2d and 3d Anderson localization in one-dimesional system with 2 and 3 incommensurate frequencies

The analysis of models like kicked rotator in the case when one of the parameters of the model is varied with a frequency incommensurate with the frequency of the kicks showed that the diffusive time scale increases exponentially with the classical diffusion rate $D$ [14]. This case corresponds to Anderson localization in 2-dimensional random potential. In the case when modulation is done with 2 frequencies the transition from localization to diffusion takes place in analogy with Anderson transition in 3d [37]. However, here all computations are done with one-dimensional system, instead of 3-dimensional. This makes such approach very efficient numerically and allows to observe indeed the transition without renormgroup assumptions and also to determine the scaling exponents near transition. More recent investigations [86] of the case with larger number of frequencies allowed to study the Anderson transition in this model for effective dimension $d \ge 4$. The numerically obtained critical exponents were found to be different from the standard scaling relation ($s = (d-2)\nu$) that can be related to a quasiperiodic nature of effective potential in $d-1$ directions.