Statistical properties of localisation--delocalisation transition in one dimension

We study a one-dimensional model of disordered electrons (also relevant for random spin chains), which exhibits a delocalisation transition at half-filling. Exact probability distribution functions for the Wigner time and transmission coefficient are calculated. We identify and distinguish those features of probability densities that are due to rare, trapping configurations of the random potential from those which are due to the proximity to the delocalisation transition.