547 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

### Asymptotics for a determinant with a confluent hypergeometric kernel

We obtain "large gap" asymptotics for a Fredholm determinant with a confluent
hypergeometric kernel. We also obtain asymptotics for determinants with two
types of Bessel kernels which appeared in random matrix theory.Comment: 34 pages, 2 figure

### Asymptotics of the Airy-kernel determinant

The authors use Riemann-Hilbert methods to compute the constant that arises
in the asymptotic behavior of the Airy-kernel determinant of random matrix
theory.Comment: 41 pages, 6 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 $N$ nonintersecting Brownian motions on the half-line as $N\to
\infty$, 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 $\textrm{Airy}_2$ 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

### First Colonization of a Spectral Outpost in Random Matrix Theory

We describe the distribution of the first finite number of eigenvalues in a
newly-forming band of the spectrum of the random Hermitean matrix model. The
method is rigorously based on the Riemann-Hilbert analysis of the corresponding
orthogonal polynomials. We provide an analysis with an error term of order
N^(-2 h) where 1/h = 2 nu+2 is the exponent of non-regularity of the effective
potential, thus improving even in the usual case the analysis of the pertinent
literature. The behavior of the first finite number of zeroes (eigenvalues)
appearing in the new band is analyzed and connected with the location of the
zeroes of certain Freud polynomials. In general all these newborn zeroes
approach the point of nonregularity at the rate N^(-h) whereas one (a stray
zero) lags behind at a slower rate of approach. The kernels for the correlator
functions in the scaling coordinate near the emerging band are provided
together with the subleading term: in particular the transition between K and
K+1 eigenvalues is analyzed in detail.Comment: 32 pages, 8 figures (typo corrected in Formula 4.13); some reference
added and minor correction

### Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent

We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of the Korteweg-de Vries equation for decaying initial data in the
soliton and similarity region. This paper can be viewed as an expository
introduction to this method.Comment: 31 page

### 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

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