Universal distributions and scaling in disordered systems

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Abstract
We study resistance distributions in disordered electronic systems and show that, in the thermodynamic limit, distributions are attracted towards some limiting universal distribution. Very few model-dependent parameters are then needed to specify this limiting distribution completely. The number of such parameters corresponds to the number of relevant scaling variables in the theory. Our numerical studies, within the Migdal-Kadanoff recursion scheme, demonstrate that, generally, there are two independent parameters in the limiting distribution (two-parameter scaling). Only in the limit of weak disorder do we find single-parameter scaling, in agreement with previous work. We also analyze some earlier results (for one-dimensional chains, multichannel wires, and genuine two- or three-dimensional systems) in terms of scaling and present evidence in support of two-parameter scaling for arbitrary disorder (and single-parameter scaling in the limit of weak disorder).

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