Scaling in the One-Dimensional Anderson Localization Problem in the Region of Fluctuation States

We numerically study the distribution function of the conductivity (transmission) in the one-dimensional tight-binding Anderson model in the region of fluctuation states. We show that while single parameter scaling in this region is not valid, the distribution can still be described within a scaling approach based upon the ratio of two fundamental quantities, the localization length, l_loc, and a new length, l_s, related to the integral density of states. In an intermediate interval of the system's length L, l_loc≪L≪l_s, the variance of the Lyapunov exponent does not follow the predictions of the central limit theorem, and may even grow with L.