104 research outputs found

### 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

### Delocalization in random polymer models

A random polymer model is a one-dimensional Jacobi matrix randomly composed
of two finite building blocks. If the two associated transfer matrices commute,
the corresponding energy is called critical. Such critical energies appear in
physical models, an example being the widely studied random dimer model. It is
proven that the Lyapunov exponent vanishes quadratically at a generic critical
energy and that the density of states is positive there. Large deviation
estimates around these asymptotics allow to prove optimal lower bounds on
quantum transport, showing that it is almost surely overdiffusive even though
the models are known to have pure-point spectrum with exponentially localized
eigenstates for almost every configuration of the polymers. Furthermore, the
level spacing is shown to be regular at the critical energy

### Mott law as lower bound for a random walk in a random environment

We consider a random walk on the support of a stationary simple point process
on $R^d$, $d\geq 2$ which satisfies a mixing condition w.r.t.the translations
or has a strictly positive density uniformly on large enough cubes. Furthermore
the point process is furnished with independent random bounded energy marks.
The transition rates of the random walk decay exponentially in the jump
distances and depend on the energies through a factor of the Boltzmann-type.
This is an effective model for the phonon-induced hopping of electrons in
disordered solids within the regime of strong Anderson localization. We show
that the rescaled random walk converges to a Brownian motion whose diffusion
coefficient is bounded below by Mott's law for the variable range hopping
conductivity at zero frequency. The proof of the lower bound involves estimates
for the supercritical regime of an associated site percolation problem

### Spectral averaging techniques for Jacobi matrices with matrix entries

A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal
matrix with invertible blocks on the off-diagonals. Averaging over boundary
conditions leads to explicit formulas for the averaged spectral measure which
can potentially be useful for spectral analysis. Furthermore another variant of
spectral averaging over coupling constants for these operators is presented

### Boundary maps for $C^*$-crossed products with R with an application to the quantum Hall effect

The boundary map in K-theory arising from the Wiener-Hopf extension of a
crossed product algebra with R is the Connes-Thom isomorphism. In this article
the Wiener Hopf extension is combined with the Heisenberg group algebra to
provide an elementary construction of a corresponding map on higher traces (and
cyclic cohomology). It then follows directly from a non-commutative Stokes
theorem that this map is dual w.r.t.Connes' pairing of cyclic cohomology with
K-theory. As an application, we prove equality of quantized bulk and edge
conductivities for the integer quantum Hall effect described by continuous
magnetic Schroedinger operators.Comment: to appear in Commun. Math. Phy

### Linear response theory for magnetic Schroedinger operators in disordered media

We justify the linear response theory for an ergodic Schroedinger operator
with magnetic field within the non-interacting particle approximation, and
derive a Kubo formula for the electric conductivity tensor. To achieve that, we
construct suitable normed spaces of measurable covariant operators where the
Liouville equation can be solved uniquely. If the Fermi level falls into a
region of localization, we recover the well-known Kubo-Streda formula for the
quantum Hall conductivity at zero temperature.Comment: Latex, 68 pages, misprints corrected, formatting change

### 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

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