arXiv subjects

P. J. Forrester

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

85 records · Page 5Linked to original sources

arXiv1996

Some exact correlations in the Dyson Brownian motion model for transitions to the CUE

The Dyson Brownian motion model for transistions to the CUE is considered. For initial eigenvalue probability density functions corresponding to the COE and CSE, the density-density correlation function between an eigenvalue at position lambda' initially (tau = 0) and an eigenvalue at position lambda for general tau, is calculated. Theoretical predictions for the asymptotic behaviour are verified, and the distribution for the initial rate of change of the eigenvalues is calculated. Also, for initial conditions corresponding to equi-spaced eigenvalues, the n-point equal parameter distribution is computed. The extension of the calculation of these quantities to a multiparameter version of the theory is given.

chao-dyn↗

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↗

arXiv1995

The two-dimensional two-component plasma plus background on a sphere : Exact results

An exact solution is given for a two-dimensional model of a Coulomb gas, more general than the previously solved ones. The system is made of a uniformly charged background, positive particles, and negative particles, on the surface of a sphere. At the special value $Γ= 2$ of the reduced inverse temperature, the classical equilibrium statistical mechanics is worked out~: the correlations and the grand potential are calculated. The thermodynamic limit is taken, and as it is approached the grand potential exhibits a finite-size correction of the expected universal form.

cond-mat↗

arXiv1995

Jack polynomials and the multi-component Calogero-Sutherland model

Using the ground state $ψ_0$ of a multicomponent generalization of the Calogero-Sutherland model as a weight function, orthogonal polynomials in the coordinates of one of the species are constructed. Using evidence from exact analytic and numerical calculations, it is conjectured that these polynomials are the Jack polynomials $J_κ^{(1+1/λ)}$, where $λ$ is the coupling constant. The value of the normalization integral for $ψ_0 J_κ^{(1+1/λ)}$ is conjectured, and some further related integrals are evaluated.

cond-mat↗

arXiv1995

Universality in some classical Coulomb systems of restricted dimension

Coulomb systems in which the particles interact through the $d$-dimensional Coulomb potential but are confined in a flat manifold of dimension $d - 1$ are considered. The Coulomb potential is defined with some boundary condition involving a characteristic macroscopic distance $W$ in the direction perpendicular to the manifold~: either it is periodic of period $W$ in that direction, or it vanishes on one ideal conductor wall parallel to the manifold at a distance $W$ from it, or it vanishes on two parallel walls at a distance $W$ from each other with the manifold equidistant from them. Under the assumptions that classical equilibrium statistical mechanics is applicable and that the system has the macroscopic properties of a conductor, it is shown that the suitably smoothed charge correlation function is universal, and that the free energy and the grand potential have universal dependences on $W$ (universal means independent of the microscopic detail). The cases $d = 2$ are discussed in detail, and the generic results are checked on an exactly solvable model. The case $d = 3$ of a plane parallel to an ideal conductor is also explicitly worked out.

cond-mat↗

arXiv1995

Two-dimensional one-component plasma in a quadrupolar field

The classical two-dimensional one-component plasma is an exactly solvable model, at some special temperature, even when the one-body potential acting on the particles has a quadrupolar term. As a supplement to a recent work of Di Francesco, Gaudin, Itzykson, and Lesage [{\it Int. J. Mod. Phys.} {\bf A9}, 4257 (1994)] about an $N$-particle system ($N$ large but finite), a macroscopic argument is given for confirming that the particles form an elliptical blob, the analogy between the classical plasma and a quantum $N$-fermion system in a magnetic field is used for the microscopic approach, and a microscopic calculation of the surface charge-surface charge correlation function is performed; an expected universal form is shown to be realized by this correlation function.

cond-mat↗

arXiv1995

Correlations in two-component log-gas systems

A systematic study of the properties of particle and charge correlation functions in the two-dimensional Coulomb gas confined to a one-dimensional domain is undertaken. Two versions of this system are considered: one in which the positive and negative charges are constrained to alternate in sign along the line, and the other where there is no charge ordering constraint. Both systems undergo a zero-density Kosterlitz-Thouless type transition as the dimensionless coupling $Γ:= q^2 / kT$ is varied through $Γ= 2$. In the charge ordered system we use a perturbation technique to establish an $O(1/r^4)$ decay of the two-body correlations in the high temperature limit. For $Γ\rightarrow 2^+$, the low-fugacity expansion of the asymptotic charge-charge correlation can be resummed to all orders in the fugacity. The resummation leads to the Kosterlitz renormalization equations.

cond-mat↗

arXiv1995

A Testbench for the Nested Dipole Hypothesis of Kosterlitz and Thouless

We consider the two-dimensional one-component plasma without a background and confined to a half plane near a metal wall. The particles are also subjected to an external potential acting perpendicular to the wall with an inverse power law Boltzmann factor. The model has a known solvable isotherm which exhibits a Kosterlitz-Thouless type transition from a conductive to an insulator phase as the power law is varied. This allows predictions of theoretical methods of analysing the Kosterlitz-Thouless transition to be compared with the exact solution. In particular we calculate the asymptotic density profile by resumming its low fugacity expansion near the zero-density critical coupling in the insulator phase, and solving a mean-field equation deduced from the first BGY equation. Agreement with the exact solution is obtained. As the former calculation makes essential use of the nested dipole hypothesis of Kosterlitz and Thouless, the validity of this hypothesis is explicitly verified.

cond-mat↗

arXiv1995

Integration formulas and exact calculations in the Calogero-Sutherland model

Some integration formulas which either occur or are implicit in Ha's recent exact calculation of some correlations in the Calogero-Sutherland model are discussed. These integration formulas include the calculation of the inner product $<0|ρ(0)|κ>$ between the density operator acting on an excited state and the ground state, and a generalization of the Selberg integral due to Dotsenko and Fateev.

cond-mat↗

arXiv1994

Normalization of the wavwfunction for the Calogero-Sutherland model with internal degrees of freedom

The exact normalization of a multicomponent generalization of the ground state wavefunction of the Calogero-Sutherland model is conjectured. This result is obtained from a conjectured generalization of Selberg's $N$-dimensional extension of the Euler beta integral, written as a trigonometric integral. A new proof of the Selberg integral is given, and the method is used to provide a proof of the mulicomponent generalization in a special two-component case.

cond-mat↗

arXiv1994

Global fluctuation formulas and universal correlations for random matrices and log-gas systems at infinite density

It is shown how the universal correlation function of Brezin and Zee, and Beenakker, for random matrix ensembles of Wigner-Dyson type with density support on a finite interval can be derived using a linear response argument and macroscopic electrostatics. The analogous formula for ensembles of unitary random matrices is derived using the same method, and the corresponding global fluctuation formula for the variance of a linear statistic is presented. The result for the universal correlation is checked using an exact result for the finite system two-point correlation in the Dyson circular ensemble at all even $β$.

cond-mat↗

arXiv1994

Exact ground state correlations in the supersymmetric $1/r^2$ exchange $t-J$ model

Some ground state static correlation functions for the $1/r^2$ exchange $t-J$ model of Kuramoto and Yokoyama are calculated exactly. The correlations calcualted are the pair distribution functions between two down spins, a down spin and a hole, and two holes, and the one body density matrix (equal time retarded Green's function) for holes

cond-mat↗

arXiv1994

Exact calculation of the ground state single-particle Green's function for the $1/r^2$ quantum many body system at integer coupling

The ground state single particle Green's function describing hole propagation is calculated exactly for the $1/r^2$ quantum many body system at integer coupling. The result is in agreement with a recent conjecture of Haldane.

cond-mat↗