20,032 research outputs found
Matrices coupled in a chain. I. Eigenvalue correlations
The general correlation function for the eigenvalues of complex hermitian
matrices coupled in a chain is given as a single determinant. For this we use a
slight generalization of a theorem of Dyson.Comment: ftex eynmeh.tex, 2 files, 8 pages Submitted to: J. Phys.
Probability density of determinants of random matrices
In this brief paper the probability density of a random real, complex and
quaternion determinant is rederived using singular values. The behaviour of
suitably rescaled random determinants is studied in the limit of infinite order
of the matrices
Density-functional theory for fermions in the unitary regime
In the unitary regime, fermions interact strongly via two-body potentials
that exhibit a zero range and a (negative) infinite scattering length. The
energy density is proportional to the free Fermi gas with a proportionality
constant . We use a simple density functional parametrized by an effective
mass and the universal constant , and employ Kohn-Sham density-functional
theory to obtain the parameters from fit to one exactly solvable two-body
problem. This yields and a rather large effective mass. Our approach
is checked by similar Kohn-Sham calculations for the exactly solvable Calogero
model.Comment: 5 pages, 2 figure
Calculation of some determinants using the s-shifted factorial
Several determinants with gamma functions as elements are evaluated. This
kind of determinants are encountered in the computation of the probability
density of the determinant of random matrices. The s-shifted factorial is
defined as a generalization for non-negative integers of the power function,
the rising factorial (or Pochammer's symbol) and the falling factorial. It is a
special case of polynomial sequence of the binomial type studied in
combinatorics theory. In terms of the gamma function, an extension is defined
for negative integers and even complex values. Properties, mainly composition
laws and binomial formulae, are given. They are used to evaluate families of
generalized Vandermonde determinants with s-shifted factorials as elements,
instead of power functions.Comment: 25 pages; added section 5 for some examples of application
Finite-difference distributions for the Ginibre ensemble
The Ginibre ensemble of complex random matrices is studied. The complex
valued random variable of second difference of complex energy levels is
defined. For the N=3 dimensional ensemble are calculated distributions of
second difference, of real and imaginary parts of second difference, as well as
of its radius and of its argument (angle). For the generic N-dimensional
Ginibre ensemble an exact analytical formula for second difference's
distribution is derived. The comparison with real valued random variable of
second difference of adjacent real valued energy levels for Gaussian
orthogonal, unitary, and symplectic, ensemble of random matrices as well as for
Poisson ensemble is provided.Comment: 8 pages, a number of small changes in the tex
The Local Semicircle Law for Random Matrices with a Fourfold Symmetry
We consider real symmetric and complex Hermitian random matrices with the
additional symmetry . The matrix elements are independent
(up to the fourfold symmetry) and not necessarily identically distributed. This
ensemble naturally arises as the Fourier transform of a Gaussian orthogonal
ensemble (GOE). It also occurs as the flip matrix model - an approximation of
the two-dimensional Anderson model at small disorder. We show that the density
of states converges to the Wigner semicircle law despite the new symmetry type.
We also prove the local version of the semicircle law on the optimal scale.Comment: 20 pages, to appear in J. Math. Phy
Initial-state randomness as a universal source of decoherence
We study time evolution of entanglement between two qubits, which are part of
a larger system, after starting from a random initial product state. We show
that, due to randomness in the initial product state, entanglement is present
only between directly coupled qubits and only for short times. Time dependence
of the entanglement appears essentially independent of the specific hamiltonian
used for time evolution and is well reproduced by a parameter-free two-body
random matrix model.Comment: 8 pages, 6 figure
Wigner surmise for Hermitian and non-Hermitian Chiral random matrices
We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral Random Matrix Theory, for both real and complex eigenvalues. Testing against known results
for zero and maximal non-Hermiticity in the microscopic large-N limit we find an excellent agreement, valid for a small number of exact zero-eigenvalues. New compact expressions are derived for real eigenvalues in the orthogonal and symplectic classes, and at intermediate non-Hermiticity for the unitary and symplectic classes. Such individual Dirac eigenvalue
distributions are a useful tool in Lattice Gauge Theory and we illustrate this by showing that our new results can describe data from two-colour QCD simulations with chemical potential in the symplectic class
Periodic orbit theory and spectral rigidity in pseudointegrable systems
We calculate numerically the periodic orbits of pseudointegrable systems of
low genus numbers that arise from rectangular systems with one or two
salient corners. From the periodic orbits, we calculate the spectral rigidity
using semiclassical quantum mechanics with reaching up to
quite large values. We find that the diagonal approximation is applicable when
averaging over a suitable energy interval. Comparing systems of various shapes
we find that our results agree well with calculated directly from
the eigenvalues by spectral statistics. Therefore, additional terms as e.g.
diffraction terms seem to be small in the case of the systems investigated in
this work. By reducing the size of the corners, the spectral statistics of our
pseudointegrable systems approaches the one of an integrable system, whereas
very large differences between integrable and pseudointegrable systems occur,
when the salient corners are large. Both types of behavior can be well
understood by the properties of the periodic orbits in the system
Glassy dynamics in granular compaction
Two models are presented to study the influence of slow dynamics on granular
compaction. It is found in both cases that high values of packing fraction are
achieved only by the slow relaxation of cooperative structures. Ongoing work to
study the full implications of these results is discussed.Comment: 12 pages, 9 figures; accepted in J. Phys: Condensed Matter,
proceedings of the Trieste workshop on 'Unifying concepts in glass physics
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