Gaussian fluctuations of products of random matrices distributed close to the identity - Drabkin, Maxim; Schulz-Baldes, Hermann (Journal of Difference Equations and Applications, 2015)
Products of random 2 × 2 matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighbourhood of order of the identity matrix. The Lyapunov exponent and the variance of the Gaussian fluctuations are calculated perturbatively in and this requires a detailed analysis of the associated random dynamical system on the unit circle and its invariant measure. The result applies to anomalies and band edges of one-dimensional random Schrödinger operators.
Keywords: products of random matrices; Lyapunov exoponent; variances; perturbation theory

Localization and Chern numbers for weakly disordered BdG operators - de Nittis, Giuseppe; Drabkin, Maxim; Schulz-Baldes, Hermann (Markov Processes and Related Fields, 2015)
After a short discussion of various random Bogoliubov-de Gennes (BdG) model operators and the associated physics, the Aizenman - Molchanov method is applied to prove Anderson localization in the weak disorder regime for the spectrum in the central gap. This allows to construct random BdG operators which have localized states in an interval centered at zero energy. Furthermore, techniques for the calculation of Chern numbers are reviewed and applied to two non-trivial BdG operators, the p + ip wave and d + id wave superconductors.
UnivIS-Import:2015-04-15:Pub.2013.nat.dma.pma22.locali
Keywords: Anderson localization; Random Bogoliubov-de Gennes operators; Weak disorder

Transport in the random Kronig-Penney model - Drabkin, Maxim; Kirsch, Werner; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 2012)
The Kronig-Penney model with random Dirac potentials on the lattice has critical energies at which the Lyapunov exponent vanishes 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. © 2012 American Institute of Physics.
UnivIS-Import:2015-03-09:Pub.2012.nat.dma.pma22.transp_3