203 research outputs found
On universality of local edge regime for the deformed Gaussian Unitary Ensemble
We consider the deformed Gaussian ensemble in which
is a hermitian matrix (possibly random) and is the Gaussian
unitary random matrix (GUE) independent of . Assuming that the
Normalized Counting Measure of converges weakly (in probability if
random) to a non-random measure with a bounded support and assuming
some conditions on the convergence rate, we prove universality of the local
eigenvalue statistics near the edge of the limiting spectrum of .Comment: 25 pages, 2 figure
Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling
The Anderson model for independent electrons in a disordered potential is
transformed analytically and exactly to a basis of random extended states
leading to a variant of augmented space. In addition to the widely-accepted
phase diagrams in all physical dimensions, a plethora of additional, weaker
Anderson transitions are found, characterized by the long-distance behavior of
states. Critical disorders are found for Anderson transitions at which the
asymptotically dominant sector of augmented space changes for all states at the
same disorder. At fixed disorder, critical energies are also found at which the
localization properties of states are singular. Under the approximation of
single-parameter scaling, this phase diagram reduces to the widely-accepted one
in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson
transition at infinitesimal disorder, there is a transition between two
localized states, characterized by a change in the nature of wave function
decay.Comment: 51 pages including 4 figures, revised 30 November 200
Spectrum of the Product of Independent Random Gaussian Matrices
We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M
independent NxN Gaussian random matrices in the large-N limit is rotationally
symmetric in the complex plane and is given by a simple expression
rho(z,\bar{z}) = 1/(M\pi\sigma^2} |z|^{-2+2/M} for |z|<\sigma, and is zero for
|z|> \sigma. The parameter \sigma corresponds to the radius of the circular
support and is related to the amplitude of the Gaussian fluctuations. This form
of the eigenvalue density is highly universal. It is identical for products of
Gaussian Hermitian, non-Hermitian, real or complex random matrices. It does not
change even if the matrices in the product are taken from different Gaussian
ensembles. We present a self-contained derivation of this result using a planar
diagrammatic technique for Gaussian matrices. We also give a numerical evidence
suggesting that this result applies also to matrices whose elements are
independent, centered random variables with a finite variance.Comment: 16 pages, 6 figures, minor changes, some references adde
Density of State in a Complex Random Matrix Theory with External Source
The density of state for a complex random matrix coupled to an
external deterministic source is considered for a finite N, and a compact
expression in an integral representation is obtained.Comment: 7 pages, late
Determining the Spectral Signature of Spatial Coherent Structures
We applied to an open flow a proper orthogonal decomposition (pod) technique,
on 2D snapshots of the instantaneous velocity field, to reveal the spatial
coherent structures responsible of the self-sustained oscillations observed in
the spectral distribution of time series. We applied the technique to 2D planes
out of 3D direct numerical simulations on an open cavity flow. The process can
easily be implemented on usual personal computers, and might bring deep
insights on the relation between spatial events and temporal signature in (both
numerical or experimental) open flows.Comment: 4 page
Weak disorder expansion for localization lengths of quasi-1D systems
A perturbative formula for the lowest Lyapunov exponent of an Anderson model on a strip is presented. It is expressed in terms of an energy-dependent doubly stochastic matrix, the size of which is proportional to the strip width. This matrix and the resulting perturbative expression for the Lyapunov exponent are evaluated numerically. Dependence on energy, strip width and disorder strength are thoroughly compared with the results obtained by the standard transfer matrix method. Good agreement is found for all energies in the band of the free operator and this even for quite large values of the disorder strength
Low density expansion for Lyapunov exponents
In some quasi-one-dimensional weakly disordered media, impurities are large
and rare rather than small and dense. For an Anderson model with a low density
of strong impurities, a perturbation theory in the impurity density is
developed for the Lyapunov exponent and the density of states. The Lyapunov
exponent grows linearly with the density. Anomalies of the Kappus-Wegner type
appear for all rational quasi-momenta even in lowest order perturbation theory
Generalized Lyapunov Exponent and Transmission Statistics in One-dimensional Gaussian Correlated Potentials
Distribution of the transmission coefficient T of a long system with a
correlated Gaussian disorder is studied analytically and numerically in terms
of the generalized Lyapunov exponent (LE) and the cumulants of lnT. The effect
of the disorder correlations on these quantities is considered in weak,
moderate and strong disorder for different models of correlation. Scaling
relations between the cumulants of lnT are obtained. The cumulants are treated
analytically within the semiclassical approximation in strong disorder, and
numerically for an arbitrary strength of the disorder. A small correlation
scale approximation is developed for calculation of the generalized LE in a
general correlated disorder. An essential effect of the disorder correlations
on the transmission statistics is found. In particular, obtained relations
between the cumulants and between them and the generalized LE show that, beyond
weak disorder, transmission fluctuations and deviation of their distribution
from the log-normal form (in a long but finite system) are greatly enhanced due
to the disorder correlations. Parametric dependence of these effects upon the
correlation scale is presented.Comment: 18 pages, 11 figure
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