N. E. Frankel

Applications and generalizations of Fisher–Hartwig asymptotics

Fisher–Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin–spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher–Hartwig formula to the asymptotic decay of the Ising correlations above Tc, while the study of the Bose gas density matrix leads us to generalize the Fisher–Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher–Hartwig asymptotic form known for Toeplitz determinants.

Finite one-dimensional impenetrable Bose systems: Occupation numbers

Bosons in the form of ultracold alkali-metal atoms can be confined to a one-dimensional (1D) domain by the use of harmonic traps. This motivates the study of the ground-state occupations {lambda}{sub i} of effective single-particle states {phi}{sub i}, in the theoretical 1D impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The {lambda}{sub i} and {phi}{sub i} are the eigenvalues and eigenfunctions, respectively, of the one-body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations {lambda}{sub i} are asymptotically proportional to {radical}(N) in both the circular and harmonically trapped cases.

A Class of Mean Field Models

A model of isotropically interacting ν‐dimensional classical spins with an infinite range potential of the molecular field‐type is solved. The partition function is represented as the integral of e−βHN over an appropriate weight function, which, for given ν, is the Pearson random walk probability distribution in ν dimensions. A molecular field‐type phase transition is obtained for all ν.

Asymptotic corrections to the eigenvalue density of the GUE and LUE

We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N×N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk.

Eigenvalue separation in some random matrix models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s>1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble an...

Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry

In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N×N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, a matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk. In addition, aspects of the asymptotic expansion of the smoothed density, which involves delta functions at the endpoints of the support, are interpreted microscopically.

Painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons

Abstract: The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in Σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations.

Response theory of particle-anti-particle plasmas

Abstract The physical motivation for our work on particle-anti-particle systems comes primarily from astrophysical objects such as pulsars and white dwarf stars. We deal first with the longitudinal dielectric response of an electron-positron or pair plasma in zero and non-zero magnetic fields. The response function must be renormalized using the standard techniques of quantum electrodynamics. For zero magnetic field, the dispersion relation and damping for plasma oscillations are given together with the screening potential about a test charge. For the case of non-zero magnetic field, the longitudinal dielectric response function is again calculated after renormalization. The response function takes on a different form depending upon whether the longitudinal oscillations are parallel or perpendicular to the direction of the magnetic field. The real and imaginary parts of the response function are then exhibited for both cases. The next topic is concerned with the plasma thought to exist in the deep interior of neutron stars or in the ephemeral plasmas of heavy ion collisions. Use of the Feshbach-Villars formalism for the Klein-Gordon equation allows for a similar approach to that previously used for the fermion-anti-fermion case. An adaption of the formalism developed here to two-dimensional systems is also given.

Classical theory of amorphous ferromagnets

A model for amorphous ferromagnets consisting of a gas of hard spheres each having a spin mu =+or-1 and interacting with a long-range Kac potential gamma nu K( gamma mod r mod ) in nu dimensions, is considered in the long-range gamma to 0+limit. The resulting free energy is a combination of the classical van der Waals and Curie-Weiss (or mean-field) expressions. Isotherms in the pure gas phase have a cusp for a range of temperatures, reflecting the transition from gas to magnetic gas, with standard mean-field exponents. The transition point on the critical isotherm is a higher-order critical point with non-classical exponents.

Lévy flights: Exact results and asymptotics beyond all orders

A comprehensive study of the symmetric Levy stable probability density function is presented. This is performed for orders both less than 2, and greater than 2. The latter class of functions are traditionally neglected because of a failure to satisfy non-negativity. The complete asymptotic expansions of the symmetric Levy stable densities of order greater than 2 are constructed, and shown to exhibit intricate series of transcendentally small terms—asymptotics beyond all orders. It is demonstrated that the symmetric Levy stable densities of any arbitrary rational order can be written in terms of generalized hypergeometric functions, and a number of new special cases are given representations in terms of special functions. A link is shown between the symmetric Levy stable density of order 4, and Pearcey’s integral, which is used widely in problems of optical diffraction and wave propagation. This suggests the existence of applications for the symmetric Levy stable densities of order greater than 2, despite their failure to define a probability density function.