141 research outputs found
Spectral fluctuations of tridiagonal random matrices from the beta-Hermite ensemble
A time series delta(n), the fluctuation of the nth unfolded eigenvalue was
recently characterized for the classical Gaussian ensembles of NxN random
matrices (GOE, GUE, GSE). It is investigated here for the beta-Hermite ensemble
as a function of beta (zero or positive) by Monte Carlo simulations. The
fluctuation of delta(n) and the autocorrelation function vary logarithmically
with n for any beta>0 (1<<n<<N). The simple logarithmic behavior reported for
the higher-order moments of delta(n) for the GOE (beta=1) and the GUE (beta=2)
is valid for any positive beta and is accounted for by Gaussian distributions
whose variances depend linearly on ln(n). The 1/f noise previously demonstrated
for delta(n) series of the three Gaussian ensembles, is characterized by
wavelet analysis both as a function of beta and of N. When beta decreases from
1 to 0, for a given and large enough N, the evolution from a 1/f noise at
beta=1 to a 1/f^2 noise at beta=0 is heterogeneous with a ~1/f^2 noise at the
finest scales and a ~1/f noise at the coarsest ones. The range of scales in
which a ~1/f^2 noise predominates grows progressively when beta decreases.
Asymptotically, a 1/f^2 noise is found for beta=0 while a 1/f noise is the rule
for beta positive.Comment: 35 pages, 10 figures, corresponding author: G. Le Cae
Differences between regular and random order of updates in damage spreading simulations
We investigate the spreading of damage in the three-dimensional Ising model
by means of large-scale Monte-Carlo simulations. Within the Glauber dynamics we
use different rules for the order in which the sites are updated. We find that
the stationary damage values and the spreading temperature are different for
different update order. In particular, random update order leads to larger
damage and a lower spreading temperature than regular order. Consequently,
damage spreading in the Ising model is non-universal not only with respect to
different update algorithms (e.g. Glauber vs. heat-bath dynamics) as already
known, but even with respect to the order of sites.Comment: final version as published, 4 pages REVTeX, 2 eps figures include
Chaotic behavior and damage spreading in the Glauber Ising model - a master equation approach
We investigate the sensitivity of the time evolution of a kinetic Ising model
with Glauber dynamics against the initial conditions. To do so we apply the
"damage spreading" method, i.e., we study the simultaneous evolution of two
identical systems subjected to the same thermal noise. We derive a master
equation for the joint probability distribution of the two systems. We then
solve this master equation within an effective-field approximation which goes
beyond the usual mean-field approximation by retaining the fluctuations though
in a quite simplistic manner. The resulting effective-field theory is applied
to different physical situations. It is used to analyze the fixed points of the
master equation and their stability and to identify regular and chaotic phases
of the Glauber Ising model. We also discuss the relation of our results to
directed percolation.Comment: 9 pages RevTeX, 4 EPS figure
New Monte Carlo method for planar Poisson-Voronoi cells
By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a
planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is
developed on the basis of earlier theoretical work; it exploits, in particular,
the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have
between four and six significant digits. Accurate n dependent averages, second
moments, and variances are obtained for the cell area and the cell perimeter.
The numerical large n behavior of these quantities is analyzed in terms of
asymptotic power series in 1/n. Snapshots are shown of typical occurrences of
extremely rare events implicating cells of up to n=1600 sides embedded in an
ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic
features of such many-sided cells and their immediate environment. Their
relevance for observable properties is stressed.Comment: 35 pages including 10 figures and 4 table
Experimental evidence of ageing and slow restoration of the weak-contact configuration in tilted 3D granular packings
Granular packings slowly driven towards their instability threshold are
studied using a digital imaging technique as well as a nonlinear acoustic
method. The former method allows us to study grain rearrangements on the
surface during the tilting and the latter enables to selectively probe the
modifications of the weak-contact fraction in the material bulk. Gradual ageing
of both the surface activity and the weak-contact reconfigurations is observed
as a result of repeated tilt cycles up to a given angle smaller than the angle
of avalanche. For an aged configuration reached after several consecutive tilt
cycles, abrupt resumption of the on-surface activity and of the weak-contact
rearrangements occurs when the packing is subsequently inclined beyond the
previous maximal tilting angle. This behavior is compared with literature
results from numerical simulations of inclined 2D packings. It is also found
that the aged weak-contact configurations exhibit spontaneous restoration
towards the initial state if the packing remains at rest for tens of minutes.
When the packing is titled forth and back between zero and near-critical
angles, instead of ageing, the weak-contact configuration exhibits "internal
weak-contact avalanches" in the vicinity of both the near-critical and zero
angles. By contrast, the stronger-contact skeleton remains stable
The perimeter of large planar Voronoi cells: a double-stranded random walk
Let be the probability for a planar Poisson-Voronoi cell to have
exactly sides. We construct the asymptotic expansion of up to
terms that vanish as . We show that {\it two independent biased
random walks} executed by the polar angle determine the trajectory of the cell
perimeter. We find the limit distribution of (i) the angle between two
successive vertex vectors, and (ii) the one between two successive perimeter
segments. We obtain the probability law for the perimeter's long wavelength
deviations from circularity. We prove Lewis' law and show that it has
coefficient 1/4.Comment: Slightly extended version; journal reference adde
Damage spreading in random field systems
We investigate how a quenched random field influences the damage spreading
transition in kinetic Ising models. To this end we generalize a recent master
equation approach and derive an effective field theory for damage spreading in
random field systems. This theory is applied to the Glauber Ising model with a
bimodal random field distribution. We find that the random field influences the
spreading transition by two different mechanisms with opposite effects. First,
the random field favors the same particular direction of the spin variable at
each site in both systems which reduces the damage. Second, the random field
suppresses the magnetization which, in turn, tends to increase the damage. The
competition between these two effects leads to a rich behavior.Comment: 4 pages RevTeX, 3 eps figure
Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results
We achieve a detailed understanding of the -sided planar Poisson-Voronoi
cell in the limit of large . Let be the probability for a cell to
have sides. We construct the asymptotic expansion of up to
terms that vanish as . We obtain the statistics of the lengths of
the perimeter segments and of the angles between adjoining segments: to leading
order as , and after appropriate scaling, these become independent
random variables whose laws we determine; and to next order in they have
nontrivial long range correlations whose expressions we provide. The -sided
cell tends towards a circle of radius (n/4\pi\lambda)^{\half}, where
is the cell density; hence Lewis' law for the average area of
the -sided cell behaves as with . For
the cell perimeter, expressed as a function of the polar
angle , satisfies , where is known Gaussian
noise; we deduce from it the probability law for the perimeter's long
wavelength deviations from circularity. Many other quantities related to the
asymptotic cell shape become accessible to calculation.Comment: 54 pages, 3 figure
High-finesse Fabry-Perot cavities with bidimensional SiN photonic-crystal slabs
Light scattering by a two-dimensional photonic crystal slab (PCS) can result in dramatic interference effects associated with Fano resonances. Such devices offer appealing alternatives to distributed Bragg reflectors or filters for various applications such as optical wavelength and polarization filters, reflectors, semiconductor lasers, photodetectors, bio-sensors, or non-linear optical components. Suspended PCSs also find natural applications in the field of optomechanics, where the mechanical modes of a suspended slab interact via radiation pressure with the optical field of a high finesse cavity. The reflectivity and transmission properties of a defect-free suspended PCS around normal incidence can be used to couple out-of-plane mechanical modes to an optical field by integrating it in a free space cavity. Here, we demonstrate the successful implementation of a PCS reflector on a high-tensile stress SiN nanomembrane. We illustrate the physical process underlying the high reflectivity by measuring the photonic crystal band diagram. Moreover, we introduce a clear theoretical description of the membrane scattering properties in the presence of optical losses. By embedding the PCS inside a high-finesse cavity, we fully characterize its optical properties. The spectrally, angular, and polarization resolved measurements demonstrate the wide tunability of the membrane's reflectivity, from nearly 0 to 99.9470~ 0.0025 \%, and show that material absorption is not the main source of optical loss. Moreover, the cavity storage time demonstrated in this work exceeds the mechanical period of low-order mechanical drum modes. This so-called resolved sideband condition is a prerequisite to achieve quantum control of the mechanical resonator with light
Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace
The degree of entanglement of random pure states in bipartite quantum systems
can be estimated from the distribution of the extreme Schmidt eigenvalues. For
a bipartition of size M\geq N, these are distributed according to a
Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a
fixed-trace constraint. We first compute the distribution and moments of the
smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary
M\geq N. Our method is based on a Laplace inversion of the recursive results
for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are
given for fixed N and M, generalizing and simplifying earlier results. In the
microscopic large-N limit with M-N fixed, the orthogonal and unitary WL
distributions exhibit universality after a suitable rescaling and are therefore
independent of the constraint. We prove that very recent results given in terms
of hypergeometric functions of matrix argument are equivalent to more explicit
expressions in terms of a Pfaffian or determinant of Bessel functions. While
the latter were mostly known from the random matrix literature on the QCD Dirac
operator spectrum, we also derive some new results in the orthogonal symmetry
class.Comment: 25 pag., 4 fig - minor changes, typos fixed. To appear in JSTA
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