2,041 research outputs found
Semi-classical Laguerre polynomials and a third order discrete integrable equation
A semi-discrete Lax pair formed from the differential system and recurrence
relation for semi-classical orthogonal polynomials, leads to a discrete
integrable equation for a specific semi-classical orthogonal polynomial weight.
The main example we use is a semi-classical Laguerre weight to derive a third
order difference equation with a corresponding Lax pair.Comment: 11 page
Applications and generalizations of Fisher-Hartwig asymptotics
Fisher-Hartwig asymptotics refers to the large 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 , 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.Comment: 25 page
Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma
The two-dimensional one-component plasma (2dOCP) is a system of mobile
particles of the same charge on a surface with a neutralising background.
The Boltzmann factor of the 2dOCP at temperature can be expressed as a
Vandermonde determinant to the power . Recent advances in
the theory of symmetric and anti-symmetric Jack polymonials provide an
efficient way to expand this power of the Vandermonde in their monomial basis,
allowing the computation of several thermodynamic and structural properties of
the 2dOCP for values up to 14 and equal to 4, 6 and 8. In this
work, we explore two applications of this formalism to study the moments of the
pair correlation function of the 2dOCP on a sphere, and the distribution of
radial linear statistics of the 2dOCP in the plane
Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles
The density function for the joint distribution of the first and second
eigenvalues at the soft edge of unitary ensembles is found in terms of a
Painlev\'e II transcendent and its associated isomonodromic system. As a
corollary, the density function for the spacing between these two eigenvalues
is similarly characterized.The particular solution of Painlev\'e II that arises
is a double shifted B\"acklund transformation of the Hasting-McLeod solution,
which applies in the case of the distribution of the largest eigenvalue at the
soft edge. Our deductions are made by employing the hard-to-soft edge
transitions to existing results for the joint distribution of the first and
second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under
of quantities specifying the latter are obtained. A Fredholm
determinant type characterisation is used to provide accurate numerics for the
distribution of the spacing between the two largest eigenvalues.Comment: 26 pages, 1 Figure, 2 Table
Solving 1d plasmas and 2d boundary problems using Jack polynomials and functional relations
The general one-dimensional ``log-sine'' gas is defined by restricting the
positive and negative charges of a two-dimensional Coulomb gas to live on a
circle. Depending on charge constraints, this problem is equivalent to
different boundary field theories. We study the electrically neutral case,
which is equivalent to a two-dimensional free boson with an impurity cosine
potential. We use two different methods: a perturbative one based on Jack
symmetric functions, and a non-perturbative one based on the thermodynamic
Bethe ansatz and functional relations. The first method allows us to compute
explicitly all coefficients in the virial expansion of the free energy and the
experimentally-measurable conductance. Some results for correlation functions
are also presented. The second method provides in particular a surprising
fluctuation-dissipation relation between the free energy and the conductance.Comment: 19 page
Large deviations of the maximal eigenvalue of random matrices
We present detailed computations of the 'at least finite' terms (three
dominant orders) of the free energy in a one-cut matrix model with a hard edge
a, in beta-ensembles, with any polynomial potential. beta is a positive number,
so not restricted to the standard values beta = 1 (hermitian matrices), beta =
1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This
model allows to study the statistic of the maximum eigenvalue of random
matrices. We compute the large deviation function to the left of the expected
maximum. We specialize our results to the gaussian beta-ensembles and check
them numerically. Our method is based on general results and procedures already
developed in the literature to solve the Pastur equations (also called "loop
equations"). It allows to compute the left tail of the analog of Tracy-Widom
laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos
corrected and preprint added ; v4 few more numbers adde
Jack vertex operators and realization of Jack functions
We give an iterative method to realize general Jack functions from Jack
functions of rectangular shapes. We first show some cases of Stanley's
conjecture on positivity of the Littlewood-Richardson coefficients, and then
use this method to give a new realization of Jack functions. We also show in
general that vectors of products of Jack vertex operators form a basis of
symmetric functions. In particular this gives a new proof of linear
independence for the rectangular and marked rectangular Jack vertex operators.
Thirdly a generalized Frobenius formula for Jack functions was given and was
used to give new evaluation of Dyson integrals and even powers of Vandermonde
determinant.Comment: Expanded versio
Exact Dynamical Correlation Functions of Calogero-Sutherland Model and One-Dimensional Fractional Statistics
One-dimensional model of non-relativistic particles with inverse-square
interaction potential known as Calogero-Sutherland Model (CSM) is shown to
possess fractional statistics. Using the theory of Jack symmetric polynomial
the exact dynamical density-density correlation function and the one-particle
Green's function (hole propagator) at any rational interaction coupling
constant are obtained and used to show clear evidences of the
fractional statistics. Motifs representing the eigenstates of the model are
also constructed and used to reveal the fractional {\it exclusion} statistics
(in the sense of Haldane's ``Generalized Pauli Exclusion Principle''). This
model is also endowed with a natural {\it exchange } statistics (1D analog of
2D braiding statistics) compatible with the {\it exclusion} statistics.
(Submitted to PRL on April 18, 1994)Comment: Revtex 11 pages, IASSNS-HEP-94/27 (April 18, 1994
Watermelon configurations with wall interaction: exact and asymptotic results
We perform an exact and asymptotic analysis of the model of vicious
walkers interacting with a wall via contact potentials, a model introduced by
Brak, Essam and Owczarek. More specifically, we study the partition function of
watermelon configurations which start on the wall, but may end at arbitrary
height, and their mean number of contacts with the wall. We improve and extend
the earlier (partially non-rigorous) results by Brak, Essam and Owczarek,
providing new exact results, and more precise and more general asymptotic
results, in particular full asymptotic expansions for the partition function
and the mean number of contacts. Furthermore, we relate this circle of problems
to earlier results in the combinatorial and statistical literature.Comment: AmS-TeX, 41 page
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