polynomials, and this theory together with the relationship between the non-symmetric, symmetric and anti-symmetric Jack polynomials is used to deduce the corresponding results for the symmetric Jack polynomials.

q-alg↗

arXiv1997

Multivariable Al-Salam & Carlitz polynomials associated with the type A q-Dunkl kernel

The Al-Salam & Carlitz polynomials are $q$-generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the $q$-analogue of the type-$A$ Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with respect to a certain Jackson integral inner product. This inner product is normalized by deriving a $q$-analogue of the Mehta integral, and the corresponding normalization of the multivariable Al-Salam & Carlitz polynomials is derived from a Pieri-type formula. Various other special properties of the polynomials are also presented, including their relationship to the shifted Macdonald polynomials and the big $q$-Jacobi polynomials.

q-alg↗

arXiv1997

A $q$-analogue of the type $A$ Dunkl operator and integral kernel

We introduce the $q$-analogue of the type $A$ Dunkl operators, which are a set of degree--lowering operators on the space of polynomials in $n$ variables. This allows the construction of raising/lowering operators with a simple action on non-symmetric Macdonald polynomials. A bilinear series of non-symmetric Macdonald polynomials is introduced as a $q$-analogue of the type $A$ Dunkl integral kernel ${\cal K}_A(x;y)$. The aforementioned operators are used to show that the function satisfies $q$-analogues of the fundamental properties of ${\cal K}_A(x;y)$.

q-alg↗

arXiv1997

Finite N Fluctuation Formulas for Random Matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic $\sum_{j=1}^N (x_j - )$ is computed exactly and shown to satisfy a central limit theorem as $N \to \infty$. For the circular random matrix ensemble the p.d.f.'s for the linear statistics ${1 \over 2} \sum_{j=1}^N (θ_j - π)$ and $- \sum_{j=1}^N \log 2|\sin θ_j/2|$ are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem as $N \to \infty$.

cond-mat.stat-mech↗

arXiv1996

Non-Symmetric Jack Polynomials and Integral Kernels

We investigate some properties of non-symmetric Jack, Hermite and Laguerre polynomials which occur as the polynomial part of the eigenfunctions for certain Calogero-Sutherland models with exchange terms. For the non-symmetric Jack polynomials, the constant term normalization ${\cal N}_η$ is evaluated using recurrence relations, and ${\cal N}_η$ is related to the norm for the non-symmetric analogue of the power-sum inner product. Our results for the non-symmetric Hermite and Laguerre polynomials allow the explicit determination of the integral kernels which occur in Dunkl's theory of integral transforms based on reflection groups of type $A$ and $B$, and enable many analogues of properties of the classical Fourier, Laplace and Hankel transforms to be derived. The kernels are given as generalized hypergeometric functions based on non-symmetric Jack polynomials. Central to our calculations is the construction of operators $\widehatΦ$ and $\widehatΨ$, which act as lowering-type operators for the non-symmetric Jack polynomials of argument $x$ and $x^2$ respectively, and are the counterpart to the raising-type operator $Φ$ introduced recently by Knop and Sahi.

q-alg↗

arXiv1996

The Calogero-Sutherland Model and Polynomials with Prescribed Symmetry

The Schrödinger operators with exchange terms for certain Calogero-Sutherland quantum many body systems have eigenfunctions which factor into the symmetric ground state and a multivariable polynomial. The polynomial can be chosen to have a prescribed symmetry (i.e. be symmetric or antisymmetric) with respect to the interchange of some specified variables. For four particular Calogero-Sutherland systems we construct an eigenoperator for these polynomials which separates the eigenvalues and establishes orthogonality. In two of the cases this involves identifying new operators which commute with the corresponding Schrödinger operators. In each case we express a particular class of the polynomials with prescribed symmetry in a factored form involving the corresponding symmetric polynomials.

solv-int↗

arXiv1996

The Calogero-Sutherland Model and Generalized Classical Polynomials

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schrödinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case.

solv-int↗

arXiv1996

Generalized weight functions and the Macdonald polynomials

A weight function which $q$-generalizes the ground state wave function of the multi-component Calogero-Sutherland quantum many body system is introduced. Conjectures, and some proofs in special cases, are given for a constant term identity involving this function. A Gram-Schmidt procedure with respect to the inner product associated with the weight function is used to define orthogonal polynomials in one of the components, which are conjectured to be the Macdonald polynomials $P_κ(w_1,\dots,w_{N_0};qt^p,t)$, and a proof is given in a special case. Conjectures are also given for an adjoint property of the Macdonald operator with respect to the inner product associated with the weight function, and the normalization of the Macdonald polynomial with respect to the same inner product.

q-alg↗