116 research outputs found
Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems
Classically, a single weight on an interval of the real line leads to
moments, orthogonal polynomials and tridiagonal matrices. Appropriately
deforming this weight with times t=(t_1,t_2,...), leads to the standard Toda
lattice and tau-functions, expressed as Hermitian matrix integrals.
This paper is concerned with a sequence of t-perturbed weights, rather than
one single weight. This sequence leads to moments, polynomials and a (fuller)
matrix evolving according to the discrete KP-hierarchy. The associated
tau-functions have integral, as well as vertex operator representations.
Among the examples considered, we mention: nested Calogero-Moser systems,
concatenated solitons and m-periodic sequences of weights. The latter lead to
2m+1-band matrices and generalized orthogonal polynomials, also arising in the
context of a Riemann-Hilbert problem.
We show the Riemann-Hilbert factorization is tantamount to the factorization
of the moment matrix into the product of a lower- times upper-triangular
matrix.Comment: 40 page
Vertex operator solutions to the discrete KP-hierarchy
Vertex operators, which are disguised Darboux maps, transform solutions of
the KP equation into new ones. In this paper, we show that the bi-infinite
sequence obtained by Darboux transforming an arbitrary KP solution recursively
forward and backwards, yields a solution to the discrete KP-hierarchy. The
latter is a KP hierarchy where the continuous space x-variable gets replaced by
a discrete n-variable. The fact that these sequences satisfy the discrete KP
hierarchy is tantamount to certain bilinear relations connecting the
consecutive KP solutions in the sequence. At the Grassmannian level, these
relations are equivalent to a very simple fact, which is the nesting of the
associated infinite-dimensional planes (flag).
It turns out that many new and old systems lead to such discrete
(semi-infinite) solutions, like sequences of soliton solutions, with more and
more solitons, sequences of Calogero-Moser systems, having more and more
particles, band matrices, etc... ; this will be developped in another paper. In
this paper, as an other example, we show that the q-KP hierarchy maps, via a
kind of Fourier transform, into the discrete KP hierarchy, enabling us to write
down a very large class of solutions to the q-KP hierarchy.Comment: 32 page
PDEs for the joint distributions of the Dyson, Airy and Sine processes
In a celebrated paper, Dyson shows that the spectrum of an n\times n random
Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves
as n noncolliding Brownian motions held together by a drift term. The universal
edge and bulk scalings for Hermitian random matrices, applied to the Dyson
process, lead to the Airy and Sine processes. In particular, the Airy process
is a continuous stationary process, describing the motion of the outermost
particle of the Dyson Brownian motion, when the number of particles gets large,
with space and time appropriately rescaled. In this paper, we answer a question
posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy
process at two different times. Similarly we find a PDE satisfied by the joint
distribution of the Sine process. This hinges on finding a PDE for the joint
distribution of the Dyson process, which itself is based on the joint
probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE
for the Dyson process is then subjected to an asymptotic analysis, consistent
with the edge and bulk rescalings. The PDEs enable one to compute the
asymptotic behavior of the joint distribution and the correlation for these
processes at different times t_1 and t_2, when t_2-t_1\to \infty, as
illustrated in this paper for the Airy process. This paper also contains a
rigorous proof that the extended Hermite kernel, governing the joint
probabilities for the Dyson process, converges to the extended Airy and Sine
kernels after the appropriate rescalings.Comment: Published at http://dx.doi.org/10.1214/009117905000000107 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Darboux transforms on Band Matrices, Weights and associated Polynomials
Classically, it is well known that a single weight on a real interval leads
to orthogonal polynomials. In "Generalized orthogonal polynomials, discrete KP
and Riemann-Hilbert problems", Comm. Math. Phys.
207, pp. 589-620 (1999), we have shown that -periodic sequences of weights
lead to "moments", polynomials defined by determinants of matrices involving
these moments and -step relations between them, thus leading to
-band matrices . Given a Darboux transformations on , which effect
does it have on the -periodic sequence of weights and on the associated
polynomials ? These questions will receive a precise answer in this paper. The
methods are based on introducing time parameters in the weights, making the
band matrix evolve according to the so-called discrete KP hierarchy.
Darboux transformations on that translate into vertex operators acting on
the -function.Comment: 43 page
- …