This review is intended to give a pedagogical and unified view on the subject of the statistics and scaling of physical quantities in disordered electron systems at very low temperatures. Quantum coherence at low temperatures and randomness of microscopic details can cause large fluctuations of physical quantities. In such mesoscopic systems a localization-delocalization transition can occur which forms a critical phenomenon. Accordingly, a one-parameter scaling theory was formulated stressing the role of conductance as the (oneparameter) scaling variable. The localized and delocalized phases are separated by a critical point determined by a critical value of conductance. However, the notion of an order parameter was not fully clarified in this theory.The one-parameter scaling theory has been questioned once it was noticed that physical quantities are broadly distributed and that average values are not characteristic for the distributions. Based on presently available analytical and numerical results we focus here on the description of the total distribution functions and their flow with increasing system size. Still, one-parameter scaling theory does work in terms of typical values of the local density of states and of the conductance which serve as order parameter and scaling variable of the localization-delocalization transition, respectively. Below a certain length scale, ξ c , related to the value of the typical conductance, local quantities are multifractally distributed. This multifractal behavior becomes universal on approaching the localization-delocalization transition with ξ c playing the role of a correlation length.

The multifractal analysis of disorder induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent β and the correlation length exponent ν is investigated. We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent ν are obtained by the typical order parameter exponent α 0 and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments.

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On the basis of the Chalker-Coddington network model, a numerical and analytical study is made of the statistics of point-contact conductances for systems in the integer quantum Hall regime. In the Hall plateau region the point-contact conductances reflect strong localization of the electrons, while near the plateau transition they exhibit strong mesoscopic fluctuations. By mapping the network model on a supersymmetric vertex model with GL(2͉2) symmetry, and postulating a two-point correlator in keeping with the rules of conformal field theory, we derive an explicit expression for the distribution of conductances at criticality. There is only one free parameter, the power law exponent of the typical conductance. Its value is computed numerically to be X t ϭ0.640Ϯ0.009. The predicted conductance distribution agrees well with the numerical data. For large distances between the two contacts, the distribution can be described by a multifractal spectrum solely determined by X t . Our results demonstrate that multifractality can show up in appropriate transport experiments.

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