12,540 research outputs found
Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach
We prove that general correlation functions of both ratios and products of
characteristic polynomials of Hermitian random matrices are governed by
integrable kernels of three different types: a) those constructed from
orthogonal polynomials; b) constructed from Cauchy transforms of the same
orthogonal polynomials and finally c) those constructed from both orthogonal
polynomials and their Cauchy transforms. These kernels are related with the
Riemann-Hilbert problem for orthogonal polynomials. For the correlation
functions we obtain exact expressions in the form of determinants of these
kernels. Derived representations enable us to study asymptotics of correlation
functions of characteristic polynomials via Deift-Zhou
steepest-descent/stationary phase method for Riemann-Hilbert problems, and in
particular to find negative moments of characteristic polynomials. This reveals
the universal parts of the correlation functions and moments of characteristic
polynomials for arbitrary invariant ensemble of symmetry class.Comment: 34page
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
Spectra of observables in the q-oscillator and q-analogue of the Fourier transform
Spectra of the position and momentum operators of the Biedenharn-Macfarlane
q-oscillator (with the main relation aa^+-qa^+a=1) are studied when q>1. These
operators are symmetric but not self-adjoint. They have a one-parameter family
of self-adjoint extensions. These extensions are derived explicitly. Their
spectra and eigenfunctions are given. Spectra of different extensions do not
intersect. The results show that the creation and annihilation operators a^+
and a of the q-oscillator for q>1 cannot determine a physical system without
further more precise definition. In order to determine a physical system we
have to choose appropriate self-adjoint extensions of the position and momentum
operators.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Moment determinants as isomonodromic tau functions
We consider a wide class of determinants whose entries are moments of the
so-called semiclassical functionals and we show that they are tau functions for
an appropriate isomonodromic family which depends on the parameters of the
symbols for the functionals. This shows that the vanishing of the tau-function
for those systems is the obstruction to the solvability of a Riemann-Hilbert
problem associated to certain classes of (multiple) orthogonal polynomials. The
determinants include Haenkel, Toeplitz and shifted-Toeplitz determinants as
well as determinants of bimoment functionals and the determinants arising in
the study of multiple orthogonality. Some of these determinants appear also as
partition functions of random matrix models, including an instance of a
two-matrix model.Comment: 24 page
Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model
We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on
the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a
double-well quartic polynomial, in the limit when n, N \to \infty. We assume
that \epsilon \le (n/N) \le \lambda_{cr} - \epsilon for some \epsilon > 0,
where \lambda_{cr} is the critical value which separates orthogonal polynomials
with two cuts from the ones with one cut. Simultaneously we derive
semiclassical asymptotics for the recursive coefficients of the orthogonal
polynomials, and we show that these coefficients form a cycle of period two
which drifts slowly with the change of the ratio n/N. The proof of the
semiclassical asymptotics is based on the methods of the theory of integrable
systems and on the analysis of the appropriate matrix Riemann-Hilbert problem.
As an application of the semiclassical asymptotics of the orthogonal
polynomials, we prove the universality of the local distribution of eigenvalues
in the matrix model with the double-well quartic interaction in the presence of
two cuts.Comment: 82 pages, published versio
Fibonacci numbers and orthogonal polynomials
We prove that the sequence of reciprocals of the
Fibonacci numbers is a moment sequence of a certain discrete probability, and
we identify the orthogonal polynomials as little -Jacobi polynomials with
. We prove that the corresponding kernel
polynomials have integer coefficients, and from this we deduce that the inverse
of the corresponding Hankel matrices have integer entries. We
prove analogous results for the Hilbert matrices.Comment: A note dated June 2007 has been added with some historical comments.
Some references have been added and complete
A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements
The basic methods of constructing the sets of mutually unbiased bases in the
Hilbert space of an arbitrary finite dimension are discussed and an emerging
link between them is outlined. It is shown that these methods employ a wide
range of important mathematical concepts like, e.g., Fourier transforms, Galois
fields and rings, finite and related projective geometries, and entanglement,
to mention a few. Some applications of the theory to quantum information tasks
are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two
more references adde
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