Localization transitions from free random variables

We motivate and use the concept of free random variables for the study of the de-pinning transition of flux lines in superconductors as recently discussed by Hatano and Nelson in one dimension. Our analysis yields naturally to a generalization of the concept of Coherent Phase Appproximation (CPA) for nonhermitean Hamiltonians, and is exact for Cauchy randomness. We derive analytical conditions for the critical points of the complex eigenvalue distribution, in very good agreement with numerical calculations. We suggest a relation between dimensionally reduced nonhermitean quantum mechanics and weak nonhermiticity.