239 research outputs found
Average characteristic polynomials in the two-matrix model
The two-matrix model is defined on pairs of Hermitian matrices of
size by the probability measure where
and are given potential functions and \tau\in\er. We study averages
of products and ratios of characteristic polynomials in the two-matrix model,
where both matrices and may appear in a combined way in both
numerator and denominator. We obtain determinantal expressions for such
averages. The determinants are constructed from several building blocks: the
biorthogonal polynomials and associated to the two-matrix
model; certain transformed functions and \Q_n(v); and finally
Cauchy-type transforms of the four Eynard-Mehta kernels , ,
and . In this way we generalize known results for the
-matrix model. Our results also imply a new proof of the Eynard-Mehta
theorem for correlation functions in the two-matrix model, and they lead to a
generating function for averages of products of traces.Comment: 28 pages, references adde
High Energy Quark-Antiquark Elastic scattering with Mesonic Exchange
We studies the high energy elastic scattering of quark anti-quark with an
exchange of a mesonic state in the channel with .
Both the normalization factor and the Regge trajectory can be calculated in
PQCD in cases of fixed (non-running) and running coupling constant. The
dependence of the Regge trajectory on the coupling constant is highly
non-linear and the trajectory is of order of in the interesting physical
range.Comment: 29 page
Logarithmic Universality in Random Matrix Theory
Universality in unitary invariant random matrix ensembles with complex matrix
elements is considered. We treat two general ensembles which have a determinant
factor in the weight. These ensembles are relevant, e.g., for spectra of the
Dirac operator in QCD. In addition to the well established universality with
respect to the choice of potential, we prove that microscopic spectral
correlators are unaffected when the matrix in the determinant is replaced by an
expansion in powers of the matrix. We refer to this invariance as logarithmic
universality. The result is used in proving that a simple random matrix model
with Ginsparg-Wilson symmetry has the same microscopic spectral correlators as
chiral random matrix theory.Comment: 16 pages, latex2
Impact of localization on Dyson's circular ensemble
A wide variety of complex physical systems described by unitary matrices have
been shown numerically to satisfy level statistics predicted by Dyson's
circular ensemble. We argue that the impact of localization in such systems is
to provide certain restrictions on the eigenvalues. We consider a solvable
model which takes into account such restrictions qualitatively and find that
within the model a gap is created in the spectrum, and there is a transition
from the universal Wigner distribution towards a Poisson distribution with
increasing localization.Comment: To be published in J. Phys.
Remarks on the Zeros of the Associated Legendre Functions with Integral Degree
We present some formulas for the computation of the zeros of the
integral-degree associated Legendre functions with respect to the order.Comment: 7 pages, 2 figure
Explicit Integration of the Full Symmetric Toda Hierarchy and the Sorting Property
We give an explicit formula for the solution to the initial value problem of
the full symmetric Toda hierarchy. The formula is obtained by the
orthogonalization procedure of Szeg\"{o}, and is also interpreted as a
consequence of the QR factorization method of Symes \cite{symes}. The sorting
property of the dynamics is also proved for the case of a generic symmetric
matrix in the sense described in the text, and generalizations of tridiagonal
formulae are given for the case of matrices with nonzero diagonals.Comment: 13 pages, Latex
Universality of a family of Random Matrix Ensembles with logarithmic soft-confinement potentials
Recently we introduced a family of invariant Random Matrix Ensembles
which is characterized by a parameter describing logarithmic
soft-confinement potentials ). We
showed that we can study eigenvalue correlations of these "-ensembles"
based on the numerical construction of the corresponding orthogonal polynomials
with respect to the weight function . In this
work, we expand our previous work and show that: i) the eigenvalue density is
given by a power-law of the form and
ii) the two-level kernel has an anomalous structure, which is characteristic of
the critical ensembles. We further show that the anomalous part, or the
so-called "ghost-correlation peak", is controlled by the parameter ;
decreasing increases the anomaly. We also identify the two-level
kernel of the -ensembles in the semiclassical regime, which can be
written in a sinh-kernel form with more general argument that reduces to that
of the critical ensembles for . Finally, we discuss the universality
of the -ensembles, which includes Wigner-Dyson universality ( limit), the uncorrelated Poisson-like behavior (
limit), and a critical behavior for all the intermediate
() in the semiclassical regime. We also comment on the
implications of our results in the context of the localization-delocalization
problems as well as the dependence of the two-level kernel of the fat-tail
random matrices.Comment: 10 pages, 13 figure
State estimation in quantum homodyne tomography with noisy data
In the framework of noisy quantum homodyne tomography with efficiency
parameter , we propose two estimators of a quantum state whose
density matrix elements decrease like , for
fixed known and . The first procedure estimates the matrix
coefficients by a projection method on the pattern functions (that we introduce
here for ), the second procedure is a kernel estimator of the
associated Wigner function. We compute the convergence rates of these
estimators, in risk
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