Finite-Size Scaling and Correlation Lengths for Disordered Systems

For a large class of d-dimensional disordered systems, we prove that if an appropriately defined finite-size scaling correlation length diverges at a nontrivial value of the disorder with an exponent, then must satisfy the bound 2d. Given the assumption that such a correlation length can be defined, the result applies to, e. g. , percolation, disordered magnets, and Anderson localization, both with and without interactions. For localization, this puts stringent constraints on scaling theories and interpretation of experiments.