7,417 research outputs found
Eigenfunction entropy and spectral compressibility for critical random matrix ensembles
Based on numerical and perturbation series arguments we conjecture that for
certain critical random matrix models the information dimension of
eigenfunctions D_1 and the spectral compressibility chi are related by the
simple equation chi+D_1/d=1, where d is the system dimensionality.Comment: 4 pages, 3 figure
Integrable random matrix ensembles
We propose new classes of random matrix ensembles whose statistical
properties are intermediate between statistics of Wigner-Dyson random matrices
and Poisson statistics. The construction is based on integrable N-body
classical systems with a random distribution of momenta and coordinates of the
particles. The Lax matrices of these systems yield random matrix ensembles
whose joint distribution of eigenvalues can be calculated analytically thanks
to integrability of the underlying system. Formulas for spacing distributions
and level compressibility are obtained for various instances of such ensembles.Comment: 32 pages, 8 figure
Periodic orbits contribution to the 2-point correlation form factor for pseudo-integrable systems
The 2-point correlation form factor, , for small values of
is computed analytically for typical examples of pseudo-integrable systems.
This is done by explicit calculation of periodic orbit contributions in the
diagonal approximation. The following cases are considered: (i) plane billiards
in the form of right triangles with one angle and (ii) rectangular
billiards with the Aharonov-Bohm flux line. In the first model, using the
properties of the Veech structure, it is shown that
where for odd ,
for even not divisible by 3, and for even
divisible by 3. For completeness we also recall informally the main
features of the Veech construction. In the second model the answer depends on
arithmetical properties of ratios of flux line coordinates to the corresponding
sides of the rectangle. When these ratios are non-commensurable irrational
numbers, where is the
fractional part of the flux through the rectangle when and it is symmetric with respect to the line when . The comparison of these results with numerical
calculations of the form factor is discussed in detail. The above values of
differ from all known examples of spectral statistics, thus confirming
analytically the peculiarities of statistical properties of the energy levels
in pseudo-integrable systems.Comment: 61 pages, 13 figures. Submitted to Communications in Mathematical
Physics, 200
Multifractal dimensions for all moments for certain critical random matrix ensembles in the strong multifractality regime
We construct perturbation series for the q-th moment of eigenfunctions of
various critical random matrix ensembles in the strong multifractality regime
close to localization. Contrary to previous investigations, our results are
valid in the region q<1/2. Our findings allow to verify, at first leading
orders in the strong multifractality limit, the symmetry relation for anomalous
fractal dimensions Delta(q)=Delta(1-q), recently conjectured for critical
models where an analogue of the metal-insulator transition takes place. It is
known that this relation is verified at leading order in the weak
multifractality regime. Our results thus indicate that this symmetry holds in
both limits of small and large coupling constant. For general values of the
coupling constant we present careful numerical verifications of this symmetry
relation for different critical random matrix ensembles. We also present an
example of a system closely related to one of these critical ensembles, but
where the symmetry relation, at least numerically, is not fulfilled.Comment: 12 pages, 12 figure
Random matrix ensembles associated with Lax matrices
A method to generate new classes of random matrix ensembles is proposed.
Random matrices from these ensembles are Lax matrices of classically integrable
systems with a certain distribution of momenta and coordinates. The existence
of an integrable structure permits to calculate the joint distribution of
eigenvalues for these matrices analytically. Spectral statistics of these
ensembles are quite unusual and in many cases give rigorously new examples of
intermediate statistics
Perturbation approach to multifractal dimensions for certain critical random matrix ensembles
Fractal dimensions of eigenfunctions for various critical random matrix
ensembles are investigated in perturbation series in the regimes of strong and
weak multifractality. In both regimes we obtain expressions similar to those of
the critical banded random matrix ensemble extensively discussed in the
literature. For certain ensembles, the leading-order term for weak
multifractality can be calculated within standard perturbation theory. For
other models such a direct approach requires modifications which are briefly
discussed. Our analytical formulas are in good agreement with numerical
calculations.Comment: 28 pages, 7 figure
The distribution of the ratio of consecutive level spacings in random matrix ensembles
We derive expressions for the probability distribution of the ratio of two
consecutive level spacings for the classical ensembles of random matrices. This
ratio distribution was recently introduced to study spectral properties of
many-body problems, as, contrary to the standard level spacing distributions,
it does not depend on the local density of states. Our Wigner-like surmises are
shown to be very accurate when compared to numerics and exact calculations in
the large matrix size limit. Quantitative improvements are found through a
polynomial expansion. Examples from a quantum many-body lattice model and from
zeros of the Riemann zeta function are presented.Comment: 5 pages, 4 figure
Open problems in nuclear density functional theory
This note describes five subjects of some interest for the density functional
theory in nuclear physics. These are, respectively, i) the need for concave
functionals, ii) the nature of the Kohn-Sham potential for the radial density
theory, iii) a proper implementation of a density functional for an "intrinsic"
rotational density, iv) the possible existence of a potential driving the
square root of the density, and v) the existence of many models where a density
functional can be explicitly constructed.Comment: 10 page
Anomaly distribution in quasar magnitudes: a test of lensing by an hypothetic Supergiant Molecular Cloud in the Galactic halo
An anomaly in the distribution of quasar magnitudes based on the SDSS survey,
has been recently reported by Longo (2012). The angular size of this anomaly is
of the order of on the sky. A low surface brightness smooth
structure in -rays, coincides with the sky location and extent of the
quasar anomaly, and is close to the Northern component of a pair of
-ray bubbles discovered in the \sl Fermi Gamma-ray Space Telescope \rm
survey. Molecular clouds are thought to be illuminated by cosmic rays. I test
the hypothesis that the magnitude anomaly in the quasar distribution, is due to
a lensing effect by an hypothetic Supergiant Molecular Cloud (SGMC) in the
Galactic halo.A series of grid lens models are built by assuming firstly that a
SGMC is a lattice with clumps of , 10 AU in size, and
assuming various filling factors of the cloud, and secondly a fractal
structure. Local amplifications are calculated for these lenses by using the
public software LensTool, and the single plane approximation. A complex network
of caustics due to the clumpy structure is present. Our best single plane lens
model capable of explaining Longo's effect, \sl at least in sparse regions, \rm
requires a mass within at a lens plane distance of 20 kpc. It is constructed
from a molecular cloud building block of within a scale
of 30 pc expanded by fractal scaling with dimension up to 5-8.6 kpc
for the SGMC. If such a Supergiant Molecular Cloud were demonstrated, it might
be part of a lens explanation for the luminous anomaly discovered in quasars
and in red galaxies. The mass budget may be varied by changing the cloud depth
and the fractal dimension.Comment: 11 pages, no Figures, 2 table
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