222 research outputs found

### Transport in the random Kronig-Penney model

The Kronig-Penney model with random Dirac potentials on the lattice \ZM has
critical energies at which the Lyapunov exponent vanishes and the density of
states has a van Hove singularity. This leads to a non-trivial quantum
diffusion even though the spectrum is known to be pure-point

### Weak Disorder in Fibonacci Sequences

We study how weak disorder affects the growth of the Fibonacci series. We
introduce a family of stochastic sequences that grow by the normal Fibonacci
recursion with probability 1-epsilon, but follow a different recursion rule
with a small probability epsilon. We focus on the weak disorder limit and
obtain the Lyapunov exponent, that characterizes the typical growth of the
sequence elements, using perturbation theory. The limiting distribution for the
ratio of consecutive sequence elements is obtained as well. A number of
variations to the basic Fibonacci recursion including shift, doubling, and
copying are considered.Comment: 4 pages, 2 figure

### Positivity of Lyapunov exponents for a continuous matrix-valued Anderson model

We study a continuous matrix-valued Anderson-type model. Both leading
Lyapunov exponents of this model are proved to be positive and distinct for all
ernergies in $(2,+\infty)$ except those in a discrete set, which leads to
absence of absolutely continuous spectrum in $(2,+\infty)$. This result is an
improvement of a previous result with Stolz. The methods, based upon a result
by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a
criterion by Goldsheid and Margulis, allow for singular Bernoulli
distributions

### 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

### 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

### A matrix-valued point interactions model

We study a matrix-valued Schr\"odinger operator with random point
interactions. We prove the absence of absolutely continuous spectrum for this
operator by proving that away from a discrete set its Lyapunov exponents do not
vanish. For this we use a criterion by Gol'dsheid and Margulis and we prove the
Zariski denseness, in the symplectic group, of the group generated by the
transfer matrices. Then we prove estimates on the transfer matrices which lead
to the H\"older continuity of the Lyapunov exponents. After proving the
existence of the integrated density of states of the operator, we also prove
its H\"older continuity by proving a Thouless formula which links the
integrated density of states to the sum of the positive Lyapunov exponents

### Exact Lyapunov Exponent for Infinite Products of Random Matrices

In this work, we give a rigorous explicit formula for the Lyapunov exponent
for some binary infinite products of random $2\times 2$ real matrices. All
these products are constructed using only two types of matrices, $A$ and $B$,
which are chosen according to a stochastic process. The matrix $A$ is singular,
namely its determinant is zero. This formula is derived by using a particular
decomposition for the matrix $B$, which allows us to write the Lyapunov
exponent as a sum of convergent series. Finally, we show with an example that
the Lyapunov exponent is a discontinuous function of the given parameter.Comment: 1 pages, CPT-93/P.2974,late

### Scalar decay in a three-dimensional chaotic flow

The decay of a passive scalar in a three-dimensional chaotic flow is studied
using high-resolution numerical simulations. The (volume-preserving) flow
considered is a three-dimensional extension of the randomised alternating sine
flow employed extensively in studies of mixing in two dimensions. It is used to
show that theoretical predictions for two-dimensional flows with small
diffusivity carry over to three dimensions even though the stretching
properties differ significantly. The variance decay rate, scalar field
structure, and time evolution of statistical moments confirm that there are two
distinct regimes of scalar decay: a locally controlled regime, which applies
when the domain size is comparable to the characteristic lengthscale of the
velocity field, and a globally controlled regime, which when applies when the
domain is larger. Asymptotic predictions for the variance decay rate in both
regimes show excellent agreement with the numerical results. Consideration of
both the forward flow and its time reverse makes it possible to compare the
scalar evolution in flows with one or two expanding directions; simulations
confirm the theoretical prediction that the decay rate of the scalar is the
same in both flows, despite the very different scalar field structures

### Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts

We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic
homeomorphism which fibers over an almost periodic dynamical system
(generalized skew-shifts). We prove that any cocycle which is not uniformly
hyperbolic can be approximated by one which is conjugate to an
$SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy
class does not display a certain obstruction to uniform hyperbolicity, then it
can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising
from Schr\"odinger operators, the obstruction vanishes and we conclude that
uniform hyperbolicity is dense, which implies that for a generic continuous
potential, the spectrum of the corresponding Schr\"odinger operator is a Cantor
set.Comment: Final version. To appear in Duke Mathematical Journa

### Localization for a matrix-valued Anderson model

We study localization properties for a class of one-dimensional,
matrix-valued, continuous, random Schr\"odinger operators, acting on
L^2(\R)\otimes \C^N, for arbitrary $N\geq 1$. We prove that, under suitable
assumptions on the F\"urstenberg group of these operators, valid on an interval
$I\subset \R$, they exhibit localization properties on $I$, both in the
spectral and dynamical sense. After looking at the regularity properties of the
Lyapunov exponents and of the integrated density of states, we prove a Wegner
estimate and apply a multiscale analysis scheme to prove localization for these
operators. We also study an example in this class of operators, for which we
can prove the required assumptions on the F\"urstenberg group. This group being
the one generated by the transfer matrices, we can use, to prove these
assumptions, an algebraic result on generating dense Lie subgroups in
semisimple real connected Lie groups, due to Breuillard and Gelander. The
algebraic methods used here allow us to handle with singular distributions of
the random parameters

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