arXiv subjects

P. J. Forrester

Publications and source records attributed to P. J. Forrester.

At least 73 records · Page 4Linked to original sources

arXiv1999

Random walks and random permutations

A connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences. Consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the GUE. The scaling of the distribution gives the maximum mean displacement $μ$ after $t$ time steps as $μ= (2t)^{1/2}$ with standard deviation proportional to $μ^{1/3}$. The exponent 1/3 is typical of a large class of two-dimensional growth problems.

math.CO↗

arXiv1999

Classical skew orthogonal polynomials and random matrices

Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the cases that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre and Jacobi cases.

solv-int↗

arXiv1999

Exact Finite Size Study of the 2dOCP at Gamma=4 and Gamma=6

An exact numerical study is undertaken into the finite $N$ calculation of the free energy and distribution functions for the two-dimensional one-component plasma. Both disk and sphere geometries are considered, with the coupling $Γ$ set equal to 4 and 6. Extrapolation of our data for the free energy is consistent with the existence of a universal term ${χ\over 12} \log N$, where $χ$ denotes the Euler characteristic of the surface, as predicted theoretically. The exact finite $N$ density profile is shown to give poor agreement with the contact theorem relating the density at contact and potential drop to the pressure in the thermodynamic limit. This is understood theoretically via a known finite $N$ version of the contact theorem. Furthermore, the ideas behind the derivation of the latter result are extended to give a sum rule for the second moment of the pair correlation in the finite disk, which in the thermodynamic limit converges to the Stillinger-Lovett result.

cond-mat.stat-mech↗

arXiv1998

Exact statistical properties of the zeros of complex random polynomials

The zeros of complex Gaussian random polynomials, with coefficients such that the density in the underlying complex space is uniform, are known to have the same statistical properties as the zeros of the coherent state representation of one-dimensional chaotic quantum systems. We extend the interpretation of these polynomials by showing that they also arise as the wave function for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest Landau level. A study of the statistical properties of the zeros is undertaken using exact formulas for the one and two point distribution functions. Attention is focussed on the moments of the two-point correlation in the bulk, the variance of a linear statistic, and the asymptotic form of the two-point correlation at the boundary. A comparison is made with the same quantities for the eigenvalues of complex Gaussian random matrices.

cond-mat.stat-mech↗

arXiv1998

Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges

For the orthogonal-unitary and symplectic-unitary transitions in random matrix theory, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion determinant. For the parameter dependent Gaussian and Laguerre ensembles the matrix elements of the determinant are expressed in terms of corresponding skew-orthogonal polynomials, and their limiting value for infinite matrix dimension are computed in the vicinity of the soft and hard edges respectively. A connection formula relating the distributions at the hard and soft edge is obtained, and a universal asymptotic behaviour of the two point correlation is identified.

cond-mat.mes-hall↗

arXiv1998

Correlations for the Dyson Brownian motion model with Poisson initial conditions

The circular Dyson Brownian motion model refers to the stochastic dynamics of the log-gas on a circle. It also specifies the eigenvalues of certain parameter-dependent ensembles of unitary random matrices. This model is considered with the initial condition that the particles are non-interacting (Poisson statistics). Jack polynomial theory is used to derive a simple exact expression for the density-density correlation with the position of one particle specified in the initial state, and the position of one particle specified at time $τ$, valid for all $β> 0$. The same correlation with two particles specified in the initial state is also derived exactly, and some special cases of the theoretical correlations are illustrated by comparison with the empirical correlations calculated from the eigenvalues of certain parameter-dependent Gaussian random matrices. Application to fluctuation formulas for time displaced linear statistics in made.

cond-mat.stat-mech↗

arXiv1998

Fluctuation formula for complex random matrices

A Gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case that the statistic is rotationally invariant. For a general linear statistic without this symmetry, Coulomb gas theory is used to predict that the distribution will again be a Gaussian, with a specific mean and variance. The variance splits naturally into a bulk and surface contibution, the latter resulting from the long range correlations at the boundary of the support of the eigenvalue density.

cond-mat.stat-mech↗

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

Exact and asymtotic formulas for overdamped Brownian dynamics

Exact and asymptotic formulas relating to dynamical correlations for overdamped Brownian motion are obtained. These formulas include a generalization of the $f$-sum rule from the theory of quantum fluids, a formula relating the static current-current correlation to the static density-density correlation, and an asymptotic formula for the small-$k$ behaviour of the dynamical structure factor. Known exact evaluations of the dynamical density-density correlation in some special models are used to illustrate and test the formulas.

cond-mat.stat-mech↗

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↗