414 research outputs found
Products and Ratios of Characteristic Polynomials of Random Hermitian Matrices
We present new and streamlined proofs of various formulae for products and
ratios of characteristic polynomials of random Hermitian matrices that have
appeared recently in the literature.Comment: 18 pages, LaTe
The Widom-Dyson constant for the gap probability in random matrix theory
In this paper we consider an asymptotic question in the theory of the
Gaussian Unitary Ensemble of random matrices. In the bulk scaling limit, the
probability that there are no eigenvalues in the interval (0,2s) is given by
P_s=det(I-K_s), where K_s is the trace-class operator with kernel
K_s(x,y)={sin(x-y)}/{\pi(x-y)} acting on L^2(0,2s). We are interested
particularly in the behavior of P_s as s tends to infinity...Comment: 31 pages, 4 figure
Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials
We study the distribution of the maximal height of the outermost path in the
model of nonintersecting Brownian motions on the half-line as , showing that it converges in the proper scaling to the Tracy-Widom
distribution for the largest eigenvalue of the Gaussian orthogonal ensemble.
This is as expected from the viewpoint that the maximal height of the outermost
path converges to the maximum of the process minus a
parabola. Our proof is based on Riemann-Hilbert analysis of a system of
discrete orthogonal polynomials with a Gaussian weight in the double scaling
limit as this system approaches saturation. We consequently compute the
asymptotics of the free energy and the reproducing kernel of the corresponding
discrete orthogonal polynomial ensemble in the critical scaling in which the
density of particles approaches saturation. Both of these results can be viewed
as dual to the case in which the mean density of eigenvalues in a random matrix
model is vanishing at one point.Comment: 39 pages, 4 figures; The title has been changed from "The limiting
distribution of the maximal height of nonintersecting Brownian excursions and
discrete Gaussian orthogonal polynomials." This is a reflection of the fact
that the analysis has been adapted to include nonintersecting Brownian
motions with either reflecting of absorbing boundaries at zero. To appear in
J. Stat. Phy
An exact formula for general spectral correlation function of random Hermitian matrices
We have found an exact formula expressing a general correlation function
containing both products and ratios of characteristic polynomials of random
Hermitian matrices. The answer is given in the form of a determinant. An
essential difference from the previously studied correlation functions (of
products only) is the appearance of non-polynomial functions along with the
orthogonal polynomials. These non-polynomial functions are the Cauchy
transforms of the orthogonal polynomials. The result is valid for any ensemble
of beta=2 symmetry class and generalizes recent asymptotic formulae obtained
for GUE and its chiral counterpart by different methods..Comment: published version, with a few misprints correcte
Exact solution of Riemann--Hilbert problem for a correlation function of the XY spin chain
A correlation function of the XY spin chain is studied at zero temperature.
This is called the Emptiness Formation Probability (EFP) and is expressed by
the Fredholm determinant in the thermodynamic limit. We formulate the
associated Riemann--Hilbert problem and solve it exactly. The EFP is shown to
decay in Gaussian.Comment: 7 pages, to be published in J. Phys. Soc. Jp
Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlev\'e-II equation
The Cauchy problem for the Korteweg de Vries (KdV) equation with small
dispersion of order \e^2, \e\ll 1, is characterized by the appearance of a
zone of rapid modulated oscillations. These oscillations are approximately
described by the elliptic solution of KdV where the amplitude, wave-number and
frequency are not constant but evolve according to the Whitham equations.
Whereas the difference between the KdV and the asymptotic solution decreases as
in the interior of the Whitham oscillatory zone, it is known to be
only of order near the leading edge of this zone. To obtain a
more accurate description near the leading edge of the oscillatory zone we
present a multiscale expansion of the solution of KdV in terms of the
Hastings-McLeod solution of the Painlev\'e-II equation. We show numerically
that the resulting multiscale solution approximates the KdV solution, in the
small dispersion limit, to the order .Comment: 20 pages, 14 figure
Dispersive Shock Wave, Generalized Laguerre Polynomials and Asymptotic Solitons of the Focusing Nonlinear Schr\"odinger Equation
We consider dispersive shock wave to the focusing nonlinear Schr\"odinger
equation generated by a discontinuous initial condition which is periodic or
quasi-periodic on the left semi-axis and zero on the right semi-axis. As an
initial function we use a finite-gap potential of the Dirac operator given in
an explicit form through hyper-elliptic theta-functions. The paper aim is to
study the long-time asymptotics of the solution of this problem in a vicinity
of the leading edge, where a train of asymptotic solitons are generated. Such a
problem was studied in \cite{KK86} and \cite{K91} using Marchenko's inverse
scattering technics. We investigate this problem exceptionally using the
Riemann-Hilbert problems technics that allow us to obtain explicit formulas for
the asymptotic solitons themselves that in contrast with the cited papers where
asymptotic formulas are obtained only for the square of absolute value of
solution. Using transformations of the main RH problems we arrive to a model
problem corresponding to the parametrix at the end points of continuous
spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is
effectively solved in terms of the generalized Laguerre polynomials which are
naturally appeared after appropriate scaling of the Riemann-Hilbert problem in
a small neighborhoods of the end points of continuous spectrum. Further
asymptotic analysis give an explicit formula for solitons at the edge of
dispersive wave. Thus, we give the complete description of the train of
asymptotic solitons: not only bearing envelope of each asymptotic soliton, but
its oscillating structure are found explicitly. Besides the second term of
asymptotics describing an interaction between these solitons and oscillating
background is also found. This gives the fine structure of the edge of
dispersive shock wave.Comment: 36 pages, 5 figure
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