M 511

The modelling of disordered solids in condensed matter physics leads to the study of random Schrödinger operators. The prototype of such an operator is the Anderson model which describes a spinless electron moving in a static random electric potential $V_\omega$ on the state space $\ell^2 (\mathbb{Z}^d)$. The potential values are assumed to be independent identically distributed random variables. With this simple model it is possible to describe the transition from metal to insulator under the presence of disorder. In a first part of the talk, we introduce the Anderson model, discuss basic properties thereof, and review classical results on localization via the so-called fractional moment method and the multiscale analysis. The second and the third part of the talk are devoted to generalizations of the classical Anderson model. First, we discuss a generalization to locally finite graphs $G = (V,E)$ instead of $\mathbb{Z}^d$. We will elaborate geometric conditions on the graph $G$, such that localization still holds in the case of sufficiently large disorder. Second, we discuss a generalization of the Anderson model to the case where the potential values at different lattice sites are correlated random variables, in particular, where $V_{\cdot} : \Omega \times \mathbb{Z}^d \to \mathbb{R}$ is a Gaussian process with sign-indefinite covariance function.

Guest of the Research Training Group (RTG) High-dimensional Phenomena in Probability.

Martin Tautenhahn

Friedrich-Schiller-Universität Jena und Technische Universität Chemnitz