Tunneling in the integer quantum Hall effect

We discuss the localization-delocalization transition in. the integer quantum Hall effect. In the so-called "semi-classical" picture of this transition, interference effects are ignored, and tunneling alone is considered responsible for deviations from a transition described by classical percolation. We discuss the (incomplete) arguments that lead from this assumption to a simple rescaling of the basic length in the problem. On the other hand, we put the rescaling assertion to a direct test by studying numerically the extended Chalker-Coddington model. We point out that this model, in the "classical" limit (i.e., the limit of large disorder) shows crossover to the so-called "smart kinetic walks". The reason why this limit previously has been identified with ordinary percolation is, presumably, that the localization length exponents nu coincide for the two problems. Other exponents, like the fractal dimension D, differ. We calculate numerically two exponents, tau and D, that characterize critical properties of the geometry of the wave function at these transitions. We find that the exponents, within our precision, are equal to those of two-dimensional percolation, as predicted by the semiclassical picture. Finally, the status of the controversy on these questions is briefly discussed.