19 research outputs found
Asymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals
In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian
matrices the probability that an interval of length contains no eigenvalues
is the Fredholm determinant of the sine kernel over
this interval. A formal asymptotic expansion for the determinant as tends
to infinity was obtained by Dyson. In this paper we replace a single interval
of length by where is a union of intervals and present a proof
of the asymptotics up to second order. The logarithmic derivative with respect
to of the determinant equals a constant (expressible in terms of
hyperelliptic integrals) times , plus a bounded oscillatory function of
(zero of , periodic if , and in general expressible in terms of the
solution of a Jacobi inversion problem), plus . Also determined are the
asymptotics of the trace of the resolvent operator, which is the ratio in the
same model of the probability that the set contains exactly one eigenvalue to
the probability that it contains none. The proofs use ideas from orthogonal
polynomial theory.Comment: 24 page
Asymptotics of block Toeplitz determinants and the classical dimer model
We compute the asymptotics of a block Toeplitz determinant which arises in
the classical dimer model for the triangular lattice when considering the
monomer-monomer correlation function. The model depends on a parameter
interpolating between the square lattice () and the triangular lattice
(), and we obtain the asymptotics for . For we apply the
Szeg\"o Limit Theorem for block Toeplitz determinants. The main difficulty is
to evaluate the constant term in the asymptotics, which is generally given only
in a rather abstract form
On Orthogonal and Symplectic Matrix Ensembles
The focus of this paper is on the probability, , that a set
consisting of a finite union of intervals contains no eigenvalues for the
finite Gaussian Orthogonal () and Gaussian Symplectic ()
Ensembles and their respective scaling limits both in the bulk and at the edge
of the spectrum. We show how these probabilities can be expressed in terms of
quantities arising in the corresponding unitary () ensembles. Our most
explicit new results concern the distribution of the largest eigenvalue in each
of these ensembles. In the edge scaling limit we show that these largest
eigenvalue distributions are given in terms of a particular Painlev\'e II
function.Comment: 34 pages. LaTeX file with one figure. To appear in Commun. Math.
Physic
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
We propose a method for computing any Gelfand-Dickey tau function living in
Segal-Wilson Grassmannian as the asymptotics of block Toeplitz determinant
associated to a certain class of symbols. Also truncated block Toeplitz
determinants associated to the same symbols are shown to be tau function for
rational reductions of KP. Connection with Riemann-Hilbert problems is
investigated both from the point of view of integrable systems and block
Toeplitz operator theory. Examples of applications to algebro-geometric
solutions are given.Comment: 35 pages. Typos corrected, some changes in the introductio
Level-Spacing Distributions and the Bessel Kernel
The level spacing distributions which arise when one rescales the Laguerre or
Jacobi ensembles of hermitian matrices is studied. These distributions are
expressible in terms of a Fredholm determinant of an integral operator whose
kernel is expressible in terms of Bessel functions of order . We derive
a system of partial differential equations associated with the logarithmic
derivative of this Fredholm determinant when the underlying domain is a union
of intervals. In the case of a single interval this Fredholm determinant is a
Painleve tau function.Comment: 18 pages, resubmitted to make postscript compatible, no changes to
manuscript conten
Non-colliding Brownian Motions and the extended tacnode process
We consider non-colliding Brownian motions with two starting points and two
endpoints. The points are chosen so that the two groups of Brownian motions
just touch each other, a situation that is referred to as a tacnode. The
extended kernel for the determinantal point process at the tacnode point is
computed using new methods and given in a different form from that obtained for
a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the
extended kernel is also different from that obtained for the extended tacnode
kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the
correlation kernel for a finite number of non-colliding Brownian motions
starting at two points and ending at arbitrary points.Comment: 38 pages. In the revised version a few arguments have been expanded
and many typos correcte
Fredholm Determinants, Differential Equations and Matrix Models
Orthogonal polynomial random matrix models of NxN hermitian matrices lead to
Fredholm determinants of integral operators with kernel of the form (phi(x)
psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm
determinants of integral operators having kernel of this form and where the
underlying set is a union of open intervals. The emphasis is on the
determinants thought of as functions of the end-points of these intervals. We
show that these Fredholm determinants with kernels of the general form
described above are expressible in terms of solutions of systems of PDE's as
long as phi and psi satisfy a certain type of differentiation formula. There is
also an exponential variant of this analysis which includes the circular
ensembles of NxN unitary matrices.Comment: 34 pages, LaTeX using RevTeX 3.0 macros; last version changes only
the abstract and decreases length of typeset versio
Finite one dimensional impenetrable Bose systems: Occupation numbers
Bosons in the form of ultra cold alkali 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 of effective single particle states
, in the theoretical 1d impenetrable Bose gas. Both the system on a
circle and the harmonically trapped system are considered. The and
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 the occupations
are asymptotically proportional to in both the circular
and harmonically trapped cases.Comment: 22 pages, 8 figures (.eps), uses REVTeX
The resultant on compact Riemann surfaces
We introduce a notion of resultant of two meromorphic functions on a compact
Riemann surface and demonstrate its usefulness in several respects. For
example, we exhibit several integral formulas for the resultant, relate it to
potential theory and give explicit formulas for the algebraic dependence
between two meromorphic functions on a compact Riemann surface. As a particular
application, the exponential transform of a quadrature domain in the complex
plane is expressed in terms of the resultant of two meromorphic functions on
the Schottky double of the domain.Comment: 44 page
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated,
e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius
of convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not for all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature
rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat