Perturbative test of single parameter scaling for 1D random media

Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance is equal to the Lyapunov exponent. We settle discussions about its validity for a wide class of models by proving that, away from anomalies, single parameter scaling holds to lowest order perturbation theory in the disorder strength. However, it is generically violated at higher order. This is explicitely exhibited for the Anderson model.Comment: minor corrections to previous version, to appear in Annales H. Poincar

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

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

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

Spectral averaging techniques for Jacobi matrices

Spectral averaging techniques for one-dimensional discrete Schroedinger operators are revisited and extended. In particular, simultaneous averaging over several parameters is discussed. Special focus is put on proving lower bounds on the density of the averaged spectral measures. These Wegner type estimates are used to analyze stability properties for the spectral types of Jacobi matrices under local perturbations

Scattering theory for lattice operators in dimension d≥3d\geq 3

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension $d\geq 3$ the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in presence of embedded eigenvalues and threshold singularities.Comment: Minor errors and misprints corrected; new result on absense of embedded eigenvalues for potential scattering; to appear in RM

Simultaneous quantization of edge and bulk Hall conductivity

The edge Hall conductivity is shown to be an integer multiple of $e^2/h$ which is almost surely independent of the choice of the disordered configuration. Its equality to the bulk Hall conductivity given by the Kubo-Chern formula follows from K-theoretic arguments. This leads to quantization of the Hall conductance for any redistribution of the current in the sample. It is argued that in experiments at most a few percent of the total current can be carried by edge states.Comment: 6 pages Latex, 1 figur